diff --git "a/Full-data/test500.json" "b/Full-data/test500.json" new file mode 100644--- /dev/null +++ "b/Full-data/test500.json" @@ -0,0 +1,3002 @@ +[ + { + "Q": "whats the difference between the square root and principal root?", + "A": "Square root can mean positive or negative value. Ex: Square root of 25 is either 5 or -5 Principal square root means you only want the positive value. Ex: The principal square root of 25 is 5", + "video_name": "sBvRJUwXJPo", + "transcript": "We're asked to divide and simplify. And we have one radical expression over another radical expression. The key to simplify this is to realize if I have the principal root of x over the principal root of y, this is the same thing as the principal root of x over y. And it really just comes out of the exponent properties. If I have two things that I take to some power-- and taking the principal root is the same thing as taking it to the 1/2 power-- if I'm raising each of them to some power and then dividing, that's the same thing as dividing first and then raising them to that power. So let's apply that over here. This expression over here is going to be the same thing as the principal root-- it's hard to write a radical sign that big-- the principal root of 60x squared y over 48x. And then we can first look at the coefficients of each of these expressions and try to simplify that. Both the numerator and the denominator is divisible by 12. 60 divided by 12 is 5. 48 divided by 12 is 4. Both the numerator and the denominator are divisible by x. x squared divided by x is just x. x divided by x is 1. Anything we divide the numerator by, we have to divide the denominator by. And that's all we have left. So if we wanted to simplify this, this is equal to the-- make a radical sign-- and then we have 5/4. And actually, we can write it in a slightly different way, but I'll write it this way-- 5/4. And we have nothing left in the denominator other than that 4. And in the numerator, we have an x and we have a y. And now we could leave it just like that, but we might want to take more things out of the radical sign. And so one possibility that you can do is you could say that this is really the same thing as-- this is equal to 1/4 times 5xy, all of that under the radical sign. And this is the same thing as the square root of or the principal root of 1/4 times the principal root of 5xy. And the square root of 1/4, if you think about it, that's just 1/2 times 1/2. Or another way you could think about it is that this right here is the same thing as-- so you could just say, hey, this is 1/2. 1/2 times 1/2 is 1/4. Or if you don't realize it's 1/2, you say, hey, this is the same thing as the square root of 1 over the square root of 4, and the square root of 1 is 1 and the principal root of 4 is 2, so you get 1/2 once again. And so if you simplify this right here to 1/2, then the whole thing can simplify to 1/2 times the principal root-- I'll just write it all in orange-- times the principal root of 5xy. And there's nothing else that you can really take out of the radical sign here. Nothing else here is a perfect square." + }, + { + "Q": "How does 2s1 suddenly change to 2p3? Where did the \"p\" come from?", + "A": "The orbitals each have a given amount of electrons, s can have 2, and then it must move up to p. The p orbital can have up to 6 electrons, and when you run out you must move on to the d orbital, which can have 10 electrons. Lastly, you have the f orbital and that orbital can have up to 14 electrons.", + "video_name": "FmQoSenbtnU", + "transcript": "In the last few videos we learned that the configuration of electrons in an atom aren't in a simple, classical, Newtonian orbit configuration. And that's the Bohr model of the electron. And I'll keep reviewing it, just because I think it's an important point. If that's the nucleus, remember, it's just a tiny, tiny, tiny dot if you think about the entire volume of the actual atom. And instead of the electron being in orbits around it, which would be how a planet orbits the sun. Instead of being in orbits around it, it's described by orbitals, which are these probability density functions. So an orbital-- let's say that's the nucleus it would describe, if you took any point in space around the nucleus, the probability of finding the electron. So actually, in any volume of space around the nucleus, it would tell you the probability of finding the electron within that volume. And so if you were to just take a bunch of snapshots of electrons -- let's say in the 1s orbital. And that's what the 1s orbital looks like. You can barely see it there, but it's a sphere around the nucleus, and that's the lowest energy state that an electron can be in. If you were to just take a number of snapshots of electrons. Let's say you were to take a number of snapshots of helium, which has two electrons. Both of them are in the 1s orbital. It would look like this. If you took one snapshot, maybe it'll be there, the next snapshot, maybe the electron is there. Then the electron is there. Then the electron is there. Then it's there. And if you kept doing the snapshots, you would have a bunch of them really close. And then it gets a little bit sparser as you get out, as you get further and further out away from the electron. But as you see, you're much more likely to find the electron close to the center of the atom than further out. Although you might have had an observation with the electron sitting all the way out there, or sitting over here. So it really could have been anywhere, but if you take multiple observations, you'll see what that probability function is describing. It's saying look, there's a much lower probability of finding the electron out in this little cube of volume space than it is in this little cube of volume space. And when you see these diagrams that draw this orbital like this. Let's say they draw it like a shell, like a sphere. And I'll try to make it look three-dimensional. So let's say this is the outside of it, and the nucleus is sitting some place on the inside. They're just saying -- they just draw a cut-off -- where can I find the electron 90% of the time? So they're saying, OK, I can find the electron 90% of the time within this circle, if I were to do the cross-section. But every now and then the electron can show up outside of that, right? Because it's all probabilistic. So this can still happen. You can still find the electron if this is the orbital we're talking about out here. Right? And then we, in the last video, we said, OK, the electrons fill up the orbitals from lowest energy state to high energy state. You could imagine it. If I'm playing Tetris-- well I don't know if Tetris is the thing-- but if I'm stacking cubes, I lay out cubes from low energy, if this is the floor, I put the first cube at the lowest energy state. And let's say I could put the second cube But I only have this much space to work with. So I have to put the third cube at the next highest energy state. In this case our energy would be described as potential energy, right? This is just a classical, Newtonian physics example. But that's the same idea with electrons. Once I have two electrons in this 1s orbital -- so let's say the electron configuration of helium is 1s2 -- the third electron I can't put there anymore, because there's only room for two electrons. The way I think about it is these two electrons are now going to repel the third one I want to add. So then I have to go to the 2s orbital. And now if I were to plot the 2s orbital on top of this one, it would look something like this, where I have a high probability of finding the electrons in this shell that's essentially around the 1s orbital, right? So right now, if maybe I'm dealing with lithium right now. So I only have one extra electron. So this one extra electron, that might be where I observed that extra electron. But every now and then it could show up there, it could show up there, it could show up there, but the high probability is there. So when you say where is it going to be 90% of the time? It'll be like this shell that's around the center. Remember, when it's three-dimensional you would kind of cover it up. So that's what they drew here. They do the 1s. It's just a red shell. And then the 2s. The second energy shell is just this blue shell over it. And you can see it a little bit better in, actually, the higher energy orbits, the higher energy shells, where the seventh s energy shell is this red area. Then you have the blue area, then the red, and the blue. And so I think you get the idea that each of those are energy shells. So you kind of keep overlaying the s energy orbitals around each other. But you probably see this other stuff here. And the general principle, remember, is that the electrons fill up the orbital from lowest energy orbital to higher energy orbital. So the first one that's filled up is the 1s. This is the 1. This is the s. So this is the 1s. It can fit two electrons. Then the next one that's filled up is 2s. It can fill two more electrons. And then the next one, and this is where it gets interesting, you fill up the 2p orbital. 2p orbital. That's this, right here. 2p orbitals. And notice the p orbitals have something, p sub z, p sub x, p sub y. What does that mean? Well, if you look at the p-orbitals, they have these dumbbell shapes. They look a little unnatural, but I think in future videos we'll show you how they're analogous to standing waves. But if you look at these, there's three ways that you can configure these dumbbells. One in the z direction, up and down. One in the x direction, left or right. And then one in the y direction, this way, forward and backwards, right? And so if you were to draw-- let's say you wanted to draw the p-orbitals. So this is what you fill next. And actually, you fill one electron here, another electron here, then another electron there. Then you fill another electron, and we'll talk about spin and things like that in the future. But, there, there, and there. And that's actually called Hund's rule. Maybe I'll do a whole video on Hund's rule, but that's not relevant to a first-year chemistry lecture. But it fills in that order, and once again, I want you to have the intuition of what this would look like. Look. I should put look in quotation marks, because it's very abstract. But if you wanted to visualize the p orbitals-- let's say we're looking at the electron configuration for, let's say, carbon. So the electron configuration for carbon, the first two electrons go into, so, 1s1, 1s2. So then it fills-- sorry, you can't see everything. So it fills the 1s2, so carbon's configuration. It fills 1s1 then 1s2. And this is just the configuration for helium. And then it goes to the second shell, which is the second period, right? That's why it's called the periodic table. We'll talk about periods and groups in the future. And then you go here. So this is filling the 2s. We're in the second period right here. That's the second period. One, two. Have to go off, so you can see everything. So it fills these two. So 2s2. And then it starts filling up the p orbitals. So then it starts filling 1p and then 2p. And we're still on the second shell, so 2s2, 2p2. So the question is what would this look like if we just wanted to visualize this orbital right here, the p orbitals? So we have two electrons. So one electron is going to be in a-- Let's say if this is, I'll try to draw some axes. That's too thin. So if I draw a three-dimensional volume kind of axes. If I were to make a bunch of observations of, say, one of the electrons in the p orbitals, let's say in the pz dimension, sometimes it might be here, sometimes it might be there, sometimes it might be there. And then if you keep taking a bunch of observations, you're going to have something that looks like this bell shape, this barbell shape right there. And then for the other electron that's maybe in the x direction, you make a bunch of observations. Let me do it in a different, in a noticeably different, color. It will look like this. You take a bunch of observations, and you say, wow, it's a lot more likely to find that electron in kind of the dumbell, in that dumbbell shape. But you could find it out there. You could find it there. You could find it there. This is just a much higher probability of finding it in here than out here. And that's the best way I can think of to visualize it. Now what we were doing here, this is called an electron configuration. And the way to do it-- and there's multiple ways that are taught in chemistry class, but the way I like to do it -- is you take the periodic table and you say, these groups, and when I say groups I mean the columns, these are going to fill the s subshell or the s orbitals. You can just write s up here, just right there. These over here are going to fill the p orbitals. Actually, let me take helium out of the picture. The p orbitals. Let me just do that. Let me take helium out of the picture. These take the p orbitals. And actually, for the sake of figuring out these, you should take helium and throw it right over there. Right? The periodic table is just a way to organize things so it makes sense, but in terms of trying to figure out orbitals, you could take helium. Let me do that. The magic of computers. Cut it out, and then let me paste it right over there. Right? And now you see that helium, you get 1s and then you get 2s, so helium's configuration is -- Sorry, you get 1s1, then 1s2. We're in the first energy shell. Right? So the configuration of hydrogen is 1s1. You only have one electron in the s subshell of the first energy shell. The configuration of helium is 1s2. And then you start filling the second energy shell. The configuration of lithium is 1s2. That's where the first two electrons go. And then the third one goes into 2s1, right? And then I think you start to see the pattern. And then when you go to nitrogen you say, OK, it has three in the p sub-orbital. So you can almost start backwards, right? So we're in period two, right? So this is 2p3. Let me write that down. So I could write that down first. 2p3. So that's where the last three electrons go into the p orbital. Then it'll have these two that go into the 2s2 orbital. And then the first two, or the electrons in the lowest energy state, will be 1s2. So this is the electron configuration, right here, of nitrogen. And just to make sure you did your configuration right, what you do is you count the number of electrons. So 2 plus 2 is 4 plus 3 is 7. And we're talking about neutral atoms, so the electrons should equal the number of protons. The atomic number is the number of protons. So we're good. Seven protons. So this is, so far, when we're dealing just with the s's and the p's, this is pretty straightforward. And if I wanted to figure out the configuration of silicon, right there, what is it? Well, we're in the third period. One, two, three. That's just the third row. And this is the p-block right here. So this is the second row in the p-block, right? One, two, three, four, five, six. Right. We're in the second row of the p-block, so we start off with 3p2. And then we have 3s2. And then it filled up all of this p-block over here. So it's 2p6. And then here, 2s2. And then, of course, it filled up at the first shell before it could fill up these other shells. So, 1s2. So this is the electron configuration for silicon. And we can confirm that we should have 14 electrons. 2 plus 2 is 4, plus 6 is 10. 10 plus 2 is 12 plus 2 more is 14. So we're good with silicon. I think I'm running low on time right now, so in the next video we'll start addressing what happens when you go to these elements, or the d-block. And you can kind of already guess what happens. We're going to start filling up these d orbitals here that have even more bizarre shapes. And the way I think about this, not to waste too much time, is that as you go further and further out from the nucleus, there's more space in between the lower energy orbitals to fill in more of these bizarro-shaped orbitals. But these are kind of the balance -- I will talk about standing waves in the future -- but these are kind of a balance between trying to get close to the nucleus and the proton and those positive charges, because the electron charges are attracted to them, while at the same time avoiding the other electron charges, or at least their mass distribution functions. Anyway, see you in the next video." + }, + { + "Q": "I'm sorry, but I don't really see what bad things deflation causes. Could someone please explain? Oh, and was it deflation that caused the Great Depression? I was under the impression that it was inflation.", + "A": "Check out the tutorial on deflation.", + "video_name": "sZRkERfzzn4", + "transcript": "I finished the last video touching on the stimulus plan and whether it's going to be big enough and what its intent was. But I was a little handwavy about things like savings and GDP and I thought it would actually be good to get a little bit more particular. This isn't just me making up things. So a good place to start is just to think about disposable income. And people talk about it without ever giving you a good definition. It's important to understand what disposable income is. So this box right here is GDP, so it's all the goods and services, essentially all the income, that we produce. The disposable income is essentially the amount that ends up in the hands of people. So the U.S. GDP, I don't know what the exact number is, it's on the order of I think $15 trillion. And so some of that money goes to taxes. I don't know. Let's say $3 trillion ends up in taxes. And these are round numbers, but it gives you a sense of Then maybe about another $1 trillion is saved by So let's say that I'm Microsoft and I make a billion Microsoft makes a lot more than that, but I make a billion dollars. And I just keep it in a bank account owned by Microsoft. I don't give it to my employees. I don't dividend it out to the shareholders. I just keep that. So corporate savings, let's say, is another trillion dollars. These are roundabout numbers. And then everything left over is, essentially, disposable income. I'm making some simplifications there, but everything left over is disposable income. And the idea is, what is in the hands of households after paying their taxes: disposable income. That check that you get after your taxes and everything else that is taken out of your paycheck, that money, that is disposable income to you. So the interesting thing is where this disposable income has been going over the last many, many years. So this right here, this chart, is a plot of the percentage of disposable income saved since 1960. I got this off the Bloomberg terminal again. And just so you get a good reference point, let me draw the line for 10%. Because I kind of view that as a reference point. That's 10% personal savings. Actually, you could go back even further than this, and you can see from the 1960s all the way until the early `80s, the go-go `80s, people were saving about 10% of their disposable income. Of what they got on that paycheck after paying taxes, they were keeping about 10% of it, putting it in the bank. And, of course, that would later be used for investment and things like that. But then starting in the early `80s, right around, let's see, about 1984, you see that people started saving less and less money. All the way to the extreme circumstance, to 2007, where, on average, people didn't save money. Where maybe I saved $5, but someone else borrowed $5. So on average, we didn't save money. Actually we went slightly negative. And all of this, and this is kind of my claim-- and it can be borne out if you look at other charts in terms of the amount of credit we took on-- is because we took on credit. I guess there's two things you could talk about. We took on credit and we started having what I call perceived savings, right? If I buy a share of IBM a lot of people say, oh, I saved I saved my money, I'm invested in the stock market. But when I bought a share of IBM, unless that dollar I paid for the share goes to IBM to build new factories-- And that's very seldom time. Most of the time, that dollar I gave to buy that IBM share, just goes to a guy who sold an IBM share. So there's no net investment. It's only investment, it's only savings, when that dollar actually goes to invest in some way, build a new factory or build a new product or something. So I think you had more and more people thinking that they were saving when they weren't. Maybe thinking that they were saving, as their home equity in their house grew, or as their stock market portfolio went up. But there wasn't actual aggregate savings going on. In fact, if anything, they were borrowing against those things, especially home equity. And the average savings rate went down and down and down. So now we're here in 2007, and maybe you could say 2008, where this is the capacity. We should probably talk about world capacity, because so much of what we consume really does come from overseas, but this is the capacity serving the U.S. And let's say, going into 2007, this was demand. Supply and demand was, let's say, pretty evenly matched. If you go to that top up here, you see that we were at 80% utilization, which isn't crazy. So if anything, you could say that demand was maybe right But that's a good level. You want to be at around 80% utilization. That's the rate at which you don't have hyperinflation, but you're utilizing things quite well. Now all of a sudden, the credit crisis hit and all this credit disappears. And I'd make the argument that the only thing that enabled us to not save this money, is more and more access to cheap credit. Every time we went through a recession, from the mid `80s, onwards, the government solution was to make financing easier. The Fed would lower interest rates. We would pump more and more money into Fannie and Freddie Mac. We would lower standards on what it We would create incentives for securitization. We would look the other way when Bear Stearns is creating collateralized debt obligations, or AIG is writing credit default swaps. And all of that enabled financing. Until we get to this point right here, where everything starts blowing up. So you essentially have forced savings. When people can't borrow anymore, the savings rate has to go to 10% because most of this is just from people taking on more credit. There were a group of people who were probably already always saving at 10% of their disposable income. But there's another group of people who were more than offsetting that by taking credit. Now that the credit crisis hits, you're going to see the savings rate-- and you already see it with this blip right there-- the savings rate is going to go back up to 10% of disposable income. Now if the savings rate goes back to 10% of disposable income, that amount of money, that 10% of disposable income, this gap, is 10% of disposable income that cannot be used for consumption. And let's think about how large of an amount that is. Right now, the U.S. disposable income, if I were to draw disposable income-- this number right here, I just looked it up-- it's around $10.7 trillion, let's say $11 trillion for roundabout. So this 10% of disposable income that we're talking about, this gap between the normal environment, the environment that was enabled from super-easy credit, that's 10% of disposable income. 10% of $11 trillion is $1.1 trillion per year of demand that will go away. That's per year. So all of a sudden, you're going to have a gap where this was the demand before and now the demand's going to drop to here. And all of this $1.1 trillion of demand is going to disappear, because credit is now not available. And then you do get a situation where you might hit a low capacity utilization. Unfortunately, the capacity utilization numbers don't go back to the Great Depression, and we could probably get a sense of, at what point does a deflationary spiral really get triggered. But because that $1.1 trillion in demand disappears, you're going to see this orange line just drop lower and lower in terms of capacity utilization. And that's going to drive prices down. And what Obama and the Fed and everyone else is trying to do, is to try to make up this gap. Now, everyone else can't borrow money. Companies can't borrow money. Homeowners can't borrow money. But the government can borrow money, because people are willing to finance it. At the bare minimum, the Fed's willing to finance the government. And so the government wants to come in and take up the slack with this demand and spend the money themselves with the stimulus. Now we just talked about, what's the magnitude of this demand shock? It's $1.1 trillion per year. While the stimulus plan is on the order of a trillion dollars. And that trillion dollars isn't per year, although I have a vague feeling that we will see more of them coming down the pipeline. This stimulus plan that just passed is expected to be spent over the next few years. So in terms of demand created over the next few years, it's going to be several hundred billion per year. So it's not going to be anywhere near large enough to make up for this demand shock. So we're still going to have low capacity utilization. So the people who argue that these stimulus plans are going to create hyperinflation, I disagree with that, at least in the near term, because in the near term it's nowhere near large enough to really soak up all of the extra capacity we have in the system. And if anything, it's going to soak up different capacity. So the stimulus plan might create inflationary-like conditions in certain markets where the stimulus plan is really focused. But in other areas, where you used to have demand, but the stimulus plan doesn't touch, like $50 spatulas from Williams Sonoma or granite countertops, that area is going to continue to see deflation. And I would say net-net, since this thing this is actually so small, even though we're talking about trillion dollars relative to the amount of demand destroyed per year, we'll probably still have deflation. And, for anyone who's paying close attention to it, and I am because I care about these things, the best thing to keep a lookout for, to know when to start running to gold maybe or being super-worried about inflation, is if you see this utilization number creeping back up into the 80% range. At least over the last 40 years, that's been the best leading indicator to say when are we And for all those goldbugs out there, who insist that all of the hyperinflation or the potential hyperinflation is caused by our not being on the gold standard, I just want to point out that you can very easily have-- we went off of Bretton Woods and completely went to a fiat currency in '73, that right around here. But our worst inflation bouts were actually while we were on the gold standard. And that's because we had very high capacity utilization. This is the war period. What happens during a war? You run your factories all-out. You run your farms all-out to feed the troops. The factories, instead of building cars, build planes. Everyone was working. Wages go up. Everything goes up and you have inflation. A lot of people say, oh, the war was the solution to the And it is true in that it got us out of that negative deflationary spiral that I talked about in the last video. And it did it by creating an unbelievable amount of inflation. And then after the war, you could argue what allowed-- I don't have GDP here, but the U.S. GDP really did do well in the postwar period-- it wasn't the war per se, although the war kind of did take us out of the deflationary spiral. So after the war, you had all of this capacity, after World War II. So you had all this capacity that was being used up by the war, and then, all of the sudden, the war ends. And you're like, well, we don't have to build planes anymore, we don't have to feed the troops anymore, and now we have all these unemployed troops who come back home. What are we going to do with them? You'd say maybe, capacity utilization would come back down there. But what a lot of people don't talk about is, the rest of the industrial world's capacity was blown to smithereens. At that point in time, the U.S. was a smaller part of GDP and the other major players were Germany, Western Europe and Japan. In the U.S., we didn't have any factories bombed. The entire war took place in these areas. And whenever people go on bombing raids, the ideal thing that they want to bomb is factories. So what you had in the postwar period, is you had all of these countries that had their capacity blown to smithereens. The U.S. was essentially the only person left with any capacity, and so all the demand from the rest of the world picked up the slack in the U.S. capacity. And that's why, even though there was a demand shock in the U.S. after that, you also had a supply shock from the war where you had all of this capacity that was blown to smithereens. In this situation that we are in now, we have a major demand shock. Financing just disappears and the savings rate's going to skyrocket because people can't borrow anymore. But there's no counteracting supply shock. Supply of factories making widgets and all the rest is So the Obama administration is trying to create a stimulus to sop some of this up, but it's not going to be enough. And my only fear is, with all the printing money and all that goes, once we do get back to the 80% capacity utilization, once we do go back here, how quickly they can unwind all of this printing press money and all of the stimulus plan. Because that's going to be the key. Because, if we do get to 80% capacity utilization or we start pushing up there, and we continue to run the printing press-- because that's what, essentially, government's incentive is to do, because they always feel better when we're flush with money-- then, and only then, you might see a hyperinflation scenario. But I don't see that at least for the next few years." + }, + { + "Q": "At 4:29 What is the exact difference between obtuse and acute triangles", + "A": "in the obtuse triangle, it has one obtuse angle (bigger than a right angle) and in an acute angle, all angles are smaller that right angle. Ex: acute:all angles 60 ( 60+60+60=180) Ex: obtuse: one angle 120 another 35 and another 25 Hope this helped! Please vote this", + "video_name": "D5lZ3thuEeA", + "transcript": "What I want to do in this video is talk about the two main ways that triangles are categorized. The first way is based on whether or not the triangle has equal sides, or at least a few equal sides. Then the other way is based on the measure of the angles of the triangle. So the first categorization right here, and all of these are based on whether or not the triangle has equal sides, is scalene. And a scalene triangle is a triangle where none of the sides are equal. So for example, if I have a triangle like this, where this side has length 3, this side has length 4, and this side has length 5, then this is going to be a scalene triangle. None of the sides have an equal length. Now an isosceles triangle is a triangle where at least two of the sides have equal lengths. So for example, this would be an isosceles triangle. Maybe this has length 3, this has length 3, and this has length 2. Notice, this side and this side are equal. So it meets the constraint of at least two of the three sides are have the same length. Now an equilateral triangle, you might imagine, and you'd be right, is a triangle where all three sides have the same length. So for example, this would be an equilateral triangle. And let's say that this has side 2, 2, and 2. Or if I have a triangle like this where it's 3, 3, and 3. Any triangle where all three sides have the same length is going to be equilateral. Now you might say, well Sal, didn't you just say that an isosceles triangle is a triangle has at least two sides being equal. Wouldn't an equilateral triangle be a special case of an isosceles triangle? And I would say yes, you're absolutely right. An equilateral triangle has all three sides equal, so it meets the constraints for an isosceles. So by that definition, all equilateral triangles are also isosceles triangles. But not all isosceles triangles are equilateral. So for example, this one right over here, this isosceles triangle, clearly not equilateral. All three sides are not the same. Only two are. But both of these equilateral triangles meet the constraint that at least two of the sides are equal. Now down here, we're going to classify based on angles. An acute triangle is a triangle where all of the angles are less than 90 degrees. So for example, a triangle like this-- maybe this is 60, let me draw a little bit bigger so I can draw the angle measures. That's a little bit less. I want to make it a little bit more obvious. So let's say a triangle like this. If this angle is 60 degrees, maybe this one right over here is 59 degrees. And then this angle right over here is 61 degrees. Notice they all add up to 180 degrees. This would be an acute triangle. Notice all of the angles are less than 90 degrees. A right triangle is a triangle that has one angle that is exactly 90 degrees. So for example, this right over here would be a right triangle. Maybe this angle or this angle is one that's 90 degrees. And the normal way that this is specified, people wouldn't just do the traditional angle measure and write 90 degrees here. They would draw the angle like this. They would put a little, the edge of a box-looking thing. And that tells you that this angle right over here is 90 degrees. And because this triangle has a 90 degree angle, and it could only have one 90 degree angle, this is a right triangle. So that is equal to 90 degrees. Now you could imagine an obtuse triangle, based on the idea that an obtuse angle is larger than 90 degrees, an obtuse triangle is a triangle that has one angle that is larger than 90 degrees. So let's say that you have a triangle that looks like this. Maybe this is 120 degrees. And then let's see, let me make sure that this would make sense. Maybe this is 25 degrees. Or maybe that is 35 degrees. And this is 25 degrees. Notice, they still add up to 180, or at least they should. 25 plus 35 is 60, plus 120, is 180 degrees. But the important point here is that we have an angle that is a larger, that is greater, than 90 degrees. Now, you might be asking yourself, hey Sal, can a triangle be multiple of these things. Can it be a right scalene triangle? Absolutely, you could have a right scalene triangle. In this situation right over here, actually a 3, 4, 5 triangle, a triangle that has lengths of 3, 4, and 5 actually is a right triangle. And this right over here would be a 90 degree angle. You could have an equilateral acute triangle. In fact, all equilateral triangles, because all of the angles are exactly 60 degrees, all equilateral triangles are actually acute. So there's multiple combinations that you could have between these situations and these situations right over here." + }, + { + "Q": "What is the meaning of AP Physics? Does AP stand for Applied Physics\nIf so the class is called Applied Physics - Physics (:<))", + "A": "Nope. AP is for A dvanced P lacement physics, not applied physics.", + "video_name": "IMC3Uqv5yYc", + "transcript": "- I have taught AP Physics classes for the last seven years, AP Physics one, AP Physics B, back in the day, and AP Physics C now. I try to make my lessons personable, relate to the students, offer them real life examples where things happen. But also focus on, instead of a bunch of little memorization facts, what's the big idea that helps you figure out all these different situations for like, allows you to be more of a robust problem solver. So we're focusing on the big core ideas of our subject, being physics, and I think you'll find that across all the different AP content subjects. We are finely tuned to what the AP exam is gonna be like, directly, and we are building in videos for instructions specific to that and also we have practice exercises that I'm really excited about, that you can assign it to students and there's hints and there's feedback why a certain problem is wrong, and why a certain problem is right, and then it shows you the solution path of how you would get to the solution. So, when you assign a student a problem, they don't have to wait a whole 24 hours until they get back to class, they can get it wrong, fail early, try another one, fail often, and do a couple times on that, and I actually think my might actually do more homework because they're getting instant feedback on the process rather than just trying a couple and getting stuck and like, ah I'm frustrated, I'm out of here. They can really get in the seat and stay in there because they're getting instant feedback and help as their working through problems. I think that's gonna be really powerful." + }, + { + "Q": "i have learned that the denser the medium, the faster the waves travel in it...", + "A": "True for sound waves and mechanical waves.", + "video_name": "yF4cvbAYjwI", + "transcript": "- [Voiceover] To change the speed of sound you have to change the properties of the medium that sound wave is traveling through. There's two main factors about a medium that will determine the speed of the sound wave through that medium. One is the stiffness of the medium or how rigid it is. The stiffer the medium the faster the sound waves will travel through it. This is because in a stiff material, each molecule is more interconnected to the other molecules around it. So any disturbance gets transmitted faster down the line. The other factor that determines the speed of a sound wave is the density of the medium. The more dense the medium, the slower the sound wave will travel through it. This makes sense because if a material is more massive it has more inertia and therefore it's more sluggish to changes in movement or oscillations. These two factors are taken into account with this formula. V is the speed of sound. Capital B is called the bulk modulus of the material. The bulk modulus is the official way physicists measure how stiff a material is. The bulk modulus has units of pascals because it's measuring how much pressure is required to compress the material by a certain amount. Stiff, rigid materials like metal would have a large bulk modulus. More compressible materials like marshmallows would have a smaller bulk modulus. Row is the density of the material since density is the mass per volume, the density gives you an idea of how massive a certain portion of the material would be. For example, let's consider a metal like iron. Iron is definitely more rigid and stiff than air so it has a much larger bulk modulus than air. This would tend to make sound waves travel faster through iron than it does through air. But iron also has a much higher density than air, which would tend to make sound waves travel slower through it. So which is it? Does sound travel faster though iron or slower? Well it turns out that the higher stiffness of iron more than compensates for the increased density and the speed of sound through iron is about 14 times faster than through air. This means that if you were to place one ear on a railroad track and someone far away struck the same railroad track with a hammer, you should hear the noise 14 times faster in the ear placed on the track compared to the ear just listening through the air. In fact, the larger bulk modulus of more rigid materials usually compensates for any larger densities. Because of this fact, the speed of sound is almost always faster through solids than it is through liquids and faster through liquids than it is through gases because solids are more rigid than liquids and liquids are more rigid than gases. Density is important in some aspects too though. For instance, if you heat up the air that a sound wave is travelling through, the density of the air decreases. This explains why sound travels faster through hotter air compared to colder air. The speed of sound at 20 degrees Celsius is about 343 meters per second, but the speed of sound at zero degrees Celsius is only about 331 meters per second. Remember, the only way to change the speed of sound is to change the properties of the medium it's travelling in and the speed of sound is typically faster through solids than it is through liquids and faster through liquids than it is through gases." + }, + { + "Q": "so if there are two supermassive black holes in the universe, and their gravitational force is the same, what would happen to the mass in between those two black holes?", + "A": "It would probably be more likely that the two super massive black holes joined each other and suck in everything in between but the black hole the matter got sucked into would depend on the distance to the black holes and the size of the black holes.", + "video_name": "DxkkAHnqlpY", + "transcript": "In the videos on massive stars and on black holes, we learned that if the remnant of a star, of a massive star, is massive enough, the gravitational contraction, the gravitational force, will be stronger than even the electron degeneracy pressure, even stronger than the neutron degeneracy pressure, even stronger than the quark degeneracy pressure. And everything would collapse into a point. And we called these points black holes. And we learned there's an event horizon around these black holes. And if anything gets closer or goes within the boundary of that event horizon, there's no way that it can never escape from the black hole. All it can do is get closer and closer to the black hole. And that includes light. And that's why it's called a black hole. So even though all of the mass is at the central point, this entire area, or the entire surface of the event horizon, this entire surface of the event horizon-- I'll do it in purple because it's supposed to be black-- this entire thing will appear black. It will emit no light. Now these type of black holes that we described, we call those stellar black holes. And that's because they're formed from collapsing massive stars. And the largest stellar black holes that we have observed are on the order of 33 solar masses, give or take. So very massive to begin with, let's just be clear. And this is what the remnant of the star has to be. So a lot more of the original star's mass might have been pushed off in supernovae. That's plural of supernova. Now there's another class of black holes here and these are somewhat mysterious. And they're called supermassive black holes. And to some degree, the word \"super\" isn't big enough, supermassive black holes, because they're not just a little bit more massive than stellar black holes. They're are a lot more massive. They're on the order of hundreds of thousands to billions of solar masses, hundred thousands to billions times the mass of our Sun, solar masses. And what's interesting about these, other than the fact that there are super huge, is that there doesn't seem to be black holes in between or at least we haven't observed black holes in between. The largest stellar black hole is 33 solar masses. And then there are these supermassive black holes that we think exist. And we think they mainly exist in the centers of galaxies. And we think most, if not all, centers of galaxies actually have one of these supermassive black holes. But it's kind of an interesting question, if all black holes were formed from collapsing stars, wouldn't we see things in between? So one theory of how these really massive black holes form is that you have a regular stellar black hole in an area that has a lot of matter that it can accrete around it. So I'll draw the-- this is the event horizon around it. The actual black hole is going to be in the center of it, or rather the mass of the black hole will be in the center of it. And then over time, you have just more and more mass just falling into this black hole. Just more and more stuff just keeps falling into this black hole. And then it just keeps growing. And so this could be a plausible reason, or at least the mass in the center keeps growing and so the event horizon will also keep growing in radius. Now this is a plausible explanation based on our current understanding. But the reason why this one doesn't gel that well is if this was the explanation for supermassive black holes, you expect to see more black holes in between, maybe black holes with 100 solar masses, or a 1,000 solar masses, or 10,000 solar masses. But we're not seeing those right now. We just see the stellar black holes, and we see the supermassive black holes. So another possible explanation-- my inclinations lean towards this one because it kind of explains the gap-- is that these supermassive black holes actually formed shortly after the Big Bang, that these are primordial black holes. These started near the beginning of our universe, primordial black holes. Now remember, what do you need to have a black hole? You need to have an amazingly dense amount of matter or a dense amount of mass. If you have a lot of mass in a very small volume, then their gravitational pull will pull them closer, and closer, and closer together. And they'll be able to overcome all of the electron degeneracy pressures, and the neutron degeneracy pressures, and the quark degeneracy pressures, to really collapse into what we think is a single point. I want to be clear here, too. We don't know it's a single point. We've never gone into the center of a black hole. Just the mathematics of the black holes, or at least as we understand it right now, have everything colliding into a single point where the math starts to break down. So we're really not sure what happens at that very small center point. But needless to say, it will be an unbelievably, maybe infinite, maybe almost infinitely, dense point in space, or dense amount of matter. And the reason why I kind of favor this primordial black hole and why this would make sense is right after the formation of the universe, all of the matter in the universe was in a much denser space because the universe was smaller. So let's say that this is right after the Big Bang, some period of time after the Big Bang. Now what we've talked about before when we talked about cosmic background is that at that point, the universe was relatively uniform. It was super, super dense but it was relatively uniform. So a universe like this, there's no reason why anything would collapse into black holes. Because if you look at a point here, sure, there's a ton of mass very close to it. But it's very close to it in every direction. So the gravitational force would be the same in every direction if it was completely uniform. But if you go shortly after the Big Bang, maybe because of slight quantum fluctuation effects, it becomes slightly nonuniform. So let's say it becomes slightly nonuniform, but it still is unbelievably dense. So let's say it looks something like this, where you have areas that are denser, but it's slightly nonuniform, but extremely dense. So here, all of a sudden, you have the type of densities necessary for a black hole. And where you have higher densities, where it's less uniform, here, all of a sudden, you will have inward force. The gravitational pull from things outside of this area are going to be less than the gravitational pull towards those areas. And the more things get pulled towards it, the less uniform it's going to get. So you could imagine in that primordial universe, that very shortly after the Big Bang when things were very dense and closely packed together, we may have had the conditions where these supermassive black holes could have formed. Where we had so much mass in such a small volume, and it was just not uniform enough, so that you could kind of have this snowballing effect, so that more and more mass would collect into these supermassive black holes that are hundreds of thousands to billions of times the mass of the Sun. And, this is maybe even the more interesting part, those black holes would become the centers of future galaxies. So you have these black holes forming, these supermassive black holes forming. And not everything would go into a black hole. Only if it didn't have a lot of angular velocity, then it might go into the black hole. But if it's going pass it fast enough, it'll just start going in orbit around the black hole. And so you could imagine that this is how the early galaxies or even our galaxy formed. And so you might be wondering, well, what about the black hole at the center of the Milky Way? And we think there is one. We think there is one because we've observed stars orbiting very quickly around something at the center of the universe-- sorry, at the center of our Milky Way. I want to be very clear, not at the center of the universe. And the only plausible explanation for it orbiting so quickly around something is that it has to have a density of either a black hole or something that will eventually turn into a black hole. And when you do the math for the middle of our galaxy, the center of the Milky Way, our supermassive black hole is on the order of 4 million times the mass of the Sun. So hopefully that gives you a little bit of food for thought. There aren't just only stellar collapsed black holes. Or maybe there are and somehow they grow into supermassive black holes and that everything in between we just can't observe. Or that they really are a different class of black holes. They're actually formed different ways. Maybe they formed near the beginning of the actual universe. When the density of things was a little uniform, things condensed into each other. And what we're going to talk about in the next video is how these supermassive black holes can help generate unbelievable sources of radiation, even though the black holes themselves aren't emitting them. And those are going to be quasars." + }, + { + "Q": "If you were to have an element in the 5th period and in the d-block, what is the reason for subtracting 1 when writing out the electon conginguration.", + "A": "Because, the period actually starts a row higher than the chart pictured.", + "video_name": "YURReI6OJsg", + "transcript": "Let's figure out the electron configuration for nickel, right there. 28 electrons. We just have to figure out what shells and orbitals they go in. 28 electrons. So the way we've learned to do it is, we defined this as the s-block. And we can just remember that helium actually belongs here when we talk about orbitals in the s-block. This is the d-block. This is the p-block. And so we could start with the lowest energy electrons. We could either work forward or work backwards. If we work forwards, first we fill up the first two electrons going to 1s2. So remember we're doing nickel. So we fill up 1s2 first with two electrons. Then we go to 2s2. And remember this little small superscript 2 just means we're putting two electrons into that subshell or into that orbital. Actually, let me do each shell in a different color. So 2s2. Then we fill out 2p6. We fill out all of these, right there. So 2p6. Let's see, so far we've filled out 10 electrons. We've configured 10. You can do it that way. Now we're on the third shell. The third shell. So now we go to 3s2. Remember, we're dealing with nickel, so we go to 3s2. Then we fill out in the third shell the p orbital. So 3p6. We're in the third period, so that's 3p6, right there. There's six of them. And then we go to the fourth shell. I'll do it in yellow. So we do 4s2. 4s2. And now we're in the d-block. And so we're filling in one, two, three, four, five, six, seven, eight in this d-block. So it's going to say d8. And remember, it's not going to be 4d8. We're going to go and backfill the third shell. So it will be 3d8. So we could write 3d8 here. So this is the order in which we fill, from lowest energy state electrons to highest energy state. But notice the highest energy state electrons, which are these that we filled in, in the end, these eight, these went into the third shell. So when you're filling the d-block, you take the period that you're in minus one. So we were in the fourth period in the periodic table, but we subtracted one, right? This is 4 minus 1. So this is the electron configuration for nickel. And of course if we remember, if we care about the valence electrons, which electrons are in the outermost shell, then you would look at these right here. These are the electrons that will react, although these are in a higher energy state. And these react because they're the furthest. Or at least, the way I visualize them is that they have a higher probability of being further from the nucleus than these right here. Now, another way to figure out the electron configuration for nickel-- and this is covered in some chemistry classes, although I like the way we just did it because you look at the periodic table and you gain a familiarity with it, which is important, because then you'll start having an intuition for how different elements react with each other -- is to just say, oK, nickel has 28 electrons, if it's neutral. It has 28 electrons, because that's the same number of protons, which is the atomic number. Remember, 28 just tells you how many protons there are. This is the number of protons. We're assuming it's neutral. So it has the same number of electrons. That's not always going to be the case. But when you do these electron configurations, that tends to be the case. So if we say nickel has 28, has an atomic number of 28, so it's electron configuration we can do it this way, too. We can write the energy shells. So one, two, three, four. And then on the top we write s, p, d. Well we're not going to get to f. But you could write f and g and h and keep going. What's going to happen is you're going to fill this one first, then you're going to fill this one, then that one, then this one, then this one. Let me actually draw it. So what you do is, these are the shells that exist, period. These are the shells that exist, in green. What I'm drawing now isn't the order that you fill them. This is just, they exist. So there is a 3d subshell. There's not a 3f subshell. There is a 4f subshell. Let me draw a line here, just so it becomes a little bit neater. And the way you fill them is you make these diagonals. So first you fill this s shell like that, then you fill this one like that. Then you do this diagonal down like that. Then you do this diagonal down like that. And then this diagonal down like that. And you just have to know that there's only two can fit in s, six in p, in this case, 10 in d. And we can worry about f in the future, but if you look at the f-block on a periodic table, you know how many there are in f. So you fill it like that. So first you just say, OK. For nickel, 28 electrons. So first I fill this one out. So that's 1s2. 1s2. Then I go, there's no 1p, so then I go to 2s2. Let me do this in a different color. So then I go right here, 2s2. That's that right there. Then I go up to this diagonal, and I come back down. And then there's 2p6. And you have to keep track of how many electrons you're dealing with, in this case. So we're up to 10 now. So we used that one up. Then the arrow tells us to go down here, so now we do the third energy shell. So 3s2. And then where do we go next? 3s2. Then we follow the arrow. We start there, there's nothing there, there's something here. So we go to 3p6. And then the next thing we fill out is 4s2. So then we go to 4s2. And then what's the very next thing we fill out? We have to go back to the top. We come here and then we fill out 3d. And then how many electrons do we have left to fill out? So we're going to be in 3d. So 3d. And how many have we used so far? 2 plus 2 is 4. 4 plus 6 is 10. 10 plus 2 is 12. 18. 20. We've used 20, so we have 8 more electrons to configure. And the 3d subshell can fit the 8 we need, so we have 3d8. And there you go, you've got the exact same answer that we had when we used the first method. Now I like the first method because you're looking at the periodic table the whole time, so you kind of understand an intuition of where all the elements are. And you also don't have to keep remembering, OK, how many have I used up as I filled the shells? Right? Here you have to say, i used two, then I used two more. And you have to draw this kind of elaborate diagram. Here you can just use the periodic table. And the important thing is you can work backwards. Here there's no way of just eyeballing this and saying, OK, our most energetic electrons are going to be and our highest energy shell is going to be 4s2. There's no way you could get that out of this without going through this fairly involved process. But when do you use this method, you can immediately say, OK, if I'm worried about element Zr, right here. If I'm worried about element Zr. I could go through the whole exercise of filling out the entire electron configuration. But usually the highest shell, or the highest energy electrons, are the ones that matter the most. So you immediately say, OK, I'm filling in 2 d there, but remember, d, you go one period below. So this is 4d2. Right? Because the period is five. So you say, 4d2. 4d2. And then, before that, you filled out the 5s2 electrons. The 5s2 electrons. And then you could keep going backwards. And you filled out the 4p6. 4p6. And then, before you filled out the 4p6. then you had 10 in the d here. But what is that? It's in the fourth period, but d you subtract one from it, so this is 3d10. So 3d10. And then you had 4s2. This is getting messy. Let me just write that. So you have 4d2. That's those two there. Then you have 5s2. 5s2. Then we had 4p6. That's over here. Then we had 3d10. Remember, 4 minus 1, so 3d10. And then you had 4s2. And you just keep going backwards like that. But what's nice about going backwards is you immediately know, OK, what electrons are in my highest energy shell? Well I have this five as the highest energy shell I'm at. And these two that I filled right there, those are actually the electrons in the highest energy shell. They're not the highest energy electrons. These are. But these are kind of the ones that have the highest probability of being furthest away from the nucleus. So these are the ones that are going to react. And these are the ones that matter for most chemistry purposes. And just a little touchpoint here, and this isn't covered a lot, but we like to think that electrons are filling these buckets, and they stay in these buckets. But once you fill up an atom with electrons, they're not just staying in this nice, well-behaved way. They're all jumping between orbitals, and doing all sorts of crazy, unpredictable things. But this method is what allows us to at least get a sense of what's happening in the electron. For most purposes, they do tend to react or behave in ways that these orbitals kind of stay to themselves. But anyway, the main point of here is really just to teach you how to do electron configurations, because that's really useful for later on knowing how things will interact. And what's especially useful is to know what electrons are in the outermost shell, or what are the valence electrons." + }, + { + "Q": "Should their be a multiplication symbol in between 4 and 5??", + "A": "Use of parentheses are one way of showing multiplication, so you do not need a second symbol, that would be extraneous (not needed).", + "video_name": "GiSpzFKI5_w", + "transcript": "We're asked to simplify 8 plus 5 times 4 minus, and then in parentheses, 6 plus 10 divided by 2 plus 44. Whenever you see some type of crazy expression like this where you have parentheses and addition and subtraction and division, you always want to keep the order of operations in mind. Let me write them down over here. So when you're doing order of operations, or really when you're evaluating any expression, you should have this in the front of your brain that the top priority goes to parentheses. And those are these little brackets over here, or however Those are the parentheses right there. That gets top priority. Then after that, you want to worry about exponents. There are no exponents in this expression, but I'll just write it down just for future reference: exponents. One way I like to think about it is parentheses always takes top priority, but then after that, we go in descending order, or I guess we should say in-- well, yeah, in descending order of how fast that computation is. When I say fast, how fast it grows. When I take something to an exponent, when I'm taking something to a power, it grows really fast. Then it grows a little bit slower or shrinks a little bit slower if I multiply or divide, so that comes next: multiply or divide. Multiplication and division comes next, and then last of all comes addition and subtraction. So these are kind of the slowest operations. This is a little bit faster. This is the fastest operation. And then the parentheses, just no matter what, always take priority. So let's apply it over here. Let me rewrite this whole expression. So it's 8 plus 5 times 4 minus, in parentheses, 6 plus 10 divided by 2 plus 44. So we're going to want to do the parentheses first. We have parentheses there and there. Now this parentheses is pretty straightforward. Well, inside the parentheses is already evaluated, so we could really just view this as 5 times 4. So let's just evaluate that right from the get go. So this is going to result in 8 plus-- and really, when you're evaluating the parentheses, if your evaluate this parentheses, you literally just get 5, and you evaluate that parentheses, you literally just get 4, and then they're next to each other, so you multiply them. So 5 times 4 is 20 minus-- let me stay consistent with the colors. Now let me write the next parenthesis right there, and then inside of it, we'd evaluate this first. Let me close the parenthesis right there. And then we have plus 44. So what is this thing right here evaluate to, this thing inside the parentheses? Well, you might be tempted to say, well, let me just go left to right. 6 plus 10 is 16 and then divide by 2 and you would get 8. But remember: order of operations. Division takes priority over addition, so you actually want to do the division first, and we could actually write it here like this. You could imagine putting some more parentheses. Let me do it in that same purple. You could imagine putting some more parentheses right here to really emphasize the fact that you're going to do the division first. So 10 divided by 2 is 5, so this will result in 6, plus 10 divided by 2, is 5. 6 plus 5. Well, we still have to evaluate this parentheses, so this results-- what's 6 plus 5? Well, that's 11. So we're left with the 20-- let me write it all down again. We're left with 8 plus 20 minus 6 plus 5, which is 11, plus 44. And now that we have everything at this level of operations, we can just go left to right. So 8 plus 20 is 28, so you can view this as 28 minus 11 plus 44. 28 minus 11-- 28 minus 10 would be 18, so this is going to be 17. It's going to be 17 plus 44. And then 17 plus 44-- I'll scroll down a little bit. 7 plus 44 would be 51, so this is going to be 61. So this is going to be equal to 61. And we're done!" + }, + { + "Q": "Aren't there 2 forces her, at least? 1. The car pushes against the wall of the loop, creating a centripetal force that pushes inward, and 2. To this we add the pull exercised by gravity? If the loop was horizontal, and the wheels on the side, the car would also rotate against the loop, without the need for gravity, so gravity is an amplifying force here, and the push inwards needs to be compounded?", + "A": "The push inward needs to be biggest at the bottom, where it has to also counteract gravity to get sufficient centripetal force, and it can be lowest at the top, where gravity is providing a lot of the centripetal force.", + "video_name": "4SQDybFjhRE", + "transcript": "What I want to do now is figure out, what's the minimum speed that the car has to be at the top of this loop de loop in order to stay on the track? In order to stay in a circular motion. In order to not fall down like this. And I think we can all appreciate that is the most difficult part of the loop de loop, at least in the bottom half right over here. The track itself is actually what's providing the centripetal force to keep it going in a circle. But when you get to the top, you now have gravity that is pulling down on the car, almost completely. And the car will have to maintain some minimum speed in order to stay in this circular path. So let's figure out what that minimum speed is. And to help figure that out, we have to figure out what the radius of this loop de loop actually is. And it actually does not look like a perfect circle, based on this little screen shot that I got here. It looks a little bit elliptical. But it looks like the radius of curvature right over here is actually smaller than the radius of the curvature of the entire loop de loop. That if you made this into a circle, it would actually be maybe even a slightly smaller circle. But let's just assume, for the sake of our arguments right over here, that this thing is a perfect circle. And it was a perfect circle, let's think about what that minimum velocity would have to be up here at the top of the loop de loop. So we know that the magnitude of your centripetal acceleration is going to be equal to your speed squared divided by the radius of the circle that you are going around. Now at this point right over here, at the top, which is going to be the hardest point, the magnitude of our acceleration, this is going to be 9.81 meters per second squared. And the radius, we can estimate-- I copied and pasted the car, and it looks like I can get it to stack on itself four times to get the radius of this circle right over here. And I looked it up on the web, and a car about this size is going to be about 1.5 meters high from the bottom of the tires to the top of the car. And so it looks like-- just eyeballing it based on these copying and pasting of the cars, that the radius of this loop de loop right over here is 6 meters. So this right over here is 6 meters. So you multiply both sides by 6 meters. Or actually, we could keep it just in the variables. So let me just rewrite it-- just to manipulate it so we can solve for v. We have v squared over r is equal to a. And then you multiply both sides by r. You get v squared is equal to a times r. And then you take the principal square root of both sides. You get v is equal to the principal square root of a times r. And then if we plug in these numbers, this velocity that we have to have in order to stay in the circle is going to be the square root of 9.81 meters per second squared, times 6 meters. And you can verify that these units work out. Meters times meters is meter squared, per second squared. You take the square root of that, you're going to get meters per second. But let's get our calculator out to actually calculate this. So we are going to get the principal square root of 9.81 times 6 meters. It gives us-- now here's our drum roll-- 7.67. I'll just round to three significant digits, 7.67 meters per second squared. And significant digits is a whole conversation, because this is just a very, very rough approximation. I'm not able to measure this that accurately at all. But I get roughly 7 point-- I'll just round, 7.7 meters per second. So this is approximately 7.7 meters per second. And just to give a sense of how that translates into units that we're used to when we're driving cars, we can convert 7.7 meters per second. If we want to say how many meters we go to an hour, well, there's 3,600 seconds in an hour. And then if you want to convert that into kilometers-- this will be in meters-- you divide by 1,000. One kilometer is equal to 1,000 meters. And you see here, the units cancel out. You have meters, meters, seconds, seconds. You're left with kilometers per hour. So let's actually calculate this. And so we get our previous answer. We want to multiply it times 3,600 to figure out how many meters in an hour. And then you divide by 1,000 to convert it to kilometers per hour. So you divide by 1,000. And we get 27.6 kilometers per hour. So this is equal to 27.6 kilometers per hour, which is surprisingly slow. I would have thought it would have to be much, much, much faster. But it turns out, it does not have to be much, much faster. Only 27.6 kilometers per hour. Now the important thing to keep in mind is this is just fast enough, at this point, to maintain the circular motion. But if this were a perfect circle right over here, and you were going at exactly 27.6 kilometers per hour, you would not have much traction with the road. And if you don't have much traction with the road, the car might slip and might not be able to actually maintain its speed. So you definitely want your speed to be a good bit larger than this in order to keep a nice margin of safety-- in order to especially have traction with the actual loop de loop, and to be able to maintain your speed. Now what I want to do in the next video is actually time the car to figure out how long does it take it to do this loop de loop. And we're going to assume that it's a circle. And we're going to figure it out. And we're going to figure out how fast it's actual average velocity was over the course of this loop de loop." + }, + { + "Q": "Why is it called quadratic if it is raised to the second power. Shouldn't it be called bidratic or something?", + "A": "It actually comes from the Latin word quadratum , which means square . The reason lies in geometry: x^2 is the surface of a square with sides of length x.", + "video_name": "eF6zYNzlZKQ", + "transcript": "In this video I want to do a bunch of examples of factoring a second degree polynomial, which is often called a quadratic. Sometimes a quadratic polynomial, or just a quadratic itself, or quadratic expression, but all it means is a second degree polynomial. So something that's going to have a variable raised to the second power. In this case, in all of the examples we'll do, it'll be x. So let's say I have the quadratic expression, x squared plus 10x, plus 9. And I want to factor it into the product of two binomials. How do we do that? Well, let's just think about what happens if we were to take x plus a, and multiply that by x plus b. If we were to multiply these two things, what happens? Well, we have a little bit of experience doing this. This will be x times x, which is x squared, plus x times b, which is bx, plus a times x, plus a times b-- plus ab. Or if we want to add these two in the middle right here, because they're both coefficients of x. We could right this as x squared plus-- I can write it as b plus a, or a plus b, x, plus ab. So in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the x term, or you could say the first degree coefficient there, that's going to be the sum of our a and b. And then the constant term is going to be the product of our a and b. Notice, this would map to this, and this would map to this. And, of course, this is the same thing as this. So can we somehow pattern match this to that? Is there some a and b where a plus b is equal to 10? And a times b is equal to 9? Well, let's just think about it a little bit. What are the factors of 9? What are the things that a and b could be equal to? And we're assuming that everything is an integer. And normally when we're factoring, especially when we're beginning to factor, we're dealing with integer numbers. So what are the factors of 9? They're 1, 3, and 9. So this could be a 3 and a 3, or it could be a 1 and a 9. Now, if it's a 3 and a 3, then you'll have 3 plus 3-- that doesn't equal 10. But if it's a 1 and a 9, 1 times 9 is 9. 1 plus 9 is 10. So it does work. So a could be equal to 1, and b could be equal to 9. So we could factor this as being x plus 1, times x plus 9. And if you multiply these two out, using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x, plus 9. So when you see something like this, when the coefficient on the x squared term, or the leading coefficient on this quadratic is a 1, you can just say, all right, what two numbers add up to this coefficient right here? And those same two numbers, when you take their product, have to be equal to 9. And of course, this has to be in standard form. Or if it's not in standard form, you should put it in that form, so that you can always say, OK, whatever's on the first degree coefficient, my a and b have to add to that. Whatever's my constant term, my a times b, the product has to be that. Let's do several more examples. I think the more examples we do the more sense this'll make. Let's say we had x squared plus 10x, plus-- well, I already did 10x, let's do a different number-- x squared plus 15x, plus 50. And we want to factor this. Well, same drill. We have an x squared term. We have a first degree term. This right here should be the sum of two numbers. And then this term, the constant term right here, should be the product of two numbers. So we need to think of two numbers that, when I multiply them I get 50, and when I add them, I get 15. And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. So what could a and b be? Let's think about the factors of 50. It could be 1 times 50. 2 times 25. Let's see, 4 doesn't go into 50. It could be 5 times 10. I think that's all of them. Let's try out these numbers, and see if any of these add up to 15. So 1 plus 50 does not add up to 15. 2 plus 25 does not add up to 15. But 5 plus 10 does add up to 15. So this could be 5 plus 10, and this could be 5 times 10. So if we were to factor this, this would be equal to x plus 5, times x plus 10. And multiply it out. I encourage you to multiply this out, and see that this is indeed x squared plus 15x, plus 10. In fact, let's do it. x times x, x squared. x times 10, plus 10x. 5 times x, plus 5x. 5 times 10, plus 50. Notice, the 5 times 10 gave us the 50. The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x, plus 50. Let's up the stakes a little bit, introduce some negative signs in here. Let's say I had x squared minus 11x, plus 24. Now, it's the exact same principle. I need to think of two numbers, that when I add them, need to be equal to negative 11. a plus b need to be equal to negative 11. And a times b need to be equal to 24. Now, there's something for you to think about. When I multiply both of these numbers, I'm getting a positive number. I'm getting a 24. That means that both of these need to be positive, or both of these need to be negative. That's the only way I'm going to get a positive number here. Now, if when I add them, I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number, so the fact that their sum is negative, and the fact that their product is positive, tells me that both a and b are negative. a and b have to be negative. Remember, one can't be negative and the other one can't be positive, because the product would be negative. And they both can't be positive, because when you add them it would get you a positive number. So let's just think about what a and b can be. So two negative numbers. So let's think about the factors of 24. And we'll kind of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6. Now, which of these when I multiply these-- well, obviously when I multiply 1 times 24, I get 24. When I get 2 times 11-- sorry, this is 2 times 12. I get 24. So we know that all these, the products are 24. But which two of these, which two factors, when I add them, should I get 11? And then we could say, let's take the negative of both of those. So when you look at these, 3 and 8 jump out. 3 times 8 is equal to 24. 3 plus 8 is equal to 11. But that doesn't quite work out, right? Because we have a negative 11 here. But what if we did negative 3 and negative 8? Negative 3 times negative 8 is equal to positive 24. Negative 3 plus negative 8 is equal to negative 11. So negative 3 and negative 8 work. So if we factor this, x squared minus 11x, plus 24 is going to be equal to x minus 3, times x minus 8. Let's do another one like that. Actually, let's mix it up a little bit. Let's say I had x squared plus 5x, minus 14. So here we have a different situation. The product of my two numbers is negative, right? a times b is equal to negative 14. My product is negative. That tells me that one of them is positive, and one of them is negative. And when I add the two, a plus b, it'd be equal to 5. So let's think about the factors of 14. And what combinations of them, when I add them, if one is positive and one is negative, or I'm really kind of taking the difference of the two, do I get 5? So if I take 1 and 14-- I'm just going to try out things-- 1 and 14, negative 1 plus 14 is negative 13. Negative 1 plus 14 is 13. So let me write all of the combinations that I could do. And eventually your brain will just zone in on it. So you've got negative 1 plus 14 is equal to 13. And 1 plus negative 14 is equal to negative 13. So those don't work. That doesn't equal 5. Now what about 2 and 7? If I do negative 2-- let me do this in a different color-- if I do negative 2 plus 7, that is equal to 5. That worked! I mean, we could have tried 2 plus negative 7, but that'd be equal to negative 5, so that wouldn't have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. So there we have it. We know it's x minus 2, times x plus 7. That's pretty neat. Negative 2 times 7 is negative 14. Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well honed this skill. So let's say we have x squared minus x, minus 56. So the product of the two numbers have to be minus 56, have to be negative 56. And their difference, because one is going to be positive, and one is going to be negative, right? Their difference has to be negative 1. And the numbers that immediately jump out in my brain-- and I don't know if they jump out in your brain, we just learned this in the times tables-- 56 is 8 times 7. I mean, there's other numbers. It's also 28 times 2. It's all sorts of things. But 8 times 7 really jumped out into my brain, because they're very close to each other. And we need numbers that are very close to each other. And one of these has to be positive, and one of these has to be negative. Now, the fact that when their sum is negative, tells me that the larger of these two should probably be negative. So if we take negative 8 times 7, that's equal to negative 56. And then if we take negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there. So when I factor this, this is going to be x minus 8, times x plus 7. This is often one of the hardest concepts people learn in algebra, because it is a bit of an art. You have to look at all of the factors here, play with the positive and negative signs, see which of those factors when one is positive, one is negative, add up to the coefficient on the x term. But as you do more and more practice, you'll see that it'll become a bit of second nature. Now let's step up the stakes a little bit more. Let's say we had negative x squared-- everything we've done so far had a positive coefficient, a positive 1 coefficient on the x squared term. But let's say we had a negative x squared minus 5x, plus 24. How do we do this? Well, the easiest way I can think of doing it is factor out a negative 1, and then it becomes just like the problems we've been doing before. So this is the same thing as negative 1 times positive x squared, plus 5x, minus 24. Right? I just factored a negative 1 out. You can multiply negative 1 times all of these, and you'll see it becomes this. Or you could factor the negative 1 out and divide all of these by negative 1. And you get that right there. Now, same game as before. I need two numbers, that when I take their product I get negative 24. So one will be positive, one will be negative. When I take their sum, it's going to be 5. So let's think about 24 is 1 and 24. Let's see, if this is negative 1 and 24, it'd be positive 23, if it was the other way around, it'd be negative 23. Doesn't work. What about 2 and 12? Well, if this is negative-- remember, one of these has to be negative. If the 2 is negative, their sum would be 10. If the 12 is negative, their sum would be negative 10. Still doesn't work. 3 and 8. If the 3 is negative, their sum will be 5. So it works! So if we pick negative 3 and 8, negative 3 and 8 work. Because negative 3 plus 8 is 5. Negative 3 times 8 is negative 24. So this is going to be equal to-- can't forget that negative 1 out front, and then we factor the inside. Negative 1 times x minus 3, times x plus 8. And if you really wanted to, you could multiply the negative 1 times this, you would get 3 minus x if you did. Or you don't have to. Let's do one more of these. The more practice, the better, I think. All right, let's say I had negative x squared plus 18x, minus 72. So once again, I like to factor out the negative 1. So this is equal to negative 1 times x squared, minus 18x, plus 72. Now we just have to think of two numbers, that when I multiply them I get positive 72. So they have to be the same sign. And that makes it easier in our head, at least in my head. When I multiply them, I get positive 72. When I add them, I get negative 18. So they're the same sign, and their sum is a negative number, they both must be negative. And we could go through all of the factors of 72. But the one that springs up, maybe you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9, doesn't work. That turns into 17. Let me show you that. Negative 9 plus negative 8, that is equal to negative 17. Close, but no cigar. We have 6 and 12. That actually seems pretty good. If we have negative 6 plus negative 12, that is equal to negative 18. Notice, it's a bit of an art. You have to try the different factors here. So this will become negative 1-- don't want to forget that-- times x minus 6, times x minus 12." + }, + { + "Q": "What if, in the second question, DE is known but CD isn't? Can you still solve this?", + "A": "You can, but it would get a little more complicated because if CD = x. then CE = x + DE. so just for fun if you put 4 for DE and all else is the same, you would get 5/8 = x/x+4, cross multiply to get 5(x+4) = 8x, distribute 5x + 20 = 8x, subtract 5x, 20= 3x, divide by 3, x = 20/3. More likely if you were given DE, the numbers would be more compatible to not end up with a fraction.", + "video_name": "R-6CAr_zEEk", + "transcript": "In this first problem over here, we're asked to find out the length of this segment, segment CE. And we have these two parallel lines. AB is parallel to DE. And then, we have these two essentially transversals that form these two triangles. So let's see what we can do here. So the first thing that might jump out at you is that this angle and this angle are vertical angles. So they are going to be congruent. The other thing that might jump out at you is that angle CDE is an alternate interior angle with CBA. So we have this transversal right over here. And these are alternate interior angles, and they are going to be congruent. Or you could say that, if you continue this transversal, you would have a corresponding angle with CDE right up here and that this one's just vertical. Either way, this angle and this angle are going to be congruent. So we've established that we have two triangles and two of the corresponding angles are the same. And that by itself is enough to establish similarity. We actually could show that this angle and this angle are also congruent by alternate interior angles, but we don't have to. So we already know that they are similar. And actually, we could just say it. Just by alternate interior angles, these are also going to be congruent. But we already know enough to say that they are similar, even before doing that. So we already know that triangle-- I'll color-code it so that we have the same corresponding vertices. And that's really important-- to know what angles and what sides correspond to what side so that you don't mess up your, I guess, your ratios or so that you do know what's corresponding to what. So we know triangle ABC is similar to triangle-- so this vertex A corresponds to vertex E over here. It's similar to vertex E. And then, vertex B right over here corresponds to vertex D. EDC. Now, what does that do for us? Well, that tells us that the ratio of corresponding sides are going to be the same. They're going to be some constant value. So we have corresponding side. So the ratio, for example, the corresponding side for BC is going to be DC. We can see it in just the way that we've written down the similarity. If this is true, then BC is the corresponding side to DC. So we know that the length of BC over DC right over here is going to be equal to the length of-- well, we want to figure out what CE is. That's what we care about. And I'm using BC and DC because we know those values. So BC over DC is going to be equal to-- what's the corresponding side to CE? The corresponding side over here is CA. It's going to be equal to CA over CE. This is last and the first. Last and the first. CA over CE. And we know what BC is. BC right over here is 5. We know what DC is. It is 3. We know what CA or AC is right over here. CA is 4. And now, we can just solve for CE. Well, there's multiple ways that you could think about this. You could cross-multiply, which is really just multiplying both sides by both denominators. So you get 5 times the length of CE. 5 times the length of CE is equal to 3 times 4, which is just going to be equal to 12. And then we get CE is equal to 12 over 5, which is the same thing as 2 and 2/5, or 2.4. So this is going to be 2 and 2/5. And we're done. We were able to use similarity to figure out this side just knowing that the ratio between the corresponding sides are going to be the same. Now, let's do this problem right over here. Let's do this one. Let me draw a little line here to show that this is a different problem now. This is a different problem. So in this problem, we need to figure out what DE is. And we, once again, have these two parallel lines like this. And so we know corresponding angles are congruent. So we know that angle is going to be congruent to that angle because you could view this as a transversal. We also know that this angle right over here is going to be congruent to that angle right over there. Once again, corresponding angles for transversal. And also, in both triangles-- so I'm looking at triangle CBD and triangle CAE-- they both share this angle up here. Once again, we could have stopped at two angles, but we've actually shown that all three angles of these two triangles, all three of the corresponding angles, are congruent to each other. And once again, this is an important thing to do, is to make sure that you write it in the right order when you write your similarity. We now know that triangle CBD is similar-- not congruent-- it is similar to triangle CAE, which means that the ratio of corresponding sides are going to be constant. So we know, for example, that the ratio between CB to CA-- so let's write this down. We know that the ratio of CB over CA is going to be equal to the ratio of CD over CE. And we know what CB is. CB over here is 5. We know what CA is. And we have to be careful here. It's not 3. CA, this entire side is going to be 5 plus 3. So this is going to be 8. And we know what CD is. CD is going to be 4. And so once again, we can cross-multiply. We have 5CE. 5 times CE is equal to 8 times 4. 8 times 4 is 32. And so CE is equal to 32 over 5. Or this is another way to think about that, 6 and 2/5. Now, we're not done because they didn't ask for what CE is. They're asking for just this part right over here. They're asking for DE. So we know that this entire length-- CE right over here-- this is 6 and 2/5. And so DE right over here-- what we actually have to figure out-- it's going to be this entire length, 6 and 2/5, minus 4, minus CD right over here. So it's going to be 2 and 2/5. 6 and 2/5 minus 4 and 2/5 is 2 and 2/5. DE is 2 and 2/5." + }, + { + "Q": "At 1:36 what does he mean by \"third variable S\"", + "A": "I think he meant to say the fourth variable (three sides of the triangle being the first three), but the ordinal number used to refer to it is irrelevant. s = the semi-perimeter of the triangle. That would be (a + b + c) / 2.", + "video_name": "-YI6UC4qVEY", + "transcript": "I think it's pretty common knowledge how to find the area of the triangle if we know the length of its base and its height. So, for example, if that's my triangle, and this length right here-- this base-- is of length b and the height right here is of length h, it's pretty common knowledge that the area of this triangle is going to be equal to 1/2 times the base times the height. So, for example, if the base was equal to 5 and the height was equal to 6, then our area would be 1/2 times 5 times 6, which is 1/2 times 30-- which is equal to 15. Now what is less well-known is how to figure out the area of a triangle when you're only given the sides of the triangle. When you aren't given the height. So, for example, how do you figure out a triangle where I just give you the lengths of the sides. Let's say that's side a, side b, and side c. a, b, and c are the lengths of these sides. How do you figure that out? And to do that we're going to apply something called Heron's Formula. And I'm not going to prove it in this video. I'm going to prove it in a future video. And really to prove it you already probably have the tools necessary. It's really just the Pythagorean theorem and a lot of hairy algebra. But I'm just going to show you the formula now and how to apply it, and then you'll hopefully appreciate that it's pretty simple and pretty easy to remember. And it can be a nice trick to impress people with. So Heron's Formula says first figure out this third variable S, which is essentially the perimeter of this triangle divided by 2. a plus b plus c, divided by 2. Then once you figure out S, the area of your triangle-- of this triangle right there-- is going to be equal to the square root of S-- this variable S right here that you just calculated-- times S minus a, times S minus b, times S minus c. That's Heron's Formula right there. This combination. Let me square it off for you. So that right there is Heron's Formula. And if that looks a little bit daunting-- it is a little bit more daunting, clearly, than just 1/2 times base times height. Let's do it with an actual example or two, and actually see this is actually not so bad. So let's say I have a triangle. I'll leave the formula up there. So let's say I have a triangle that has sides of length 9, 11, and 16. So let's apply Heron's Formula. S in this situation is going to be the perimeter divided by 2. So 9 plus 11 plus 16, divided by 2. Which is equal to 9 plus 11-- is 20-- plus 16 is 36, divided by 2 is 18. And then the area by Heron's Formula is going to be equal to the square root of S-- 18-- times S minus a-- S minus 9. 18 minus 9, times 18 minus 11, times 18 minus 16. And then this is equal to the square root of 18 times 9 times 7 times 2. Which is equal to-- let's see, 2 times 18 is 36. So I'll just rearrange it a bit. This is equal to the square root of 36 times 9 times 7, which is equal to the square root of 36 times the square root of 9 times the square root of 7. The square root of 36 is just 6. This is just 3. And we don't deal with the negative square roots, because you can't have negative side lengths. And so this is going to be equal to 18 times the square root of 7. So just like that, you saw it, it only took a couple of minutes to apply Heron's Formula, or even less than that, to figure out that the area of this triangle right here is equal to 18 square root of seven. Anyway, hopefully you found that pretty neat." + }, + { + "Q": "Cant mushy or sticky go in material?\nThey are both physical traits.....", + "A": "I m a bit confused about these categories myself, but I think that the material category refers more to what something is made of, such as the wooden spoon or the glass vase. Mushy and sticky are more relative terms, and so they would fit in the opinion category. But I totally understand your confusion, though. I had never even heard of DOSASCOMP before watching this video. Also, WOW! 5 million energy points? That s impressive.", + "video_name": "OfxiZdsqGeA", + "transcript": "- [David] Hey grammarians, hey Paige. - [Paige] Hi David. - [David] Hey, so Paige, I went to the grocery store yesterday and I got this apple and I put it in the fridge. And this morning, when I opened the fridge, the apple was all gross and sticky and mushy and I really wanna write a letter to the grocery store and say, \"Hey, you sold me a gross apple.\" - [Paige] Yeah. - [David] But I'm stuck. - [Paige] Okay? - [David] I feel like I ought to put a comma in here. So, here's the sentence I've got so far. \"You sold me a mushy, sticky apple!\" - [Paige] Gross. - [David] Gross. - [David] Unacceptable. - [Paige] Totally. - [David] \"So, I would like my $1.38 back.\" (Paige laughs) But in another draft of the letter, I wrote, \"You sold me a mushy green apple!\" - [Paige] Okay, so mushy, sticky or mushy green. - [David] Yeah, and I feel like, and I don't know how to put this into words yet, but we're gonna put it into words 'cause that's our job, (Paige laughs) whether or not there should be a comma between mushy and green or mushy and sticky, 'cause these feel different to me. - [Paige] Right, right. Yeah, so there's actually a couple tests that you can do on these sentences to see if you need a comma between the adjectives or not. - [David] So, Paige, what distinguishes a pair like mushy and sticky from mushy and green? - [Paige] There's this sort of hierarchy or order that you put adjectives in when you have multiple of them in a sentence, and it is called DOSASCOMP. - [David] DOSASCOMP? All together now, D - [Both] Determiner! - [David] O. - [Both] Opinion! - [David] S. - [Both] Size! - [David] A. - [Both] Age! - [David] S again. - [Both] Shape! - [David] C. - [Both] Color! - [David] O. - [Both] Origin! - [David] M. - [Both] Material! - [David] And P. - [Both] Purpose! - [David] Oh, okay. So, mushy and sticky are both opinion adjectives. - [Paige] Yeah. - [David] So, these are kinda in the same class, whereas mushy and green, that's an opinion and a color. - [Paige] Yeah, they're in two different classes. - [David] Right, so DOSASCOMP helps determine adjective order, right? - [David] So, it's the order in which you would use, if in which you were stacking these adjectives you would use them in this DOSASCOMP order. - [Paige] Right, so like in the second sentence, mushy is an opinion, like you said, and green is a color. And in DOSASCOMP, opinion comes before color, so that's the order that you write them in. So, like in the first sentence, when you have mushy and sticky, which are both opinion adjectives, these are called coordinate adjectives. They're in the same category of DOSASCOMP. And if you wanted to, you could reverse the order. You could say, \"Sticky, mushy apple.\" - [David] Well, let's try that. - [Paige] Yeah. - [David] Looks good to me. Sticky, mushy, mushy, sticky. - [Paige] Same thing. - [David] Okay, so if we're trying to determine whether or not we have coordinate adjectives, I guess the first thing we would do is consult DOSASCOMP, right? So that's step one. And what is a dosa, Paige? - [Paige] Oh, it's like a pancake, right? - [David] Yeah, it's like a South Indian pancake. - [Paige] That's pretty cool. - [David] And to scomp is a word we made up that means, \"to eat.\" - [Paige] Sure, eat some pancakes. - [David] Scomp on 'em. So, that's step one. Step two, try the reversal method. - [Paige] Right, and that's like changing it from mushy, sticky to sticky, mushy. - [David] And step three is stick an \"and\" in there. - [Paige] Mm-hmm. If you can reverse the order of the adjectives and you can put \"and\" in between the two of them, then they're coordinate adjectives. - [David] And if they're coordinate adjectives, you need to separate them with a comma. - [Paige] Exactly. And that's why, in the case of the second sentence, with mushy green apple, you don't put a comma between them. They're in different categories in DOSASCOMP, and so they must stay in that order and there's no need for a comma. - [David] Thanks, Paige. I'm gonna get back to drafting my angry letter. (Paige laughs) You can learn anything. - [David] David out. - [Paige] Paige out." + }, + { + "Q": "How can you tell if an angle is going clockwise or counterclockwise if there is no angle arc there to show?", + "A": "You can t unless there is some indication of the direction it is going. Later on when you deal with more complicated mathematics angles usually go counterclockwise.", + "video_name": "wJ37GJyViU8", + "transcript": "This is the video for the measuring angles module because, clearly, at the time that I'm doing this video, there is no video for the measuring angles module. And this is a pretty neat module. This was made by Omar Rizwan, one of our amazing high school interns that we had this past summer. This is the summer of 2011. And what it really is, is it makes you measure angles. And he made this really cool protractor tool here so that you actually use this protractor to measure the angles there. And so the trick here is you would actually measure it the way you would measure any angle using an actual physical protractor. You'd want to put the center of the protractor right at the vertex of where the two lines intersect. You can view it as the vertex of the angle. And then you'd want to rotate it so that, preferably, this edge, this edge at 0 degrees, is at one of these sides. So let's do it so that this edge right over here is right along this line. So let me rotate it. So then-- I've got to rotate it a little bit further, maybe No, that's too far. So that looks about right. And then if you look at it this way, you can see that the angle-- and I don't have my Pen tool here. I'm just using my regular web browser-- if you look at the angle here, you see that the other line goes to 130 degrees. So this angle that we need to measure here is 130 degrees, assuming you can read sideways. So that is 130 degrees. Let me check my answer. Very good, I got it right. It would have been embarrassing if I didn't. Let's do the next question. I'll do a couple of examples like this. So once again, let us put the center of the protractor right at the vertex right over there. And let's get this 0 degrees side to be on one of these sides so that this angle will be within the protractor. So let me rotate it this way. And this really is pretty cool what Omar did with this module. So let's see. Let's do it one more time. That's too far. And so that looks about right. And then you can see that the angle right over here, if we look at where the other line points to, it is 40 degrees. Check answer-- very good. Let's do another one. This is fun. So let's get our protractor right over there. And you don't always have to do it in that same order. You could rotate it first so that the 0 degrees is-- and what you want to do is you want to rotate the 0 degrees to one of the sides so that the angle is still within the protractor. So let's rotate it around. So if you did it like that-- so you don't always have to do it in that same order. Although I think it's easier to rotate it when you have the center of the protractor at the vertex of the angle. So we have to rotate it a little bit more. So 0 degrees is this line. And then as we go further and further up, I guess, since this is on its side, it looks like this other line gets us to 150 degrees. And hopefully you're noticing that the higher the degrees, the more open this angle is. And so this one right over here is 150 degrees. And so let's do that-- 150. Let's do one more. Now let me show you what not to do. So what not to do is-- so you could put the center right over there. And you might say, OK, let me make the 0 go right over on this side, right over here. So if you did that, notice your angle would not be within the protractor. So you won't be able to measure it. And what you're attempting to do is measure this outer angle over here, which is an angle, but that's not the angle that this question is asking us to measure. This little arc over here is telling us that that's the angle that we need to measure. So that arc has to be within the protractor. So let's rotate this protractor a little bit more. I overdid it. And so this looks like this is 0 degrees, and then this right over here is 60 degrees. 60 degrees-- we got that one right, too. So hopefully that helps you with this module. It's kind of fun." + }, + { + "Q": "Can we simply break the numerator (X^3 -1) as X^2 (X-1) ??\nThat way also we can get a X-1 to cancel out with the denominator right?", + "A": "Consider your suggestion, and do the math. You want to rewrite (x\u00c2\u00b3-1) as x\u00c2\u00b2(x-1). That means you are asserting that (x\u00c2\u00b3-1) = x\u00c2\u00b2(x-1). When you distribute the x\u00c2\u00b2 in x\u00c2\u00b2(x-1) you get x\u00c2\u00b3-x\u00c2\u00b2 . . . Oops x\u00c2\u00b3-1 \u00e2\u0089\u00a0 x\u00c2\u00b3-x\u00c2\u00b2. It was an interesting observation, and an opportunity to test your intuition, which you should have tried. Getting into the habit of investigating your ideas and trying them out is an important aspect to getting really good at math! Keep Studying!", + "video_name": "rU222pVq520", + "transcript": "Let's try to find the limit as x approaches 1 of x to the third minus 1 over x squared minus 1. And at first when you just try to substitute x equals 1, you get 0/0 1 minus 1 over 1 minus 1. So that doesn't help us. So let's see if we can try to simplify this in some way. So you might immediately recognize-- so let's rewrite this expression right over here so it's x to the third minus 1 over x squared minus 1. This on the bottom immediately jumps out as a difference of squares. So we know on the bottom that this could be factored as x minus 1 times x plus 1. And so if somehow this thing on the top also has an x minus 1 as a factor, then that x minus 1 will cancel with this, and then we're not going to have an issue of dividing by 0. The reason why I care about the x minus 1 term is that this is what's making our denominator equal 0. When you say x equals 1, you have 1 minus 1 times 1 plus 1. So 0 times 2, it's this 0 that's making our denominator 0. So if we can have an x minus 1 up here, then we can cancel these out for any x not equal to 1. And then we might have a much simpler thing to find the limit of. So let's think about whether x to the third minus 1 is the product of x minus 1 and something else. And to do that we can do a little bit of algebraic long division. Some of you guys might already recognize a pattern here, but we'll try to do-- well, let's divide x minus 1 into it to see whether it divides evenly into x to the third minus 1. So x minus 1-- we just look at the highest degree term-- x goes into x to the third x squared times. Goes x squared times. Actually, let me do it this way so that way we can keep track of the place. So this would be x-- this would be the second degree place, first degree place, and this would be the constant. So x to the third minus 1. x goes into x to the third x squared times. x squared times x is x to the third. x squared times negative 1 is minus x squared. And now we're going to want to subtract this. So we are then left with x squared. x goes into x squared x times plus x. x times x is x squared. x times minus 1 is minus x. And once again we're going to subtract this. We'll swap the signs, negative and positive. And so these cancel out, and we're left with x. And then we bring down a minus 1. x minus 1 goes into x minus 1 exactly one time. 1 times x minus 1 is x minus 1. And then you subtract, and then you have no remainder. So this numerator right over here can be factored as x minus 1 times x squared plus x plus 1. And so we can say that this is the same exact thing. We can have these cancel out if we assume x does not equal 1. So that is equal to x squared plus x plus 1 over x plus 1, for x does not equal 1. And that's completely fine, because we're not evaluating x equals 1. We're evaluating as x approaches 1. So this is going to be the same thing as the limit as x approaches 1 of x squared plus x plus 1 over x plus 1. And now this is much easier to find. You could literally just say, well, what happens as we get right to x equals 1? Then you have 1 squared, which is 1 plus 1 plus 1, which is 3, over 1 plus 1, which is 2. So we get that equaling 3/2." + }, + { + "Q": "I have a question about demand curves in general. If the quantity supplied cannot be increased, is the price more likely to go up?", + "A": "I don t know a lot about microeconomics, but i do know that if the demand for a product is large, and the quantity of the product cannot be increased as you said, the product will become more expensive. For example, if a new Samsung phone was released and it became crazy popular, but there were only 20,000 of them manufactured, the price would skyrocket because of the limited quantity. Not to mention there would be huge fights at stores.", + "video_name": "do1HDIdfQkU", + "transcript": "So we've been going through all of the other things that we were assuming are held constant in order to be moving along one demand curve. And now let's list a few other. And before I do any more of them, let's talk about the ones we already talked about. So one, we said that one of the things we held constant-- let me write this down. So held constant. One of the things that we held constant to move along one demand curve for the demand itself to not shift, for the curve to not shift, is price of related goods. The other thing we assumed that's being held constant is price expectations for our good. And now we'll list a couple of them that are fairly intuitive, but you'll see in the next few videos that there are often special cases even to this. So the other thing that we've been holding constant to stay on one demand curve is income. And this one is fairly intuitive. What happens if everyone's income were to increase? And in real terms, it were to actually increase. Well then, all of a sudden, they have more disposable income, maybe to spend on something like e-books. And so for any given price point, the demand would increase. And so it would increase the demand. And once again, when we talk about increasing demand, we're talking about shifting the entire curve. We're not talking about a particular quantity of demand. So income goes up, then it increases demand. Demand goes up. And remember, when we're talking about when demand goes up, we're talking about the whole curve shifting to the right. At any given price point, we are going to have a larger quantity demanded. So the whole curve, this whole demand schedule would change. And likewise if income went down, demand would go down. And we're going to see in a future video-- it's actually quite interesting-- that's not always the case. This is only true for normal goods. And in a future video we'll see goods called inferior goods where this is not necessarily the case. Or by definition for an inferior good, it would not be the case. Now the other ones that are somewhat intuitive are population-- once again, if population goes up, obviously, at any given price point, more people will want it. So it would shift the demand curve to the right, or it would increase demand. If population were to go down, it would decrease demand, which means shifting the whole curve to the left. And then the last one we'll talk about-- and remember, we're holding all of these things constant in order for demand not to change. The last thing is just preferences. We're assuming that people's tastes and preferences don't change while we move along a specific demand curve. If preferences actually change, then it will change the curve. So for example, if all of a sudden, the author of the book is on some very popular talk show that tells everyone that this is the best book that was ever written, then preferences would go up, and that would increase the total demand. At any given price point, more people will be willing to buy the book. If, on the other hand, on that same talk show, it turns out that they do an expose on the author having this sordid past, and the author plagiarized the whole book, then the demand will go down. The entire curve, regardless of the price point-- at any given price point, the quantity demanded will actually go down." + }, + { + "Q": "So the video 'Price of Related Products and Demand' comes before 'Change in Expected Future Prices and Demand'?", + "A": "yes change in expected future prices and demand will effect price of related products and demand", + "video_name": "-oClpRv7msg", + "transcript": "We've talked a little bit about the law of demand which tells us all else equal, if we raise the price of a product, then the quantity demanded for that product will go down. Common sense. If we lower the price, than the quantity demanded will go up, and we'll see a few special cases for this. But what I want to do in this video is focus on these other things that we've been holding equal, the things that allow us to make this statement, that allow us to move along this curve, and think about if we were to change one of those things, that we were otherwise considering equal, how does that change the actual curve? How does that actually change the whole quantity demanded price relationship? And so the first of these that I will focus on, the first is the price of competing products. So if you assume that the price of-- actually I shouldn't say competing products, I'll say the price of related products, because we'll see that they're not competing. The price of related products is one of the things that we're assuming is constant when we, it's beheld equal when we show this relationship. We're assuming that these other things aren't changing. Now, what would happen if these things changed? Well, imagine we have, say, other ebooks-- books is price-- price goes up. The price of other ebooks go up. So what will that do to our price quantity demanded relationship? If other ebooks prices go up, now all of a sudden, my ebook, regardless of what price point we're at, at any of the price points, my ebook is going to look more desirable. At $2, it's more likely that people will want it, because the other stuff's more expensive. At $4 more people will want it, at $6 more people will want it, $8 more people will want it, at $10 more people will want it. So if this were to happen, that would actually shift the entire demand curve to the right. So it would start to look something like this. That is scenario one. And these other ebooks, we can call them substitutes for my product. So this right over here, these other ebooks, these are substitutes. People might say, oh, you know, that other book looks kind of comparable, if one is more expensive or one is cheaper, maybe I'll read one or the other. So in order to make this statement, in order to stay along this curve, we have to assume that this thing is constant. If this thing changes, this is going to move the curve. If other ebooks prices go up, it'll probably shift our curve to the right. If other ebooks prices go down, that will shift our entire curve to the left. So this is actually changing our demand. It's changing our whole relationship. So it's shifting demand to the right. So let me write that. So this is going to shift demand. So the entire relationship, demand, to the right. I really want to make sure that you have this point clear. When we hold everything else equal, we're moving along a given demand curve. We're essentially saying the demand, the price quantity demanded relationship, is held constant, and we can pick a price and we'll get a certain quantity demanded. We're moving along the curve. If we change one of those things, we might actually shift the curve. We'll actually change this demand schedule, which will change this curve. Now, there other related products, they don't just have to be substitutes. So, for example, let's think about scenario two. Or maybe the price of a Kindle goes up. Let me write this this way. Kindle's price goes up. Now, the Kindle is not a substitute. People don't either buy an ebook or they won't either buy my ebook or a Kindle. Kindle is a compliment. You actually need a Kindle or an iPad or something like it in order to consume my ebook. So this right over here is a complement. So if a complement's price becomes more expensive, and this is one of the things people might use to buy my book, then it would actually, for any given price, lower the quantity demanded. So in this situation, if my book is $2, since fewer people are going to have Kindles, or since maybe they used some of their money already to buy the Kindle, they're going to have less to buy my book or just fewer people will have the Kindle, for any given price is going to lower the quantity demanded. And so it'll essentially will shift, it'll change the entire demand curve will shift the demand curve to the left. So this right over here is scenario two. And you could imagine the other way, if the Kindle's price went down, then that would shift my demand curve to the right. If the price of substitutes went down, then that would shift my entire curve to the left. So you can think about all the scenarios, and actually I encourage you to. Think about drawing yourself, think about for products, that could be an ebook or could be some other type of product, and think about what would happen. Well, one, think about what the related products are, the substitutes and potentially complements, and then think about what happen as those prices change. And always keep in mind the difference between demand, which is this entire relationship, the entire curve that we can move along if we hold everything else equal and only change price, and quantity demanded, which is a particular quantity for our particular price holding everything else equal." + }, + { + "Q": "why is the soviet union so dangerous in the winter", + "A": "It is because the climate of Soviet Union is so cold that the army can t live.", + "video_name": "X3bqQI7-sCg", + "transcript": "- [Voiceover] As we've already seen in the last few videos, with the war officially starting in September of 1939, the Axis powers get momentum through the end of 1939, all the way into 1940. That was the last video that we covered and that takes us to 1941 and what we're gonna see in 1941 which is the focus of this video is that the Axis powers only seem to gain more momentum. Because of all of that momentum they perhaps gets a little bit overconfident and stretch themselves or begin to stretch themselves too thin. So let's think about what happens in 1941. So, if we talk about early 1941 or the Spring of 1941, in March, Bulgaria decides to join the Axis powers. You can imagine there's a lot of pressuring applied to them and they kind of see where the momentum is. Let's be on that side. Bulgaria joins the Axis and then in North Africa you might remember that in 1940, the Allies, in particular, the British, were able to defeat the Italians and push them back into Libya but now in March of 1941, the Italians get reinforcements, Italian reinforcements and also German reinforcements under the command of Rommel the Desert Fox, famous desert commander and they are able to push the British back to the Egyptian border and they also take siege of the town of Tobruk. Now, you might have noticed something that I just drew. The supply lines in the North Africa campaign are very, very, very long and that's part of the reason why there's one side. One side has supply lines and as they start to make progress and as the Allies make progress and push into Libya, their supply lines got really long and so the other side has an easier timely supply. Then as the Axis pushes the Allies back into Egypt, then their supply lines get really long and the other side...it makes it easier for them to resupply and so North Africa is kind of defined by this constant back and forth. But, by early 1941, it looks like the Axis is on the offensive, able to push the British back into Egypt lay siege to the town of Tobruk. So, let me write this down as North Africa. So, I'll just say North Africa over here or I'll could say Rommel in North Africa pushing the British back. And then we can start talking about what happens in the Balkans and this is still in Spring as we go into April of 1941 and just as a little bit of background here, and frankly I should have covered it a couple of videos ago. As far back as 1939, actually before World War II officially started, in Spring of 1939, Italy actually occupies Albania so this actually should have already been red. This is in 1939 that this happens and then at the end of 1940, Italy uses Albania as a base of operation to try to invade Greece but they are pushed back. Actually one of the reasons why the British we able to be pushed back in North Africa is after they were successful against the Italians, most of the bulk of the British forces we sent to Greece to help defend Greece at the end of 1940. So, in 1939, Albania gets taken over by Italy and at the end of 1940...October 1940, Greece is invaded by Italy but they are then pushed back but to help the Greeks, the Allies send many of the forces that were in North Africa after they were successful against the Italians in Libya. Now, as we go into April of 1941... that was all background, remember Albania before the war started in April 1939, October 1940 was Italy's kind of first push into Greece and it was unsuccessful. Then the Greeks get support from the Allies in North Africa and now as we go into 1941, the Germans start supporting and really take charge in Balkans and in Greece and so with the help of the Germans the Axis is able to take over Yugoslavia and Greece and start aerial bombardment of Crete. So, once again, we're not even halfway through the year in 1941 and we see a huge swath of Europe is under the control of the Axis powers. And now we go into the summer of 1941. This is actually a pivotal move, what's about to happen. Now you can imagine that the Axis powers, in particular, Hitler, are feeling pretty confident. We are only about that far into the war. So we're not even two years into the war yet and it looks like the Axis is going to win. Now you might remember that they have a pact with the Soviet Union. Hey, we're gonna split a lot of Eastern Europe into our spheres of influence so to speak, but now Hitler's like, well, I think I'm ready to attack and when you attack the Soviet Union really matters. You do not want to attack the Soviet Union in the winter...or Russia in the winter. Russia's obviously at the heart of the Soviet Union. That something that Napoleon learned. Many military commanders have learnt. You do not want to be fighting in Russia over the winter, so summer of 1941, Hitler figures, hey, this is the Axis chance. And so, in June, he decides to attack the Soviet Union. So, this is a very, very, very bold move because now they're fighting the British. Remember, the British are kind of not a joke to be battling out here in Western Europe and now they're going to be taking on the Soviet Union in the east, a major, major world power. But at first, like always, it seems like it's going well for the Germans. By September, they're able to push up all the way to Leningrad. So, this is September of 1941 and lay lay siege and begin laying siege to that town. This is kind of a long bloody siege that happens there. So, we're right now, right about there. And most historians would tell you that this was one of the mistakes of Adolph Hitler because now he is stretched very, very, very thin. He has to fight two world powers, Soviet Union and Great Britain and the United States hasn't entered into the war yet and that's what we're about to get into because if we go into Asia it was still in 1941 what happens in July. So, little bit after Hitler decides to start invading the Soviet Union going back on the pact, the non- aggression pact. In July, you could imagine the US, they were never pleased with what's been happening, what the Empire of Japan has been doing in the Pacific, what they've been doing in China, in Manchuria or even in terms of the war in China, the second Sino Japanese War. They weren't happy of the Japanese taking over French Indochina. There's a big world power here, the Empire of Japan. There's a big world power here, the United States, that has a lot of possessions in the Pacific and so, the United States in July of 1941...So remember, this is still all 1941, this is the same year...decides to freeze the assets of Japan and probably the most important part of that was an oil embargo of Japan. This is a big, big deal. Japan is fighting a major conflict with the Chinese. It's kind of flexing it's imperial muscles but it does not have many natural resources in and of itself and in fact, that's one of the reasons why it's trying to colonize other places to get more control of natural resources. And now if it's fighting a war it doesn't have it's own oil resources and now there's an oil embargo of Japan and the United States at the time was major oil producer and even today, it's major oil producer. This was a big deal to the Japanese because some estimates say they only had about two years of reserves and they were fighting a war where they might have to touch their reserves even more. So, you could imagine the Japanese, they want to have their imperial ambitions. They probably want, especially now with this oil embargo, they probably want to take over more natural resources and they probably want to knock out the US or at least keep the US on its heels so the US can't stop Japan from doing what it wants to do. So, all it wants in December 1941, that's over the course of December 7th and 8th, and it gets a little confusing because a lot of this happens across the International Date Line. But over the course of December 7th and 8th, Japan goes on the offensive in a major way in the Pacific. Over the course of several hours, at most, a day, Japan is able to attack Malaya, which is a British possession. It's able to attack Pearl Harbor, where the US Pacific fleet is in hope to knock out the US Pacific fleet so the US will have trouble stopping Japan from doing whatever Japan wants to do. In the US, we focus a lot on Pearl Harbor but this was just one of the attacks in this whole kind of several hours of attacks where Japan went on the offensive. So, we have Malaya, we have Pearl Harbor, we have Singapore, we have Guam, we have (which was the US military base), Wake Island. was a US possession ever since the Spanish American war. You have Hong Kong, which is a British possession and then shortly after that as you get further into December, so this is kind of when you have Japan offensive. Then as you go on into later Decemeber, the kind of real prize for Japan was what we would now call Indonesia but the Dutch East Indies. On this map it says Netherlands East Indies. You have to remember the Netherlands had been overrun. They're the low countries they were already overrun by German forces so the Japanese say hey, look there are a lot of resources here, natural resources, especially oil. Let's go for this and so by the end of 1941, they're also going for the Dutch East Indies and for Burma so you could imagine it's a very aggressive, very, very bold move on Japan but they kind of had imperial ambitions. They were afraid of they access to natural resources so they went for it but obviously one of the major consequence of this is the United States was not happy about this and they were already sympathetic to the Allies. They didn't like what was going on in Europe either. They didn't like what was going on in China and so that causes the United States to enter into World War II on the side of the Allies and then the Axis powers to declare war on the United States, which was a big deal." + }, + { + "Q": "At 7:48, it says there's a decrease in Sn2 mechanism. I understand that, but if a strong nucleophile is put in a polar protic solvent, it becomes a weak nucleophile. Then, shouldn't Sn1 be favoured over Sn2 as a minor product ?", + "A": "A polar protic solvent doesn t make a strong nucleophile into a weak nucleophile.It makes it into a weaker nucleophile. Ethoxide ion is a strong nucleophile in an aprotic solvent. It is weaker in a protic solvent, but it is still a pretty strong nucleophile even there.", + "video_name": "vFSZ5PU0dIY", + "transcript": "- [Instructor] Let's look at elimination versus substitution for a secondary substrate. And these are harder than for a primary or tertiary substrate because all four of these are possible to start with. So, if we look at the structure of our substrate and we say it's secondary, we next need to look at the reagent. So, we have NaCl which we know is Na plus and Cl minus and the chloride anion functions only as a nucleophile. So, we would expect a substitution reaction, nucleophilic substitution. So, E1 and E2 are out. Between SN1 and SN2 with the secondary substrate, we're not sure until we look at the solvent and DMSO is a polar aprotic solvent, which we saw in an earlier video, favors an SN2 mechanism. So, SN1 is out and we're gonna think about our chloride anion functioning as a nucleophile. So, let me draw it in over here. So, this is with a negative one formal charge. And an SN2 mechanism are nucleophile attacks the same time we get loss of a leaving group and our nucleophile is going to attack this carbon in red. So, we're gonna form a bond between the chlorine and this carbon in red and when the nucleophile attacks, we also get loss of our leaving group. So, these electrons come off onto the oxygen and we know that tosylate is a good leaving group. So, when we draw our product, let's draw this in here, and the carbon in red is this one, we know an SN2 mechanism means inversion of configuration. The nucleophile has to attack from the side opposite of the leaving group. So, we had a wedge here for our leaving groups, so that means we're gonna have a dash for our chlorines. We're gonna put the chlorine right here and that's the product of our SN2 reaction. For our next problem, we have a secondary alkyl halide. So, just looking at our reactions, we can't really rule any out here. So, all four are possible, until we look at our reagent. Now, we saw in an earlier video, that DBN is a strong base, it does not act like a nucleophile. So SN1 and SN2 are out. And a strong base means an E2 reaction. So, E1 is out. Now that we know we're doing an E2 mechanism, let's analyze the structure of our alkyl halide. The carbon that's directly bonded to our halogen is our alpha carbon and the carbons directly bonded to the alpha carbon are the beta carbons. So, I'll just do the beta carbon on the right since they are the same essentially. And we know that our base is gonna take a proton from that beta carbon. So, let me just draw in a hydrogen here. And DBN is a neutral base, so I'll just draw a generic base here. Our base is going to take this proton at the same time these electrons move in to form a double bond and these electrons come off to form our bromide anion. So, our final product is an alkyne and our electrons in magenta in here moved in to form our double bond. we have another secondary alkyl halide, so right now all four of these are possible until we look at our reagent which is sodium hydroxide, Na plus, OH minus, and we know that the hydroxide ion can function as a strong nucleophile or a strong base. So a strong nucleophile makes us think an SN2 reaction and not an SN1. The strong base makes us think about an E2 reaction and not an E1 reaction. Since we have heat, heat favors an elimination reaction over a substitution, so E2 should be the major reaction here. So, when we analyze our alkyl halide, the carbon bonded to the halogen is our alpha carbon and the carbons directly bonded to that would be our beta carbons. So, we have two beta carbons here and let me number this ring. I'm gonna say the alpha carbon is carbon one, I'm gonna go round clockwise, so that's one, two, three, carbon four, carbon five and then carbon six. And next we're going to translate this to our chair confirmation over here. So, carbon one would be this carbon and then carbon two would be this one. This'll be carbon three, four, five and six. The bromine is coming out at us in space at carbon one which means it's going up. So, if I look at carbon one, we would have the bromine going up, which would be up axial. At carbon two, I have a methyl group going away from me in space, so that's going down, so at carbon two we must have a methyl group going down which makes it down axial. So, we care about carbon two. Let me highlight these again. So, we care about carbon two which is a beta carbon. We also care about carbon six which is another beta carbon. So, let's put in the hydrogens on those beta carbons. At carbon two, we would have a hydrogen that's up equatorial and at carbon six we would have a hydrogen that's down axial and one that is up equatorial. So, when we think about our E2 mechanism, we know our strong base is going to take a proton and that proton must be antiperiplanar to our halogen. So, our halogen, let me highlight our halogen here which is bromine, that is in the axial position, so we need to take a proton that is antiperiplanar to that bromine, so that carbon two, and let's look at carbon two first. At carbon two I do not have a hydrogen that's antiperiplanar to my halogen but I do have one at carbon six. It's the one that is down axial. So, our base is gonna take that proton, so let's draw in the hydroxide ion which is a strong base and the hydroxide ion is going to take this proton and then these electrons are gonna move in to form a double bond at the same time we get these electrons coming off onto the bromine to form the bromide ion. So, let's draw the product for this reaction. We would have our ring and a double bond forms between carbon one and carbon six. So, that means a double bond forms in here. And then at carbon two, we still have a methyl group going away from us in space. So, let me draw that in like that. So, the electrons in red, hard to see, but if you think about these electrons in red back here, are gonna move in to form our double bond between what I've labeled as carbon one and carbon six. Let me label those again here. So, carbon one and carbon six. Again not IUPAC nomenclature just so we can think about our product compared to our starting material. So, this would be the major product of our reaction which is an E2 reaction. It would also be possible to get some products from an SN2 mechanism, but since heat is here, an elimination reaction is favored over a substitution. Next we have a secondary alcohol with phosphoric acid and heat. And we saw a lot of these types of problems in the videos on elimination reactions. So, it's not gonna be SN1 or SN2 and we don't have a strong base, so don't think E2, think E1. And our first step would be to protonate our alcohol to form a better leaving group. So phosphoric acid is a source of protons and we're going to protonate this oxygen for our first step. So, let's draw in our ring and we protonate our oxygen, so now our oxygen has two bonds to hydrogen, one lone pair of electrons and a plus one formal charge on the oxygen. So, this lone pair of electrons on the oxygen picked up a proton from phosphoric acid to form this bond. And now we have a better leaving group than the hydroxide ion. These electrons come off onto the oxygen and we remove a bond from this carbon in red which would give us a secondary carbocation. So, let's draw in our secondary carbocation and the carbon in red is this one and that carbon would have a plus one formal charge. So, let me draw in a plus one formal charge here. And now we have water which can function as a weak base in our E1 reaction and take a proton from a carbon next to our carbon with a positive charge. So, let's say this carbon right here. It has two hydrogens on it. I'll just draw one hydrogen in and water functions as a base, takes this proton and these electrons move in to form a double bond. So, let's draw our final product here. We would have a ring, we would have a double bond between these two carbons, so our electrons in, let's use magenta, electrons in magenta moved in to form our double bond. So, our product is cyclohexane. So, a secondary alcohol undergoes an E1 reaction if you use something like sulfuric acid or phosphoric acid and you heat it up. For this reaction we have this secondary alkyl halide reacting with an aqueous solution of formic acid. Formic acid is a weak nucleophile and water is a polar protic solvent. A weak nucleophile and a polar protic solvent should make us think about an SN1 type mechanism because water as a polar protic solvent can stabilize the formation of a carbocation. So, let's draw the carbocation that would result. These electrons would come off onto our bromine and we're taking a bond away from this carbon in red. So, the carbon in red gets a plus one formal charge and let's draw our carbocation. So, we have our benzine ring here. I'll put in my pi electrons and the carbon in red is this one, so that carbon gets a plus one formal charge. This is a secondary carbocation but it's also a benzylic carbocation. So, the positive charge is actually de-localized because of the pi electrons on the ring. So, this is more stable than most secondary carbocations. Next, if we're thinking an SN1 type mechanism, this would be our electrophile, our carbocation is our electrophile and our nucleophile would be formic acid. And we saw in an earlier video how the carbonyl oxygen is actually more nucleophilic than this oxygen. So, a lone pair of electrons on the carbonyl oxygen would attack our carbon in red. And we would end up with, let's go ahead and draw in the result of our nucleophilic attack, and I won't go through all the steps of the mechanisms since I cover this in great detail in an earlier video. So, this is from our SN1, SN2 final summary video. So, let me draw in what we would form in here. So, this would be carbon double bonded to an oxygen and this would be a hydrogen. So, this is our product and this carbon is a chiral center and because this is an SN1 type mechanism and we have planer geometry in our carbocation, our nucleophile can attack from either side and we're gonna end up with a mix of enantiomers. So again, for more details on this mechanism, I skipped a few steps here, please watch the SN1, SN2 final summary video. Next, let's think about what else could possibly happen. So, SN2 is out, we formed a carbocation, E1 is possible because we have a carbocation here and we also have a weak base present. So, our weak base could be something like water, and I'll just draw a generic base in here, and let's draw in a proton on this carbon. So, our base could take this proton here and these electrons would move in to form a double bond. So, another possible product, we would have our benzine ring, so I'll draw that in, and then we would have a double bond. So, another possibility is an E1 mechanism." + }, + { + "Q": "Why do they have so many \"spheres\" like the asthenosphere or the atmosphere", + "A": "sphere is not only the round thing, but this term is used frequently to mean something like realm", + "video_name": "f2BWsPVN7c4", + "transcript": "What I want to do in this video is talk a little bit about plate tectonics. And you've probably heard the word before, and are probably, or you might be somewhat familiar with what it discusses. And it's really just the idea that the surface of the Earth is made up of a bunch of these rigid plates. So it's broken up into a bunch of rigid plates, and these rigid plates move relative to each other. They move relative to each other and take everything that's on them for a ride. And the things that are on them include the continents. So it literally is talking about the movement of these plates. And over here I have a picture I got off of Wikipedia of the actual plates. And over here you have the Pacific Plate. Let me do that in a darker color. You have the Pacific Plate. You have a Nazca Plate. You have a South American Plate. I could keep going on. You have an Antarctic Plate. It's actually, obviously whenever you do a projection onto two dimensions of a surface of a sphere, the stuff at the bottom and the top look much bigger than they actually are. Antarctica isn't this big relative to say North America or South America. It's just that we've had to stretch it out to fill up the rectangle. But that's the Antarctic Plate, North American Plate. And you can see that they're actually moving relative to each other. And that's what these arrows are depicting. You see right over here the Nazca Plate and the Pacific Plate are moving away from each other. New land is forming here. We'll talk more about that in other videos. You see right over here in the middle of the Atlantic Ocean the African Plate and the South American Plate meet each other, and they're moving away from each other, which means that new land, more plate material I guess you could say, is somehow being created right here-- we'll talk about that in future videos-- and pushing these two plates apart. Now, before we go into the evidence for plate tectonics or even some of the more details about how plates are created and some theories as to why the plates might move, what I want to do is get a little bit of the terminology of plate tectonics out of the way. Because sometimes people call them crustal plates, and that's not exactly right. And to show you the difference, what I want to do is show you two different ways of classifying the different layers of the Earth and then think about how they might relate to each other. So what you traditionally see, and actually I've made a video that goes into a lot more detail of this, is a breakdown of the chemical layers of the Earth. And when I talk about chemical layers, I'm talking about what are the constituents of the different layers? So when you talk of it in this term, the top most layer, which is the thinnest layer, is the crust. Then below that is the mantle. Actually, let me show you the whole Earth, although I'm not going to draw it to scale. So if I were to draw the crust, the crust is the thinnest outer layer of the Earth. You can imagine the blue line itself is the crust. Then below that, you have the mantle. So everything between the blue and the orange line, this over here is the mantle. So let me label the crust. The crust you can literally view as the actual blue pixels over here. And then inside of the mantle, you have the core. And when you do this very high level division, these are chemical divisions. This is saying that the crust is made up of different types of elements. Its makeup is different than the stuff that's in the mantle, which is made up of different things than what's inside the core. It's not describing the mechanical properties of it. And when I talk about mechanical properties I'm talking about whether something is solid and rigid. Or maybe it's so hot and melted it's kind of a magma, or kind of a plastic solid. So this would be the most brittle stuff. If it gets warmed up, if rock starts to melt a little bit, then you have something like a magma, or you can view it as like a deformable or a plastic solid. When we talk about plastic, I'm not talking about the stuff that the case of your cellphone is made of. I'm talking about it's deformable. This rock is deformable because it's so hot and it's somewhat melted. It kind of behaves like a fluid. It actually does behave like a fluid, but it's much more viscous. It's much thicker and slower moving than what we would normally associate with a fluid like water. So this a viscous fluid. And then the most fluid would, of course, be the liquid state. This is what we mean when we talk about the mechanical properties. And when you look at this division over here, the crust is solid. The mantle actually has some parts of it that are solid. So the uppermost part of the mantle is solid. Then below that, the rest of the mantle is kind of in this magma, this deformable, somewhat fluid state, and depending on what depth you go into the mantle there are kind of different levels of fluidity. And then the core, the outer level layer of the core, the outer core is liquid, because the temperature is so high. The inner core is made up of the same things, and the temperature is even higher, but since the pressure is so high it's actually solid. So that's why the mantle, crust, and core differentiations don't tell you whether it's solid, whether it's magma, or whether it's really a liquid. It just really tells you what the makeup is. Now, to think about the makeup, and this is important for plate tectonics, because when we talk about these plates we're not talking about just the crust. We're talking about the outer, rigid layer. Let me just zoom in a little bit. Let's say we zoomed in right over there. So now we have the crust zoomed in. This right here is the crust. And then everything below here we're actually talking about the upper mantle. We haven't gotten too deep in the mantle right here. So that's why we call it the upper mantle. Now, right below the crust, the mantle is cool enough that it is also in real solid form. So this right here is solid mantle. And when we talk about the plates were actually talking about the outer solid layer. So that includes both the crust and the solid part of the mantle. And we call that the lithosphere. When people talk about plate tectonics, they shouldn't say crustal plates. They should call these lithospheric plates. And then below the lithosphere you have the least viscous part of the mantle, because the temperature is high enough for the rock to melt, but the pressure isn't so large as what will happen when you go into the lower part of the mantle that the fluid can actually kind of move past each other, although still pretty viscous. This still a magma. So this is still kind of in its magma state. And this fluid part of the mantle, we can't quite call it a liquid yet, but over large periods of time it does have fluid properties. This, that essentially the lithosphere is kind of riding on top of, we call this the asthenosphere. So when we talk about the lithosphere and asthenosphere we're really talking about mechanical layers. The outer layer, the solid layers, the lithosphere sphere. The more fluid layer right below that is the asthenosphere. When we talk about the crust, mantle, and core, we are talking about chemical properties, what are the things actually made up of." + }, + { + "Q": "Why is it not ln|ln|x||?", + "A": "The natural log on the inside is just u , which came from the original expression. You can see that one didn t have an absolute value on it.", + "video_name": "OLO64d4Y1qI", + "transcript": "We are faced with a fairly daunting-looking indefinite integral of pi over x natural log of x dx. Now, what can we do to address this? Is u-substitution a possibility here? Well for u-substitution, we want to look for an expression and its derivative. Well, what happens if we set u equal to the natural log of x? Now what would du be equal to in that scenario? du is going to be the derivative of the natural log of x with respect to x, which is just 1/x dx. This is an equivalent statement to saying that du dx is equal to 1/x. So do we see a 1/x dx anywhere in this original expression? Well, it's kind of hiding. It's not so obvious, but this x in the denominator is essentially a 1/x. And then that's being multiplied by a dx. Let me rewrite this original expression to make a little bit more sense. So the first thing I'm going to do is I'm going to take the pi. I should do that in a different color since I've already used-- let me take the pi and just stick it out front. So I'm going to stick the pi out in front of the integral. And so this becomes the integral of-- and let me write the 1 over natural log of x first. 1 over the natural log of x times 1/x dx. Now it becomes a little bit clearer. These are completely equivalent statements. But this makes it clear that, yes, u-substitution will work over here. If we set our u equal to natural log of x, then our du is 1/x dx. Let's rewrite this integral. It's going to be equal to pi times the indefinite integral of 1/u. Natural log of x is u-- we set that equal to natural log of x-- times du. Now this becomes pretty straightforward. What is the antiderivative of all of this business? And we've done very similar things like this multiple times already. This is going to be equal to pi times the natural log of the absolute value of u so that we can handle even negative values of u. The natural log of the absolute value of u plus c, just in case we had a constant factor out here. And we're almost done. We just have to unsubstitute for the u. u is equal to natural log of x. So we end up with this kind of neat-looking expression. The anti of this entire indefinite integral we have simplified. We have evaluated it, and it is now equal to pi times the natural log of the absolute value of u. But u is just the natural log of x. And then we have this plus c right over here. And we could have assumed that, from the get go, this original expression was only defined for positive values of x because you had to take the natural log here, and it wasn't an absolute value. So we can leave this as just a natural log of x, but this also works for the situations now because we're doing the absolute value of that where the natural log of x might have been a negative number. For example, if it was a natural log of 0.5 or, who knows, whatever it might be. But then we are all done. We have simplified what seemed like a kind of daunting expression." + }, + { + "Q": "Sal uses the world orthogonal, could someone define it for me?", + "A": "Orthogonal is a generalisation of the geometric concept of perpendicular. When dealing with vectors it means that the vectors are all at 90 degrees from each other.", + "video_name": "9kW6zFK5E5c", + "transcript": "I want to bring everything we've learned about linear independence and dependence, and the span of a set of vectors together in one particularly hairy problem, because if you understand what this problem is all about, I think you understand what we're doing, which is key to your understanding of linear algebra, these two concepts. So the first question I'm going to ask about the set of vectors s, and they're all three-dimensional vectors, they have three components, Is the span of s equal to R3? It seems like it might be. If each of these add new information, it seems like maybe I could describe any vector in R3 by these three vectors, by some combination of these three vectors. And the second question I'm going to ask is are they linearly independent? And maybe I'll be able to answer them at the same time. So let's answer the first one. Do they span R3? To span R3, that means some linear combination of these three vectors should be able to construct any vector in R3. So let me give you a linear combination of these vectors. I could have c1 times the first vector, 1, minus 1, 2 plus some other arbitrary constant c2, some scalar, times the second vector, 2, 1, 2 plus some third scaling vector times the third vector minus 1, 0, 2. I should be able to, using some arbitrary constants, take a combination of these vectors that sum up to any vector in R3. And I'm going to represent any vector in R3 by the vector a, b, and c, where a, b, and c are any real numbers. So if you give me any a, b, and c, and I can give you a formula for telling you what your c3's, your c2's and your c1's are, then than essentially means that it spans R3, because if you give me a vector, I can always tell you how to construct that vector with these three. So Let's see if I can do that. Just from our definition of scalar multiplication of a vector, we know that c1 times this vector, I could rewrite it if I want. I normally skip this step, but I really want to make it clear. So c1 times, I could just rewrite as 1 times c-- it's each of the terms times c1. Similarly, c2 times this is the same thing as each of the terms times c2. And c3 times this is the same thing as each of the terms times c3. I want to show you that everything we do it just formally comes from our definition of multiplication of a vector times a scalar, which is what we just did, or vector addition, which is what we're about to do. So vector addition tells us that this term plus this term plus this term needs to equal that term. So let me write that down. We get c1 plus 2c2 minus c3 will be equal to a. Likewise, we can do the same thing with the next row. Minus c1 plus c2 plus 0c3 must be equal to b. So we get minus c1 plus c2 plus 0c3-- so we don't even have to write that-- is going to be equal to b. And then finally, let's just do that last row. 2c1 plus 3c2 plus 2c3 is going to be equal to c. Now, let's see if we can solve for our different constants. I'm going to do it by elimination. I think you might be familiar with this process. I think I've done it in some of the earlier linear algebra videos before I started doing a formal presentation of it. And I'm going to review it again in a few videos from now, but I think you understand how to solve it this way. What I'm going to do is I'm going to first eliminate these two terms and then I'm going to eliminate this term, and then I can solve for my various constants. If I want to eliminate this term right here, what I could do is I could add this equation to that equation. Or even better, I can replace this equation with the sum of these two equations. Let me do that. I'm just going to add these two equations to each other and replace this one with that sum. So minus c1 plus c1, that just gives you 0. I can ignore it. Then c2 plus 2c2, that's 3c2. And then 0 plus minus c3 is equal to minus c3. Minus c3 is equal to-- and I'm replacing this with the sum of these two, so b plus a. It equals b plus a. Let me write down that first equation on the top. So the first equation, I'm not doing anything to it. So I get c1 plus 2c2 minus c3 is equal to a. Now, in this last equation, I want to eliminate this term. Let's take this equation and subtract from it 2 times this top equation. You can also view it as let's add this to minus 2 times this top equation. Since we're almost done using this when we actually even wrote it, let's just multiply this times minus 2. So this becomes a minus 2c1 minus 4c2 plus 2c3 is equal to minus 2a. If you just multiply each of these terms-- I want to be very careful. I don't want to make a careless mistake. Minus 2 times c1 minus 4 plus 2 and then minus 2. And now we can add these two together. And what do we get? 2c1 minus 2c1, that's a 0. I don't have to write it. 3c2 minus 4c2, that's a minus c2. And then you have your 2c3 plus another 2c3, so that is equal to plus 4c3 is equal to c minus 2a. All I did is I replaced this with this minus 2 times that, and I got this. Now I'm going to keep my top equation constant again. I'm not going to do anything to it, so I'm just going to move it to the right. So I get c1 plus 2c2 minus c3 is equal to a. I'm also going to keep my second equation the same, so I get 3c2 minus c3 is equal to b plus a. Let me scroll over a good bit. And then this last equation I want to eliminate. My goal is to eliminate this term right here. What I want to do is I want to multiply this bottom equation times 3 and add it to this middle equation to eliminate this term right here. So if I multiply this bottom equation times 3-- let me just do-- well, actually, I don't want to make things messier, so this becomes a minus 3 plus a 3, so those cancel out. This becomes a 12 minus a 1. So this becomes 12c3 minus c3, which is 11c3. And then this becomes a-- oh, sorry, I was already done. When I do 3 times this plus that, those canceled out. And then when I multiplied 3 times this, I get 12c3 minus a c3, so that's 11c3. And I multiplied this times 3 plus this, so I get 3c minus 6a-- I'm just multiplying this times 3-- plus this, plus b plus a. So what can I rewrite this by? Actually, I want to make something very clear. This c is different than these c1's, c2's and c3's that I had up here. I think you realize that. But I just realized that I used the letters c twice, and I just didn't want any confusion here. So this c that doesn't have any subscript is a different constant then all of these things over here. Let's see if we can simplify this. We have an a and a minus 6a, so let's just add them. So let's get rid of that a and this becomes minus 5a. If we divide both sides of this equation by 11, what do we get? We get c3 is equal to 1/11 times 3c minus 5a. So you give me any a or c and I'll already tell you what c3 is. What is c2? c2 is equal to-- let me simplify this equation right here. Let me do it right there. So if I just add c3 to both sides of the equation, I get 3c2 is equal to b plus a plus c3. And if I divide both sides of this by 3, I get c2 is equal to 1/3 times b plus a plus c3. I'll just leave it like that for now. Then what is c1 equal to? I could just rewrite this top equation as if I subtract 2c2 and add c3 to both sides, I get c1 is equal to a minus 2c2 plus c3. What have I just shown you? You can give me any vector in R3 that you want to find. So you can give me any real number for a, any real number for b, any real number for c. And if you give me those numbers, I'm claiming now that I can always tell you some combination of these three vectors that will add up to those. And I've actually already solved for what I have to multiply each of those vectors by to add up to this third vector. So you give me your a's, b's and c's, I just have to substitute into the a's and the c's right here. Oh, sorry. I forgot this b over here. There's also a b. It was suspicious that I didn't have to deal with a b. So there was a b right there. So this is 3c minus 5a plus b. Let me write that. There's a b right there in a parentheses. But I think you get the general idea. You give me your a's, b's and c's, any real numbers can apply. There's no division over here, so I don't have to worry about dividing by zero. So this is just a linear combination of any real numbers, so I can clearly get another real number. So you give me your a's, b's and c's, I'm going to give you a c3. Now, you gave me a's, b's and c's. I got a c3. This is just going to be another real number. I'm just going to take that with your former a's and b's and I'm going to be able to give you a c2. We were already able to solve for a c2 and a c3, and then I just use your a as well, and then I'm going to give you a c1. Hopefully, you're seeing that no matter what a, b, and c you give me, I can give you a c1, c2, or c3. There's no reason that any a's, b's or c's should break down these formulas. We're not doing any division, so it's not like a zero would break it down. I can say definitively that the set of vectors, of these three vectors, does indeed span R3. Let me ask you another question. I already asked it. Are these vectors linearly independent? We said in order for them to be linearly independent, the only solution to c1 times my first vector, 1, minus 1, 2, plus c2 times my second vector, 2, 1, 3, plus c3 times my third vector, minus 1, 0, 2. If something is linearly independent that means that the only solution to this equation-- so I want to find some set of combinations of these vectors that add up to the zero vector, and I did that in the previous video. If they are linearly dependent, there must be some non-zero solution. One of these constants, at least one of these constants, would be non-zero for this solution. You can always make them zero, no matter what, but if they are linearly dependent, then one of these could be non-zero. If they're linearly independent then all of these have to be-- the only solution to this equation would be c1, c2, c3. All have to be equal to 0. c1, c2, c3 all have to be equal to 0. Linear independence implies this, this implies linear independence. Now, this is the exact same thing we did here, but in this case, I'm just picking my a's, b's and c's to be zero. This is a, this is b and this is c, right? I can pick any vector in R3 for my a's, b's and c's. I'm now picking the zero vector. So let's see what our c1's, c2's and c3's are. So my a equals b is equal to c is equal to 0. I'm setting it equal to the zero vector. What linear combination of these three vectors equal the zero vector? Well, if a, b, and c are all equal to 0, that term is 0, that is 0, that is 0. You have 1/11 times 0 minus 0 plus 0. That's just 0. So c3 is equal to 0. Now, if c3 is equal to 0, we already know that a is equal to 0 and b is equal to 0. C2 is 1/3 times 0, so it equals 0. Well, it's c3, which is 0. c2 is 0, so 2 times 0 is 0. So c1 is just going to be equal to a. I just said a is equal to 0. So the only solution to this equation right here, the only linear combination of these three vectors that result in the zero vector are when you weight all of them by zero. So I just showed you that c1, c2 and c3 all have to be zero. And because they're all zero, we know that this is a linearly independent set of vectors. Or that none of these vectors can be represented as a combination of the other two. I have exactly three vectors that span R3 and they're linearly independent. And linearly independent, in my brain that means, look, I don't have any redundant vectors, anything that could have just been built with the other vectors, and I have exactly three vectors, and it's spanning R3. So in general, and I haven't proven this to you, but I could, is that if you have exactly three vectors and they do span R3, they have to be linearly independent. If they weren't linearly independent, then one of these would be redundant. Let's say that that guy was a redundant one. I always pick the third one, but let's say this guy would be redundant, which means that the span of this would be equal to the span of these two, right? Because if this guy is redundant, he could just be part of the span of these two guys. And the span of two of vectors could never span R3. Or the other way you could go, if you have three linear independent-- three tuples, and they're all independent, then you can also say that that spans R3. I haven't proven that to you, but hopefully, you get the sense that each of these is contributing new directionality, right? One is going like that. They're not completely orthogonal to each other, but they're giving just enough directionality that you can add a new dimension to what's going on. Hopefully, that helped you a bit, and I'll see you in the next video." + }, + { + "Q": "is it possible to borrow the stock and then sell it without buying it later when it goes down. You will get the entire price of the stock without buying it in the beginning.", + "A": "No, that would be stealing, not borrowing.", + "video_name": "jAOtWm_WZiE", + "transcript": "Let's say you don't like company ABCD very much and you're convinced that the stock is going to go down. So in that situation, you can actually short the stock, which in a very high level is a bet that the stock is going to go down. And the way that you do that mechanically is that you borrow the stock from someone else who owns it, and then you immediately sell that stock that you don't even own. You sell the stock that you borrowed from someone else, and you'll sell it at the current price. So, for example, in this situation, you would sell it at the current trading price of $50. You would then hope that the stock price goes down. Let's think about the situation where the stock price goes down. So if you shorted it right over here, you borrowed the stock and you sold it for $50. And then if the stock were to actually go down-- let's say it goes all the way down to $20, and you think that's about how far it's going to go down, then you can buy back the stock for $20 in this situation. And then give it back to the owner. You had borrowed the stock. Now you can hand back the stock to the owner. So you could give it back and you've essentially unwound it. And what it allowed you to do is it allowed you to do the buying and selling in reverse order. Normally before you sell something you have to buy something. But here you were able to sell it and buy it later for a lower price. So the situation where the end price is $20 you had sold it for $50. So you got $50. And then you had to use $20 of it to buy it back. So in that situation, shorting the stock, you would have made $30. Let me write this column here. This is the short option. You sold at $50. You borrowed and sold at $50. Then when it went down to $20 you bought it back for $20. So you had $50 of proceeds. You had to use $20 of it to buy it back. So you had a $30 profit. But that's only in the good scenario. What happens if your bet is wrong? What happens if the stock price goes up to $80? And over here you get so scared. You're like, oh my god, I have to buy the stock back by $80. What if it keeps going up? I could lose an unlimited amount of money. So over here you get scared and you unwind your situation. You say, OK, I'll go and buy the stock for $80, so I can give it back. So in this situation where the stock goes up you actually could lose a lot of money. You had sold it for $50. So you only have $50 that you have from the transaction. But now you have to buy the stock for $80. So if you sold for $50 and you buy for $80 you've now lost $30. You're $30 in the hole. So now you are at negative $30. And really shorting is the riskiest of all of the things you can do, because a stock price and go unbelievably high. What happens if the stock price goes to $800 or goes to $8,000? All of a sudden, you've sold something for $50 and you have the obligation at some point in the future, because you have to give the stock back, of paying $500, or $800, or $8,000. You don't know how much you'll have to lose. So it's really the riskiest thing you can do. But it is one way to bet that a stock price will go down, or profit from a stock price going down." + }, + { + "Q": "What is the difference between Equilateral and Isosceles triangles?", + "A": "An equilateral triangle has 3 congruent sides. An isosceles triangle has 2 congruent sides. A scalene triangle has no congruent sides.", + "video_name": "7FTNWE7RTfQ", + "transcript": "Let's do some example problems using our newly acquired knowledge of isosceles and equilateral triangles. So over here, I have kind of a triangle within a triangle. And we need to figure out this orange angle right over here and this blue angle right over here. And we know that side AB or segment AB is equal to segment BC, which is equal to segment CD. Or we could also call that DC. So first of all, we see that triangle ABC is isosceles. And because it's isosceles, the two base angles are going to be congruent. This is one leg. This is the other leg right over there. So the two base angles are going to be congruent. So we know that this angle right over here is also 31 degrees. Well, if we know two of the angles in a triangle, we can always figure out the third angle. They have to add up to 180 degrees. So we could say 31 degrees plus 31 degrees plus the measure of angle ABC is equal to 180 degrees. You can subtract 62. This right here is 62 degrees. You subtract 62 from both sides. You get the measure of angle ABC is equal to-- let's see. 180 minus 60 would be 120. You subtract another 2. You get 118 degrees. So this angle right over here is 118 degrees. Let me just write it like this. This is 118 degrees. Well, this angle right over here is supplementary to that 118 degrees. So that angle plus 118 is going to be equal to 180. We already know that that's 62 degrees. 62 plus 118 is 180. So this right over here is 62 degrees. Now, this angle is one of the base angles for triangle BCD. I didn't draw it that way, but this side and this side are congruent. BC has the same length as CD. Those are the two legs of an isosceles triangle. You can kind of imagine it was turned upside down. This is the vertex. This is one base angle. This is the other base angle. Well, the base angles are going to be congruent. So this is going to be 62 degrees, as well. And then finally, if you want to figure out this blue angle, the blue angle plus these two 62-degree angles are going to have to add up to 180 degrees. So you get 62 plus 62 plus the blue angle, which is the measure of angle BCD, is going to have to be equal to 180 degrees. These two characters-- let's see. 62 plus 62 is 124. You subtract 124 from both sides. You get the measure of angle BCD is equal to-- let's see. If you subtract 120, you get 60, and then you have to subtract another 4. So you get 56 degrees. So this is equal to 56 degrees. And we're done. Now, we could do either of these. Let's do this one right over here. So what is the measure of angle ABE? So they haven't even drawn segment BE here. So let me draw that for us. And so we have to figure out the measure of angle ABE. So we have a bunch of congruent segments here. And in particular, we see that triangle ABD, all of its sides are equal. So it's an equilateral triangle, which means all of the angles are equal. And if all of the angles are equal in a triangle, they all have to be 60 degrees. So all of these characters are going to be 60 degrees. Well, that's part of angle ABE, but we have to figure out this other part right over here. And to do that, we can see that we're actually dealing with an isosceles triangle kind of tipped over to the left. This is the vertex angle. This is one base angle. This is the other base angle. And the vertex angle right here is 90 degrees. And once again, we know it's isosceles because this side, segment BD, is equal to segment DE. And once again, these two angles plus this angle right over here are going to have to add up to 180 degrees. So you call that an x. You call that an x. You've got x plus x plus 90 is going to be 180 degrees. So you get 2x plus-- let me just write it out. Don't want to skip steps here. We have x plus x plus 90 is going to be equal to 180 degrees. x plus x is the same thing as 2x, plus 90 is equal to 180. And then we can subtract 90 from both sides. You get 2x is equal to 90. Or divide both sides by 2. You get x is equal to 45 degrees. And then we're done because angle ABE is going to be equal to the 60 degrees plus the 45 degrees. So it's going to be this whole angle, which is what we care about. Angle ABE is going to be 60 plus 45, which is 105 degrees. And now we have this last problem over here. This one looks a little bit simpler. I have an isosceles triangle. This leg is equal to that leg. This is the vertex angle. I have to figure out B. And the trick here is like, wait, how do I figure out one side of a triangle if I only know one other side? Don't I need to know two other sides? And we'll do it the exact same way we just did that second part of that problem. If this is an isosceles triangle, which we know it is, then this angle is going to be equal to that angle there. And so if we call this x, then this is x as well. And we get x plus x plus 36 degrees is equal to 180. The two x's, when you add them up, you get 2x. And then-- I won't skip steps here. 2x plus 36 is equal to 180. Subtract 36 from both sides, we get 2x-- that 2 looks a little bit funny. We get 2x is equal to-- 180 minus 30 is 150. And then you want to subtract another 6 from 150, gets us to 144. Did I do that right? 180 minus 30 is 150, yep, 144. Divide both sides by 2. You get x is equal to 72 degrees. So this is equal to 72 degrees. And we are done." + }, + { + "Q": "When we say \"surface of the sun\" is there actually a surface? Let's say, hypothetically, if a spaceship could survive the heat of the sun would it just travel straight through the sun without obstacle? It is just gas, after all.", + "A": "If you re thinking of a solid surface, like that or Earth, then no. The sun s surface is made up of gas. So it s a gaseous surface.", + "video_name": "EdYyuUUY-nc", + "transcript": "In the last video, we started with a star in its main sequence, like the sun. And inside the core of that star, you have hydrogen fusion going on. So that is hydrogen fusion, and then outside of the core, you just had hydrogen. You just hydrogen plasma. And when we say plasma, it's the electrons and protons of the individual atoms have been disassociated because the temperatures and pressures are so high. So they're really just kind of like this soup of electrons and protons, as opposed to proper atoms that we associate with at lower temperatures. So this is a main sequence star right over here. And we saw in the last video that this hydrogen is fusing into helium. So we start having more and more helium here. And as we have more and more helium, the core becomes more and more dense, because helium is a more massive atom. It is able to pack more mass in a smaller volume. So this gets more and more dense. So core becomes more dense. And so while the core is becoming more and more dense, that actually makes the fusion happen faster and faster. Because it's more dense, more gravitational pressure, more mass wanting to get to it, more pressure on the hydrogen that's fusing, so it starts to fuse hotter. So let me write this, so the fusion, so hydrogen fuses faster. And actually, we even see this in our sun. Our sun today is brighter and hotter. It's fusing faster than it was when it was born 4.5 or 4.6 billion years ago. But eventually you're going to get to the point so that the core, you only have helium. So there's going to be some point where the entire core is all helium. And it's going to be way denser than this core over here. All of that mass over there has now been turned into helium. A lot of it has been turned into energy. But most of it is now in helium, and it's going to be at a much, much smaller volume. And the whole time, the temperature is increasing, the fusion is getting faster and faster. And now there's this dense volume of helium that's not fusing. You do have, and we saw this in this video, a shell around it of hydrogen that is fusing. So this right here is hydrogen fusion going on. And then this over here is just hydrogen plasma. Now the unintuitive thing, or at least this was unintuitive to me at first, is what's going on the core is that the core is getting more and more dense. It's fusing at a faster rate. And so it's getting hotter and hotter. So the core is hotter, fusing faster, getting more and more dense. I kind of imagine it's starting to collapse. Every time it collapses, it's getting hotter and more dense. But at the same time that's happening, the star itself is getting bigger. And this is actually not drawn to scale. Red giants are much, much larger than main sequence stars. But the whole time that this is getting more dense, the rest of the star is, you could kind of view it as getting less dense. And that's because this is generating so much energy that it's able to more than offset, or better offset the gravitational pull into it. So even though this is hotter, it's able to disperse the rest of the material in the sun over a larger volume. And so that volume is so big that the surface, and we saw this in the last video, the surface of the red giant is actually cooler-- let me write that a little neater-- is actually cooler than the surface of a main sequence star. This right here is hotter. And just to put things in perspective, when the sun becomes a red giant, and it will become a red giant, its diameter will be 100 times the diameter that it is today. Or another way to be put it, it will have the same diameter as the Earth's orbit around the current sun. Or another way to view it is, where we are right now will be on the surface or near the surface or maybe even inside of that future sun. Or another way to put it, when the sun becomes a red giant, the Earth's going to be not even a speck out here. And it will be liquefied and vaporized at that point in time. So this is super, super huge. And we've even thought about it. Just for light to reach the current sun to our point in orbit, it takes eight minutes. So that's how big one of these stars are. To get from one side of the star to another side of the star, it'll take 16 minutes for light to travel, if it was traveling that diameter, and even slightly longer if it was to travel it in a circumference. So these are huge, huge, huge stars. And we'll talk about other stars in the future. They're even bigger than this when they become supergiants. But anyway, we have the hydrogen in the center-- sorry. We have the helium in the center. Let me write this down. We have a helium core in the center. We're fusing faster and faster and faster. We're now a red giant. The core is getting hotter and hotter and hotter until it gets to the temperature for ignition of helium. So until it gets to 100 million Kelvin-- remember the ignition temperature for hydrogen was 10 million Kelvin. So now we're at 100 million Kelvin, factor of 10. And now, all of a sudden in the core, you actually start to have helium fusion. And we touched on this in the last video, but the helium is fusing into heavier elements. And some of those heavier elements, and predominately, it will be carbon and oxygen. And you may suspect this is how heavier and heavier elements form in the universe. They form, literally, due to fusion in the core of stars. Especially when we're talking about elements up to iron. But anyway, the core is now experiencing helium fusion. It has a shell around it of helium that is not quite there, does not quite have the pressures and temperatures to fuse yet. So just regular helium. But then outside of that, we do have the pressures and temperatures for hydrogen to continue to fuse. So out here, you do have hydrogen fusion. And then outside over here, you just have the regular hydrogen plasma. So what just happened here? When you have helium fusion all of a sudden-- now this is, once again, providing some type of energetic outward support for the core. So it's going to counteract the ever-increasing contraction of the core as it gets more and more dense, because now we have energy going outward, energy pushing things outward. But at the same time that that is happening, more and more hydrogen in this layer is turning into helium, is fusing into helium. So it's making this inert part of the helium core even larger and larger and denser, even larger and larger, and putting even more pressure on this inside part. And so what's actually going to happen within a few moments, I guess, especially from a cosmological point of view, this helium fusion is going to be burning super-- I shouldn't use-- igniting or fusing at a super-hot level. But it's contained due to all of this pressure. But at some point, the pressure won't be able to contain it, and the core is going to explode. But it's not going to be one of these catastrophic explosions where the star is going to be destroyed. It's just going to release a lot of energy all of a sudden into the star. And that's called a helium flash. But once that happens, all of a sudden, then now the star is going to be more stable. And I'll use that in quotes without writing it down because red giants, in general, are already getting to be less stable than a main sequence star. But once that happens, you now will have a slightly larger volume. So it's not being contained in as small of a tight volume. That helium flash kind of took care of that. So now you have helium fusing into carbon and oxygen. And there's all sorts of other combinations of things. Obviously, there's many elements in between helium and carbon and oxygen. But these are the ones that dominate. And then outside of that, you have helium forming. You have helium that is not fusing. And then outside of that, you have your fusing hydrogen. Over here, you have hydrogen fusing into helium. And then out here in the rest of the radius of our super-huge red giant, you just have your hydrogen plasma out here. Now what's going to happen as this star ages? Well, if we fast forward this a bunch-- and remember, as a star gets denser and denser in the core, and the reactions happen faster and faster, and this core is expelling more and more energy outward, the star keeps growing. And the surface gets cooler and cooler. So if we fast forward a bunch, and this is what's going to happen to something the mass of our sun, if it's more massive, then at some point, the core of carbon and oxygen that's forming can start to fuse into even heavier elements. But in the case of the sun, it will never get to that 600 million Kelvin to actually fuse the carbon and the oxygen. And so eventually you will have a core of carbon and oxygen, or mainly carbon and oxygen surrounded by fusing helium surrounded by non-fusing helium surrounded by fusing hydrogen, which is surrounded by non-fusing hydrogen, or just the hydrogen plasma of the sun. But eventually all of this fuel will run out. All of the hydrogen will run out in the stars. All of this hydrogen, all of this fusing hydrogen will run out. All of this fusion helium will run out. This is the fusing hydrogen. This is the inert helium, which will run out. It'll be used in kind of this core, being fused into the carbon and oxygen, until you get to a point where you literally just have a really hot core of carbon and oxygen. And it's super-dense. This whole time, it will be getting more and more dense as heavier and heavier elements show up in the course. So it gets denser and denser and denser. But the super dense thing will not, in the case of the sun-- and if it was a more massive star, it would get there-- but in the case of the sun, it will not get hot enough for the carbon and the oxygen to form. So it really will just be this super-dense ball of carbon and oxygen and all of the other material in the sun. Remember, it was superenergetic. It was releasing tons and tons of energy. The more that we progressed down this, the more energy was releasing outward, and the larger the radius of the star became, and the cooler the outside of the star became, until the outside just becomes this kind of cloud, this huge cloud of gas around what once was the star. And in the center-- so I could just draw it as this huge-- this is now way far away from the star, much even bigger than the radius or the diameter of a red giant. And all we'll have left is a mass, a superdense mass of, I would call it, inert carbon or oxygen. This is in the case of the sun. And at first, when it's hot, and it will be releasing radiation because it's so hot. We'll call this a white dwarf. This right here is called a white dwarf. And it'll cool down over many, many, many, many, many, many, many, years, until it becomes, when it's completely cooled down, lost all of its energy-- it'll just be this superdense ball of carbon and oxygen, at which point, we would call it a black dwarf. And these are obviously very hard to observe because they're not emitting light. And they don't have quite the mass of something like a black hole that isn't even emitting light, but you can see how it's affecting things around it. So that's what's going to happen to the sun. In the next few videos, we're going to talk about what would happen to things less massive than the sun and what would happen to things more massive can imagine the more massive. There would be so much pressure on these things, because you have so much mass around it, that these would begin to fuse into heavier and heavier elements until we get to iron." + }, + { + "Q": "At 6:16 why did he subtract (36-9)!?", + "A": "36! means 36*35*34*33*32*31*30*29*28*27*26*25*24*23*22....*1 But since we want it to stop at 28, we have to cancel the rest out by dividing with 27!(27*26*25*24*23*22*21*20*19*18*17....*1)", + "video_name": "SbpoyXTpC84", + "transcript": "A card game using 36 unique cards, four suits, diamonds, hearts, clubs and spades-- this should be spades, not spaces-- with cards numbered from 1 to 9 in each suit. A hand is chosen. A hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards-- and I won't worry about, you know, there's nine numbers in each suit, and there are four suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick nine of them. So at first we'll say, well look, I have nine slots in my hand, right? 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick nine cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked one, so there will only be 35 to pick from. And then for the third slot, 34, and then Then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28 possible hands. Now, this would be true if order mattered. This would be true if I have card 15 here. Maybe I have a-- let me put it here-- maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have-- and that's one hand. And then I have another. So then I have cards one, two, three, four, five, six, seven, eight. I have eight other cards. Or maybe another hand is I have the eight cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses, so order doesn't matter. So we're overcounting. We're counting all of the different ways that the same number of cards can be arranged. So in order to not overcount, we have to divide this by the ways in which nine cards can be rearranged. So we have to divide this by the way nine cards can be rearranged. So how many ways can nine cards be rearranged? If I have nine cards and I'm going to pick one of nine to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took one to put it in the first, so I have 8 left. Then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be 1 card left to put in it. So this number right here, where you take 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1, or 9-- you start with 9 and then you multiply it by every number less than 9. Every, I guess we could say, natural number less than 9. This is called 9 factorial, and you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer and this will be the correct answer. Now this is a super, super duper large number. Let's figure out how large of a number this is. We have 36-- let me scroll to the left a little bit-- 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28, divided by 9. Well, I can do it this way. I can put a parentheses-- divided by parentheses, 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1. Now, hopefully the calculator can handle this. And it gave us this number, 94,143,280. Let me put this on the side, so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9 card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is to say, look, we have 36 things and we are going to choose 9 of them. And we don't care about order, so sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only nine away from 36. So this is 36 factorial, so this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36, minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial-- so let's think about this-- 36 factorial, it'd be 36 times 35, you keep going all the way, times 28 times 27, going all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial, that's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26, so that and that would cancel out. So if you do 36 divided by 36, minus 9 factorial, you just get the first, the largest nine terms of 36 factorial, which is exactly what we have over there. And then we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this, n choose k. And they'll write the formula as equal to n factorial over n minus k factorial, and also in the denominator, k factorial. And this is a general formula that if you have n things, and you want to find out all of the possible ways you can pick k things from those n things, and you don't care about the order. All you care is about which k things you picked, you don't care about the order in which you picked those k things. So that's what we did here." + }, + { + "Q": "hi Khan Academy,\nI was just thinking if there a subset for the X which can make this statement true\nA \u00e2\u0088\u00aa x = B when B\u00e2\u008a\u0084 A\nThank you!", + "A": "What if x=B where A is a proper subset of B?", + "video_name": "1wsF9GpGd00", + "transcript": "Let's define ourselves some sets. So let's say the set A is composed of the numbers 1. 3. 5, 7, and 18. Let's say that the set B-- let me do this in a different color-- let's say that the set B is composed of 1, 7, and 18. And let's say that the set C is composed of 18, 7, 1, and 19. Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset-- and this is a notation right over here, this is a subset-- B is a subset of A. B is a subset. So let me write that down. B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further and we could say that B is a strict or sometimes said a proper subset of A. And the way you do that is, you could almost imagine that this is kind of a less than or equal sign, and then you kind of cross out this equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member A, but everything that's in A is not a member of B. So let me write this. This is B. B is a strict or proper subset. So, for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see. C contains a 1, it contains a 7, it contains an 18. So every member of B is indeed a member C. So this right over here is true. Now, can we write that C is a subset? Can we write that C is a subset of A? Can we write C is a subset of A? Let's see. Every element of C needs to be in A. So A has an 18, it has a 7, it has a 1. But it does not have a 19. So once again, this right over here is false. Now we could have also added-- we could write B is a subset of C. Or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this. And then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B, and this is just another way of saying that B is a subset of A. But the way you could think about this is, A contains every element that is in B. And it might contain more. It might contain exactly every element. So you can kind of view this as you kind of have the equals symbol there. If you were to view this as greater than or equal. They're note quite exactly the same thing. But we know already that we could also write that A is a strict superset of B, which means that A contains everything B has and then some. A is not equivalent to B. So hopefully this familiarizes you with the notions of subsets and supersets and strict subsets." + }, + { + "Q": "At 2:30, when he says that the absolute value could also work in ensuring that the value is positive, can someone explain to me why the absolute value isn't used in the equation?", + "A": "In the video, Sal does some algebraic manipulation to achieve the formula of the parabola. It is much easier to derive the parabola if he were to square the expression and take the square root than to take the absolute value of the expression. Both methods yield the same value of the expression; however, the latter method (squaring then taking the square root) allows for more easier manipulation. Hope this helps!", + "video_name": "okXVhDMuGFg", + "transcript": "- [Voiceover] What I have attempted to draw here in yellow is a parabola, and as we've already seen in previous videos, a parabola can be defined as the set of all points that are equidistant to a point and a line, and the point is called the focus of the parabola, and the line is called the directrix of the parabola. What I want to do in this video, it's gonna get a little bit of hairy algebra, but given that definition, I want to see, and given that definition, and given a focus at the point x equals a, y equals b, and a line, a directrix, at y equals k, to figure out what is the equation of that parabola actually going to be, and it's going to be based on a's, b's, and k's, so let's do that. So let's take a arbitrary point on the parabola. Let's say we take this point right over here, and its x-coordinate is x, and its y-coordinate is y, and by definition, in order for this to be a parabola, it has to be equidistant to its focus and its directrix, so what does that mean? That means that the distance to the directrix, which I'm drawing here in blue, has to be the same as the distance to the focus, which I am drawing in magenta, and when we take the distance to the directrix, we literally just drop a perpendicular, that is, that's going to be the shortest distance to that line, but the distance to the focus, well we see that's at a bit of an angle, and we might have to use the distance formula, which is really just the Pythagorean Theorem. So let's do that. This distance has to be the same as that distance. So, what's this blue distance? Well, that's just gonna be our change in y. It's going to be this y, minus k. It's just this distance. So it's going to be y minus k. Now we have to be careful. The way I've just drawn it, yes, y is greater than k, so this is going to give us a positive value, and you need a non-negative value if you're talking about distances, but you can definitely have a parabola where the y-coordinate of the focus is lower than the y-coordinate of the directrix, in which case this would be negative. So what we really want is the absolute value of this, or, we could square it, and then we could take the square root, the principle root, which would be equivalent to taking the absolute value of y minus k. So that's this distance right over here, and by the definition of a parabola, in order for (x,y) to be sitting on the parabola, that distance needs to be the same as the distance from (x,y) to (a,b), to the focus. So what's that going to be? Well, we just apply the distance formula, or really, just the Pythagorean Theorem. It's gonna be our change in x, so, x minus a, squared, plus the change in y, y minus b, squared, and the square root of that whole thing, the square root of all of that business. Now, this right over here is an equation of a parabola. It doesn't look like it, it looks really hairy, but it IS the equation of a parabola, and to show you that, we just have to simplify this, and if you get inspired, I encourage you to try to simplify this on your own, it's just gonna be a little bit of hairy algebra, but it really is not too bad. You're gonna get an equation for a parabola that you might recognize, and it's gonna be in terms of a general focus, (a,b), and a gerneral directrix, y equals k, so let's do that. So the simplest thing to start here, is let's just square both sides, so we get rid of the radicals. So if you square both sides, on the left-hand side, you're gonna get y minus k, squared is equal to x minus a, squared, plus y minus b, squared. Fair enough? Now what I want to do is, I just want to end up with just a y on the left-hand side, and just x's, ab's, and k's on the right-hand side, so the first thing I might want to do, is let's expand each of these expressions that involve with y, so this blue one on the left-hand side, that is going to be y squared minus 2yk, plus k squared, and that is going to be equal to, I'm gonna keep this first one the same, so it's gonna be x minus a, squared, and now let me expand, I'm gonna find a color, expand this in green, so plus y squared, minus 2yb, plus b squared. All I did, is I multiplied y minus b, times y minus b. Now let's see if we can simplify things. So, I have a y squared on the left, I have a y squared on the right, well, if I subtract y squared from both sides, so I can do that. Well, that simplified things a little bit, and now I can, let's see what I can do. Well let's get the k squared on this side, so let's subtract k squared from both sides, so, subtract k squared from both sides, so that's gonna get rid of it on the left-hand side, and now let's add 2yb to both sides, so we have all the y's on the left-hand side, so, plus 2yb, that's gonna give us a 2yb on the left-hand side, plus 2yb. So what is this going to be equal to? And I'm starting to run into my graph, so let me give myself a little bit more real estate over here. So on the left-hand side, what am I going to have? This is the same thing as 2yb minus 2yk, which is the same thing, actually let me just write that down. That's going to be 2y-- Do it in green, actually, well, yeah, why not green? That's going to be-- Actually, let me start a new color. (chuckles) That's going to be 2yb minus 2yk. You can factor out a 2y, and it's gonna be 2y times, b minus k. So let's do that. So we could write this as 2 times, b minus k, y if you factor out a 2 and a y, so that's the left-hand side, so that's that piece right over there. These things cancel out. Now, on our right-hand side, I promised you a little bit of hairy algebra, so hopefully you see that I'm delivering on that promise. On the right-hand side, you have x minus a, squared, and then, let's see, these characters cancel out, and you're left with b squared minus k squared, so these two are gonna be b squared minus k squared, plus b squared minus k squared. Now, I said all I want is a y on the left-hand side, so let's divide everything by two times, b minus k. So, let's divide everything, two times, b minus k, so, two times, b minus k. And I'm actually gonna divide this whole thing by two times, b minus k. Now, obviously on the left-hand side, this all cancels out, you're left with just a y, and then it's going to be y equals, y is equal to one over, two times, b minus k, and notice, b minus k is the difference between the y-coordinate of the focus, and the y-coordinate, I guess you could say, of the line, y equals k, so it's one over, two times that, times x minus a, squared. So if you knew what b minus k was, this would just simplify to some number, some number that's being multiplied times x minus a, squared, so hopefully this is starting to look like the parabolas that you remember from your childhood, (chuckles) if you do remember parabolas from your childhood. Alright, so then let's see if we could simplify this thing on the right, and you might recognize, b squared minus k squared, that's a difference of squares, that's the same thing as b plus k, times b minus k, so the b minus k's cancel out, and we are just left with, and we deserve a little bit of a drum roll, we are just left with 1/2 times, b plus k. So, there you go. Given a focus at a point (a,b), and a directrix at y equals k, we now know what the formula of the parabola is actually going to be. So, for example, if I had a focus at the point, I don't know, let's say the point (1,2), and I had a directrix at y is equal to, I don't know, let's make it y is equal to -1, what would the equation of this parabola be? Well, it would be y is equal to one over, two times, b minus k, so two minus -1, that's the same thing as two plus one, so that's just three, two minus -1 is three, times x minus one, squared, plus 1/2 times, b plus k. Two plus -1 is one, so one, and so what is this going to be? You're gonna get y is equal to 1/6, x minus one, squared, plus 1/2. There you go. That is the parabola with a focus at (1,2) and a directrix at y equals -1. Fascinating." + }, + { + "Q": "difference between arcsin and inverse sin", + "A": "inverse sin x, arcsin x and sin\u00e2\u0081\u00bb\u00c2\u00b9 x all mean exactly the same thing.", + "video_name": "JGU74wbZMLg", + "transcript": "If I were to walk up to you on the street and say you, please tell me what-- so I didn't want to write that thick --please tell me what sine of pi over 4 is. And, obviously, we're assuming we're dealing in radians. You either have that memorized or you would draw the unit circle right there. That's not the best looking unit circle, but you get the idea. You'd go to pi over 4 radians, which is the same thing as 45 degrees. You would draw that unit radius out. And the sine is defined as a y-coordinate on the unit circle. So you would just want to know this value right here. And you would immediately say OK. This is a 45 degrees. Let me draw the triangle a little bit larger. The triangle looks like this. This is 45. That's 45. This is 90. And you can solve a 45 45 90 triangle. The hypotenuse is 1. This is x. They're going to be the same values. This is an isosceles triangle, right? Their base angles are the same. So you say, look. x squared plus x squared is equal to 1 squared, which is just 1. 2x squared is equal to 1. x squared is equal to 1/2. x is equal to the square root of 1/2, which is one over the square root of 2. I can put that in rational form by multiplying that by the square root of 2 over 2. And I get x is equal to the square root of 2 over 2. So the height here is square root of 2 over 2. And if you wanted to know this distance too, it would also be the same thing. But we just cared about the height. Because the sine value, the sine of this, is just this height right here. The y-coordinate. And we got that as the square root of 2 over 2. This is all review. We learned this in the unit circle video. But what if someone else-- Let's say on another day, I come up to you and I say you, please tell me what the arcsine of the square root of 2 over 2 is. What is the arcsine? You're like I know what the sine of an angle is, but this is some new trigonometric function that Sal has devised. And all you have to realize, when they have this word arc in front of it-- This is also sometimes referred to as the inverse sine. This could have just as easily been written as: what is the inverse sine of the square root of 2 over 2? All this is asking is what angle would I have to take the sine of in order to get the value square root of 2 over 2. This is also asking what angle would I have to take the sine of in order to get square root of 2 over 2. I could rewrite either of these statements as saying square-- Let me do it. I could rewrite either of these statements as saying sine of what is equal to the square root of 2 over 2. And this, I think, is a much easier question for you to answer. Sine of what is square root of 2 over 2? Well I just figured out that the sine of pi over 4 is square root of 2 over 2. So, in this case, I know that the sine of pi over 4 is equal to square root of 2 over 2. So my question mark is equal to pi over 4. Or, I could have rewritten this as, the arcsine-- sorry --arcsine of the square root of 2 over 2 is equal to pi over 4. Now you might say so, just as review, I'm giving you a value and I'm saying give me an angle that gives me, when I take the sine of that angle that gives me that value. But you're like hey Sal. Let me go over here. You're like, look pi over 2 worked. 45 degrees worked. But I could just keep adding 360 degrees or I could keep just adding 2 pi. And all of those would work because those would all get me to that same point of the unit circle, right? And you'd be correct. And so all of those values, you would think, would be valid answers for this, right? Because if you take the sine of any of those angles-- You could just keep adding 360 degrees. If you take the sine of any of them, you would get square root of 2 over 2. And that's a problem. You can't have a function where if I take the function-- I can't have a function, f of x, where it maps to multiple values, right? Where it maps to pi over 4, or it maps to pi over 4 plus 2 pi or pi over 4 plus 4 pi. So in order for this to be a valid function-- In order for the inverse sine function to be valid, I have to restrict its range. And the way that-- We'll just restrict its range to the most natural place. So let's restrict its range. Actually, just as a side note, what's its domain restricted to? So if I'm taking the arcsine of something. So if I'm taking the arcsine of x, and I'm saying that that is equal to theta, what's the domain restricted to? What are the valid values of x? x could be equal to what? Well if I take the sine of any angle, I can only get values between 1 and negative 1, right? So x is going to be greater than or equal to negative 1 and then less than or equal to 1. That's the domain. Now, in order to make this a valid function, I have to restrict the range. The possible values. I have to restrict the range. Now for arcsine, the convention is to restrict it to the first and fourth quadrants. To restrict the possible angles to this area right here along the unit circle. So theta is restricted to being less than or equal to pi over 2 and then greater than or equal to minus pi over 2. So given that, we now understand what arcsine is. Let's do another problem. Clear out some space here. Let me do another arcsine. So let's say I were to ask you what the arcsine of minus the square root of 3 over 2 is. Now you might have that memorized. And say, I immediately know that sine of x, or sine of theta is square root of 3 over 2. And you'd be done. But I don't have that memorized. So let me just draw my unit circle. And when I'm dealing with arcsine, I just have to draw the first and fourth quadrants of my unit circle. That's the y-axis. That's my x-axis. x and y. And where am I? If the sine of something is minus square root of 3 over 2, that means the y-coordinate on the unit circle is minus square root of 3 over 2. So it means we're right about there. So this is minus the square root of 3 over 2. This is where we are. Now what angle gives me that? Let's think about it a little bit. My y-coordinate is minus square root of 3 over 2. This is the angle. It's going to be a negative angle because we're going below the x-axis in the clockwise direction. And to figure out-- Let me just draw a little triangle here. Let me pick a better color than that. That's a triangle. Let me do it in this blue color. So let me zoom up that triangle. Like that. This is theta. That's theta. And what's this length right here? Well that's the same as the y-height, I guess Which is square root of 3 over 2. It's a minus because we're going down. But let's just figure out this angle. And we know it's a negative angle. So when you see a square root of 3 over 2, hopefully you recognize this is a 30 60 90 triangle. The square root of 3 over 2. This side is 1/2. And then, of course, this side is 1. Because this is a unit circle. So its radius is 1. So in a 30 60 90 triangle, the side opposite to the square root of 3 over 2 is 60 degrees. This side over here is 30 degrees. So we know that our theta is-- This is 60 degrees. That's its magnitude. But it's going downwards. So it's minus 60 degrees. So theta is equal to minus 60 degrees. But if we're dealing in radians, that's not good enough. So we can multiply that times 100-- sorry --pi radians for every 180 degrees. Degrees cancel out. And we're left with theta is equal to minus pi over 3 radians. And so we can say-- We can now make the statements that the arcsine of minus square root of 3 over 2 is equal to minus pi over 3 radians. Or we could say the inverse sign of minus square root of 3 over 2 is equal to minus pi over 3 radians. And to confirm this, let's just-- Let me get a little calculator out. I put this in radian mode already. You can just check that. Per second mode. I'm in radian mode. So I know I'm going to get, hopefully, the right answer. And I want to figure out the inverse sign. So the inverse sine-- the second and the sine button --of the minus square root of 3 over 2. It equals minus 1.04. So it's telling me that this is equal to minus 1.04 radians. So pi over 3 must be equal to 1.04. Let's see if I can confirm that. So if I were to write minus pi divided by 3, what do I get? I get the exact same value. So my calculator gave me the exact same value, but it might have not been that helpful because my calculator doesn't tell me that this is minus pi over 3." + }, + { + "Q": "How do you know weather to multiply, divide, add, or subtract?", + "A": "The rule is that you do the opposite function. Examples: 1) y+5 = 7. The 5 is being added to the y . To move it, we do the opposite operation which is subtraction. 2) 5x = 20. The 5 is being multiplied with the 20. To eliminate the 5, we do the opposite of multiplication, which is division.", + "video_name": "kbqO0YTUyAY", + "transcript": "So once again, we have three equal, or we say three identical objects. They all have the same mass, but we don't know what the mass is of each of them. But what we do know is that if you total up their mass, it's the same exact mass as these nine objects And each of these nine objects have a mass of 1 kilograms. So in total, you have 9 kilograms on this side. And over here, you have three objects. They all have the same mass. And we don't know what it is. We're just calling that mass x. And what I want to do here is try to tackle this a little bit more symbolically. In the last video, we said, hey, why don't we just multiply 1/3 of this and multiply 1/3 of this? And then, essentially, we're going to keep things balanced, because we're taking 1/3 of the same mass. This total is the same as this total. That's why the scale is balanced. Now, let's think about how we can represent this symbolically. So the first thing I want you to think about is, can we set up an equation that expresses that we have these three things of mass x, and that in total, their mass is equal to the total mass Can we express that as an equation? And I'll give you a few seconds to do it. Well, let's think about it. Over here, we have three things with mass x. So their total mass, we could write as-- we could write their total mass as x plus x plus x. And over here, we have nine things with mass of 1 kilogram. I guess we could write 1 plus 1 plus 1. That's 3. Plus 1 plus 1 plus 1 plus 1. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9. And actually, this is a mathematical representation. If we set it up as an equation, it's an algebraic representation. It's not the simplest possible way we can do it, but it is a reasonable way to do it. If we want, we can say, well, if I have an x plus another x plus another x, I have three x's. So I could rewrite this as 3x. And 3x will be equal to? Well, if I sum up all of these 1's right over here-- 1 We're doing that. We have 9 of them, so we get 3x is equal to 9. And let me make sure I did that. 1, 2, 3, 4, 5, 6, 7, 8, 9. So that's how we would set it up. And so the next question is, what would we do? What can we do mathematically? Actually, to either one of these equations, but we'll focus on this one right now. What can we do mathematically in order to essentially solve for the x? In order to figure out what that mystery mass actually is? And I'll give you another second or two to think about it. Well, when we did it the last time with just the scales we said, OK, we've got three of these x's here. We want to have just one x here. So we can say, whatever this x is, if the scale stays balanced, it's going to be the same as whatever we have there. There might be a temptation to subtract two of the x's maybe from this side, but that won't help us. And we can even see it mathematically over here. If we subtract two x's from both sides, on the left-hand side you're going to have 3x minus 2x. And on the right-hand side, you're going to have 9 minus 2x. And you're just going to be left with 3 of something minus 2 something is just 1 of something. So you will just have an x there if you get rid of two of them. But on the right-hand side, you're going to get 9 minus 2 So the x's still didn't help you out. You still have a mystery mass on the right-hand side. So that doesn't help. So instead, what we say is-- and we did this the last time. We said, well, what if we took 1/3 of these things? If we take 1/3 of these things and take 1/3 of these things, we should still get the same mass on both sides because the original things had the same mass. And the equivalent of doing that mathematically is to say, why don't we multiply both sides by 1/3? Or another way to say it is we could divide both sides by 3. Multiplying by 1/3 is the same thing as dividing by 3. So we're going to multiply both sides by 1/3. When you multiply both sides by 1/3-- visually over here, if you had three x's, you multiply it by 1/3, you're only going to have one x left. If you have nine of these one-kilogram boxes, you multiply it by 1/3, you're only going to have three left. And over here, you can even visually-- if you divide by 3, which is the same thing as multiplying by 1/3, you divide by 3. So you divide by 3. You have an x is equal to a 1 plus 1 plus 1. An x is equal to 3. Or you see here, an x is equal to 3. Over here you do the math. 1/3 times 3 is 1. You're left with 1x. So you're left with x is equal to 9 times 1/3. Or you could even view it as 9 divided by 3, which is equal to 3." + }, + { + "Q": "If 6.0 g of NO2 came in contact with a cloud containing 2.0 g of H2O, what mass in grams of acid would be produced?", + "A": "You need to take the molar masses of the two substances, find the number of moles of both and then look at which of the substances is the limiting reagent, depending on the stoichiometry of the reaction. Once you have the limiting reagent, the molar ratio will give you the number of moles of reactants and products. Moles of product is easily converted into mass (grams) by using the molar mass formula. Your reaction (I assume) is 8NO2 + 3H2O = 6HNO3 + N2O", + "video_name": "SjQG3rKSZUQ", + "transcript": "We know what a chemical equation is and we've learned how to balance it. Now, we're ready to learn about stoichiometry. And this is an ultra fancy word that often makes people think it's difficult. But it's really just the study or the calculation of the relationships between the different molecules in a reaction. This is the actual definition that Wikipedia gives, stoichiometry is the calculation of quantitative, or measurable, relationships of the reactants and the products. And you're going to see in chemistry, sometimes people use the word reagents. For most of our purposes you can use the word reagents and reactants interchangeably. They're both the reactants in a reaction. The reagents are sometimes for special types of reactions where you want to throw a reagent in and see if something happens. And see if your belief about that substance is true or things like that. But for our purposes a reagent and reactant is the same thing. So it's a relationship between the reactants and the products in a balanced chemical equation. So if we're given an unbalanced one, we know how to get to the balanced point. A balanced chemical equation. So let's do some stoichiometry. Just so we get practice balancing equations, I'm always going to start with unbalanced equations. Let's say we have iron three oxide. Two iron atoms with three oxygen atoms. Plus aluminum, Al. And it yields Al2 O3 plus iron. So remember when we're doing stoichiometry, first of all we want to deal with balanced equations. A lot of stoichiometry problems will give you a balanced equation. But I think it's good practice to actually balance the equations ourselves. So let's try to balance this one. We have two iron atoms here in this iron three oxide. How many iron atoms do we have on the right hand side? We only have one. So let's multiply this by 2 right here. All right, oxygen, we have three on this side. We have three oxygens on that side. That looks good. Aluminum, on the left hand side we only have one aluminum atom. On the right hand side we have two aluminum atoms. So we have to put a 2 here. And we have balanced this equation. So now we're ready to do some stoichiometry. So the stoichiometry essentially ... If I give you... There's not just one type of stoichiometry problem, but they're all along the lines of, if I give you x grams of this how many grams of aluminum do I need to make this reaction happen? Or if I give you y grams of this molecule and z grams of this molecule which one's going to run out first? That's all stoichiometry. And we'll actually do those exact two types of problems in this video. So let's say that we were given 85 grams of the iron three oxide. So 85 grams. So my question to you is how many grams of aluminum do we need? How many grams of aluminum? Well you look at the equation, you immediately see the mole ratio. So for every mole of this, so for every one atom we use of iron three oxide we need two aluminums. So what we need to do is figure out how many moles of this molecule there are in 85 grams. And then we need to have twice as many moles of aluminum. Because for every mole of the iron three oxide, we have two moles of aluminum. And we're just looking at the coefficients, we're just looking at the numbers. One molecule of iron three oxide combines with two molecule of aluminum to make this reaction happen. So lets first figure out how many moles 85 grams are. So what's the atomic mass or the mass number of this entire molecule? Let me do it down here. So we have two irons and three oxygens. So let me go down and figure out the atomic masses of iron and oxygen. So iron is right here, 55.85. I think it's fair enough to round to 56. Let's say we're dealing with the version of iron, the isotope of iron, that has 30 neutrons. So it has an atomic mass number of 56. So iron has 56 atomic mass number. And then oxygen, we already know, is 16. Iron was 56. This mass is going to be 2 times 56 plus 3 times 16. We can do that in our heads. But this isn't a math video, so I'll get the calculator out. Is that right? That's 48 plus 112, right, 160. So one molecule of iron three oxide is going to be 160 atomic mass units. So one mole or 6.02 times 10 to the 23 molecules of iron oxide is going to have a mass of 160 grams. So in our reaction we said we're starting off with 85 grams of iron oxide. How many moles is that? Well 85 grams of iron three oxide is equal to 85 over 160 moles. So that's equal to, 85 divided by 160 equals 0.53125. Equals 0.53 moles. So everything we've done so far in this green and light blue, we figured out how many moles 85 grams of iron three oxide is. And we figured out it's 0.53 moles. Because a full mole would have been 160 grams. But we only have 85. So it's .53 moles. And we know from this balanced equation, that for every mole of iron three oxide we have, we need to have two moles of aluminum. So if we have 0.53 moles of the iron molecule, iron three oxide, then we're going to need twice as many aluminum. So we're going to need 1.06 moles of aluminum. I just took 0.53 times 2. Because the ratio is 1:2. For every molecule of this, we need two molecules of that. So for every mole of this, we need two moles of this. If we have 0.53 moles, you multiply that by 2, and you have 1.06 moles of aluminum. All right, so we just have to figure out how many grams is a mole of aluminum and then multiply that times 1.06 and we're done. So aluminum, or aluminium as some of our friends across the pond might say. Aluminium, actually I enjoy that more. Aluminium has the atomic weight or the weighted average is 26.98. But let's just say that the aluminium that we're dealing with has a mass of 27 atomic mass units. So one aluminum is 27 atomic mass units. So one mole of aluminium is going to be 27 grams. Or 6.02 times 10 to 23 aluminium atoms is going to be 27 grams. So if we need 1.06 moles, how many is that going to be? So 1.06 moles of aluminium is equal to 1.06 times 27 grams. And what is that? What is that? Equals 28.62. So we need 28.62 grams of aluminium, I won't write the whole thing there, in order to essentially use up our 85 grams of the iron three oxide. And if we had more than 28.62 grams of aluminium, then they'll be left over after this reaction happens. Assuming we keep mixing it nicely and the whole reaction happens all the way. And we'll talk more about that in the future. And in that situation where we have more than 28.63 grams of aluminium, then this molecule will be the limiting reagent. Because we had more than enough of this, so this is what's going to limit the amount of this process from happening. If we have less than 28.63 grams of, I'll start saying aluminum, then the aluminum will be the limiting reagent, because then we wouldn't be able to use all the 85 grams of our iron molecule, or our iron three oxide molecule. Anyway, I don't want to confuse you in the end with that limiting reagents. In the next video, we'll do a whole problem devoted to limiting reagents." + }, + { + "Q": "is there such thing as a 400 degree angle?", + "A": "yes, but the reference angle is 40 degrees. Hope this helps", + "video_name": "92aLiyeQj0w", + "transcript": "Now that we know what an angle is, let's think about how we can measure them. And we already hinted at one way to think about the measure of angle in the last video where we said, look, this angle XYZ seems more open than angle BAC. So maybe the measure of angle XYZ should be larger than the angle of the BAC, and that is exactly the way we think about the measures of angles. But what I want to do in this video is come up with an exact way to measure an angle. So what I've drawn over here is a little bit of a half-circle, and it looks very similar to a tool that you can buy at your local school supplies store to measure angles. So this is actually a little bit of a drawing of a protractor. And what we do in something like a protractor-- you could even construct one with a piece of paper-- is we've taken a half-circle right here, and we've divided it into a 180 sections, and each of these marks marks 10 of those sections. And what you do for any given angle is you put one of the sides of the angle. So each of the rays of an angle are considered one of its sides. So you put the vertex of the angle at the center of this half-circle-- or if you're dealing with an actual protractor, at the center of that protractor-- and then you put one side along the 0 mark. So I'm going redraw this angle right over here at the center of this protractor. So if we said this is Y, then the Z goes right over here. And then the other ray, ray YX in this circumstance, will go roughly in that direction. And so it is pointing on the protractor to the-- let's see. This looks like this is the 70th of section. This is the 80th section. So maybe this is, I would guess, the 77th section. So this is pointing to 77 right over here. Assuming that I drew it the right way right over here, we could say the measure of angle XYZ-- sometimes they'll just say angle XYZ is equal to, but this is a little bit more formal-- the measure of angle XYZ is equal to 77. Each of these little sections, we call them \"degrees.\" So it's equal to 77-- sometimes it's written like that, the same way you would write \"degrees\" for the temperature outside. So you could write \"77 degrees\" like that or you could actually write out the word right over there. So each of these sections are degrees, so we're measuring in degrees. And I want to be clear, degrees aren't the only way to measure angles. Really, anything that measures the openness. So when you go into trigonometry, you'll learn that you can measure angles, not only in degrees, but also using something called \"radians.\" But I'll leave that to another day. So let's measure this other angle, angle BAC. So once again, I'll put A at the center, and then AC I'll put along the 0 degree edge of this half-circle or of this protractor. And then I'll point AB in the-- well, assuming that I'm drawing it exactly the way that it's Normally, instead of moving the angle, you could actually move the protractor to the angle. So it looks something like that, and you could see that it's pointing to right about the 30 degree mark. So we could say that the measure of angle BAC is equal to 30 degrees. And so you can look just straight up from evaluating these numbers that 77 degrees is clearly larger than 30 degrees, and so it is a larger angle, which makes sense because it is a more open angle. And in general, there's a couple of interesting angles to think about. If you have a 0 degree angle, you actually have something that's just a closed angled. It really is just a ray at that point. As you get larger and larger or as you get more and more open, you eventually get to a point where one of the rays is completely straight up and down while the other one is left to right. So you could imagine an angle that looks like this where one ray goes straight up down like that and the other ray goes straight right and left. Or you could imagine something like an angle that looks like this where, at least, the way you're looking at it, one doesn't look straight up down or one does it look straight right left. But if you rotate it, it would look just like this thing right over here where one is going straight up and down and one is going straight right and left. And you can see from our measure right over here that that gives us a 90 degree angle. It's a very interesting angle. It shows up many, many times in geometry and trigonometry, and there's a special word for a 90 degree angle. It is called a \"right angle.\" So this right over here, assuming if you rotate it around, would look just like this. We would call this a \"right angle.\" And there is a notation to show that it's a right angle. You draw a little part of a box right over there, and that tells us that this is, if you were to rotate it, exactly up and down while this is going exactly right and left, if you were to rotate it properly, or vice versa. And then, as you go even wider, you get wider and wider and wider and wider until you get all the way to an angle that looks like this. So you could imagine an angle where the two rays in that angle form a line. So let's say this is point X. This is point Y, and this is point Z. You could call this angle ZXY, but it's really so open that it's formed an actual line here. Z, X, and Y are collinear. This is a 180 degree angle where we see the measure of angle ZXY is 180 degrees. And you can actually go beyond that. So if you were to go all the way around the circle so that you would get back to 360 degrees and then you could keep going around and around and around, and you'll start to see a lot more of that when you enter a trigonometry class. Now, there's two last things that I want to introduce in this video. There are special words, and I'll talk about more types of angles in the next video. But if an angle is less than 90 degrees, so, for example, both of these angles that we started our discussion with are less than 90 degrees, we call them \"acute angles.\" So this is acute. So that is an acute angle, and that is an acute angle right over here. They are less than 90 degrees. What does a non-acute angle look like? And there's a word for it other than non-acute. Well, it would be more than 90 degrees. So, for example-- let me do this in a color I haven't used-- an angle that looks like this, and let me draw it a little bit better than that. An angle it looks like this. So that's one side of the angle or one of the rays and then I'll put the other one on the baseline right Clearly, this is larger than 90 degrees. If I were to approximate, let's see, that's 100, 110, 120, almost 130. So let's call that maybe a 128-degree angle. We call this an \"obtuse angle.\" The way I remember it as acute, it's kind of \"a cute\" angle. It's nice and small. I believe acute in either Latin or Greek or maybe both means something like \"pin\" or \"sharp.\" So that's one way to think about it. An acute angle seems much sharper. Obtuse, I kind of imagine something that's kind of lumbering and large. Or you could think it's not acute. It's not nice and small and pointy. So that's one way to think about it, but this is just general terminology for different types of angles. Less than 90 degrees, you have an acute angle. At 90 degrees, you have a right angle. Larger than 90 degrees, you have an obtuse angle. And then, if you get all the way to 180 degrees, your angle actually forms a line." + }, + { + "Q": "how is direct and inverse variation the same or how is different ?", + "A": "They are both functions that make two parameters increase or decrease. The first makes both decrease or increase, while the second makes one parameter increase while the other decreases.", + "video_name": "92U67CUy9Gc", + "transcript": "I want to talk a little bit about direct and inverse variations. So I'll do direct variation on the left over here. And I'll do inverse variation, or two variables that vary inversely, on the right-hand side over here. So a very simple definition for two variables that vary directly would be something like this. y varies directly with x if y is equal to some constant with x. So we could rewrite this in kind of English as y varies directly with x. And if this constant seems strange to you, just remember this could be literally any constant number. So let me give you a bunch of particular examples of y varying directly with x. You could have y is equal to x. Because in this situation, the constant is 1. We didn't even write it. We could write y is equal to 1x, then k is 1. We could write y is equal to 2x. We could write y is equal to 1/2 x. We could write y is equal to negative 2x. We are still varying directly. We could have y is equal to negative 1/2 x. We could have y is equal to pi times x. We could have y is equal to negative pi times x. I don't want to beat a dead horse now. I think you get the point. Any constant times x-- we are varying directly. And to understand this maybe a little bit more tangibly, let's think about what happens. And let's pick one of these scenarios. Well, I'll take a positive version and a negative version, just because it might not be completely intuitive. So let's take the version of y is equal to 2x, and let's explore why we say they vary directly with each other. So let's pick a couple of values for x and see what the resulting y value would have to be. So if x is equal to 1, then y is 2 times 1, or is 2. If x is equal to 2, then y is 2 times 2, which is going to be equal to 4. So when we doubled x, when we went from 1 to 2-- so we doubled x-- the same thing happened to y. We doubled y. So that's what it means when something varies directly. If we scale x up by a certain amount, we're going to scale up y by the same amount. If we scale down x by some amount, we would scale down y by the same amount. And just to show you it works with all of these, let's try the situation with y is equal to negative 2x. I'll do it in magenta. y is equal to negative-- well, let me do a new example that I haven't even written here. Let's try y is equal to negative 3x. So once again, let me do my x and my y. When x is equal to 1, y is equal to negative 3 times 1, which is negative 3. When x is equal to 2, so negative 3 times 2 is negative 6. So notice, we multiplied. So if we scaled-- let me do that in that same green color. If we scale up x by 2-- it's a different green color, but it serves the purpose-- we're also scaling up y by 2. To go from 1 to 2, you multiply it by 2. To go from negative 3 to negative 6, you're also multiplying by 2. So we grew by the same scaling factor. And if you wanted to go the other way-- let's try, I don't know, let's go to x is 1/3. If x is 1/3, then y is going to be-- negative 3 times 1/3 is negative 1. So notice, to go from 1 to 1/3, we divide by 3. To go from negative 3 to negative 1, we also divide by 3. We also scale down by a factor of 3. So whatever direction you scale x in, you're going to have the same scaling direction as y. That's what it means to vary directly. Now, it's not always so clear. Sometimes it will be obfuscated. So let's take this example right over here. y is equal to negative 3x. And I'm saving this real estate for inverse variation in a second. You could write it like this, or you could algebraically manipulate it. You could maybe divide both sides of this equation by x, and then you would get y/x is equal to negative 3. Or maybe you divide both sides by x, and then you divide both sides by y. So from this, so if you divide both sides by y now, you could get 1/x is equal to negative 3 times 1/y. These three statements, these three equations, are all saying the same thing. So sometimes the direct variation isn't quite in your face. But if you do this, what I did right here with any of these, you will get the exact same result. Or you could just try to manipulate it back to this form over here. And there's other ways we could do it. We could divide both sides of this equation by negative 3. And then you would get negative 1/3 y is equal to x. And now, this is kind of an interesting case here because here, this is x varies directly with y. Or we could say x is equal to some k times y. And in general, that's true. If y varies directly with x, then we can also say that x varies directly with y. It's not going to be the same constant. It's going to be essentially the inverse of that constant, but they're still directly varying. Now with that said, so much said, about direct variation, let's explore inverse variation a little bit. Inverse variation-- the general form, if we use the same variables. And it always doesn't have to be y and x. It could be an a and a b. It could be a m and an n. If I said m varies directly with n, we would say m is equal to some constant times n. Now let's do inverse variation. So if I did it with y's and x's, this would be y is equal to some constant times 1/x. So instead of being some constant times x, it's some constant times 1/x. So let me draw you a bunch of examples. It could be y is equal to 1/x. It could be y is equal to 2 times 1/x, which is clearly the same thing as 2/x. It could be y is equal to 1/3 times 1/x, which is the same thing as 1 over 3x. it could be y is equal to negative 2 over x. And let's explore this, the inverse variation, the same way that we explored the direct variation. So let's pick-- I don't know/ let's pick y is equal to 2/x. And let me do that same table over here. So I have my table. I have my x values and my y values. If x is 1, then y is 2. If x is 2, then 2 divided by 2 is 1. So if you multiply x by 2, if you scale it up by a factor of 2, what happens to y? y gets scaled down by a factor of 2. You're dividing by 2 now. Notice the difference. Here, however we scaled x, we scaled up y by the same amount. Now, if we scale up x by a factor, when we have inverse variation, we're scaling down y by that same. So that's where the inverse is coming from. And we could go the other way. If we made x is equal to 1/2. So if we were to scale down x, we're going to see that it's going to scale up y. Because 2 divided by 1/2 is 4. So here we are scaling up y. So they're going to do the opposite things. They vary inversely. And you could try it with the negative version of it, as well. So here we're multiplying by 2. And once again, it's not always neatly written for you like this. It can be rearranged in a bunch of different ways. But it will still be inverse variation as long as they're algebraically equivalent. So you can multiply both sides of this equation right here by x. And you would get xy is equal to 2. This is also inverse variation. You would get this exact same table over here. You could divide both sides of this equation by y. And you could get x is equal to 2/y, which is also the same thing as 2 times 1/y. So notice, y varies inversely with x. And you could just manipulate this algebraically to show that x varies inversely with y. So y varies inversely with x. This is the same thing as saying-- and we just showed it over here with a particular example-- that x varies inversely with y. And there's other things. We could take this and divide both sides by 2. And you would get y/2 is equal to 1/x. There's all sorts of crazy things. And so in general, if you see an expression that relates to variables, and they say, do they vary inversely or directly or maybe neither? You could either try to do a table like this. If you scale up x by a certain amount and y gets scaled up by the same amount, then it's direct variation. If you scale up x by some-- and you might want to try a couple different times-- and you scale down y, you do the opposite with y, then it's probably inverse variation. A surefire way of knowing what you're dealing with is to actually algebraically manipulate the equation so it gets back to either this form, which would tell you that it's inverse variation, or this form, which would tell you that it is direct variation." + }, + { + "Q": "why does liquid change to steam?", + "A": "because when molecules absorb energy (thermal energy ) its kinetic energy increase and it collide faster then spaces between molecules increase and its density decrease and it turns into steam", + "video_name": "pKvo0XWZtjo", + "transcript": "I think we're all reasonably familiar with the three states of matter in our everyday world. At very high temperatures you get a fourth. But the three ones that we normally deal with are, things could be a solid, a liquid, or it could be a gas. And we have this general notion, and I think water is the example that always comes to at least my mind. Is that solid happens when things are colder, relatively colder. And then as you warm up, you go into a liquid state. And as your warm up even more you go into a gaseous state. So you go from colder to hotter. And in the case of water, when you're a solid, you're ice. When you're a liquid, some people would call ice water, but let's call it liquid water. I think we know what that is. And then when it's in the gas state, you're essentially vapor or steam. So let's think a little bit about what, at least in the case of water, and the analogy will extend to other types of molecules. But what is it about water that makes it solid, and when it's colder, what allows it to be liquid. And I'll be frank, liquids are kind of fascinating because you can never nail them down, I guess is the best way to view them. Or a gas. So let's just draw a water molecule. So you have oxygen there. You have some bonds to hydrogen. And then you have two extra pairs of valence electrons in the oxygen. And a couple of videos ago, we said oxygen is a lot more electronegative than the hydrogen. It likes to hog the electrons. So even though this shows that they're sharing electrons here and here. At both sides of those lines, you can kind of view that hydrogen is contributing an electron and oxygen is contributing an electron on both sides of that line. But we know because of the electronegativity, or the relative electronegativity of oxygen, that it's hogging these electrons. And so the electrons spend a lot more time around the oxygen than they do around the hydrogen. And what that results is that on the oxygen side of the molecule, you end up with a partial negative charge. And we talked about that a little bit. And on the hydrogen side of the molecules, you end up with a slightly positive charge. Now, if these molecules have very little kinetic energy, they're not moving around a whole lot, then the positive sides of the hydrogens are very attracted to the negative sides of oxygen in other molecules. Let me draw some more molecules. When we talk about the whole state of the whole matter, we actually think about how the molecules are interacting with Not just how the atoms are interacting with each other within a molecule. I just drew one oxygen, let me copy and paste that. But I could do multiple oxygens. And let's say that that hydrogen is going to want to be near this oxygen. Because this has partial negative charge, this has a partial positive charge. And then I could do another one right there. And then maybe we'll have, and just to make the point clear, you have two hydrogens here, maybe an oxygen wants to hang out there. So maybe you have an oxygen that wants to be here because it's got its partial negative here. And it's connected to two hydrogens right there that have their partial positives. But you can kind of see a lattice structure. Let me draw these bonds, these polar bonds that start forming between the particles. These bonds, they're called polar bonds because the molecules themselves are polar. And you can see it forms this lattice structure. And if each of these molecules don't have a lot of kinetic energy. Or we could say the average kinetic energy of this matter is fairly low. And what do we know is average kinetic energy? Well, that's temperature. Then this lattice structure will be solid. These molecules will not move relative to each other. I could draw a gazillion more, but I think you get the point that we're forming this kind of fixed structure. And while we're in the solid state, as we add kinetic energy, as we add heat, what it does to molecules is, it just makes them vibrate around a little bit. If I was a cartoonist, they way you'd draw a vibration is to put quotation marks there. That's not very scientific. But they would vibrate around, they would buzz around a little bit. I'm drawing arrows to show that they are vibrating. It doesn't have to be just left-right it could be up-down. But as you add more and more heat in a solid, these molecules are going to keep their structure. So they're not going to move around relative to each other. But they will convert that heat, and heat is just a form of energy, into kinetic energy which is expressed as the vibration of these molecules. Now, if you make these molecules start to vibrate enough, and if you put enough kinetic energy into these molecules, what do you think is going to happen? Well this guy is vibrating pretty hard, and he's vibrating harder and harder as you add more and more heat. This guy is doing the same thing. At some point, these polar bonds that they have to each other are going to start not being strong enough to contain the vibrations. And once that happens, the molecules-- let me draw a couple more. Once that happens, the molecules are going to start moving past each other. So now all of a sudden, the molecule will start shifting. But they're still attracted. Maybe this side is moving here, that's moving there. You have other molecules moving around that way. But they're still attracted to each other. Even though we've gotten the kinetic energy to the point that the vibrations can kind of break the bonds between the polar sides of the molecules. Our vibration, or our kinetic energy for each molecule, still isn't strong enough to completely separate them. They're starting to slide past each other. And this is essentially what happens when you're in a liquid state. You have a lot of atoms that want be touching each other but they're sliding. They have enough kinetic energy to slide past each other and break that solid lattice structure here. And then if you add even more kinetic energy, even more heat, at this point it's a solution now. They're not even going to be able to stay together. They're not going to be able to stay near each other. If you add enough kinetic energy they're going to start looking like this. They're going to completely separate and then kind of bounce around independently. Especially independently if they're an ideal gas. But in general, in gases, they're no longer touching They might bump into each other. But they have so much kinetic energy on their own that they're all doing their own thing and they're not touching. I think that makes intuitive sense if you just think about what a gas is. For example, it's hard to see a gas. Why is it hard to see a gas? Because the molecules are much further apart. So they're not acting on the light in the way that a liquid or a solid would. And if we keep making that extended further, a solid-- well, I probably shouldn't use the example with ice. Because ice or water is one of the few situations where the solid is less dense than the liquid. That's why ice floats. And that's why icebergs don't just all fall to the bottom of the ocean. And ponds don't completely freeze solid. But you can imagine that, because a liquid is in most cases other than water, less dense. That's another reason why you can see through it a little Or it's not diffracting-- well I won't go into that too much, than maybe even a solid. But the gas is the most obvious. And it is true with water. The liquid form is definitely more dense than the gas form. In the gas form, the molecules are going to jump around, not touch each other. And because of that, more light can get through the substance. Now the question is, how do we measure the amount of heat that it takes to do this to water? And to explain that, I'll actually draw a phase change diagram. Which is a fancy way of describing something fairly straightforward. Let me say that this is the amount of heat I'm adding. And this is the temperature. We'll talk about the states of matter in a second. So heat is often denoted by q. Sometimes people will talk about change in heat. They'll use H, lowercase and uppercase H. They'll put a delta in front of the H. Delta just means change in. And sometimes you'll hear the word enthalpy. Let me write that. Because I used to say what is enthalpy? It sounds like empathy, but it's quite a different concept. At least, as far as my neural connections could make it. But enthalpy is closely related to heat. It's heat content. For our purposes, when you hear someone say change in enthalpy, you should really just be thinking of change in heat. I think this word was really just introduced to confuse chemistry students and introduce a non-intuitive word into their vocabulary. The best way to think about it is heat content. Change in enthalpy is really just change in heat. And just remember, all of these things, whether we're talking about heat, kinetic energy, potential energy, enthalpy. You'll hear them in different contexts, and you're like, I thought I should be using heat and they're talking about enthalpy. These are all forms of energy. And these are all measured in joules. And they might be measured in other ways, but the traditional way is in joules. And energy is the ability to do work. And what's the unit for work? Well, it's joules. Force times distance. But anyway, that's a side-note. But it's good to know this word enthalpy. Especially in a chemistry context, because it's used all the time and it can be very confusing and non-intuitive. Because you're like, I don't know what enthalpy is in my everyday life. Just think of it as heat contact, because that's really But anyway, on this axis, I have heat. So this is when I have very little heat and I'm increasing my heat. And this is temperature. Now let's say at low temperatures I'm here and as I add heat my temperature will go up. Temperature is average kinetic energy. Let's say I'm in the solid state here. And I'll do the solid state in purple. No I already was using purple. I'll use magenta. So as I add heat, my temperature will go up. Heat is a form of energy. And when I add it to these molecules, as I did in this example, what did it do? It made them vibrate more. Or it made them have higher kinetic energy, or higher average kinetic engery, and that's what temperature is a measure of; average kinetic energy. So as I add heat in the solid phase, my average kinetic energy will go up. And let me write this down. This is in the solid phase, or the solid state of matter. Now something very interesting happens. Let's say this is water. So what happens at zero degrees? Which is also 273.15 Kelvin. Let's say that's that line. What happens to a solid? Well, it turns into a liquid. Ice melts. Not all solids, we're talking in particular about water, about H2O. So this is ice in our example. All solids aren't ice. Although, you could think of a rock as solid magma. Because that's what it is. I could take that analogy a bunch of different ways. But the interesting thing that happens at zero degrees. Depending on what direction you're going, either the freezing point of water or the melting point of ice, something interesting happens. As I add more heat, the temperature does not to go up. As I add more heat, the temperature does not go up for a little period. Let me draw that. For a little period, the temperature stays constant. And then while the temperature is constant, it stays a solid. We're still a solid. And then, we finally turn into a liquid. Let's say right there. So we added a certain amount of heat and it just stayed a solid. But it got us to the point that the ice turned into a liquid. It was kind of melting the entire time. That's the best way to think about it. And then, once we keep adding more and more heat, then the liquid warms up too. Now, we get to, what temperature becomes interesting again for water? Well, obviously 100 degrees Celsius or 373 degrees Kelvin. I'll do it in Celsius because that's what we're familiar with. That's the temperature at which water will vaporize or which water will boil. But something happens. And they're really getting kinetically active. But just like when you went from solid to liquid, there's a certain amount of energy that you have to contribute to the system. And actually, it's a good amount at this point. Where the water is turning into vapor, but it's not getting any hotter. So we have to keep adding heat, but notice that the temperature didn't go up. We'll talk about it in a second what was happening then. And then finally, after that point, we're completely vaporized, or we're completely steam. Then we can start getting hot, the steam can then get hotter as we add more and more heat to the system. So the interesting question, I think it's intuitive, that as you add heat here, our temperature is going to go up. But the interesting thing is, what was going on here? We were adding heat. So over here we were turning our heat into kinetic energy. Temperature is average kinetic energy. But over here, what was our heat doing? Well, our heat was was not adding kinetic energy to the system. The temperature was not increasing. But the ice was going from ice to water. So what was happening at that state, is that the kinetic energy, the heat, was being used to essentially break these bonds. And essentially bring the molecules into a higher energy state. So you're saying, Sal, what does that mean, higher energy state? Well, if there wasn't all of this heat and all this kinetic energy, these molecules want to be very close to each other. For example, I want to be close to the surface of the earth. When you put me in a plane you have put me in a higher energy state. I have a lot more potential energy. I have the potential to fall towards the earth. Likewise, when you move these molecules apart, and you go from a solid to a liquid, they want to fall towards each other. But because they have so much kinetic energy, they never quite are able to do it. But their energy goes up. Their potential energy is higher because they want to fall towards each other. By falling towards each other, in theory, they could do some work. So what's happening here is, when we're contributing heat-- and this amount of heat we're contributing, it's called the heat of fusion. Because it's the same amount of heat regardless how much direction we go in. When we go from solid to liquid, you view it as the heat of melting. It's the head that you need to put in to melt the ice into liquid. When you're going in this direction, it's the heat you have to take out of the zero degree water to turn it into ice. So you're taking that potential energy and you're bringing the molecules closer and closer to each other. So the way to think about it is, right here this heat is being converted to kinetic energy. Then, when we're at this phase change from solid to liquid, that heat is being used to add potential energy into the system. To pull the molecules apart, to give them more potential energy. If you pull me apart from the earth, you're giving me potential energy. Because gravity wants to pull me back to the earth. And I could do work when I'm falling back to the earth. A waterfall does work. It can move a turbine. You could have a bunch of falling Sals move a turbine as well. And then, once you are fully a liquid, then you just become a warmer and warmer liquid. Now the heat is, once again, being used for kinetic energy. You're making the water molecules move past each other faster, and faster, and faster. To some point where they want to completely disassociate from each other. They want to not even slide past each other, just completely jump away from each other. And that's right here. This is the heat of vaporization. And the same idea is happening. Before we were sliding next to each other, now we're pulling apart altogether. So they could definitely fall closer together. And then once we've added this much heat, now we're just heating up the steam. We're just heating up the gaseous water. And it's just getting hotter and hotter and hotter. But the interesting thing there, and I mean at least the interesting thing to me when I first learned this, whenever I think of zero degrees water I'll say, oh it must be ice. But that's not necessarily the case. If you start with water and you make it colder and colder and colder to zero degrees, you're essentially taking heat out of the water. You can have zero degree water and it hasn't turned into ice yet. And likewise, you could have 100 degree water that hasn't turned into steam yeat. You have to add more energy. You can also have 100 degree steam. You can also have zero degree water. Anyway, hopefully that gives you a little bit of intuition of what the different states of matter are. And in the next problem, we'll talk about how much heat exactly it does take to move along this line. And maybe we can solve some problems on how much ice we might need to make our drink cool." + }, + { + "Q": "in moon 1day=how many hours", + "A": "That is an interesting fact. Approximately, 29.5 days on earth is equal to 1 moon day. So 1 moon day = 708 hours", + "video_name": "TvPsESzCYis", + "transcript": "Do you feel that? Do you feel the weight of the atmosphere pushing down on you? Actually, you, like most people, have probably not thought about how heavy the atmosphere actually is and about the pressure it pushes on you every day. In fact, you have about 14.7 pounds of force pushing on you per every square inch of you. But where does this pressure come from? For the answer to that question, science had to wait till the middle of the 17th century, when Pascal had an hypothesis. You see he thought that the atmospheric pressure that we experience every day is actually due to the massive column of air that towers above us and the weight that it exerts by gravity. To test this, he devised a very simple experiment. He took an instrument called a barometer that measures atmospheric pressure and made a measurement at several locations in the lowest part of town, at the top of a church tower. And he even climbed a local mountain to measure the pressure at the top with the idea that if the height of the column of air stayed the same pressure, the pressure should stay the same in each of those locations. But if the height of the column decreases at each location, then the pressure should decrease as well. I'm going to show you how to build a simple pressure measuring device called a manometer. First, I'll need two relatively stiff tubes. I used two plastic pipe heads I had lying around the lab. I got one end off to make room for the plug. You'll also need a plug to plug one end and a piece of relatively stiff tube. Now to make the manometer, you need to connect the tubing to each of the pipe heads or plastic tubes. Once they're connected, you'll have a U-shaped device. You can then fill it with water, add some food coloring if you like, and then plug one end to fix the pressure. You have a manometer, and you're ready to make pressure measurements. At the beginning, the pressure exerted on each column of liquid is equal so the liquid levels in tube A and B are equal as well. If the pressure on tube A increases, the water in tube A will fall, and the water in tube B will rise. If the pressure over tube A decreases, then the water in tube B will fall, and the water in tube A will rise. For our first measurement, we went to Constitution Park Here we show that the pressure is greater than it was in the lab but using our manometer to show that column A is 12 millimeters below column B. But what's the pressure going to be up on Bunker Hill Monument? Well, let's find out. After climbing 275 stairs, we were able to take a measurement at the top of Bunker Hill Monument. Here we are able to show that the column A is just six millimeters below column B. For our final measurement, we drove about 20 miles south to the great Blue Hill. After climbing for about a mile and a half, we reached the top with our stunning views of Boston and the ocean. Here we get our lowest measurement yet. Column A is actually three millimeters above column B. Now for some resulting calculations-- getting from changes in water levels to pressure calculations. For starters, the air in Tube B is constrained by the plug. Because of this, it's an ideal gas and obeys the ideal gas law, which states the pressure times the volume equals nRT. Now nRT remains constant through all measurements. Because of this, we can then produce a nice relationship, and that is, P1 times V1 equals P2 times V2. Or in other words, the pressure times the volume at one state must equal the pressure and the volume at another state. But how do we get from differences in the water levels that we measured in the field to changes in the volume of Tube B, which is what we need for our actual calculations. Well, from my device, I've calculated that for each 1 millimeter change in the water level that that corresponds to a change in volume of 0.133 ml, with half of that volume change occurring in column A and the other half occurring in column B. So to calculate the change in volume of Tube B only, we can take my conversion factor of 0.133 ml per millimeter and multiply it by half of the height change. Therefore, the final volume in Tube B that we saw in the field is V2 equals the initial volume, V1 of 22 ml minus the change in volume. So using this formula, I went ahead and calculated the final volume to be at each of our locations. At Constitution Park, the final volume of Tube B was 20.4 ml, at Bunker Hill Monument, 21.2 ml, and at Great Blue Hill, 22.4 ml. Now we can take this relationship and rearrange it to actually solve for the pressure in each of these locations. Using the initial conditions of the air in Tube B, of P1 equals 101.6 kilopascals, which is the pressure the day that I built my manometer and the initial volume of Tube B, that is the air volume of 22 miles. Using this equation, we get an actual atmospheric pressure at Constitution Park of 109.6 kilopascals, at Bunker Hill Monument 105.5 pascals, and at the Great Blue Hill of 99.8% kilopascals. But how do these atmospheric pressures relate to the elevations that we measured them. Well just as pascal showed, we see that the pressure decreases with increased elevation at a very linear relationship over the elevations that we measured. Finally, I'd like to leave you with some tips when you make your own measurement. First, be patient. it can take time for the manometer to reach equilibrium as you change pressures. Second, take multiple measurements. This will greatly increase your accuracy. And third, pay attention to weather as things like changes in temperature. Or incoming weather systems can have a dramatic effect on the pressures that you measure. Some final questions to think about based on what you've learned today. First, what would happen to the water in the manometer if it went to 5,000 meters, or what about the moon? And second, what would happen if you made a measurement on an airplane at 10,000 meters? Think about these things as you think about pressure." + }, + { + "Q": "At 1:48 what he's doing looks a little like Fermat's Little Theory. Am I correct?", + "A": "Kind of, but it s not exactly Fermat s Little Theorem. It certainly looks like it, though. It s more about redundancy and how consecutive a s are more common than consecutive b s or c s in this language.", + "video_name": "WyAtOqfCiBw", + "transcript": "Voiceover: Shannon had just finished developing his theories related to cryptography and therefore was well aware that human communication was a mix of randomness and statistical dependencies. Letters in our messages were obviously dependent on previous letters to some extent. In 1949, he published a groundbreaking paper, \"A Mathematical Theory of Communication\". In it, he uses Markov models as the basis for how we can think about communication. He starts with a toy example. Imagine you encounter a bunch of text written in an alphabet of A, B, and C. Perhaps you know nothing about this language, though you notice As seem to clump together, while Bs and Cs do not. He then shows that you could design a machine to generate similar-looking text, using a Markov chain. He starts off with a zeroth-order approximation, which means we just independently select each symbol A, B, or C at random, and form a sequence However, notice that this sequence doesn't look like the original. He shows then you could do a bit better with a first-order approximation, where the letters are chosen independently, but according to the probability of each letter in the original sequence. This is slightly better as As are now more likely, but it still doesn't capture much structure. The next step is key. A second-order approximation takes into account each pair of letters which can occur. In this case, we need three states. The first state represents all pairs which begin with A, the second all pairs that begin with B, and the third state all pairs that begin with C. Notice now that the A cup has many AA pairs, which makes sense, since the conditional probability of an A after an A is higher in our original message. We can generate a sequence using this second-order model easily as follows. We start anywhere and pick a tile, and we write down our output the first letter, and move to the cup defined by the second letter. Then we pick a new tile, and repeat this process indefinitely Notice that this sequence is starting to look very similar to the original message, because this model is capturing the conditional dependencies between letters. If we wanted to do even better, we could move to a third-order approximation, which takes into account groups of three letters, or \"trigrams\". In this case, we would need nine states. But next, Shannon applies this exact same logic to actual English text, using statistics that were known for letters, pairs, and trigrams, etc. He shows the same progression from zeroth-order random letters to first-order, second-order and third-order sequences. He then goes on and tries the same thing using words instead of letters, and he writes \"the resemblance to ordinary English text \"increases quite noticeably at each depth.\" Indeed, these machines were producing meaningless text, though they contained approximately the same statistical structure you'd see in actual English. Shannon then proceeds to define a quantitative measure of information, as he realizes that the amount of information in some message must be tied up in the design of the machine which could be used to generate similar-looking sequences. Which brings us to his concept of entropy." + }, + { + "Q": "velocity = displacement/ time\nvelocity inversely proportional to time\nthis implies that velocity-time graph should be a rectangular hyperbola but here it is a straight line curve..?why?", + "A": "Just because you have a function of the form F(t) = y/t doesn t always indicate a hyperbolic curve. If in the function above y = 2t it would simplify to F(t) = 2.", + "video_name": "MAS6mBRZZXA", + "transcript": "The goal of this video is to explore some of the concept of formula you might see in introductional physics class but more importantly to see they are really just common sense ideas So let's just start with a simple example Let's say that and for the sake of this video keep things that magnitudes and velocities that's the direction of velocity etc. let's just assume that if I have a positive number that it means for example postive velocity that it means I'm going to the right let's say I have a negative number we won't see in this video let's assume we are going to the left In that way I can just write a number down only operating in one dimension you know that by specifying the magnitude and the direction if I say velocity is 5m/s that means 5m/s to the right if I say negative 5m/s that means 5m/s to the left let's say just for simplicitiy, say that we start with initial velocity we start with an initial velocity of 5m/s once again I specify the magnitude and the direction because of this convention here, we know it is to the right let's say we have a constant acceleration we have a constant acceleration 2m/s^2 or 2m per second square and once again since this is positive it is to the right and let's say that we do this for a duration so my change in time, let's say we do this for a duration of 4 I will just use s, second and s different places so s for this video is seconds So I want to do is to think about how far do we travel? and there is two things how fast are we going? after 4 seconds and how far have we travel over the course of those 4 seconds? so let's draw ourselves a little diagram here So this is my velocity axis, and this over here is time axis we have to draw a straighter line than that So that is my time axis, time this is velocity This is my velocity right over there and I'm starting off with 5m/s, so this is 5m/s right over here So vi is equal to 5m/s And every second goes by it goes 2m/s faster that's 2m/s*s every second that goes by So after 1 second when it goes 2m/s faster it will be at 7 another way to think about it is the slope of this velocity line is my constant accleration, my constant slope here so it might look something like that So what has happend after 4s? So 1 2 3 4 this is my delta t So my final velocity is going to be right over there I'm writing it here because this get into the way of veloctiy so this is v this is my final velocity what would it be? Well I'm starting at 5m/s So we are doing this both using the variable and concretes Some starting with some initial velocity I'm starting with some initial velocity Subscript i said i for initial and then each second that goes by I'm getting this much faster so if I gonna see how much faster have I gone I multiply the number of second, I will just multiply the number second it goes by times my acceleration, times my acceleration and once again, this right here, subscript c saying that is a constant acceleration, so that will tell my how fast I have gone If I started at this point and multiply the duration time with slope I will get this high, I will get to my final velocity just to make it clear with the numbers, this number can really be anything I'm just taking this to make it concrete in your mind you have 5m/s plus 4s plus, I wanna do it in yellow plus 4s times our acceleration with 2m per second square and what is this going to be equal to? you have a second that is cancelling out one of the second down here You have 4 times, so you have 5m/s plus 4 times 2 is 8 this second gone, we just have 8m/s or this is the same thing as 13m/s which is going to be our final velocity and I wanna take a pause here, you can pause and think about it yourself this whole should be intuitive, we are starting by going with 5m/s every second goes by we are gonna going 2m/s faster so after 1s it would be 7 m/s, after 2s we will be 9m/s after 3s we will be 11m/s, and after 4s we will be at 13 m/s so you multiply how much time pass times acceleration this is how much faster we are gonna be going, we are already going 5m/s 5 plus how much faster? 13 m/s so this right up here is 13m/s So I will take a little pause here hopefully intuitive and the whole play of that is to show you this formula you will see in many physics book is not something that randomly pop out of there it just make complete common sense Now the next thing I wanna talk about is what is the total distance that we would have travel? and we know from the last video that distance is just the area under this curve right over here, so it's just the area under this curve you see this is kind of a strange shape here how do I caculate this area? and we can use a little symbol of geometry to break it down into two different areas, it's very easy to calculate their areas two simple shapes, you can break it down to two, blue part is the rectangle right over here, easy to figure out the area of a rectangle and we can break it down to this purple part, this triangle right here easy to figure out the area of a triangle and that will be the total distance we travel even this will hopefully make some intuition because this blue area is how far we would have travel if we are not accelerated, we just want 5m/s for 4s so you goes 5m/s 1s 2s 3s 4s so you are going from 0 to 4 you change in time is 4s so if you go 5m/s for 4s you are going to go 20 m this right here is 20m that is the area of this right here 5 times 4 this purple or magentic area tells you how furthur than this are you going because you are accelerating because kept going faster and faster and faster it's pretty easy to calculate this area the base here is still 5(4) because that's 5(4) second that's gone by what's the height here? The height here is my final velocity minus my initial velocity minus my initial velocity or it's the change in velocity due to the accleration 13 minus 5 is 8 or this 8 right over here it is 8m/s so this height right over here is 8m/s the base over here is 4s that's the time that past what's this area of the triangle? the area of this triangle is one half times the base which is 4s 4s times the height which is 8m/s times 8m/s second cancel out one half time 4 is 2 times 8 is equal to 16m So the total distance we travel is 20 plus 16 is 36m that is the total I could say the total displacement and once again is to the right, since it's positive so that is our displacement What I wanna do is to do the exact the same calculation keep it in variable form, that will give another formula many people often memorize You might understand this is completely intuitive formula and that just come out of the logical flow of reasoning that we went through this video what is the area once again if we just think about the variables? well the area of this rectangle right here is our initial velocity times our change in time, times our change in time So that is the blue rectangle right over here, and plus what do we have to do? we have the change in time once again we have the change in time times this height which is our final velocity which is our final velocity minus our initial velocity these are all vectors, they are just positive if going to the right we just multiply the base with the height that will just be the area of the entire rectangle I will take it by half because triangle is just half of that rectangle so times one half, so times one half so this is the area, this is the purple area right over here this is the area of this, this is the area of that and let's simplify this a little bit let's factor out the delta t, so you factor out the delta t you get delta t times a bunch of stuff v sub i your initial velocity we factor this out plus this stuff, plus this thing right over here and we can distribute the one half we factor the one half, we factor the delta t out, taking it out and let's multiply one half by each of these things so it's gonna be plus one half times vf, times our final velocity that's not the right color, I will use the right color so you would understand what I am doing, so this is the one half so plus one half times our final velocity final velocity minus one half, minus one half times our initial velocity I'm gonna do that in blue, sorry I have trouble in changing color today minus one half times our initial velocity, times our initial velocity and what is this simplify do? we have something plus one half times something else minus one half of the original something so what is vi minus one half vi? so anything minus its half is just a positive half left so these two terms, this term and this term will simplify to one half vi one half initial velocity plus one half times the final velocity plus one half times the final velocity and all of that is being multiplied with the change in time the time that has gone by and this tells us the distance, the distance that we travel another way to think about it, let's factor out this one half you get distance that is equal to change in time times factoring out the one half vi plus vf, vi no that's not the right color vi plus vf so this is interesting, the distance we travel is equal to one half of the initial velocity plus the final velocity so this is really if you just took this quantity right over here it's just the arithmetic, I have trouble saying that word it's the arithmetic mean of these two numbers, so I'm gonna define, this is something new, I'm gonna call this the average velocity we have to be very careful with this this right here is the average velocity but the only reason why I can just take the starting velocity and ending velocity and adding them together and divide them by two since you took an average of two thing it's some place over here and I take that as average velocity it's because my acceleration is constant which is usually an assumption in introductory physics class but it's not always the assumption but if you do have a constant acceleration like this you can assume that the average velocity is gonna be the average of the initial velocity and the final velocity if this is a curve and the acceleration is changing you could not do that but what is useful about this is if you wanna figure out the distance that was travelled, you just need to know the initial velocity and the final velocity, average their two and multiply the times it goes by so in this situation our final velocity is 13m/s our initial velocity is 5m/s so you have 13 plus 5 is equal to 18 you divided that by 2, you average velocity is 9m/s if you take the average of 13 and 5 and 9m/s times 4s gives you 36m so hopefully it doesn't confuse you I just wannt show you some of these things you will see in your physics class but you shouldn't memorize they can all be deduced" + }, + { + "Q": "can you multiply 3 digit number times 2 digit number", + "A": "Yes, you can, the method is just the same as that of a 3 digit multiplication by another 3 digit multiplication.", + "video_name": "t8m0NalQtEk", + "transcript": "Let's start with a warm-up problem to avoid getting any mental cramps as we learn new things. So this is a problem that hopefully, if you understood what we did in the last video, you can kind of understand what we're about to do right now. And I'm going to escalate it even more. In the last video I think we finished what a four-digit number times a one-digit number. Let's up the stakes to a five-digit number. Let's do 64,329 times-- let me think of a nice number. Times 4. I'm going to show you right now that we're going to do the exact same process that we did in the last video. We just have to do it a little bit longer than we did before. So we just start off saying, OK, what's 4 times 9? 4 times 9 is equal to 36. 18 times 2. Yep, 36. So we write the 6 down here, carry the 3 up there. Just put the 3 up there, then you got 4 times 2. And they're going to have to add the 3. So let me just write that there. Plus 3 is equal to-- you do the multiplication first. You can even think of it as order of operations, but you just should know that you do the multiplication first. So 4 times 2 is 8. Plus 3 is equal to 11. Put this 1 down here and put the one 10 and 11 up there. Then you got 4 times 3. You got that one up there, so you're going to have to add that plus 1 is equal to-- that's going to equal 12. Plus 1, which is equal to 13. So it's 13. Then you have 4 times 4. You have this little one hanging out here from the previous multiplications, so you're going to to have to add that. And that's equal to 16 plus 1. It's equal to 17. Stick the 7 down here, put the 1 up there. We're almost done. And then we have 4 times 6. Plus 1. What is that? 4 times 6 is 24. Plus 1 is 25. Put the 5 down here. There's no where to put the 2. There's no more multiplications to do, so we just put the 2 down there. So 64,329 is 257,316. And in case you're wondering, these commas don't mean much. They just help me read the number. So I put it after every 3 digits, so I know that for example, that everything after this-- so s is 7,000. If I had another comma here, this is millions. So it just helps me read the problem a bit. So if you got that you're now ready to escalate to a slightly more complicated situation. Although the first way that we're going to do it it's actually not going to look any more complicated. It's just going to involve one more step. So everything we've done so far are a bunch of digits times a one-digit number. Now let's do a bunch of digits times a two-digit number. So let's say we want to multiply 36 times-- instead of putting a one-digit number here I'm going to put a two-digit number. So times 23. So you start off doing this problem exactly the way you would have done it if there was just a 3 down here. You can kind of ignore the 2 for a little bit. So 3 times 6 is equal to 18. So you just put the 8 here, put the 10 there, or the 1 there because it's 10 plus 8. 3 times 3 is 9. Plus 1, so 3 times 3 plus 1 is equal to-- that's 9 plus 1 is equal to 10. So you put the 10 there. There's nothing left. You put the 0 there. There's nothing left to put the 1, so you put the 10 there. So you essentially have solved the problem that 36-- let me do this is another color. That 36 times 3 is equal to 108. That's what we've solved so far, but we have this 20 sitting out here. We have this 20. We have to figure out what 20 times 360 is. Or sorry, what 20 times 36 is. This 2 is really a 20. And to make it all work out like that, what we do is we throw a 0 down here. We throw a 0 right there. In a second I'm going to explain why exactly we did that. So let's just do the same process as we did before with the 3. Now we do it with a 2, but we start filling up here and move to the left. So 2 times 6. That's 12. So 2 times 6 is 12. We put the 1 up here and we have to be very careful because we had this 1 from our previous problem, which doesn't apply anymore. So we could erase it or that 1 we could get rid of. If you have an eraser get rid of it, or you can just keep track in your head that the 1 you're about to write is a different 1. So what were we doing? We wrote 2 times 6 is 12. Put the 2 here. Put the 1 up here. And I got rid of the previous 1 because that would've just messed me up. Now I have 2 times 3. 2 times 3 is equal to 6. But then I have this plus 1 up here, so I have to add plus 1. So I get 7. So that is equal to 7. 2 times 3 plus 1 is equal to 7. So this 720 we just solved, that's literally-- let me write that down. That is 36 times 20. 36 times 20 is equal to 720. And hopefully that should explain why we had to throw this 0 here. If we didn't throw that 0 here we would have just a 2-- we would just have a 72 here, instead of 720. And 72 is 36 times 2. But this isn't a 2. This is a 2 in the 10's place. This is a 20. So we have to multiply 36 times 20, and that's why we got 720 there. So 36 times 23. Let's write it this way. Let me get some space up here. So we could write 30-- well, actually, let me just finish the problem and then I'll explain to you why it worked. So now to finish it up we just add 108 to 720. So 8 plus 0 is 8. 0 plus 2 is 2. 1 plus 7 is 8. So 36 times 23 is 828. Now you're saying Sal, why did that work? why were we able to figure out separately 36 times 3 is equal to 108, and then 36 times 20 is equal to 720, and then add them up like that? Because we could have rewritten the problem like this. We could have rewritten the problem as 36-- the original problem was this. We could have rewritten this as 36 times 20 plus 3. And this, and I don't know if you've learned the distributive property yet, but this is just the distributive property. This is just the same thing as 36 times 20 plus 36 times 3. If that confuses you, then you don't have to worry about it. But if it doesn't, then this is good. It's actually teaching you something. 36 times 20 we saw was 720. We learned that 36 times 3 was 108. And when you added them together we got what? 828? We got 828. And you could expand it even more like we did in the previous video. You could write this out as 30 plus 6 times 20 plus 3. Actually, let me just do it that way because I think that could help you out a little bit. If it confuses you, ignore it. If it doesn't, that's good. So we could do it 3 times 6. 3 times 6 is 18. 18 is just 10 plus 8. So it's 8, then we put a 10 up here. And ignore all this up here. 3 times 30. 3 times 30 is 90. 90 plus 10 is 100. So 100 is zero 10's plus 100. I don't know if this confuses you or not. If it does, ignore it. If it doesn't, well I don't want to complicate the issue. And now we can multiply 20. We can ignore this thing that we had before. 20 times 6 is 120. So that's 20 plus 100. So I'll put that 100 up here. 20 times 30 is 2 times 3 and you have two 0's there. And I think I'm maybe jumping the gun a little bit, assuming a little bit too much of what you may or may not know. But 20 times 30 is going to be 600. and you add another hundred there, that's 700. And then you add them all up. You get 800. 100 plus 700. Plus 20 plus 8, which is equal to 828. My point here is to show you why that system we did worked. Why we added a 0 here to begin with. But if it confuses you, don't worry about that right now. Learn how to do it and then maybe re-watch this video. Let's just do a bunch of more examples because I think the examples are what really, hopefully, explain the situation. So let's do 77. 77 times 77. 7 times 7 is 49. Put the 1 up here. 7 times 7, well, that's 49. Plus 4 is 53. There's no where to put the 5, so we put it down here. 7 times 7 is 49. Plus 4 is 53. Stick a 0 here. Now we're going to do this 7. So stick a 0 here. Let's get rid of this right there because that'll just mess us up. 7 times 7 is 49. Stick a 9 there. Put a 4 there. 7 times 7 is 49. Plus 4, which is 53. So notice, when we multiplied 7 times 77 we got 539. When we multiplied 70 times 77 we got 5,390. And it makes sense. They just differ by a 0. By a factor of 10. And now we can just add them up, and what do we get? 9 plus 0 is 9. 3 plus 9 is 12. Carry the 1. 1 plus 5 is 6. 6 plus 3 is 9. So it's 5,929." + }, + { + "Q": "In 5:02 he says AB is congruent to segment AC but he wrote AC before AB", + "A": "In Geometry the order only matters when you say AC and AB individually. a is congruent to a, while C is congruent to B. saying CA is congruent to AB would be incorrect. Tell me if it is still unclear, I m not the best explainer", + "video_name": "7UISwx2Mr4c", + "transcript": "So we're starting off with triangle ABC here. And we see from the drawing that we already know that the length of AB is equal to the length of AC, or line segment AB is congruent to line segment AC. And since this is a triangle and two sides of this triangle are congruent, or they have the same length, we can say that this is an isosceles triangle. Isosceles triangle, one of the hardest words for me to spell. I think I got it right. And that just means that two of the sides are equal to each other. Now what I want to do in this video is show what I want to prove. So what I want to prove here is that these two-- and they're sometimes referred to as base angles, these angles that are between one of the sides, and the side that isn't necessarily equal to it, and the other side that is equal and the side that's not equal to it. I want to show that they're congruent. So I want to prove that angle ABC, I want to prove that that is congruent to angle ACB. And so for an isosceles triangle, those two angles are often called base angles. And this might be called the vertex angle over here. And these are often called the sides or the legs of the isosceles triangle. And these are-- obviously they're sides. These are the legs of the isosceles triangle and this one down here, that isn't necessarily the same as the other two, you would call the base. So let's see if we can prove that. So there's not a lot of information here, just that these two sides are equal. But we have, in our toolkit, a lot that we know about triangle congruency. So maybe we can construct two triangles here that are congruent. And then we can use that information to figure out whether this angle is congruent to that angle there. And the first step, if we're going to use triangle congruency, is to actually construct two triangles. So one way to construct two triangles is let's set up another point right over here. Let's set up another point D. And let's just say that D is the midpoint of B and C. So it's the midpoint. So the distance from B to D is going to be the same thing as the distance-- let me do a double slash here to show you it's not the same as that distance. So the distance from B to D is going to be the same thing as the distance from D to C. And obviously, between any two points, you have a midpoint. And so let me draw segment AD. And what's useful about that is that we have now constructed two triangles. And what's even cooler is that triangle ABD and triangle ACD, they have this side is congruent, this side is congruent, and they actually share this side right over here. So we know that triangle ABD we know that it is congruent to triangle ACD. And we know it because of SSS, side-side-side. You have two triangles that have three sides that are congruent, or they have the same length. Then the two triangles are congruent. And what's useful about that is if these two triangles are congruent, then their corresponding angles are congruent. And so we've actually now proved our result. Because the corresponding angle to ABC in this triangle is angle ACD in this triangle right over here. So that we then know that angle ABC is congruent to angle ACB. So that's a pretty neat result. If you have an isosceles triangle, a triangle where two of the sides are congruent, then their base angles, these base angles, are also going to be congruent. Now let's think about it the other way. Can we make the other statement? If the base angles are congruent, do we know that these two legs are going to be congruent? So let's try to construct a triangle and see if we can prove it the other way. So I'll do another triangle right over here. Let me draw another one just like that. That's not that pretty of a triangle, so let me draw it a little nicer. I'm going to draw it like this. Let me do that in a different color. So I'll call that A. I will call this B. I will call that C right over there. And now we're going to start off with the idea that this angle, angle ABC, is congruent to angle ACB. So they have the same exact measure. And what we want to do in this case-- we want to prove-- so let me draw a little line here to show that we're doing a different idea. Here we're saying if these two sides are the same, then the base angles are going to be the same. We've proved that. Now let's go the other way. If the base angles are the same, do we know that the two sides are the same? So we want to prove that segment AC is congruent to AB. Or you could say that the length of segment AC, which we would denote that way, is equal to the length of segment AB. These are essentially equivalent statements. Once again in our toolkit, we have our congruency theorems. But in order to apply them, you really do need to have two triangles. So let's construct two triangles here. And this time, instead of defining another point as the midpoint, I'm going to define D this time as the point that if I were to go straight up, the point that is essentially-- if you view BC as straight horizontal, the point that goes straight down from A. And the reason why I say that is there's some point-- you could call it an altitude-- that intersects BC at a right angle. And there will definitely be some point like that. And so if it's a right angle on that side, if that's 90 degrees, then we know that this is 90 degrees as well. Now, what's interesting about this? And let me write this down. So I've constructed AD such that AD is perpendicular to BC. And you can always construct an altitude. Essentially, you just have to make BC lie flat on the ground. And then you just have to drop something from A, and that will give you point D. You can always do that with a triangle like this. So what does this give us? So over here, we have an angle, an angle, and then a side in common. And over here, you have an angle that corresponds to that angle, an angle that corresponds to this angle, and the same side in common. And so we know that these triangles are congruent by AAS, angle-angle-side, which we've shown is a valid congruent postulate. So we can say now that triangle ABD is congruent to triangle ACD. And we know that by angle-angle-side. This angle and this angle and this side. This angle and this angle and this side. And once we know these two triangles are congruent, we know that every corresponding angle or side of the two triangles are also going to be congruent. So then we know that AB is a corresponding side to AC. So these two sides must be congruent. And so you get AB is going to be congruent to AC, and that's because these are congruent triangles. And we've proven what we wanted to show. If the base angles are equal, then the two legs are going to be equal. If the two legs are equal, then the base angles are equal. It's a very, very, very useful tool in geometry. And in case you're curious, for this specific isosceles triangle, over here we set up D so it was the midpoint. Over here we set up D so it was directly below A. We didn't say whether it was the midpoint. But here, we can actually show that it is the midpoint just as a little bit of a bonus result, because we know that since these two triangles are congruent, BD is going to be congruent to DC because they are the corresponding sides. So it actually turns out that point D for an isosceles triangle, not only is it the midpoint but it is the place where, it is the point at which AD-- or we could say that AD is a perpendicular bisector of BC. So not only is AD perpendicular to BC, but it bisects it. That D is the midpoint of that entire base." + }, + { + "Q": "How would I do a cube root on a scientific calculator?", + "A": "Typically there is a root symbol with a y in the valley of the symbol.", + "video_name": "87_qIofPwhg", + "transcript": "- [Voiceover] We already know a little bit about square roots. For example, if I were to tell you that seven squared is equal to 49, that's equivalent to saying that seven is equal to the square root of 49. The square root essentially unwinds taking the square of something. In fact, we could write it like this. We could write the square root of 49, so this is whatever number times itself is equal to 49. If I multiply that number times itself, if I square it, well I'm going to get 49. And that's going to be true for any number, not just 49. If I write the square root of X and if I were to square it, that's going to be equal to X and that's going to be true for any X for which we can evaluate the square root, evaluate the principle root. Now typically and as you advance in math you're going to see that this will change, but typically you say, okay if I'm going to take the square root of something, X has to be non-negative. X has to be non-negative. This is going to change once we start thinking about imaginary and complex numbers, but typically for the principle square root, we assume that whatever's under the radical, whatever's under here, is going to be non-negative because it's hard to square a number at least the numbers that we know about, it's hard to square them and get a negative number. So for this thing to be defined, for it to make sense, it's typical to say that, okay we need to put a non-negative number in here. But anyway, the focus of this video is not on the square root, it's really just to review things so we can start thinking about the cube root. And as you can imagine, where does the whole notion of taking a square of something or a square root come from? Well it comes from the notion of finding the area of a square. If I have a square like this and if this side is seven, well if it's a square, all the sides are going to be seven. And if I wanted to find the area of this, it would be seven times seven or seven squared. That would be the area of this. Or if I were to say, well what is if I have a square, if I have, and that doesn't look like a perfect square, but you get the idea, all the sides are the same length. If I have a square with area X. If the area here is X, what are the lengths of the sides going to be? Well it's going to be square root of X. All of the sides are going to be the square root of X, so it's going to be the square root of X by the square root of X and this side is going to be the square root of X as well and that's going to be the square root of X as well. So that's where the term square root comes from, where the square comes from. Now what do you think cube root? Well same idea. If I have a cube. If I have a cube. Let me do my best attempt at drawing a cube really fast. If I have a cube and a cube, all of it's dimensions have the same length so this is a two, by two, by two cube, what's the volume over here. Well the volume is going to be two, times two, times two, which is two to the third power or two cubed. This is two cubed. That's why they use the word cubed because this would be the volume of a cube where each of its sides have length two and this of course is going to be equal to eight. But what if we went the other way around? What if we started with the cube? What if we started with this volume? What if we started with a cube's volume and let's say the volume here is eight cubic units, so volume is equal to eight and we wanted to find the lengths of the sides. So we wanted to figure out what X is cause that's X, that's X, and that's X. It's a cube so all the dimensions have the same length. Well there's two ways that we could express this. We could say that X times X times X or X to the third power is equal to eight or we could use the cube root symbol, which is a radical with a little three in the right place. Or we could write that X is equal to, it's going to look very similar to the square root. This would be the square root of eight, but to make it clear, they were talking about the cube root of eight, we would write a little three over there. In theory for square root, you could put a little two over here, but that'd be redundant. If there's no number here, people just assume that it's the square root. But if you're figuring out the cube root or sometimes you say the third root, well then you have to say, well you have to put this little three right over here in this little notch in the radical symbol right over here. And so this is saying X is going to be some number that if I cube it, I get eight. So with that out of the way, let's do some examples. Let's say that I have... Let's say that I want to calculate the cube root of 27. What's that going to be? Well if say that this is going to be equal to X, this is equivalent to saying that X to the third or that 27 is equal to X to the third power. So what is X going to be? Well X times X times X is equal to 27, well the number I can think of is three, so we would say that X, let me scroll down a little bit, X is equal to three. Now let me ask you a question. Can we write something like... Can we pick a new color? The cube root of, let me write negative 64. I already talked about that if we're talking the square root, it's fairly typical that hey you put a negative number in there at least until we learn about imaginary numbers, we don't know what to do with it. But can we do something with this? Well if I cube something, can I get a negative number? Sure. So if I say this is equal to X, this is the same thing as saying that negative 64 is equal to X to the third power. Well what could X be? Well what happens if you take negative four times negative four times negative four? Negative four times four is positive 16, but then times negative four is negative 64 is equal to negative 64. So what could X be here? Well X could be equal to negative four. X could be equal to negative four. So based on the math that we know so far you actually can take the cube root of a negative number. And just so you know, you don't have to stop there. You could take a fourth root and in this case you'd have a four here, a fifth root, a sixth root, a seventh root of numbers and we'll talk about that later in your mathematical career. But most of what you're going to see is actually going to be square root and every now and then you're going to see a cube root. Now you might be saying, well hey look, you know, you just knew that three to the third power is 27, you took the cube root, you get X, is there any simple way to do this? And like you know if i give you an arbitrary number. If I were to just say, I don't know, if I were to say cube root of 125. And the simple answer is, well the easiest way to actually figure this out is actually just to do a factorization and particular prime factorization of this thing right over here and then you would figure it out. So you would say, okay well 125 is five times 25, which is five times five. Alright, so this is the same thing as the cube root of five to the third power, which of course, is going to be equal to five. If you have a much larger number here, yes, there's no very simple way to compute what a cube root or a fourth root or a fifth root might be and even square root can get quite difficult. There's no very simple way to just calculate it the way that you might multiply things or divide it." + }, + { + "Q": "What happens when the steam gets hotter and hotter ?", + "A": "Water molecules move, vibrate, and rotate in several ways. As temperatures increase, their kinetic, vibrational, and rotational energies increase. At extremely high temperatures, these energies of motion are greater than the strength of the covalent O-H bonds that hold the molecules together. The molecules fall apart. We end up with a plasma, a gas that consists of H\u00e2\u0081\u00ba and O\u00c2\u00b2\u00e2\u0081\u00bb ions.", + "video_name": "pKvo0XWZtjo", + "transcript": "I think we're all reasonably familiar with the three states of matter in our everyday world. At very high temperatures you get a fourth. But the three ones that we normally deal with are, things could be a solid, a liquid, or it could be a gas. And we have this general notion, and I think water is the example that always comes to at least my mind. Is that solid happens when things are colder, relatively colder. And then as you warm up, you go into a liquid state. And as your warm up even more you go into a gaseous state. So you go from colder to hotter. And in the case of water, when you're a solid, you're ice. When you're a liquid, some people would call ice water, but let's call it liquid water. I think we know what that is. And then when it's in the gas state, you're essentially vapor or steam. So let's think a little bit about what, at least in the case of water, and the analogy will extend to other types of molecules. But what is it about water that makes it solid, and when it's colder, what allows it to be liquid. And I'll be frank, liquids are kind of fascinating because you can never nail them down, I guess is the best way to view them. Or a gas. So let's just draw a water molecule. So you have oxygen there. You have some bonds to hydrogen. And then you have two extra pairs of valence electrons in the oxygen. And a couple of videos ago, we said oxygen is a lot more electronegative than the hydrogen. It likes to hog the electrons. So even though this shows that they're sharing electrons here and here. At both sides of those lines, you can kind of view that hydrogen is contributing an electron and oxygen is contributing an electron on both sides of that line. But we know because of the electronegativity, or the relative electronegativity of oxygen, that it's hogging these electrons. And so the electrons spend a lot more time around the oxygen than they do around the hydrogen. And what that results is that on the oxygen side of the molecule, you end up with a partial negative charge. And we talked about that a little bit. And on the hydrogen side of the molecules, you end up with a slightly positive charge. Now, if these molecules have very little kinetic energy, they're not moving around a whole lot, then the positive sides of the hydrogens are very attracted to the negative sides of oxygen in other molecules. Let me draw some more molecules. When we talk about the whole state of the whole matter, we actually think about how the molecules are interacting with Not just how the atoms are interacting with each other within a molecule. I just drew one oxygen, let me copy and paste that. But I could do multiple oxygens. And let's say that that hydrogen is going to want to be near this oxygen. Because this has partial negative charge, this has a partial positive charge. And then I could do another one right there. And then maybe we'll have, and just to make the point clear, you have two hydrogens here, maybe an oxygen wants to hang out there. So maybe you have an oxygen that wants to be here because it's got its partial negative here. And it's connected to two hydrogens right there that have their partial positives. But you can kind of see a lattice structure. Let me draw these bonds, these polar bonds that start forming between the particles. These bonds, they're called polar bonds because the molecules themselves are polar. And you can see it forms this lattice structure. And if each of these molecules don't have a lot of kinetic energy. Or we could say the average kinetic energy of this matter is fairly low. And what do we know is average kinetic energy? Well, that's temperature. Then this lattice structure will be solid. These molecules will not move relative to each other. I could draw a gazillion more, but I think you get the point that we're forming this kind of fixed structure. And while we're in the solid state, as we add kinetic energy, as we add heat, what it does to molecules is, it just makes them vibrate around a little bit. If I was a cartoonist, they way you'd draw a vibration is to put quotation marks there. That's not very scientific. But they would vibrate around, they would buzz around a little bit. I'm drawing arrows to show that they are vibrating. It doesn't have to be just left-right it could be up-down. But as you add more and more heat in a solid, these molecules are going to keep their structure. So they're not going to move around relative to each other. But they will convert that heat, and heat is just a form of energy, into kinetic energy which is expressed as the vibration of these molecules. Now, if you make these molecules start to vibrate enough, and if you put enough kinetic energy into these molecules, what do you think is going to happen? Well this guy is vibrating pretty hard, and he's vibrating harder and harder as you add more and more heat. This guy is doing the same thing. At some point, these polar bonds that they have to each other are going to start not being strong enough to contain the vibrations. And once that happens, the molecules-- let me draw a couple more. Once that happens, the molecules are going to start moving past each other. So now all of a sudden, the molecule will start shifting. But they're still attracted. Maybe this side is moving here, that's moving there. You have other molecules moving around that way. But they're still attracted to each other. Even though we've gotten the kinetic energy to the point that the vibrations can kind of break the bonds between the polar sides of the molecules. Our vibration, or our kinetic energy for each molecule, still isn't strong enough to completely separate them. They're starting to slide past each other. And this is essentially what happens when you're in a liquid state. You have a lot of atoms that want be touching each other but they're sliding. They have enough kinetic energy to slide past each other and break that solid lattice structure here. And then if you add even more kinetic energy, even more heat, at this point it's a solution now. They're not even going to be able to stay together. They're not going to be able to stay near each other. If you add enough kinetic energy they're going to start looking like this. They're going to completely separate and then kind of bounce around independently. Especially independently if they're an ideal gas. But in general, in gases, they're no longer touching They might bump into each other. But they have so much kinetic energy on their own that they're all doing their own thing and they're not touching. I think that makes intuitive sense if you just think about what a gas is. For example, it's hard to see a gas. Why is it hard to see a gas? Because the molecules are much further apart. So they're not acting on the light in the way that a liquid or a solid would. And if we keep making that extended further, a solid-- well, I probably shouldn't use the example with ice. Because ice or water is one of the few situations where the solid is less dense than the liquid. That's why ice floats. And that's why icebergs don't just all fall to the bottom of the ocean. And ponds don't completely freeze solid. But you can imagine that, because a liquid is in most cases other than water, less dense. That's another reason why you can see through it a little Or it's not diffracting-- well I won't go into that too much, than maybe even a solid. But the gas is the most obvious. And it is true with water. The liquid form is definitely more dense than the gas form. In the gas form, the molecules are going to jump around, not touch each other. And because of that, more light can get through the substance. Now the question is, how do we measure the amount of heat that it takes to do this to water? And to explain that, I'll actually draw a phase change diagram. Which is a fancy way of describing something fairly straightforward. Let me say that this is the amount of heat I'm adding. And this is the temperature. We'll talk about the states of matter in a second. So heat is often denoted by q. Sometimes people will talk about change in heat. They'll use H, lowercase and uppercase H. They'll put a delta in front of the H. Delta just means change in. And sometimes you'll hear the word enthalpy. Let me write that. Because I used to say what is enthalpy? It sounds like empathy, but it's quite a different concept. At least, as far as my neural connections could make it. But enthalpy is closely related to heat. It's heat content. For our purposes, when you hear someone say change in enthalpy, you should really just be thinking of change in heat. I think this word was really just introduced to confuse chemistry students and introduce a non-intuitive word into their vocabulary. The best way to think about it is heat content. Change in enthalpy is really just change in heat. And just remember, all of these things, whether we're talking about heat, kinetic energy, potential energy, enthalpy. You'll hear them in different contexts, and you're like, I thought I should be using heat and they're talking about enthalpy. These are all forms of energy. And these are all measured in joules. And they might be measured in other ways, but the traditional way is in joules. And energy is the ability to do work. And what's the unit for work? Well, it's joules. Force times distance. But anyway, that's a side-note. But it's good to know this word enthalpy. Especially in a chemistry context, because it's used all the time and it can be very confusing and non-intuitive. Because you're like, I don't know what enthalpy is in my everyday life. Just think of it as heat contact, because that's really But anyway, on this axis, I have heat. So this is when I have very little heat and I'm increasing my heat. And this is temperature. Now let's say at low temperatures I'm here and as I add heat my temperature will go up. Temperature is average kinetic energy. Let's say I'm in the solid state here. And I'll do the solid state in purple. No I already was using purple. I'll use magenta. So as I add heat, my temperature will go up. Heat is a form of energy. And when I add it to these molecules, as I did in this example, what did it do? It made them vibrate more. Or it made them have higher kinetic energy, or higher average kinetic engery, and that's what temperature is a measure of; average kinetic energy. So as I add heat in the solid phase, my average kinetic energy will go up. And let me write this down. This is in the solid phase, or the solid state of matter. Now something very interesting happens. Let's say this is water. So what happens at zero degrees? Which is also 273.15 Kelvin. Let's say that's that line. What happens to a solid? Well, it turns into a liquid. Ice melts. Not all solids, we're talking in particular about water, about H2O. So this is ice in our example. All solids aren't ice. Although, you could think of a rock as solid magma. Because that's what it is. I could take that analogy a bunch of different ways. But the interesting thing that happens at zero degrees. Depending on what direction you're going, either the freezing point of water or the melting point of ice, something interesting happens. As I add more heat, the temperature does not to go up. As I add more heat, the temperature does not go up for a little period. Let me draw that. For a little period, the temperature stays constant. And then while the temperature is constant, it stays a solid. We're still a solid. And then, we finally turn into a liquid. Let's say right there. So we added a certain amount of heat and it just stayed a solid. But it got us to the point that the ice turned into a liquid. It was kind of melting the entire time. That's the best way to think about it. And then, once we keep adding more and more heat, then the liquid warms up too. Now, we get to, what temperature becomes interesting again for water? Well, obviously 100 degrees Celsius or 373 degrees Kelvin. I'll do it in Celsius because that's what we're familiar with. That's the temperature at which water will vaporize or which water will boil. But something happens. And they're really getting kinetically active. But just like when you went from solid to liquid, there's a certain amount of energy that you have to contribute to the system. And actually, it's a good amount at this point. Where the water is turning into vapor, but it's not getting any hotter. So we have to keep adding heat, but notice that the temperature didn't go up. We'll talk about it in a second what was happening then. And then finally, after that point, we're completely vaporized, or we're completely steam. Then we can start getting hot, the steam can then get hotter as we add more and more heat to the system. So the interesting question, I think it's intuitive, that as you add heat here, our temperature is going to go up. But the interesting thing is, what was going on here? We were adding heat. So over here we were turning our heat into kinetic energy. Temperature is average kinetic energy. But over here, what was our heat doing? Well, our heat was was not adding kinetic energy to the system. The temperature was not increasing. But the ice was going from ice to water. So what was happening at that state, is that the kinetic energy, the heat, was being used to essentially break these bonds. And essentially bring the molecules into a higher energy state. So you're saying, Sal, what does that mean, higher energy state? Well, if there wasn't all of this heat and all this kinetic energy, these molecules want to be very close to each other. For example, I want to be close to the surface of the earth. When you put me in a plane you have put me in a higher energy state. I have a lot more potential energy. I have the potential to fall towards the earth. Likewise, when you move these molecules apart, and you go from a solid to a liquid, they want to fall towards each other. But because they have so much kinetic energy, they never quite are able to do it. But their energy goes up. Their potential energy is higher because they want to fall towards each other. By falling towards each other, in theory, they could do some work. So what's happening here is, when we're contributing heat-- and this amount of heat we're contributing, it's called the heat of fusion. Because it's the same amount of heat regardless how much direction we go in. When we go from solid to liquid, you view it as the heat of melting. It's the head that you need to put in to melt the ice into liquid. When you're going in this direction, it's the heat you have to take out of the zero degree water to turn it into ice. So you're taking that potential energy and you're bringing the molecules closer and closer to each other. So the way to think about it is, right here this heat is being converted to kinetic energy. Then, when we're at this phase change from solid to liquid, that heat is being used to add potential energy into the system. To pull the molecules apart, to give them more potential energy. If you pull me apart from the earth, you're giving me potential energy. Because gravity wants to pull me back to the earth. And I could do work when I'm falling back to the earth. A waterfall does work. It can move a turbine. You could have a bunch of falling Sals move a turbine as well. And then, once you are fully a liquid, then you just become a warmer and warmer liquid. Now the heat is, once again, being used for kinetic energy. You're making the water molecules move past each other faster, and faster, and faster. To some point where they want to completely disassociate from each other. They want to not even slide past each other, just completely jump away from each other. And that's right here. This is the heat of vaporization. And the same idea is happening. Before we were sliding next to each other, now we're pulling apart altogether. So they could definitely fall closer together. And then once we've added this much heat, now we're just heating up the steam. We're just heating up the gaseous water. And it's just getting hotter and hotter and hotter. But the interesting thing there, and I mean at least the interesting thing to me when I first learned this, whenever I think of zero degrees water I'll say, oh it must be ice. But that's not necessarily the case. If you start with water and you make it colder and colder and colder to zero degrees, you're essentially taking heat out of the water. You can have zero degree water and it hasn't turned into ice yet. And likewise, you could have 100 degree water that hasn't turned into steam yeat. You have to add more energy. You can also have 100 degree steam. You can also have zero degree water. Anyway, hopefully that gives you a little bit of intuition of what the different states of matter are. And in the next problem, we'll talk about how much heat exactly it does take to move along this line. And maybe we can solve some problems on how much ice we might need to make our drink cool." + }, + { + "Q": "What are capillaries and where are they found?", + "A": "Capillaries are the smallest blood vessels that bring nutriments and oxygen to the tissues and absorbs carbon dioxide and waste products. They are located in many parts of the body such as lungs, muscles, endocrine glands, liver, central nervous system, etc. Hope this helped! :)", + "video_name": "QhiVnFvshZg", + "transcript": "Where I left off in the last video, we talked about how the hemoglobin in red blood cells is what sops up all of the oxygen so that it increases the diffusion gradient-- or it increases the incentive, we could say, for the oxygen to go across the membrane. We know that the oxygen molecules don't know that there's less oxygen here, but if you watch the video on diffusion you know how that process happens. If there's less concentration here than there, the oxygen will diffuse across the membrane and there's less inside the plasma because the hemoglobin is sucking it all up like a sponge. Now, one interesting question is, why does the hemoglobin even have to reside within the red blood cells? Why aren't hemoglobin proteins just freely floating in the blood plasma? That seems more efficient. You don't have to have things crossing through, in and out of, these red blood cell membranes. You wouldn't have to make red blood cells. What's the use of having these containers of hemoglobin? It's actually a very interesting idea. If you had all of the hemoglobin sitting in your blood plasma, it would actually hurt the flow of the blood. The blood would become more viscous or more thick. I don't want to say like syrup, but it would become thicker than blood is right now-- and by packaging the hemoglobin inside these containers, inside the red blood cells, what it allows the blood to do is flow a lot better. Imagine if you wanted to put syrup in water. If you just put syrup straight into water, what's going to happen? The water's going to become a little syrupy, a little bit more viscous and not flow as well. So what's the solution if you wanted to transport syrup in water? Well, you could put the syrup inside little containers or inside little beads and then let the beads flow in the water and then the water wouldn't be all gooey-- and that's exactly what's happening inside of our blood. Instead of having the hemoglobin sit in the plasma and make it gooey, it sits inside these beads that we call red blood cells that allows the flow to still be non-viscous. So I've been all zoomed in here on the alveolus and these capillaries, these pulmonary capillaries-- let's zoom out a little bit-- or zoom out a lot-- just to understand, how is the blood flowing? And get a better understanding of pulmonary arteries and veins relative to the other arteries and veins that are in the body. So here-- I copied this from Wikipedia, this diagram of the human circulatory system-- and here in the back you can see the lungs. Let me do it in a nice dark color. So we have our lungs here. You can see the heart is sitting right in the middle. And what we learned in the last few videos is that we have our little alveoli and our lungs. Remember, we get to them from our bronchioles, which are branching off of the bronchi, which branch off of the trachea, which connects to our larynx, which connects to our pharynx, which connects to our mouth and nose. But anyway, we have our little alveoli right there and then we have the capillaries. So when we go away from the heart-- and we're going to delve a little bit into the heart in this video as well-- so when blood travels away from the heart, it's de-oxygenated. It's this blue color. So this right here is blood. This right here is blood traveling away from the heart. It's going behind these two tubes right there. So this is the blood going away from the heart. So this blue that I've been highlighting just now, these are the pulmonary arteries and then they keep splitting into arterials and all of that and eventually we're in capillaries-- super, super small tubes. They run right past the alveoli and then they become oxygenated and now we're going back to the heart. So we're talking about pulmonary veins. So we go back to the heart. So these capillaries-- in the capillaries we get oxygen. Now we're going to go back to the heart. Hope you can see what I'm doing. And we're going to enter the heart on this side. You actually can't even see where we're entering the heart. We're going to enter the heart right over here-- and I'm going to go into more detail on that. Now we have oxygenated blood. And then that gets pumped out to the rest of the body. Now this is the interesting thing. When we're talking about pulmonary arteries and veins-- remember, the pulmonary artery was blue. As we go away from the heart, we have de-oxygenated blood, but it's still an artery. Then as we go towards the heart from the lungs, we have a vein, but it's oxygenated. So that's this little loop here that we start and I'm going to keep going over the circulation pattern because the heart can get a little confusing, especially because of its three-dimensional nature. But what we have is, the heart pumps de-oxygenated blood from the right ventricle. You're saying, hey, why is it the right ventricle? That looks like the left side of the drawing, but it's this dude's right-hand side, right? This is this guy's right hand. And this is this dude's left hand. He's looking at us, right? We don't care about our right or left. We care about this guy's right and left. And he's looking at us. He's got some eyeballs and he's looking at us. So this is his right ventricle. Actually, let me just start off with the whole cycle. So we have de-oxygenated blood coming from the rest of the body, right? The name for this big pipe is called the inferior vena cava-- inferior because it's coming up below. Actually, you have blood coming up from the arms and the head up here. They're both meeting right here, in the right atrium. Let me label that. I'm going to do a big diagram of the heart in a second. And why are they de-oxygenated? Because this is blood returning from our legs if we're running, or returning from our brain, that had to use respiration-- or maybe we're working out and it's returning from our biceps, but it's de-oxygenated blood. It shows up right here in the right atrium. It's on our left, but this guy's right-hand side. From the right atrium, it gets pumped into the right ventricle. It actually passively flows into the right ventricle. The ventricles do all the pumping, then the ventricle contracts and pumps this blood right here-- and you don't see it, but it's going behind this part right here. It goes from here through this pipe. So you don't see it. I'm going to do a detailed diagram in a second-- into the pulmonary artery. We're going away from the heart. This was a vein, right? This is a vein going to the heart. This is a vein, inferior vena cava vein. This is superior vena cava. They're de-oxygenated. Then I'm pumping this de-oxygenated blood away from the heart to the lungs. Now this de-oxygenated blood, this is in an artery, right? This is in the pulmonary artery. It gets oxygenated and now it's a pulmonary vein. And once it's oxygenated, it shows up here in the left-- let me do a better color than that-- it shows up right here in the left atrium. Atrium, you can imagine-- it's kind of a room with a skylight or that's open to the outside and in both of these cases, things are entering from above-- not sunlight, but blood is entering from above. On the right atrium, the blood is entering from above. And in the left atrium, the blood is entering-- and remember, the left atrium is on the right-hand side from our point of view-- on the left atrium, the blood is entering from above from the lungs, from the pulmonary veins. Veins go to the heart. Then it goes into-- and I'll go into more detail-- into the left ventricle and then the left ventricle pumps that oxygenated blood to the rest of the body via the non-pulmonary arteries. So everything pumps out. Let me make it a nice dark, non-blue color. So it pumps it out through there. You don't see it right here, the way it's drawn. It's a little bit of a strange drawing. It's hard to visualize, but I'll show it in more detail and then it goes to the rest of the body. Let me show you that detail right now. So we said, we have de-oxygenated blood. Let's label it right here. This is the superior vena cava. This is a vein from the upper part of our body from our arms and heads. This is the inferior vena vaca. This is veins from our abdomen and from our legs and the rest of our body. So it it first enters the right atrium. Remember, we call the right atrium because this is someone's heart facing us, even though this is on the left-hand side. It enters through here. It's de-oxygenated blood. It's coming from veins. the body used the oxygen. Then it shows up in the right ventricle, right? These are valves in our heart. And it passively, once the right ventricle pumps and then releases, it has a vacuum and it pulls more blood from the It pumps again and then it pushes it through here. Now this blood right here-- remember, this one still is de-oxygenated blood. De-oxygenated blood goes to the lungs to become oxygenated. So this right here is the pulmonary-- I'm using the word pulmonary because it's going to or from the lungs. It's dealing with the lungs. And it's going away from the heart. It's the pulmonary artery and it is de-oxygenated. Then it goes to the heart, rubs up against some alveoli and then gets oxygenated and then it comes right back. Now this right here, we're going to the heart. So that's a vein. It's in the loop with the lungs so it's a pulmonary vein and it rubbed up against the alveoli and got the oxygen diffused into it so it is oxygenated. And then it flows into your left atrium. Now, the left atrium, once again, from our point of view, is on the right-hand side, but from the dude looking at it, it's his left-hand side. So it goes into the left atrium. Now in the left ventricle, after it's done pumping, it expands and that oxygenated blood flows into the left ventricle. Then the left ventricle-- the ventricles are what do all the pumping-- it squeezes and then it pumps the blood into the aorta. This is an artery. Why is it an artery? Because we're going away from the heart. Is it a pulmonary artery? No, we're not dealing with the lungs anymore. We dealt with the lungs when we went from the right ventricle, went to the lungs in a loop, back to the left atrium. Now we're in the left ventricle. We pump into the aorta. Now this is to go to the rest of the body. This is an artery, a non-pulmonary artery-- and it is oxygenated. So when we're dealing with non-pulmonary arteries, we're oxygenated, but a pulmonary artery has no oxygen. It's going away from the heart to get the oxygen. Pulmonary vein comes from the lungs to the heart with oxygen, but the rest of the veins go to the heart without oxygen because they want to go into that loop on the pulmonary loop right there. So I'll leave you there. Hopefully that gives-- actually, let's go back to that first diagram. I think you have a sense of how the heart is dealing, but let's go look at the rest of the body and just get a sense of things. You can look this up on Wikipedia if you like. All of these different branching points have different names to them, but you can see right here you have kind of a branching off, a little bit below the heart. This is actually the celiac trunk. Celiac, if I remember correctly, kind of refers to an abdomen. So this blood that-- your hepatic artery. Hepatic deals with the liver. Your hepatic artery branches off of this to get blood flow to the liver. It also gives blood flow to your stomach so it's very important in digestion and all that. And then let's say this is the hepatic trunk. Your liver is sitting like that. Hepatic trunk-- it delivers oxygen to the liver. The liver is doing respiration. It takes up the oxygen and then it gives up carbon dioxide. So it becomes de-oxygenated and then it flows back in and to the inferior vena cava, into the vein. I want to make it clear-- it's a loop. It's a big loop. The blood doesn't just flow out someplace and then come back someplace else. This is just one big loop. And if you want to know at any given point in time, depending on your size, there's about five liters of blood. And I looked it up-- it takes the average red blood cell to go from one point in the circulatory system and go through the whole system and come back, 20 seconds. That's an average because you can imagine there might be some red blood cells that get stuck someplace and take a little bit more time and some go through the completely perfect route. Actually, the 20 seconds might be closer to the perfect route. I've never timed it myself. But it's an interesting thing to look at and to think about what's connected to what. You have these these arteries up here that they first branch off the arteries up here from the aorta into the head and the neck and the arm arteries and then later they go down and they flow blood to the rest of the body. So anyway, this is a pretty interesting idea. In the next video, what I want to do is talk about, how does the hemoglobin know when to dump the oxygen? Or even better, where to dump the oxygen-- because maybe I'm running so I need a lot of oxygen in the capillaries around my thigh muscles. I don't need them necessarily in my hands. How does the body optimize where the oxygen is actually It's actually fascinating." + }, + { + "Q": "Is there an actuall unit that can measure our weight and length accurately the way we are,like we never know how big the universe is,it may can be just some dust out there in a bigger world,we cant never know what exactly how big we are right?", + "A": "Excellent question! We can measure things very, very precisely according to the definitions we make up. But if I know that something is 1 meter, how can I say if that is big or small? The only way is by comparing it to other things, right? You have just discovered the essence of the theory of relativity. Good thinking!", + "video_name": "5FEjrStgcF8", + "transcript": "The purpose of this video is to just begin to appreciate how vast and enormous the universe is. And frankly, our brains really can't grasp it. What we'll see in this video is that we can't even grasp things that are actually super small compared to the size of the universe. And we actually don't even know what the entire size of the universe is. But with that said, let's actually just try to appreciate how small we are. So this is me right over here. I am 5 foot 9 inches, depending on whether I'm wearing shoes-- maybe 5 foot 10 with shoes. But for the sake of this video, let's just roughly approximate around 6 feet, or around roughly-- I'm not to go into the details of the math-- around 2 meters. Now, if I were to lie down 10 times in a row, you'd get about the length of an 18-wheeler. That's about 60 feet long. So this is times 10. Now, if you were to put an 18-wheeler-- if you were to make it tall, as opposed to long-- somehow stand it up-- and you were to do that 10 times in a row, you'll get to the height of roughly a 60-story skyscraper. So once again, if you took me and you piled me up 100 times, you'll get about a 60-story skyscraper. Now, if you took that skyscraper and if you were to lie it down 10 times in a row, you'd get something of the length of the Golden Gate Bridge. And once again, I'm not giving you the exact numbers. It's not always going to be exactly 10. But we're now getting to about something that's a little on the order of a mile long. So the Golden Gate Bridge is actually longer than a mile. But if you go within the twin spans, it's roughly about It's actually a little longer than that. But that gives you a sense of a mile. Now, if you multiply that by 10, you get to the size of a large city. And this right here is a satellite photograph of San Francisco. This is the actual Golden Gate Bridge here. And when I copy and pasted this picture, I tried to make it roughly 10 miles by 10 miles just so you appreciate the scale. And what's interesting here-- and this picture's interesting. Because this is the first time we can relate to cities. But when you look at a city on this scale, it's starting to get larger than what we're used to processing on a daily basis. A bridge-- we've been on a bridge. We know what a bridge looks like. We know that a bridge is huge. But it doesn't feel like something that we can't comprehend. Already, a city is something that we can't comprehend all at once. We can drive across a city. We can look at satellite imagery. But if I were to show a human on this, it would be unbelievably, unbelievably small. You wouldn't actually be able to see it. It would be less than a pixel on this image. A house is less than a pixel on this image. But let's keep multiplying by 10. If you multiply by 10 again, you get to something roughly the size of the San Francisco Bay Area. This whole square over here is roughly that square right over there. Let's multiply by 10 again. So this square is about 100 miles by 100 miles. So this one would be about 1,000 miles by 1,000 miles. And now you're including a big part of the Western United States. You have California here. You Nevada here. You have Arizona and New Mexico-- so a big chunk of a big continent we're already including. And frankly, this is beyond the scale that we're used to operating. We've seen maps, so maybe we're a little used to it. But if you ever had to walk across this type of distance, it would take you a while. To some degree, the fact that planes goes so fast-- almost unimaginably fast for us-- that it's made it feel like things like continents aren't as big. Because you can fly across them in five or six hours. But these are already huge, huge, huge distances. But once again, you take this square that's about 1,000 miles by 1,000 miles, and you multiply that by 10. And you get pretty close-- a little bit over-- the diameter of the Earth-- a little bit over the diameter of the Earth. But once again, we're on the Earth. We kind of relate to the Earth. If you look carefully at the horizon, you might see a little bit of a curvature, especially if you were to get into the plane. So even though this is, frankly, larger than my brain can really grasp, we can kind of relate to the Earth. Now you multiply the diameter of Earth times 10. And you get to the diameter of Jupiter. And so if you were to sit Earth right next to Jupiter-- obviously, they're nowhere near that close. That would destroy both of the planets. Actually, it would definitely destroy Earth. It would probably just be merged into Jupiter. So if you put Earth next to Jupiter, it would look something like that right over there. So I would say that Jupiter is definitely-- on this diagram that I'm drawing here-- is definitely the first thing that I have I can't comprehend. The Earth, itself, is so vastly huge. Jupiter is-- it's 10 times bigger in diameter. It's much larger in terms of mass, and volume, and all the rest. But just in terms of diameter, it is 10 times bigger. But let's keep going. 10 times Jupiter gets us to the sun. This is times 10. So if this is the Sun-- and if I were to draw Jupiter, it would look something like-- I'll do Jupiter in pink-- Jupiter would be around that big. And then the Earth would be around that big if you were to put them all next to each other. So the Sun, once again, is huge. Even though we see it almost every day, it is unimaginably huge. Even the Earth is unimaginably huge. And the Sun is 100 times more unimaginably bigger. Now we're going to start getting really, really, really wacky. You multiply the diameter of the Sun, which is already 100 times the diameter of the Earth-- you multiply that times 100. And that is the distance from the Earth to the Sun. So I've drawn the Sun here as a little pixel. And I didn't even draw the Earth as a pixel. Because a pixel would be way too large. It would have to be a hundredth of a pixel in order to draw the Earth properly. So this is a unbelievable distance between the Earth and the Sun. It's 100 times the distance of the diameter of the Sun itself. So it's massive, massive. But once again, these things are relatively close compared to where we're about to go. Because if we want to get to the nearest star-- so remember, the Sun is 100 times the diameter of the Earth. The distance between the Sun and the Earth is 100 times that. Or you could say it's 10,000 times the diameter of the Earth. So these are unimaginable distances. But to get to the nearest star, which is 4.2 light years away, it's 200,000 times-- and once again, unimaginable. It's 200,000 times the distance between the Earth and the Sun. And to give you a rough sense of how far apart these things are, if the Sun was roughly the size of a basketball-- if the average star was about the size of a basketball-- in our part of the galaxy in a volume the size of the Earth-- so if you had a big volume the size of the Earth, if the stars were the sizes of basketballs, in our part of the galaxy, you would only have a handful of basketballs per that volume. So unbelievably sparse. Even though, when you look at the galaxy-- and this is just an artist's depiction of it-- it looks like something that has the spray of stars, and it looks reasonably dense, there is actually a huge amount of space that the great, great, great, great, great majority of the volume in the galaxy is just empty, empty space. There's no stars, no planets, no nothing. I mean, this is a huge jump that I'm talking about. And then if you really want to realize how large a galaxy, itself, can be, you take this distance between the Sun, or between our solar system and the nearest star-- so that's 200,000 times the distance between the Earth and the Sun-- and you multiply that distance by 25,000. So if the Sun is right here, our nearest star will be in that same pixel. They'll actually be within-- you'd actually get a ton of stars within that one pixel, even though they're so far apart. And then this whole thing is 100,000 light years. It's 25,000 times the distance than the distance between the Sun and the nearest star. So we're talking about unimaginable, unfathomable distances, just for a galaxy. And now we're going to get our-- frankly, my brain is already well beyond anything that it can really process. At this point, it almost just becomes abstract thinking. It just becomes playing with numbers and mathematics. But to get a sense of the universe, itself, the observable universe-- and we have to be clear. Because we can only observe light that started leaving from its source 13.7 billion years ago. Because that's how old the universe is. The observable universe is about 93 billion light years across. And the reason why it's larger than 13.7 billion is that the points in space that emitted light 13.7 billion years ago, those have been going away from us. So now they're on the order of 40 billion light years away. But this isn't about cosmology. This is just about scale and appreciating how huge the universe is. Just in the part of the universe that we can theoretically observe, you have to get-- and that we can observe, just because we're getting electromagnetic radiation from those parts of the universe-- you would have to multiply this number. So let me make this clear. 100,000 light years-- that's the diameter of the Milky Way. You would have to multiply not by 1,000. 1,000 would get you to 100 million light years. This is 100,000 times 1,000 is 100 million. You have to multiply by 1,000 again to get to 100 billion light years. And the universe, for all we know, might be much, much, much, much, larger. It might even be infinite. Who knows? But to get from just the diameter of the Milky Way to the observable universe, you have to multiply by a million. And already, this is an unfathomable distance. So in the whole scheme of things, not only are we pretty small, and not only are the things we build pretty small, and not only is our planet ultra small, and not only is our Sun ultra small, and our solar system ultra small, but our galaxy is really nothing compared to the vastness of the universe." + }, + { + "Q": "So I was working on a christmas themed hangman game, and wanted to make a candycane. Is there a way to make a striped pattern without making a ton of rectangles right next to eachother? I don't really like the idea of long that would take.", + "A": "You can try using quads and for loops.", + "video_name": "4VqHGULLA4o", + "transcript": "Voiceover: Congratulations! You now understand the JavaScript language. Variables, loops, strings, functions, objects, arrays, even object-oriented design. But what good is a language if you can't make something cool with it? There are lots of ways you can use your new knowledge, but one of the most popular ways to use JavaScript with Processing.js is to make games and visualizations, which you probably know, if you've ever looked at our hot programs list. A game is something interactive, where you get some reward. There is usually a win state, a lose state, a score. A visualization is also highly interactive, but without the game mechanics. Let's look at some of the common components we'll need. We'll need UI controls for anything we're doing. Like buttons, and sliders, and menus. And some of these will be simple buttons, other times we'll need multiple buttons, other times we'll need sliders, and drop-downs, and it all builds upon the same basic principles, though. Besides interaction with a mouse, we'll also want keyboard control, like being able to use the arrows to move our characters up and down, or to change the angle of our visualization. We often also want the notion of scenes. A scene is like your start screen, and your options screen, and your main screen, and your end screen. And they're usually very different, and at any given point we want to be showing one of them or the other, so you have to really organize your code in order to know the difference between the scenes, and have a good way of switching between them. And now let's talk about a few things specific to games. The environment of a game. Is it a side-scroller, which means it's kinda a character moving forward through a space? Is it a bird's eye view, like going through a maze? Is it a 3D environment? It's crazy, but you can do it. Are there multiple levels, and each of them have different environments? What are the characters in the game? They'll probably have different behaviors, and emotions, and states, like a happy state and a dead state. And they'll might be user controlled or sometimes they'll be programatically controlled. And your program gives it some sort of logic to follow. There could be one of them, there could be lots of them. They could get spawned during the game as it's played. Now once we've got characters in an environment, we usually also add in some items, and then we have a lot of things colliding with each other. And we usually want to know when things collide, because things are typically trying to attract each other, or avoid each other, like when you're picking up gems, or avoiding nasty turtles. So we need to be able to detect collision between objects, and sometimes it's very simple collision, and other times it's more complex if the objects are all different shapes and sizes. Finally, if it's a game, it's usually got a score. So how do you measure how well the user is doing? When do you tell them if they've lost or won? How spectacular can you make the win screen or the lose screen? So, as you can see, there are a lot of aspects to think about when making games and visualizations. We'll walk through some of them here, but we don't know what's in your head, and most likely, you'll have to just combine the knowledge you learned here to make whatever really cool thing is in your head right now." + }, + { + "Q": "What happens when one of the things is negative? example: 3x - 45\nWouldn't the answer be negative/ That wouldn't make sense.", + "A": "3x-45=0 3x = 45 (-ive becomes +ive) x = 45/3 x= 15", + "video_name": "Ld7Vxb5XV6A", + "transcript": "So I've got two parallel lines. So that's the first line right over there and then the second line right over here. Let me denote that these are parallel. These are parallel lines. Actually, I can do that a little bit neater. And let me draw a transversal, so a line that intersects both of these parallel lines, so something like that. And now let's say that we are told that this angle right over here is 9x plus 88. And this is in degrees. And we're also told that this angle right over here is 6x plus 182, once again, in degrees. So my goal here and my question for you is, can we figure out what these angles actually are, given that these are parallel lines and this is a transversal line? And I encourage you to pause this video to try this on your own. Well, the key here to realize is that these right over here are related by the fact that they're formed from a transversal intersecting parallel lines. And we know, for example, that this angle corresponds to this angle right over here. They're going to be congruent angles. And so this is 6x plus 182. This is also going to be 6x plus 182. And then that helps us realize that this blue angle and this orange angle are actually going to be supplementary. They're going to add up to 180 degrees, because put together, when you make them adjacent, their outer rays form a line right over here. So we know that 6x plus 182 plus 9x plus 88 is going to be equal to 180 degrees. And now we just have to simplify this thing. So 6x plus 9x is going to give us 15x. And then we have 182 plus 88. Let's see, 182 plus 8, would get us to 190. And then we add another 80. It gets us to 270-- plus 270-- is equal to 180. If we subtract 270 from both sides, we get 15x is equal to negative 90. And now we can divide both sides by 15. And we get x is equal to-- what is this? Let's see, 6 times 15 is 60 plus 30 is 90. So x is going to be equal to negative 6. So far, we've made a lot of progress. We figured out what x is equal to. x is equal to negative 6, but we still haven't figured out what these angles are equal to. So this angle right over here, 9x plus 88, this is going to be equal to 9 times negative 6 plus 88. 9 times negative 6 is negative 54. Let me write this down before I make a mistake. Negative 54 plus 88 is going to be-- let's see, to go from 88 minus 54 will give us 34 degrees. So this is equal to 34, and it's in degrees. So this orange angle right here is 34 degrees. The blue angle is going to be 180 minus that. But we can verify that by actually evaluating 6x plus 182. So this is going to be equal to 6 times negative 6 is negative 36 plus 182. So this is going to be equal to-- let's see, if I subtract the 6 first, I get to 176. So this gets us to 146 degrees. And you can verify-- 146 plus 34 is equal to 180 degrees. Now, we could also figure out the other angles from this as well. We know that if this is 34 degrees, then this must be 34 degrees as well. Those are opposite angles. This angle also corresponds to this angle so it must also be 34 degrees, which is opposite to this angle, which is going to be 34 degrees. Similarly, if this one right over here is 146 degrees, we already know that this one is going to be 146. This one's going to be 146 since it's opposite. And that's going to be 146 degrees as well." + }, + { + "Q": "why is the magnetic field always perpendicular to velocity? are they related?", + "A": "Yes, they are related. Go to youtube and type veritasium how do magnets work Watch both videos.", + "video_name": "NnlAI4ZiUrQ", + "transcript": "We know a little bit about magnets now. Let's see if we can study it further and learn a little bit about magnetic field and actually the effects that they have on moving charges. And that's actually really how we define magnetic field. So first of all, with any field it's good to have a way to visualize it. With the electrostatic fields we drew field lines. So let's try to do the same thing with magnetic fields. Let's say this is my bar magnet. This is the north pole and this is the south pole. Now the convention, when we're drawing magnetic field lines, is to always start at the north pole and go towards the south pole. And you can almost view it as the path that a magnetic north monopole would take. So if it starts here-- if a magnetic north monopole, even though as far as we know they don't exist in nature, although they theoretically could, but let's just say for the sake of argument that we do have a magnetic north monopole. If it started out here, it would want to run away from this north pole and would try to get to the south pole. So it would do something, its path would look something like this. If it started here, maybe its path would look something like this. Or if it started here, maybe its path would look something like this. I think you get the point. Another way to visualize it is instead of thinking about a magnetic north monopole and the path it would take, you could think of, well, what if I had a little compass here? Let me draw it in a different color. Let's say I put the compass here. That's not where I want to do it. Let's say I do it here. The compass pointer will actually be tangent to the field line. So the pointer could look something like this at this point. It would look something like this. And this would be the north pole of the pointer and this would be the south pole of the pointer. Or you could-- that's how north and south were defined. People had compasses, they said, oh, this is the north seeking pole, and it points in that direction. But it's actually seeking the south pole of the larger magnet. And that's where we got into that big confusing discussion of that the magnetic geographic north pole that we're used to is actually the south pole of the magnet that we call Earth. And you could view the last video on Introduction to Magnetism to get confused about that. But I think you see what I'm saying. North always seeks south the same way that positive seeks negative, and vice versa. And north runs away from north. And really the main conceptual difference-- although they are kind of very different properties-- although we will see later they actually end up being the same thing, that we have something called an electromagnetic force, once we start learning about Maxwell's equations and relativity and all that. But we don't have to worry about that right now. But in classical electricity and magnetism, they're kind of a different force. And the main difference-- although you know, these field lines, you can kind of view them as being similar-- is that magnetic forces always come in dipoles, soon. while you could have electrostatic forces that are monopoles. You could have just a positive or a negative charge. So that's fine, you say, Sal, that's nice. You drew these field lines. And you've probably seen it before if you've ever dropped metal filings on top of a magnet. They kind of arrange themselves But you might say, well, that's kind of useful. But how do we determine the magnitude of a magnetic field at any point? And this is where it gets interesting. The magnitude of a magnetic field is really determined, or it's really defined, in terms of the effect that it has on a moving charge. So this is interesting. I've kind of been telling you that we have this different force called magnetism that is different than the electrostatic force. But we're defining magnetism in terms of the effect that it has on a moving charge. And that's a bit of a clue. And we'll learn later, or hopefully you'll learn later as you advance in physics, that magnetic force or a magnetic field is nothing but an electrostatic field moving at a very high speed. At a relativistic speed. Or you could almost view it as they are the same thing, just from different frames of reference. I don't want to confuse you right now. But anyway, back to what I'll call the basic physics. So if I had to find a magnetic field as B-- so B is a vector and it's a magnetic field-- we know that the force on a moving charge could be an electron, a proton, or some other type of moving charged particle. And actually, this is the basis of how they-- you know, when you have supercolliders-- how they get the particles to go in circles, and how they studied them by based on how they get deflected by the magnetic field. But anyway, the force on a charge is equal to the magnitude of the charge-- of course, this could be positive or negative-- times, and this is where it gets interesting, the velocity of the charge cross the magnetic field. So you take the velocity of the charge, you could either multiply it by the scalar first, or you could take the cross product then multiply it by the scalar. Doesn't matter because it's just a number, this isn't a vector. But you essentially take the cross product of the velocity and the magnetic field, multiply that times the charge, and then you get the force vector on that particle. Now there's something that should immediately-- if you hopefully got a little bit of intuition about what the cross product was-- there's something interesting going on here. The cross product cares about the vectors that are perpendicular to each other. So for example, if the velocity is exactly perpendicular to the magnetic field, then we'll actually get a number. If they're parallel, then the magnetic field has no impact on the charge. That's one interesting thing. And then the other interesting thing is when you take the cross product of two vectors, the result is perpendicular to both of these vectors. So that's interesting. A magnetic field, in order to have an effect on a charge, has to be perpendicular to its you velocity. And then the force on it is going to be perpendicular to both the velocity of the charge and the magnetic field. I know I'm confusing you at this point, so let's play around with it and do some problems. But before that, let's figure out what the units of the magnetic field are. So we know that the cross product is the same thing as-- so let's say, what's the magnitude of the force? The magnitude of the force is equal to? Well, the magnitude of the charge-- this is just a scalar quantity, so it's still just the charge-- times the magnitude of the velocity times the magnitude of the field times the sine of the angle between them. This is the definition of a cross product and then we could put-- if we wanted the actual force vector, we can just multiply this times the vector we get using the We'll do that in a second. Anyway we're just focused on units. Sine of theta has no units so we can ignore it for this discussion. We're just trying to figure out the units of the magnetic field. So force is newtons-- so we could say newtons equals-- charge is coulombs, velocity is meters per second, and then this is times the-- I don't know what we'll call this-- the B units. We'll call it unit sub B. So let's see. If we divide both sides by coulombs and meters per second, we get newtons per coulomb. And then if we divide by meters per second, that's the same thing as multiplying by seconds per meter. Equals the magnetic field units. So the magnetic field in SI terms, is defined as newton seconds per coulomb meter. And that might seem a little disjointed, and they've come up with a brilliant name. And it's named after a deserving fellow, and that's Nikolai Tesla. And so the one newton second per coulomb meter is equal to one tesla. And I'm actually running out of time in this video, because I want to do a whole problem here. But I just want you to sit and think about it for a second. Even though in life we're used to dealing with magnets as we have these magnets-- and they're fundamentally maybe different than what at least we imagine electricity to be-- but the magnitude or actually the units of magnetism is actually defined in terms of the effect that it would have on a moving charge. And that's why the unit-- one tesla, or a tesla-- is defined as a newton second per coulomb. So the electrostatic charge per coulomb meter. Well, I'll leave you now in this video. Maybe you can sit and ponder that. But it'll make a little bit more sense when we do some actual problems with some actual numbers in the next video. See" + }, + { + "Q": "Why did Hitler hate the Jews, weren't they good people", + "A": "He believed that the Jewish leaders during WW1 had lost the war by surrendering. He was a soldier on the German side in WW1, so that devastated him greatly.", + "video_name": "EtZnPoYbRyA", + "transcript": "Narrator: Where we left off in the last video, in 1924, Hitler was in jail, his famous coup d'\u00e9tat in 1923, his famous Beer Hall Putsched in Munich had failed. He's now in jail, he's writing Mein Kampf. When he gets out of jail, so this is when he's in jail, the Nazi party is banned and a lot of the economic turmoil that made the possibility of overthrowing the government more likely, that we saw in the early 20's, that hyperinflation in Weimar Germany, this was now under control by the time Hitler comes out of jail. They had issued new currency, it was far more stable. To a certain degree the Nazi's and Hitler were starting from scratch, although even at this point Hitler continues to be an ever growing influence. He's a famous speaker, there are more and more people who are knowing about him and who are following him. Over the next few years his book does get published and it sells, actually, tens of thousands of copies over the next several years, but for the most part he's still a relatively small actor in German politics. But then we fast forward as we get to the late 20s, the Nazi's are gaining some influence, but then in 1929, (writing) in 1929, you have a global change for the economy of the world and that's the beginning of The Great Depression. In particular, what's often the first sign that The Great Depression was at hand is you have the U.S. stock market crashes in October of 1929, famous Black Tuesday. That was the mark of the beginning of a, not just American Depression, but a global depression. So you have the whole world going into a depression. Anytime you have economic turmoil it tends give more energy to the more extreme parties, whether it is the parties like the Nazi's, who one could consider maybe to be on the extreme right, or often considered to be on the extreme right, or maybe you could say very nationalistic, or even the extreme left parties who are obviously against capitalist systems and whatever else. So, by the election of 1930, now we're talking about Parliamentary elections and the Parliament in Germany is the Reichstag. (writing) The Reichstag, and I know I'm mispronouncing it. In the Reichstag elections, the Nazi party, for the first time is able to have a significant showing. It gets 18, it gets roughly 18% of the vote and a proportional representation in the Parliament. Now all of a sudden, this kind of marks the beginning of the Nazi's being significant, significant players in German politics. Then we get to 1932 and the economy is not improving, it is only getting worse. (writing) 1932. Adolf Hitler actually makes a run for President. The current President at that point is Paul von Hindenburg, famous for the Hindenburg line, later for the Hindenburg, the Zeppelin, the famous exploding Zeppelin disaster. He was, with Ludendorff, one of the two leaders of the German military effort during World War I. He's President of the Weimar Republic since 1925 and in 1932 he is able to get re-election, but Hitler has a fairly good showing. Hitler is able to get 35% of the vote. (writing) Hitler gets 35% of the presidential election votes, (writing) of the vote. The Weimar Republic had this strange system. It wasn't quite a Presidential system like the U.S. and it wasn't quite a pure Parliamentary system like the current-day Germany. The President was independently elected and had some powers, and then the Parliament was also independently elected and then they would try to build coalitions to have a ruling government. Needless to say, 1932 Hitler is now a major actor, the Nazi's also have a many, many, many seats in Parliament. Now, you have several Parliamentary elections as well in 1932 and as we just talked about two in particular. In order for a government to form in Parliament, in order to find the Cabinet and the Chancellor, who essentially is the Prime Minister, you have an election and the different parties get different amounts of votes. If no party has a majority, the parties have to form a coalition that can make a majority. There's a lot of horsetrading going on with parties negotiating, hey why don't we form a coalition with each other, if we do that maybe someone from my party can be Minister of the Interior, someone of your party could be the Chancellor and maybe we can get a coalition together to rule over the government. But you have two Parliamentary elections and no majority coalition forms. (writing) So, two, two elections. So this is Parliamentary. So this is in the Presidential election, Hindenburg is still President, but Hitler has a good showing and then you have two Parliamentary elections. (writing) Parliament elections, or Reichstag elections where you have no majority, no coalition. (writing) no majority, majority coalition. The Nazi's continue to be a major actor here, they continue to have more and more of a showing inside the Reichstag. Then by 1933 it's a bit of crisis. So as we get in to early 1933 we have a little bit of a crisis. We have no government, we have no Chancellor, we have no Cabinet to essentially be the executive, the government of the country because there's been no major coalitions. The Weimar Constitution allowed a strange thing, it allowed the President to appoint a government, appoint a Cabinet, a Chancellor that might not even be representative of what's going on in Parliament. So, Paul von Hindenburg is convinced that ... hey look, he was no fan, he was no fan of Adolf Hitler but he's convinced that look, Adolf Hitler was your opponent if you make Adolf Hitler the head of an interim Government, the head of an interim Cabinet then that might be a way to create some national unity and then maybe we could have some Parliamentary elections that there can be a majority coalition and you could have, I guess you could say, a more legitimate government take hold. So, Paul von Hindenburg is convinced and so he does, even though the Nazi's are still a minority party, even though they weren't part of any type of a majority coalition, Paul von Hindenburg who is not a fan of Adolf Hitler appoints him as Chancellor. This is in January. So in January, Hitler, (writing) Hitler is appointed Chancellor, Chancellor, which is essentially the Prime Minister of the Reichstag of Germany. Then we get to February and events get really, really, really interesting. In February of 1933 you have a fire in the Reichstag building in Berlin. This is the Reichstag building right over here and it is on fire. They find this gentleman here on the scene, Marinus van der Lubbe, he is a Dutch communist. It is essentially the blame is placed as this was some type of a, the beginning of some type of a communist revolution. This is used as a pretext. Hitler then advises Paul von Hindenburg to essentially use some of his emergency powers as President, which is another strange thing that the Weimar Constitution allowed for, it allowed the President under emergency conditions to start to suspend civil rights. This was an emergency situation and so Paul von Hindenburg does that. He essentially issues ... once you have the Reichstag fire (writing) Reichstag fire, and then Hindenburg is convinced by the Nazi's to pass the Reichstag Fire Decree. (writing) Fire decree, which essentially suspends, it gives the government emergency powers and it suspends civil liberties, which everything up to this point now is actually legal, this was actually allowed for in the Weimar Constitution. (writing) Suspends, suspends civil, civil liberties. And since there's no coalition, the whole point that Hitler's Cabinet was going to be an interim one, you have another Parliamentary election coming in March with the hope of maybe a majority coalition forms, but that March election, especially with civil liberties suspended you could imagine that the Nazi's ... and they have their paramilitary troopers started intimidating other parties, making sure that they had a better showing at the polls, they started intimidating other candidates. The March election start to swing hugely in the Nazi's favor, so in the March election they're able to get 44% of the vote, which is still not enough, by themselves, to form a government. It's still not a majority, but they're able now ... they're now the largest part in the Reichstag, in the Parliament. They're able to now form a majority coalition, and I guess you could say more legitimately ... although this was a election of intimidation, they were able to now form a government, they're able to now form a government based on a majority coalition and Hitler remains Chancellor. But then, this new Parliament passes the Enabling Act in March. (writing) Enabling Act, Enabling Act, which is essentially an amendment to the Weimar Constitution which gives the Cabinet, especially the Chancellor, effectively the Chancellor who's the head of the Cabinet, legislative powers, unlimited legislative powers for the next four years. So, it gives legislative powers and remember we already have suspended civil rights. So, the Reichstag is essentially giving over the legislative powers, (writing) legislative powers, to the Chancellor who happens to be, who happens to be Hitler. There was some check on this by the President, but then we have Hindenburg dying the next year. After this, after the suspension of civil rights and then the Enabling Act shortly afterwards, Hitler is essentially in full control, Hitler and the Nazi's are essentially in full control of the German government. At this point, Hitler is the dictator, (writing) the dictator of, he is the dictator of Germany. They start to act fast, they start to intimidate other parties, they use violence, they start to imprison people and by July of 1933 ... so they're acting very, very fast, by July of 1933 Nazi's are the only legal party. (writing) only legal Pot party and they essentially have full control. Now, this is how Hitler came to power and the question that's probably circling in your mind is, \"Who did this fire?\" This fire was the catalyst, although Hitler was already Chancellor and maybe he would have found some way to get to power regardless, but this fire, even though there was evidence that it looked like maybe Marinus van der Lubbe did it, it was blamed on the communist, it was the pretext that was used to give the government even more power, especially the Nazi's even more power. This is an open question, one of those great open questions, one of those great open questions of history. Some people feel that maybe it was just a communist plot, maybe it was Marinus van der Lubbe acting on his own and maybe it just happened to fall into the hands of Hitler and they were able to use it, while other historians think that this was actually a plot by the Nazi's to create this emergency state and Marinus van der Lubbe was kind of a puppet in this whole plot. So, open question of history, but needless to say as we go from 1919 to 1933, Hitler goes from a fairly unknown individual to full dictator of Germany." + }, + { + "Q": "How to use vector addition in real life or application development? In general, I understand how it is done but why should it is important?", + "A": "Understanding vectors is essential to the understanding of more advanced topics which also have no application! ; )", + "video_name": "r4bH66vYjss", + "transcript": "In the last video I was a little formal in defining what Rn is, and what a vector is, and what vector addition or scalar multiplication is. In this video I want to kind of go back to basics and just give you a lot of examples. And give you a more tangible sense for what vectors are and how we operate with them. So let me define a couple of vectors here. And I'm going to do, most of my vectors I'm going to do in this video are going to be in R2. And that's because they're easy to draw. Remember R2 is the set of all 2-tuples. Ordered 2-tuples where each of the numbers, so you know you could have x1, my 1 looks like a comma, x1 and x2, where each of these are real numbers. So you each of them, x1 is a member of the reals, and x2 is a member of the reals. And just to give you a sense of what that means, if this right here is my coordinate axes, and I wanted a plot all my x1's, x2's. You know you could view this as the first coordinate. We always imagine that as our x-axis. And then our second coordinate we plotted on the vertical axis. That traditionally is our y-axis, but we'll just call that the second number axis, whatever. You could visually represent all of R2 by literally every single point on this plane if we were to continue off to infinity in every direction. That's what R2 is. R1 would just be points just along one of these number lines. That would be R1. So you could immediately see that R2 is kind of a bigger space. But anyway, I said that I wouldn't be too abstract, that I would show you examples. So let's get some vectors going in R2. So let me define my vector a. I'll make it nice and bold. My vector a is equal to, I'll make some numbers up, negative 1, 2. And my vector b, make it nice and bold, let me make that, I don't know, 3, 1. Those are my two vectors. Let's just add them up and see what we get. Just based on my definition of vector addition. I'll just stay in one color for now so I don't have to keep switching back and forth. So a, nice deep a, plus bolded b is equal to, I just add up each of those terms. Negative 1 plus 3. And then 2 plus 1. That was my definition of vector addition. So that is going to be equal to 2 and 3. Fair enough that just came out of my definition of vector addition. But how can we represent this vector? So we already know that if we have coordinates, you know, if I have the coordinate, and this is just a convention. It's just the way that we do it. The way we visualize things. If I wanted to plot the point 1, 1, I go to my coordinate axes. The first point I go along the horizontal, what we traditionally call our x-axis. And I go 1 in that direction. And then convention is, the second point I go 1 in the vertical direction. So the point 1, 1. Oh, sorry, let me be very clear. This is 2 and 2, so one is right here, and one is right there. So the point 1, 1 would be right there. That's just the standard convention. Now our convention for representing vectors are, you might be tempted to say, oh, maybe I just represent this vector at the point minus 1, 2. And on some level you can do that. I'll show you in a second. But the convention for vectors is that you can start at any point. Let's say we're dealing with two dimensional vectors. You can start at any point in R2. So let's say that you're starting at the point x1, and x2. This could be any point in R2. To represent the vector, what we do is we draw a line from that point to the point x1. And let me call this, let's say that we wanted to draw a. So x1 minus 1. So this is, I'm representing a. So this is, I want to represent the vector a. x1 minus 1, and then x1 plus 2. Now if that seems confusing to you, when I draw it, it'll be very obvious. So let's say I just want to start at the point, let's just say for quirky reasons, I just pick a random point here. I just pick a point. That one right there. That's my starting point. So minus 4, 4. Now if I want to represent my vector a, what I just said is that I add the first term in vector a to my first So x1 plus minus 1 or x1 minus 1. So my new one is going to be, so this is my x1 minus 4. So now it's going to be, let's see, I'm starting at the point minus 4 comma 4. If I want to represent a, what I do is, I draw an arrow to minus 4 plus this first term, minus 1. And then 4 plus the second term. 4 plus 2. And so this is what? This is minus 5 comma 6. So I go to minus 5 comma 6. So I go to that point right there and I just draw a line. So my vector will look like this. I draw a line from there to there. And I draw an arrow at the end point. So that's one representation of the vector minus 1, 2. Actually let me do it a little bit better. Because minus 5 is actually more, a little closer to right here. Minus 5 comma 6 Is right there, so I draw my vector like that. But remember this point minus 4 comma 4 was an arbitrary place to draw my vector. I could have started at this point here. I could have started at the point 4 comma 6 and done the same thing. I could have gone minus 1 in the horizontal direction, that's my movement in the horizontal direction. And then plus 2 in the vertical direction. So I could have drawn, so minus 1 in the horizontal and plus 2 in the vertical gets me right there. So I could have just as easily drawn my vector like that. These are both interpretations of the same vector a. I should draw them in the color of vector a. So vector a was this light blue color right there. So this is vector a. This is vector a. Sometimes there'll be a little arrow notation over the vector. I could draw an infinite number of vector a's. I could draw vector a here. I could draw it like that. Vector a, it goes back 1 and up 2. So vector a could be right there. Similarly vector b. What does vector b do? I could pick some arbitrary point for vector b. It goes to the right 3, so it goes to the right 1, 2, 3 and then it goes up 1. So vector b, one representation of vector b, looks like this. Another represention. I can start it right here. I could go to the right 3, 1, 2, 3, and then up 1. This would be another representation of my vector b. There's an infinite number of representations of them. But the convention is to often put them in what's called the standard position. And that's to start them off at 0, 0. So your initial point, let me write this down. Standard position is just to start the vectors at 0, 0 and then draw them. So vector a in standard position, I'd start at 0, 0 like that and I would go back 1 and then up 2. So this is vector a in standard position right there. And then vector b in standard position. Let me write that. That's a. And then vector b in standard position is 3, go to the 3 right and then up 1. These are the vectors in standard position, but any of these other things we drew are just as valid. Now let's see if we can get an interpretation of what happened when we added a plus b. Well if I draw that vector in standard position, I just calculated, it's 2, 3. So I go to the right 2 and I go up 3. So if I just draw it in standard position it looks like this. This vector right there. And at first when you look at it, this vector right here is the vector a plus b in standard position. When you draw it like that, it's not clear what the relationship is when we added a and b. But to see the relationship what you do is, you put a and b head to tails. What that means is, you put the tail end of b to the front end of a. Because remember, all of these are valid representations of b. All of the representations of the vector b. They all have, they're all parallel to each other, but they can start from anywhere. So another equally valid representation of vector b is to start at this point right here, kind of the end point of vector a in standard position, and then draw vector b So you go 3 to the right. So you go 1, 2, 3. And then you go up 1. So vector b could also be drawn just like that. And then you should see something interesting had happened. And remember, this vector b representation is not in standard position, but it's just an equally valid way to represent my vector. Now what do you see? When I add a, which is right here, to b what do I get if I connect the starting point of a with the end point of b? I get the addition. I have added the two vectors. And I could have done that anywhere. I could have started with a here. And then I could have done the end point. I could have started b here and gone 3 to the right, 1, 2, 3 and then up 1. And I could have drawn b right there like that. And then if I were to add a plus b, I go to the starting point of a, and then the end point of b. And that should also be the visual representation of a plus b. Just to make sure it confirms with this number, what I did here was I went 2 to the right, 1, 2 and then I went 3 up. 1, 2, 3 and I got a plus b. Now let's think about what happens when we scale our vectors. When we multiply it times some scalar factor. So let me pick new vectors. Those have gotten monotonous. Let me define vector v. v for vector. Let's say that it is equal to 1, 2. So if I just wanted to draw vector v in standard position, I would just go 1 to the horizontal and then 2 to the vertical. That's the vector in standard position. If I wanted to do it in a non standard position, I could do it right here. 1 to the right up 2, just like that. Equally valid way of drawing vector v. Equally valid way of doing it. Now what happens if I multiply vector v. What if I have, I don't know, what if I have 2 times v? 2 times my vector v is now going to be equal to 2 times each of these terms. So it's going to be 2 times 1 which is 2, and then 2 times 2 which is 4. Now what does 2 times vector v look like? Well let me just start from an arbitrary position. Let me just start right over here. So I'm going to go 2 to the right, 1, 2. And I go up 4. 1, 2, 3, 4. So this is what 2 times vector v looks like. This is 2 times my vector v. And if you look at it, it's pointing in the exact same direction but now it's twice as long. And that makes sense because we scaled it by a factor of 2. When you multiply it by a scalar, or you're not changing its direction. Its direction is the exact same thing as it was before. You're just scaling it by that amount. And I could draw this anywhere. I could have drawn it right here. I could have drawn 2v right on top of v. Then you would have seen it, I don't want to cover it. You would have seen that it goes, it's exactly, in this case when I draw it in standard position, it's colinear. It's along the same line, it's just twice as far. it's just twice as long but they have the exact same direction. Now what happens if I were to multiply minus 4 times our vector v? Well then that will be equal to minus 4 times 1, which is minus 4. And then minus 4 times 2, which is minus 8. So this is on my new vector. Minus 4, minus 8. This is minus 4 times our vector v. So let's just start at some arbitrary point. Let's just do it in standard position. So you go to the right 4. Or you go to the left 4. So so you go to the left 4, 1, 2, 3, 4. And then down 8. Looks like that. So this new vector is going to look like this. Let me try and draw a relatively straight line. There you go. So this is minus 4 times our vector v. I'll draw a little arrow on it to make sure you know it's a vector. Now what happened? Well we're kind of in the same direction. Actually we're in the exact opposite direction. But we're still along the same line, right? But we're just in the exact opposite direction. And it's this negative right there that flipped us around. If we just multiplied negative 1 times this, we would have just flipped around to right there, right? But we multiplied it by negative 4. So we scaled it by 4, so you make it 4 times as long, and then it's negative, so then it flips around. It flips backwards. So now that we have that notion, we can kind of start understanding the idea of subtracting vectors. Let me make up 2 new vectors right now. Let's say my vector x, nice and bold x, is equal to, and I'm doing everything in R2, but in the last part of this video I'll make a few examples in R3 or R4. Let's say my vector x is equal to 2, 4. And let's say I have a vector y. y, make it nice and bold. And then that is equal to negative 1, minus 2. And I want to think about the notion of what x minus y is equal to. Well we can say that this is the same thing as x plus minus 1 times our vector y. So x plus minus 1 times our vector y. Now we can use our definitions. We know how to multiply by a scalar. So we'll say that this is equal to, let me switch colors. I don't like this color. This is equal to our x vector is 2, 4. And then what's minus 1 times y? So minus 1 times y is minus 1 times minus 1 is 1. And then minus 1 times minus 2 is 2. So x minus y is going to be these two vectors added to I'm just adding the minus of y. This is minus vector y. So this x minus y is going to be equal to 3 and 3 and 6. So let's see what that looks like when we visually represent them. Our vector x was 2, 4. So 2, 4 in standard position it looks like this. That's my vector x. And then vector y in standard position, let me do it in a different color, I'll do y in green. Vector y is minus 1, minus 2. It looks just like this. And actually I ended up inadvertently doing collinear vectors, but, hey, this is interesting too. So this is vector y. So then what's their difference? This is 3, 6. So it's the vector 3, 6. So it's this vector. Let me draw it someplace else. If I start here I go 1, 2, 3. And then I go up 6. So then up 6. It's a vector that looks like this. That's the difference between the two vectors. So at first you say, this is x minus y. Hey, how is this the difference of these two? Well if you overlay this. If you just shift this over this, you could actually just start here and go straight up. And you'll see that it's really the difference between the end points. You're kind of connecting the end points. I actually didn't want to draw collinear vectors. Let me do another example. Although that one's kind of interesting. You often don't see that one in a book. Let me to define vector x in this case to be 2, 3. And let me define vector y to be minus 4, minus 2. So what would be x in standard position? It would be 2, 3. It'd look like that. That is our vector x if we start at the origin. So this is x. And then what does vector y look like? I'll do y in orange. Minus 4, minus 2. So vector y looks like this. Now what is x minus y? Well you know, we could view this, 2 plus minus 1 times this. We could just say 2 minus minus 4. I think you get the idea now. But we just did it the first way the last time because I wanted to go from my basic definitions of scalar multiplication. So x minus y is just going to be equal to 2 plus minus 1 times minus 4, or 2 minus minus 4. That's the same thing as 2 plus 4, so it's 6. And then it's 3 minus minus 2, so it's 5. So the difference between the two is the vector 6, 5. So you could draw it out here again. So you could go, add 6 to 4, go up there, then to 5, you'd go like that. So the vector would look something like this. It shouldn't curve like that, so that's x minus y. But if we drew them between, like in the last example, I showed that you could draw it between their two heads. So if you do it here, what does it look like? Well if you start at this point right there and you go 6 to the right and then up 5, you end up right there. So the difference between the two vectors, let me make sure I get it, the difference between the two vectors looks like that. It looks just like that. Which kind of should make sense intuitively. x minus y. That's the difference between the two vectors. You can view the difference as, how do you get from one vector to another vector, right? Like if, you know, let's go back to our kind of second grade world of just scalars. If I say what 7 minus 5 is, and you say it's equal to 2, well that just tells you that 5 plus 2 is equal to 7. Or the difference between 5 and 7 is 2. And here you're saying, look the difference between x and y is this vector right there. It's equal to that vector right there. Or you could say look, if I take 5 and add 2 I get 7. Or you could say, look, if I take vector y, and I add vector x minus y, then I get vector x. Now let's do something else that's interesting. Let's do what y minus x is equal to. y minus x. What is that equal to? Do it in another color right here. Well we'll take minus 4, minus 2 which is minus 6. And then you have minus 2, minus 3. It's minus 5. So y minus x is going to be, let's see, if we start here we're going to go down 6. 1, 2, 3, 4, 5, 6. And then back 5. So back 2, 4, 5. So y minus x looks like this. It's really the exact same vector. Remember, it doesn't matter where we start. It's just pointing in the opposite direction. So if we shifted it here. I could draw it right on top of this. It would be the exact as x minus y, but just in the opposite direction. Which is just a general good thing to know. So you can kind of do them as the negatives of each other. And actually let me make that point very clear. You know we drew y. Actually let me draw x, x we could draw as 2, 3. So you go to the right 2 and then up 3. I've done this before. This is x in non standard position. That's x as well. What is negative x? Negative x is minus 2 minus 3. So if I were to start here, I'd go to minus 2, then I'd go minus 3. So minus x would look just like this. Minus x. It looks just like x. It's parallel. It has the same magnitude. It's just pointing in the exact opposite direction. And this is just a good thing to kind of really get seared into your brain is to have an intuition for these things. Now just to kind of finish up this kind of idea of adding and subtracting vectors. Everything I did so far was in R2. But I want to show you that we can generalize them. And we can even generalize them to vector spaces that aren't normally intuitive for us to actually visualize. So let me define a couple of vectors. Let me define vector a to be equal to 0, minus 1, 2, and 3. Let me define vector b to be equal to 4, minus 2, 0, 5. We can do the same addition and subtraction operations with them. It's just it'll be hard to visualize. We can keep them in just vector form. So that it's still useful to think in four dimensions. So if I were to say 4 times a. This is the vector a minus 2 times b. What is this going to be equal to? This is a vector. What is this going to be equal to? Well we could rewrite this as 4 times this whole column vector, 0, minus 1, 2, and 3. Minus 2 times b. Minus 2 times 4, minus 2, 0, 5. And what is this going to be equal to? This term right here, 4 times this, you're going to get, the pen tablet seems to not work well there, so I'm going to do 4 times this, you're going to get 4 times 0, 0, minus 4, 8. 4 times 3 is 12. And then minus, I'll do it in yellow, minus 2 times 4 is 8. 2 times minus 2 is minus 4. 2 times 0 is 0. 2 times 5 is 10. This isn't a good part of my board, so let me just. It doesn't write well right over there. I haven't figured out the problem, but if I were just right it over here, what do we get? With 0 minus 8? Minus 8. Minus 4, minus 4. Minus negative 4. So that's minus 4 plus 4, so that's 0. 8 minus 0 is 8. 12 minus, what was this? I can't even read it, what it says. Oh, this is a 10. Now you can see it again. Something is very bizarre. 2 times 5 is 10. So it's 12 minus 10, so it's 2. So when we take this vector and multiply it by 4, and subtract 2 times this vector, we just get this vector. And even though you can't represent this in kind of an easy kind of graph-able format, this is a useful concept. And we're going to see this later when we apply some of these vectors to multi-dimensional spaces." + }, + { + "Q": "What's the difference between a prophage and a provirus?\n\nIs it that a prophage is the integration of a chromosome in a bacteria, whereas, a provirus is the integration of a chromosome in an animal cell.", + "A": "A prophage is the genetic material of a bacteriophage, that becomes incorporated into the genome of a bacterium, and is able to produce more phages if stimulated for the specific activity. A provirus is similar to a prophage, in that it is the genetic material of a virus that becomes incorporated into the genome of a host cell, and has the ability to replicate with that cell.", + "video_name": "0h5Jd7sgQWY", + "transcript": "Considering that I have a cold right now, I can't imagine a more appropriate topic to make a video on than a virus. And I didn't want to make it that thick. A virus, or viruses. And in my opinion, viruses are, on some level, the most fascinating thing in all of biology. Because they really blur the boundary between what is an inanimate object and what is life? I mean if we look at ourselves, or life as one of those things that you know it when you see it. If you see something that, it's born, it grows, it's constantly changing. Maybe it moves around. Maybe it doesn't. But it's metabolizing things around itself. It reproduces and then it dies. You say, hey, that's probably life. And in this, we throw most things that we see-- or we throw in, us. We throw in bacteria. We throw in plants. I mean, I could-- I'm kind of butchering the taxonomy system here, but we tend to know life when we see it. But all viruses are, they're just a bunch of genetic information inside of a protein. Inside of a protein capsule. So let me draw. And the genetic information can come in any form. So it can be an RNA, it could be DNA, it could be single-stranded RNA, double-stranded RNA. Sometimes for single stranded they'll write these two little S's in front of it. Let's say they are talking about double stranded DNA, they'll put a ds in front of it. But the general idea-- and viruses can come in all of these forms-- is that they have some genetic information, some chain of nucleic acids. Either as single or double stranded RNA or single or double stranded DNA. And it's just contained inside some type of protein structure, which is called the capsid. And kind of the classic drawing is kind of an icosahedron type looking thing. Let me see if I can do justice to it. It looks something like this. And not all viruses have to look exactly like this. There's thousands of types of viruses. And we're really just scratching the surface and understanding even what viruses are out there and all of the different ways that they can essentially replicate themselves. We'll talk more about that in the future. And I would suspect that pretty much any possible way of replication probably does somehow exist in the virus world. But they really are just these proteins, these protein capsids, are just made up of a bunch of little proteins put together. And inside they have some genetic material, which might be DNA or it might be RNA. So let me draw their genetic material. The protein is not necessarily transparent, but if it was, you would see some genetic material inside of there. So the question is, is this thing life? It seems pretty inanimate. It doesn't grow. It doesn't change. It doesn't metabolize things. This thing, left to its own devices, is just It's just going to sit there the way a book on a table just sits there. It won't change anything. But what happens is, the debate arises. I mean you might say, hey Sal, when you define it that way, just looks like a bunch of molecules put together. That isn't life. But it starts to seem like life all of a sudden when it comes in contact with the things that we normally consider life. So what viruses do, the classic example is, a virus will attach itself to a cell. So let me draw this thing a little bit smaller. So let's say that this is my virus. I'll draw it as a little hexagon. And what it does is, it'll attach itself to a cell. And it could be any type of cell. It could be a bacteria cell, it could be a plant cell, it could be a human cell. Let me draw the cell here. Cells are usually far larger than the virus. In the case of cells that have soft membranes, the virus figures out some way to enter it. Sometimes it can essentially fuse-- I don't want to complicate the issue-- but sometimes viruses have their own little membranes. And we'll talk about in a second where it gets their membranes. So a virus might have its own membrane like that. That's around its capsid. And then these membranes will fuse. And then the virus will be able to enter into the cell. Now, that's one method. And another method, and they're seldom all the same way. But let's say another method would be, the virus convinces-- just based on some protein receptors on it, or protein receptors on the cells-- and obviously this has to be kind of a Trojan horse type of thing. The cell doesn't want viruses. So the virus has to somehow convince the cell that it's a non-foreign particle. We could do hundreds of videos on how viruses work and it's a continuing field of research. But sometimes you might have a virus that just gets consumed by the cell. Maybe the cell just thinks it's something that it needs to consume. So the cell wraps around it like this. And these sides will eventually merge. And then the cell and the virus will go into it. This is called endocytosis. I'll just talk about that. It just brings it into its cytoplasm. It doesn't happen just to viruses. But this is one mechanism that can enter. And then in cases where the cell in question-- for example in the situation with bacteria-- if the cell has a very hard shell-- let me do it in a good color. So let's say that this is a bacteria right here. And it has a hard shell. The viruses don't even enter the cell. They just hang out outside of the cell like this. Not drawing to scale. And they actually inject their genetic material. So there's obviously a huge-- there's a wide variety of ways of how the viruses get into cells. But that's beside the point. The interesting thing is that they do get into the cell. And once they do get into the cell, they release their genetic material into the cell. So their genetic material will float around. If their genetic material is already in the form of RNA-- and I could imagine almost every possibility of different ways for viruses to work probably do exist in nature. We just haven't found them. But the ones that we've already found really do kind of do it in every possible way. So if they have RNA, this RNA can immediately start being used to essentially-- let's say this is the nucleus of the cell. That's the nucleus of the cell and it normally has the DNA in it like that. Maybe I'll do the DNA in a different color. But DNA gets transcribed into RNA, normally. So normally, the cell, this a normal working cell, the RNA exits the nucleus, it goes to the ribosomes, and then you have the RNA in conjunction with the tRNA and it produces these proteins. The RNA codes for different proteins. And I talk about that in a different video. So these proteins get formed and eventually, they can form the different structures in a cell. But what a virus does is it hijacks this process here. Hijacks this mechanism. This RNA will essentially go and do what the cell's own RNA would have done. And it starts coding for its own proteins. Obviously it's not going to code for the same things there. And actually some of the first proteins it codes for often start killing the DNA and the RNA that might otherwise compete with it. So it codes its own proteins. And then those proteins start making more viral shells. So those proteins just start constructing more and more viral shells. At the same time, this RNA is replicating. It's using the cell's own mechanisms. Left to its own devices it would just sit there. But once it enters into a cell it can use all of the nice machinery that a cell has around to replicate itself. And it's kind of amazing, just the biochemistry of it. That these RNA molecules then find themselves back in these capsids. And then once there's enough of these and the cell has essentially all of its resources have been depleted, the viruses, these individual new viruses that have replicated themselves using all of the cell's mechanisms, will find some way to exit the cell. The most-- I don't want to say, typical, because we haven't even discovered all the different types of viruses there are-- but one that's, I guess, talked about the most, is when there's enough of these, they'll release proteins or they'll construct proteins. Because they don't make their own. That essentially cause the cell to either kill itself or its membrane to dissolve. So the membrane dissolves. And essentially the cell lyses. Let me write that down. The cell lyses. And lyses just means that the cell's membrane just And then all of these guys can emerge for themselves. Now I talked about before that have some of these guys, that they have their own membrane. So how did they get there, these kind of bilipid membranes? Well some of them, what they do is, once they replicate inside of a cell, they exit maybe not even killing-- they don't have to lyse. Everything I talk about, these are specific ways that a virus might work. But viruses really kind of explore-- well different types of viruses do almost every different combination you could imagine of replicating and coding for proteins and escaping from cells. Some of them just bud. And when they bud, they essentially, you can kind of imagine that they push against the cell wall, or the membrane. I shouldn't say cell wall. The cell's outer membrane. And then when they push against it, they take some of the membrane with them. Until eventually the cell will-- when this goes up enough, this'll pop together and it'll take some of the membrane with it. And you could imagine why that would be useful thing to have with you. Because now that you have this membrane, you kind of look like this cell. So when you want to go infect another cell like this, you're not going to necessarily look like a foreign particle. So it's a very useful way to look like something that you're not. And if you don't think that this is creepy-crawly enough, that you're hijacking the DNA of an organism, viruses can actually change the DNA an organism. And actually one of the most common examples is HIV virus. Let me write that down. HIV, which is a type of retrovirus, which is fascinating. Because what they do is, so they have RNA in them. And when they enter into a cell, let's say that they got into the cell. So it's inside of the cell like this. They actually bring along with them a protein. And every time you say, where do they get this protein? All of this stuff came from a different cell. They use some other cell's amino acids and ribosomes and nucleic acids and everything to build themselves. So any proteins that they have in them came from another cell. But they bring with them, this protein reverse transcriptase. And the reverse transcriptase takes their RNA and codes it into DNA. So its RNA to DNA. Which when it was first discovered was, kind of, people always thought that you always went from DNA to RNA, but this kind of broke that paradigm. But it codes from RNA to DNA. And if that's not bad enough, it'll incorporate that DNA into the DNA of the host cell. So that DNA will incorporate itself into the DNA of the host cell. Let's say the yellow is the DNA of the host cell. And this is its nucleus. So it actually messes with the genetic makeup of what it's infecting. And when I made the videos on bacteria I said, hey for every one human cell we have twenty bacteria cells. And they live with us and they're useful and they're part of us and they're 10% of our dry mass and all of that. But bacteria are kind of along for the ride. They don't change who we are. But these retroviruses, they're actually changing our I mean, my genes, I take very personally. They define who I am. But these guys will actually go in and change my genetic makeup. And then once they're part of the DNA, then just the natural DNA to RNA to protein process will code their actual proteins. Or their-- what they need to-- so sometimes they'll lay dormant and do nothing. And sometimes-- let's say sometimes in some type of environmental trigger, they'll start coding for themselves again. And they'll start producing more. But they're producing it directly from the organism's cell's DNA. They become part of the organism. I mean I can't imagine a more intimate way to become part of an organism than to become part of its DNA. I can't imagine any other way to actually define an organism. And if this by itself is not eerie enough, and just so you know, this notion right here, when a virus becomes part of an organism's DNA, this is called a provirus. But if this isn't eerie enough, they estimate-- so if this infects a cell in my nose or in my arm, as this cell experiences mitosis, all of its offspring-- but its offspring are genetically identical-- are going to have this viral DNA. And that might be fine, but at least my children won't get it. You know, at least it won't become part of my species. But it doesn't have to just infect somatic cells, it could infect a germ cell. So it could go into a germ cell. And the germ cells, we've learned already, these are the ones that produce gametes. For men, that's sperm and for women it's eggs. But you could imagine, once you've infected a germ cell, once you become part of a germ cell's DNA, then I'm passing on that viral DNA to my son or my daughter. And they are going to pass it on to their children. And just that idea by itself is, at least to my mind. vaguely creepy. And people estimate that 5-8%-- and this kind of really blurs, it makes you think about what we as humans really are-- but the estimate is 5-8% of the human genome-- so when I talked about bacteria I just talked about things that were along for the ride. But the current estimate, and I looked up this a lot. I found 8% someplace, 5% someplace. I mean people are doing it based on just looking at the DNA and how similar it is to DNA in other organisms. But the estimate is 5-8% of the human genome is from viruses, is from ancient retroviruses that incorporated themselves into the human germ line. So into the human DNA. So these are called endogenous retroviruses. Which is mind blowing to me, because it's not just saying these things are along for the ride or that they might help us or hurt us. It's saying that we are-- 5-8% of our DNA actually comes from viruses. And this is another thing that speaks to just genetic variation. Because viruses do something-- I mean this is called horizontal transfer of DNA. And you could imagine, as a virus goes from one species to the next, as it goes from Species A to B, if it mutates to be able to infiltrate these cells, it might take some-- it'll take the DNA that it already has, that makes it, it with it. But sometimes, when it starts coding for some of these other guys, so let's say that this is a provirus right here. Where the blue part is the original virus. The yellow is the organism's historic DNA. Sometimes when it codes, it takes up little sections of the other organism's DNA. So maybe most of it was the viral DNA, but it might have, when it transcribed and translated itself, it might have taken a little bit-- or at least when it translated or replicated itself-- it might take a little bit of the organism's previous DNA. So it's actually cutting parts of DNA from one organism and bringing it to another organism. Taking it from one member of a species to another member of But it can definitely go cross-species. So you have this idea all of a sudden that DNA can jump between species. It really kind of-- I don't know, for me it makes me appreciate how interconnected-- as a species, we kind of imagine that we're by ourselves and can only reproduce with each other and have genetic variation within But viruses introduce this notion of horizontal transfer via transduction. Horizontal transduction is just the idea of, look when I replicate this virus, I might take a little bit of the organism that I'm freeloading off of, I might take a little bit of their DNA with me. And infect that DNA into the next organism. So you actually have this DNA, this jumping, from organism to organism. So it kind of unifies all DNA-based life. Which is all the life that we know on the planet. And if all of this isn't creepy enough-- and actually maybe I'll save the creepiest part for the end. But there's a whole-- we could talk all about the different classes of viruses. But just so you're familiar with some of the terminology, when a virus attacks bacteria, which they often do. And we study these the most because this might be a good alternative to antibiotics. Because viruses that attack bacteria might-- sometimes the bacteria is far worse for the virus-- but these are called bacteriaphages. And I've already talked to you about how they have their DNA. But since bacteria have hard walls, they will just inject the DNA inside of the bacteria. And when you talk about DNA, this idea of a provirus. So when a virus lyses it like this, this is called the lytic cycle. This is just some terminology that's good to know if you're going to take a biology exam about this stuff. And when the virus incorporates it into the DNA and lays dormant, incorporates into the DNA of the host organism and lays dormant for awhile, this is called the lysogenic cycle. And normally, a provirus is essentially experiencing a lysogenic cycle in eurkaryotes, in organisms that have a nuclear membrane. Normally when people talk about the lysogenic cycle, they're talking about viral DNA laying dormant in the DNA of bacteria. Or bacteriophage DNA laying dormant in the DNA of bacteria. But just to kind of give you an idea of what this, quote unquote, looks like, right here. I got these two pictures from Wikipedia. One is from the CDC. These little green dots you see right here all over the surface, this big thing you see here, this is a white blood cell. Part of the human immune system. This is a white blood cell. And what you see emerging from the surface, essentially budding from the surface of this white blood cell-- and this gives you a sense of scale too-- these are HIV-1 viruses. And so you're familiar with the terminology, the HIV is a virus that infects white blood cells. AIDS is the syndrome you get once your immune system is weakened to the point. And then many people suffer infections that people with a strong immune system normally won't suffer from. But this is creepy. These things went inside this huge cell, they used the cell's own mechanism to reproduce its own DNA or its own RNA and these protein capsids. And then they bud from the cell and take a little bit of the membrane with it. And they can even leave some of their DNA behind in this cell's own DNA. So they really change what the cell is all about. This is another creepy picture. These are bacteriaphages. And these show you what I said before. This is a bacteria right here. This is its cell wall. And it's hard. So it's hard to just emerge into it. Or you can't just merge, fuse membranes with it. So they hang out on the outside of this bacteria. And they are essentially injecting their genetic material into the bacteria itself. And you could imagine, just looking at the size of these things. I mean, this is a cell. And it looks like a whole planet or something. Or this is a bacteria and these things are so much smaller. Roughly 1/100 of a bacteria. And these are much less than 1/100 of this cell we're talking about. And they're extremely hard to filter for. To kind of keep out. Because they are such, such small particles. If you think that these are exotic things that exist for things like HIV or Ebola , which they do cause, or SARS, you're right. But they're also common things. I mean, I said at the beginning of this video that I have a cold. And I have a cold because some viruses have infected the tissue in my nasal passage. And they're causing me to have a runny nose and whatnot. And viruses also cause the chicken pox. They cause the herpes simplex virus. Causes cold sores. So they're with us all around. I can almost guarantee you have some virus with you as you speak. They're all around you. But it's a very philosophically puzzling question. Because I started with, at the beginning, are these life? And at first when I just showed it to you, look they are just this protein with some nucleic acid molecule in it. And it's not doing anything. And that doesn't look like life to me. It's not moving around. It doesn't have a metabolism. It's not eating. It's not reproducing. But then all of a sudden, when you think about what it's doing to cells and how it uses cells to kind of reproduce. It kind of like-- in business terms it's asset light. It doesn't need all of the machinery because it can use other people's machinery to replicate itself. You almost kind of want to view it as a smarter form of life. Because it doesn't go through all of the trouble of what every other form of life has. It makes you question what life is, or even what we are. Are we these things that contain DNA or are we just transport mechanisms for the DNA? And these are kind of the more important things. And these viral infections are just battles between different forms of DNA and RNA and whatnot. Anyway I don't want to get too philosophical on you. But hopefully this gives you a good idea of what viruses are and why they really are, in my mind, the most fascinating pseudo organism in all of biology." + }, + { + "Q": "Wait... at 0:43 Sal says that 0 to the first power is 0*1, but I thought it was 0*0, can anyone explain why? Thanks!", + "A": "Ah, 0^1 = 0 and 0^0 is undefined. Does that help?", + "video_name": "PwDnpb_ZJvc", + "transcript": "So let's think a little bit about powers of 0. So what do you think 0 to the first power is going to be? And I encourage you to pause this video. Well, let's just think about it. One definition of exponentiation is that you start with a 1, and then, you multiply this number times a 1 one time. So this is literally going to be 1 times-- let me do it in the right color-- it's 1 times 0. You're multiplying the 1 by 0 one time. 1 times 0, well, that's just going to be equal to 0. Now, what do you think 0 squared or 0 to the second power is going to be equal to? Well, once again, one way of thinking about this is that you start with a 1, and we're going to multiply it by 0 two times. So times 0 times 0. Well, what's that going to be? Well, you multiply anything times 0, once again, you are going to get 0. And I think you see a pattern here. If I take 0 to any non-zero number-- so to the power of any non-zero, so this is some non-zero number, then this is going to be equal to 0. Now, this raises a very interesting question. What happens at 0 to the 0-th power? So here, 0 to the millionth power is going to be 0. 0 to the trillionth power is going to be 0. Even negative or fractional exponents, which we haven't talked about yet, as long as they're non-zero, this is just going to be equal to 0, kind of makes sense. But now, let's think about what 0 to the 0-th power is, because this is actually a fairly deep question. And I'll give you a hint. Well, actually, why don't you pause the video and think a little bit about what 0 to the 0-th power should be. Well, there's two trains of thought here. You could say, look, 0 to some non-zero number is 0. So why don't we just extend this to all numbers and say 0 to any number should be 0. And so maybe you should say that 0 to the 0-th power is 0. But then, there was another train of logic that we've already learned, that any non-zero number, if you take any non-zero number, and you raise it to the 0-th power. We've already established that you start with a 1, and you multiply it times that non-zero number 0 times. So this is always going to be equal to 1 for non-zero numbers. So maybe say, hey, maybe we should extend this to all numbers, including 0. So maybe 0 to the 0-th power should be 1. So we could make the argument that 0 to the 0-th power should be equal to 1. So you see a conundrum here, and there's actually really good cases, and you can get actually fairly sophisticated with your mathematics. And there's really good cases for both of these, for 0 to 0-th being 0, and 0 to the 0-th power being 1. And so when mathematicians get into this situation, where they say, well, there's good cases for either. There's not a completely natural one. Either of these definitions would lead to difficulties in mathematics. And so what mathematicians have decided to do is, for the most part-- and you'll find people who will dispute this; people will say, no, I like one more than the other-- but for the most part, this is left undefined. 0 to the 0-th is not defined by at least just kind of more conventional mathematics. In some use cases, it might be defined to be one of these two things. So 0 to any non-zero number, you're going to get 0. Any non-zero number to the 0-th power, you're going to get 1. But 0 to the 0, that's a little bit of a question mark." + }, + { + "Q": "When I learn how to use negative numbers, would it be strange for me to not regroup, but to allow the ones place to go negative?", + "A": "Yes and no. If you were doing 86 - 19, you would not want to write the answer as 7-3 or 7negative3. However, because 80 - 10 = 70 and 6 - 9 = -3, you can think of the answer as 3 less than 70, which would be 67. Have a blessed, wonderful day!", + "video_name": "9T3AAn-Cw3g", + "transcript": "Let's try to subtract 659 from 971. And as soon as you start trying to do it, you face a problem. You go to the ones place, and you say, how am I going to subtract a 9 from a 1? And the answer lies in regrouping, taking value from one of the other places here and giving it to the ones place. And to understand that a little bit better, let me rewrite these two numbers. Let me expand it out. So this 9 is in the hundreds place, so it represents 900. The 7 is in the tens place, so it represents 7 tens. And then, this 1 is in the ones place, so it just represents 1. And then down here, this 6 represents 600. This 5 represents 5 tens, or 50. And then, this 9-- well, it still just represents 9 ones, or 9. And we're subtracting this. We're subtracting 600 plus 50 plus 9. Or another way of thinking about it, we're subtracting 600, we're subtracting 50, we are subtracting 9. So let's work it out over here. So this is the exact same problem, just written a little bit differently. And we still have the same issue. How do we subtract a larger number from a smaller number? And the solution lies in trying to take value from one of the other places. And the easiest place to go is-- look, we've got 70 here. Why don't we take 10 from here, and we'll be left with 60, and give that 10 to the ones place. So if you add 10 to 1, what do we have? Well, then we're going to have 11. Notice, I have not changed the value of the number. 971 is the same thing as 900 plus 60 plus 11. It's still 971. And now we can actually subtract. 11 minus 9 is 2. 60 minus 50 is 10. And 900 minus 600 is 300. So this subtraction should result in 300 plus 10 plus 2, which is 312. Now, let's do the exact same thing here, but we're going to do it without expanding it out. So same issue-- how do we subtract a 9 from a 1? Well, let's take a 10 from the tens place. We're going to regroup. So we're going to get rid of one of these tens, so we're only going to have 6 tens left in the tens place. And we're going to give that 10 to the ones place. So 10 plus 1 is 11. Now we are ready to subtract. 11 minus 9 is 2. 6 minus 5 is 1. 9 minus 6 is 3. We get-- let me do that same color-- 312." + }, + { + "Q": "so any number can follow this law?", + "A": "Yes that is correct", + "video_name": "5RzDVNob0-0", + "transcript": "Use the associative law of multiplication to write-- and here they have 12 times 3 in parentheses, and then they want us to multiply that times 10-- in a different way. Simplify both expressions to show they have identical results. So the way that they wrote it is-- let me just rewrite it. So they have 12 times 3 in parentheses, and then they multiply that times 10. Now whenever something is in parentheses, that means do that first. So this literally says let's do the 12 times 3 first. Now, what is 12 times 3? It's 36. So this evaluates to 36, and then we still have that times 10 over there. Whenever we multiply something times a power of ten, we just add the number of zeroes that we have at the back of it, so this is going to be 360. This is going to be equal to 360. Now, the associative law of multiplication, once again, it sounds like a very fancy thing. All that means is it doesn't matter how we associate the multiplication or it doesn't matter how we put the parentheses, we're going to get the same answer, so let me write it down again. If we were to do 12 times 3 times 10, if we just wrote it like this without parentheses, if we just went left to right, that would essentially be exactly what we just did here But the associative law of multiplication says, you know what? We can multiply the 3 times 10 first and then multiply the 12, and we're going to get the exact same answer as if we multiplied the 12 times the 3 and then the 10. So let's just verify it for ourselves. So 3 times 10 is 30, and we still want to multiply the 12 times that. Now, what's 12 times 30? And we've seen this several times before. You can view it as a 12 times 3, which is 36, but we still have this 0 here. So that is also equal to 360. So it didn't matter how we associated the multiplication. You can do the 12 times 3 first or you can do the 3 times 10 first. Either way, they both evaluated to 360." + }, + { + "Q": "What would happen if the shape of the car changed when it hit the truck and the system didn't move?", + "A": "If the car were to change shape, there would be energy lost in the collision. This would mean that the final velocity after the collision would be a bit less.", + "video_name": "XFhntPxow0U", + "transcript": "Welcome back. I will now introduce you to the concept of momentum. And the letter for momentum is, in physics, or at least in mechanics, it's the letter P. P for momentum. And I assume that's because the letter M has already been used for mass, which is I guess an even more fundamental idea. So P for momentum. So what is momentum? Well, you probably have a general idea of it. If you see a big guy running really fast, they'll say, he has a lot of momentum. And if there's a big guy running really fast and a small guy running really fast, most people would say, well, the big guy has more momentum. Maybe they don't have a quantitative sense of why they're saying that, but they just feel that And if we look at the definition of momentum, it'll The definition of momentum is equal to mass times velocity. So something with, say, a medium mass and a huge velocity is going to have a big momentum. Or something with maybe a medium mass, but-- the other I forgot what I just said. So medium mass and big velocity, huge momentum, or Huge mass, medium velocity would have maybe the same momentum, but it would still have a big momentum. Or another way of doing momentum is how little you would like to be in the way of that object as it passes by. How unpleasant would it be to be hit by that object? That's a good way of thinking about momentum. So momentum is mass times velocity. So how does it relate to everything we've been learning so far? So we know that force is equal to mass times acceleration. And what's acceleration? Well acceleration is just change in velocity. So we also know that force is equal to mass times change in velocity per unit of time, right? Per change in time. T for time. So force is also equal to-- well, mass times change in velocity. Mass, let's assume that mass doesn't change. So that could also be viewed as the change in mass times velocity in the unit amount of time. And this is a little tricky here, I said, you know, the mass times the change in velocity, that's the same thing as the change in the mass times the velocity, assuming the mass doesn't change. And here we have mass times velocity, which is momentum. So force can also be viewed as change in momentum per unit of time. And I'll introduce you to another concept called impulse. And impulse kind of means that you think it means. An impulse is defined as force times time. And I just want to introduce this to you just in case you see it on the exam or whatever, show you it's not a difficult concept. So force times change in time, or time, if you assume time starts at time 0. But force times change in time is equal to impulse. I actually don't know-- I should look up what letters they use for impulse. But another way of viewing impulse is force times change in time. Well that's the same thing as change in momentum over change in time times change in time. Right? Because this is just the same thing as force. And that's just change in momentum, so that's impulse as well. And the unit of impulse is the joule. And we'll go more into the joule when we do work in all of that. And if this confuses you, don't worry about it too much. The main thing about momentum is that you realize it's mass times velocity. And since force is change in momentum per unit of time, if you don't have any external forces on a system or, on say, on a set of objects, their combined, or their net momentum won't change. And that comes from Newton's Laws. The only way you can get a combined change in momentum is if you have some type of net force acting on the system. So with that in mind, let's do some momentum problems. Whoops. Invert colors. OK. So let's say we have a car. Say it's a car. Let me do some more interesting colors. A car with a magenta bottom. And it is, let's see, what does this problem say? It's 1,000 kilograms. So a little over a ton. And it's moving at 9 meters per second east. So its velocity is equal to 9 meters per second east, or to the right in this example. And it strikes a stationary 2, 000 kilogram truck. So here's my truck. Here's my truck and this is a 2,000 kilogram truck. And it's stationary, so the velocity is 0. And when the car hits the truck, let's just say that it somehow gets stuck in the truck and they just both keep moving together. So they get stuck together. The question is, what is the resulting speed of the combination truck and car after the collision? Well, all we have to do is think about what is the combined momentum before the collision? The momentum of the car is going to be the mass times the car-- mass of the car. Well the total momentum is going to the mass of the car times the velocity of the car plus the mass of the truck times the velocity of the truck. And this is before they hit each other. So what's the mass of the car? That's 1,000. What's the velocity of the car? It's 9 meters per second. So as you can imagine, a unit of momentum would be kilogram meters per second. So it's 1,000 times 9 kilogram meters per second, but I won't write that right now just to keep things simple, or so I save space. And then the mass of the truck is 2,000. And what's its velocity? Well, it's 0. It's stationary initially. So the initial momentum of the system-- this is 2,000 times 0-- is 9,000 plus 0, which equals 9,000 kilogram meters per second. That's the momentum before the car hits the back of the truck. Now what happens after the car hits the back of the truck? So let's go to that situation. So we have the truck. I'll draw it a little less neatly. And then you have the car and it's probably a little bit-- well, I won't go into whether it's banged up and whether it released heat and all of that. Let's assume that there was nothing-- if this is a simple problem that we can do. So if we assume that, there would be no change in momentum. Because we're saying that there's no net forces acting on the system. And when I say system, I mean the combination of the car and the truck. So what we're saying is, is this combination, this new vehicle called a car truck, its momentum will have to be the same as the car and the truck's momentum when they were separate. So what do we know about this car truck object? Well we know its new mass. The car truck object, it will be the combined mass of the two. So it's 1,000 kilograms plus 2,000 kilograms. So it's 3,000 kilograms. And now we can use that information to figure out its velocity. How? Well, its momentum-- this 3,000 kilogram object's momentum-- has to be the same as the momentum of the two objects before the collision. So it still has to be 9,000 kilogram meters per second. So once again, mass times velocity. So mass is 3,000 times the new velocity. So we could call that, I don't know, new velocity, v sub n. That will equal 9,000. Because momentum is conserved. That's what you always have to remember. Momentum doesn't change unless there's a net force acting on the system. Because we saw a force is change in momentum per time. So if you have no force in it, you have no change in momentum. So let's just solve. Divide both sides of this by 3,000 and you get the new velocity is 3 meters per second. And that kind of makes sense. You have a relatively light car moving at 9 meters per second and a stationary truck. Then it smacks the truck and they move together. The combined object-- and it's going to be to the east. And we'll do more later, but we assume that a positive velocity is east. If somehow we ended up with a negative, it would have been west. But it makes sense because we have a light object and a stationery, heavy object. And when the light object hits the stationery, heavy object, the combined objects still keeps moving to the right, but it moves at a relatively slower speed. So hopefully that gives you a little bit of intuition for momentum, and that was not too confusing of a problem. And in the next couple of videos, I'll do more momentum problems and then I'll introduce you to momentum problems in two dimensions. I will see you soon." + }, + { + "Q": "I always wonder how does a company like SpaceX make money? It certainly does not sell a product or provide people with services...Could someone explain how it works? :D", + "A": "In the interview, Mr. Musk mentioned that the company was started using a portion of his resources accumulated from the success of PayPal. He went on to say that SpaceX now stays in business providing services in space. They launch everything from communication and Global Positioning System (GPS) satellites to supplies for the International Space Station (ISS) into orbit around Earth. SpaceX makes money through the efficient transportation services they offer. I hope this helps!", + "video_name": "vDwzmJpI4io", + "transcript": "0:00 0:01 0:04 0:05 0:06 0:08 0:10 36:32 0:12 0:14 0:15 0:16 0:20 0:23 0:26 0:27 0:29 0:31 0:34 0:39 0:40 0:42 0:45 26:02 0:49 0:53 0:56 0:57 0:59 1:01 1:04 1:10 1:12 1:14 1:17 1:22 1:27 1:29 1:35 1:36 1:38 1:39 1:39 1:40 1:42 1:44 1:47 1:50 1:52 1:57 2:00 2:03 2:07 2:11 2:13 2:15 2:20 2:21 2:25 2:27 2:30 2:32 2:34 2:36 2:39 2:43 2:46 2:49 2:52 2:54 2:55 3:00 3:02 3:05 3:08 48:11 3:11 3:13 3:16 3:17 3:22 3:23 34:58 3:25 3:27 3:32 3:34 3:35 3:37 3:39 3:44 3:45 3:47 3:49 3:53 3:56 3:57 3:59 4:04 4:06 4:07 4:10 4:13 4:16 34:06 4:17 4:19 4:22 4:25 4:28 4:30 4:32 4:34 4:38 4:40 4:42 4:44 4:48 4:49 4:51 4:55 5:00 5:02 5:04 5:07 5:09 5:11 5:12 5:15 It's been 700,000, Right. SAL KHAN: Super volcano for those of you who don't know. It would envelop, but well-- I know exactly what you're talking about. SAL KHAN: We read the same books. I can tell. ELON MUSK: Absolutely. I mean something bad is bound to happen if you give it enough time. And civilization has been around for such a very short period of time that these time scales seem like very long, but on an evolutionary time scale, they're very short. A million years on an evolutionary time scale is really not very much. And Earth's been around for four and a half billion years, so that's a very tiny, tiny amount of time, But for us that would be-- can you can imagine if human civilization continued at anything remotely like the current pace of technology ad advancement for a million years? Where would we be? I think we're either extinct or on a lot of planets. We should-- SAL KHAN: But given that-- I mean, one, that's kind of as epic as one can think about things, literally. How did you make that concrete? How does that turn into SpaceX, Tesla and Paypal? 6:32 6:35 6:36 6:38 6:40 6:45 6:47 6:49 6:52 6:55 6:58 6:59 7:00 7:02 7:03 7:04 7:05 7:07 7:09 7:12 7:14 7:16 7:18 7:21 7:23 7:25 7:29 7:31 7:37 7:38 7:38 7:40 7:41 25:43 7:44 7:45 7:45 7:48 7:50 7:52 7:55 7:57 7:59 8:02 8:06 And they actually were pretty good. They had like the energy density of a lead-acid battery, which for a capacitor, that's a big deal. But they used ruthenium tantalum oxide. And I think at the time, there was maybe like one or two tons of ruthenium mined per year in the world. So it's not a scalable solution. But I thought there could be some solid-state solution, like just using chip-making equipment. That was going to be the basic idea. But it was one of those things where I wasn't sure if success was one of possible outcomes. 8:43 8:46 8:48 8:50 8:50 8:51 8:52 8:54 8:56 35:06 8:59 9:02 9:06 9:08 9:09 9:11 9:13 9:16 9:22 9:25 9:28 9:32 9:34 9:36 9:39 9:42 9:43 9:46 9:50 9:56 9:58 And there goes seven years of my life. So that was one path. And I was prepared to do that. But then the internet came along. And it was like, oh, OK, the Internet, I'm pretty sure success is one of the outcomes, and it seemed like I could either do a PhD and watch the Internet happen, or I could participate and help build in some fashion. Like, I was just concerned with the idea of watching it happen. So I decided to put things on hold and start an Internet company. And we worked on internet publishing software, maps and directions, yellow pages, those kind of things. And we had as investors and customers the media companies. So like the New York Times Company, Knight Ridder. SAL KHAN: And this is just at the early stages. I mean this was like-- So it's really early stages, so it's really out the gate. Absolutely. And so then we-- the reason we worked with the media companies was because we needed to have money. There was no advertising money in '95. In fact, the idea of advertising on the internet seemed like a ridiculous idea to people. Obviously, not so ridiculous anymore. But, at the time, it seemed like a very unlikely proposition. And a lot of the media companies weren't even sure that they should be online. Like, what's the point of that? SAL KHAN: And did you all think that PayPal was just going to be a simple, little internet way to-- or did you think it was going to turn into the major kind of transaction processing engine that it is right now? ELON MUSK: I didn't expect PayPal's growth rate to be what it was. And that actually created major problems. So we started Paypal on University Avenue. After the first month or so of the website being active, we 100,000 customers. SAL KHAN: Really? Wow, I didn't realize it was-- ELON MUSK: Yeah, it was nutty. SAL KHAN: And how did it start? How did people just even know to use it? I mean, obviously, both buyer and seller have to be involved. ELON MUSK: Yeah. Well, we started off first by offering people $20 if they opened an account. And $20 if they referred anyone. And then we dropped it to $10. And we dropped it to $5. As the network got bigger and bigger, the value of the network itself exceeded any sort of carrot SAL KHAN: So much money did you all spend with that kind of $5, $10, $20 incentive to get that critical mass going? ELON MUSK: It was a fair amount. I think it was probably $60 or $70 million. SAL KHAN: Oh, wow, OK. So it was substantial. OK. So we're not talking peanuts here. ELON MUSK: It depends on your relative scale. 12:34 12:35 12:37 12:37 12:39 12:41 12:44 12:45 12:47 12:49 12:50 12:52 12:55 12:56 12:57 12:57 13:00 39:06 13:05 13:06 It's just like bacteria in a Petri dish. So what you want to do is try to have one customer generate like two customers. Or something like that. Maybe three customers, ideally. And then you want that to happen really fast. And you could probably model it just like bacteria growth in a Petri dish. And then it'll just expand very quickly until it hits the side of the Petri dish and then it slows down. SAL KHAN: And then after Paypal, then I mean-- to some degree, especially us in Silicon Valley, we kind of understand the Internet. We know people. PayPal's obviously of the scale that is noteworthy, but then SpaceX just seems really, you know-- well, one, how did you decide that I'm definitely going to do that? And then like what's the first thing that you do? How do you even go out-- I don't even know how to start trying to make a rocket company. ELON MUSK: Well, neither did I really. And in fact, the first three launches failed. So it's not as though it was like spot on. It's like, did not hit the bull's eye. But you're launching rockets. I don't even how do you get there? One, how did you decide? And then what did you do on day one? Did you write a plan? Did you start-- I don't even know. ELON MUSK: Actually, the origin of SpaceX is that I was trying to figure out why we'd not sent any people to Mars. Because the obvious next step after Apollo was to send people to Mars. But what in fact happened was that we sent a few people to the moon and then we didn't send anyone after that to the moon or Mars or anything. But if you'd asked people in 1969, what would 2013 look like, they would have said, there will be a base on the moon. We would have least sent some people to Mars. And maybe there'd even be a base on Mars. There'd be like orbiting space hotels. And there'd be all this awesome stuff in space. And that's what people expected. And if you said, well, actually, the United States in 2013 will not be able to send anyone to orbit. But I'll tell you what will exist is that there'll be this device in your pocket that's like the size of-- smaller than a deck of cards that has access to all the world's information, and you can talk to any one on planet Earth. And even if you're like in some remote village somewhere so long as there's something called the Internet-- they wouldn't know what that means, of course-- then you would you be able to communicate with anyone instantly and have access to all of humanity's knowledge. They would have said, like bullshit. There's no way that that's going to be true. Right. ELON MUSK: And yet we all have that. And space is not happening. So I was trying to figure out like what was the deal here. And this was 2001. And it was just a friend of mine asked me, what am I going to do after Paypal. And I said, well, you know, I've always been interested in space, but I don't think there's anything that an individual could do in space, because it's the province of government, and usually a large government. But, I am curious as to when we're going to send some one to Mars. So I went to the NASA website to try to figure out where is the place that tells you that. And I couldn't find that. So I was like, either I'm bad at looking at the website, or they have a terrible website, because surely there must be a date. SAL KHAN: That should be a big date. ELON MUSK: Yeah. This should be on the front page. And then I discovered actually that NASA had no plans to send people to Mars, or even really back to the moon. So this was really was disappointing. I thought well, maybe this is a question of national will. Like do we to get people excited about space again? And try to get NASA a bigger budget, and then we would send people to Mars. And so I started researching the area, becoming more familiar with space, reading lots of books. And I came up with this idea to do so-called Mars oasis, which was to send a small greenhouse with seeds in dehydrated gel that upon landing, you hydrate the gel. You have green plants on a red background. The public responses to precedents and superlatives. So it would be the first life on Mars. The furthest that life's ever traveled. And you'd have this money shot of green plants on a red background. So that seemed like it would get people pretty excited. 17:34 17:36 17:39 17:42 17:45 17:47 17:51 17:53 17:54 17:57 17:58 18:04 18:08 18:09 18:13 18:15 18:19 18:20 18:22 18:24 18:27 18:29 18:32 18:37 18:39 18:41 18:42 18:44 And there were just lots of people that thought it was a really crazy idea. And there was some people that had tried to start rocket companies, not succeeded. And they tried to talk me out of it. But the thing is that-- their premise for talking me out of it was, well, we think you're going to lose the money that you invest. I was like, well, that was my expectation anyway, so I don't really mind if I lose-- you I mean, I mind, but I mean it's not like I was trying to figure out the rank-ordered best way to invest money and on that basis chose space. It's not like that's-- I thought, wow-- SAL KHAN: You weren't looking at like money-market bonds, AAA bonds, rocket company. You weren't like-- ELON MUSK: I could do real estate. I could invest in shoe making. Anything. And, whoa, space is the highest ROI. That is not what-- it wasn't the premise. I just thought that it was important that humanity expand beyond Earth, and we weren't doing that, so maybe there was something I could do to spur that on. And then I was able to compress the costs of the spacecraft and everything down to a relatively manageable number. And I got stuck on the rocket. The US rockets were way too expensive. I ended up going to Russia-- I flew to Russia three times to negotiate a purchase of an ICBM. I tried to buy two of the biggest ICBMs in the Russian fleet in 2001 and 2002. And I actually negotiated a price. SAL KHAN: I'll just let that statement stand. I'm not even going to-- Well, actually, I have to-- like who did you call? ELON MUSK: You open the yellow pages. Go to ICBMs. SAL KHAN: How does this-- I don't want to get too much in to it but I'm curious about this one particular thing. You decide at some point you need to buy an ICBM? ELON MUSK: Yeah. Well, actually at first I tried to buy just a normal launch program that they use to launch satellites, but those are too expensive. I see. ELON MUSK: The Boeing Delta II would have cost $65 million each, so two would have been $130 million. And then I was like, woah, OK, that breaks my budget right there. 20:54 20:56 20:57 20:59 21:01 21:02 21:04 21:05 21:07 21:11 21:13 21:18 21:20 21:24 21:29 21:32 21:34 21:37 21:38 21:39 21:45 21:50 21:52 21:53 21:58 22:01 22:03 22:05 22:06 22:08 22:10 22:12 22:16 22:17 22:20 22:22 22:24 22:25 22:28 22:33 22:35 22:38 22:39 22:40 Anyways, so I thought, OK, it's not really going to maybe matter that much if I do this mission, because what really matters is having a way. So I was wrong-- I thought there wasn't enough will, but there actually was plenty of will, if people thought there was a way. So then I said, OK, well, I need to work on the way. How hard is it really to make a rocket? Historically, all rockets have been expensive, so therefore, in the future, all rockets will be expensive. But actually that's not true. If you say, what is a rocket made of. And say, OK, it's made of aluminum, titanium, copper, carbon fiber, if you want to go that direction. And you can break down and say, what is the raw material cost of all these components. And if you have them stacked on the floor and could wave a magic wand so that the cost of rearranging the atoms was zero, then what would the cost of the rocket And I was like, wow, OK, it's really small. It's like 2% of what a rocket costs. So clearly it would be in how the atoms are arranged. 23:50 23:51 23:56 23:59 24:04 24:06 24:08 24:10 24:11 24:13 24:14 24:16 24:18 24:21 24:23 24:27 24:28 24:31 24:33 24:38 24:43 24:45 24:50 24:54 25:00 25:02 25:04 25:06 25:07 But there's an even better step beyond that which is to make rockets reusable. Right now that is around what our comparison price is-- excluding the refurbished ICBMs. So, if you say building a rocket from new, how does the SpaceX rocket compare to a rocket from Boeing or Lockheed? It's about a quarter of the price. 25:34 25:37 25:39 25:41 25:42 For you. SAL KHAN: Only today. Memorial day sale. 25:52 25:55 25:59 26:01 26:04 26:06 26:08 26:10 26:12 26:13 26:15 26:17 26:18 26:18 26:21 26:22 26:24 26:27 We've been working on it for a long time. I should say, SpaceX has been around for 11 years, and thus far we have not recovered any rockets. We recovered the spacecraft from orbit. So that was great. But none of our attempts to recover the rocket stages have been successful. The rocket stages have always blown up essentially on reentry. Now, we think we've figured out why that was the case. And it's a tricky thing, because Earth's gravity is really quite strong. And with an advanced rocket, you can do maybe 2% to 3% of your lift-off mass to orbit, typically. And then reusability subtracts 2% to 3% So then you've got like nothing to orbit or negative. And that's obviously not helpful. And so the trick is to try to shift that from say 2%, 3% in an expendable configuration to make the rocket mass efficiency, engine efficiency, and so forth, so much better that it moves to maybe around 3.5% to 4% in expendable configuration. And then try to get clever about the reusability elements and try to drop that to around the 1.5% to 2% level. So you have a net payload to orbit of about 2%. SAL KHAN: But you're doing it at one, two orders of magnitude cheaper. Absolutely, because our Falcon 9 rocket cost about $60 million. But the propellant cost-- which is mostly oxygen-- it's two-thirds oxygen, one-third fuel-- is only about $200,000. SAL KHAN: Wow. ELON MUSK: And it's much like a 747. It costs about as much to refuel our rocket as it does to refuel a 747 within-- well, pretty close, essentially. SAL KHAN: So assuming you all are successful, and you all have proven yourself to be successful on these audacious things in the past, I mean, what happens? I mean that seems like it's-- what happens in the next 5, 10 years in the space industry, if you all are successful there? I mean do we get to Mars? Do we have kind of market forces, commercialization of space starting to happen? ELON MUSK: Yeah. Let's see. Well, the first step is that we need to earn enough money to keep going as a company. So we have to make sure that we're launching satellites. 28:55 29:00 29:06 29:09 29:09 29:11 29:13 29:15 29:16 29:20 29:26 29:29 29:34 29:39 29:42 29:45 29:47 29:49 29:51 29:52 29:53 29:56 29:59 30:01 It's not for sure. SAL KHAN: I could talk about this for-- people know, I'm-- ELON MUSK: Aspirational it'd be a round trip. SAL KHAN: This is mind blowing. And then on Tesla. I mean Tesla's obviously, from my vantage, it's a huge success. What do you think in that industry-- well, one, I'll ask kind of the same question. What did you think-- this is something that GM and Toyota and these massive multi-billion dollar organizations have been trying. What gave you the confidence to pursue it? And now that it seems to be a huge success, where do you think this industry's going to be the next 5, 10 years? ELON MUSK: Yes. So with Tesla, the goal is try to accelerate the advent of sustainable transport. I think it would happen anyway, just out of necessity. But because we have an un-priced externality in the cost of gasoline. We weren't pricing in the environmental effects of CO2 in the oceans and atmosphere. That's causing the normal market forces to not function properly. And so the goal of Tesla is to try to act as a catalyst to accelerate those sort of normal forces. The normal sort of market reaction that would occur. We're trying to have a catalytic effect on that. And try to make it happen, I don't know, maybe 10 years sooner than it would otherwise occur. That's the goal of Tesla. So that's the reason we're making electric cars and not any other kind of car. And we also supply power trains to Toyota and to Mercedes and maybe to other car companies in the future to accelerate their production of electric vehicles. So that's the goal there. And so far, it's working out pretty well. SAL KHAN: I mean, I just saw a news report earlier today that you all sold more Model S's than-- you all are leading that segment of the industry. The Mercedes S class, the BMW 7 Series, or the Lexus LS400, or whatever it is. ELON MUSK: Yeah, actually, that seems to be the case. I didn't realize they sold so few cars in that segment. 32:16 32:17 32:19 32:22 32:26 32:29 32:31 32:34 32:35 32:37 32:38 32:42 32:45 32:47 32:49 32:53 32:59 33:01 33:05 33:08 33:13 33:15 33:17 33:18 33:19 33:22 33:23 33:27 33:28 33:31 33:33 33:38 33:42 33:44 33:47 33:49 33:54 34:02 34:05 34:10 34:12 34:12 34:13 34:15 The nature of new technology adoption is it tends to follow an S-curve. So what usually happens is people under-predict it in in the beginning, because people tend to extrapolate in a straight line. And then they'll over-predict it at the midpoint, because there's late adopters. And then it'll actually take longer than people think at the mid-point, but much shorter than people think at the beginning. 34:44 34:47 34:51 34:55 34:57 35:00 35:01 35:02 35:03 35:03 35:05 35:08 35:09 35:11 35:12 35:14 35:16 35:17 ELON MUSK: Yeah. It seems like you're doing an amazing job of-- really super leveraged. I mean, obviously, a small team, and you're having a dramatic effect-- SAL KHAN: Yeah, half these people don't even work here. There just like-- so it's like it's even-- ELON MUSK: Right. So it's, I think very impressive thing you're doing to spread knowledge and understanding throughout the world. SAL KHAN: The universe soon, if you hold up your end of the bargain. ELON MUSK: It's actually kind of funny. If you think, what is education? Like you're basically downloading data and algorithms into your brain. And it's actually amazingly bad in conventional education. Because like it shouldn't be like this huge chore. So you're making it way, way better. 36:10 36:12 36:16 36:19 36:23 36:25 36:27 36:28 36:32 36:35 36:37 36:38 ELON MUSK: So to the degree that you can make somehow learning like a game, then it's better. And I think, unfortunately, a lot of education is very vaudevillian. You've got someone standing up there kind of lecturing at people. And they've done the same lecture 20 years in a row, and they're not very excited about it. And that lack of enthusiasm is conveyed to the students. They're not very excited about it. They don't know why they're there. Like why are we letting this stuff. We don't even know why. In fact, I think a lot of things that people learn that probably there's no point in learning them. Because they never use them in the future. SAL KHAN: Because who's going to launch a rocket into space? I mean, that's just like-- exactly, that never happens. ELON MUSK: Well, you have to say-- people don't stand back and say, well, why are we teaching people these things. And we should tell them, probably, why we're teaching these things. Because a lot of kids are probably just in school, probably puzzled as to why they're there. 37:40 37:43 37:44 37:46 37:48 So I think that's pretty important. And just make it entertaining. But I think just in general conventional education should be massively overhauled. And I'm sure you pretty much agree with that. I mean the analogy I sometimes use is, have you seen like Batman, the Chris Nolan movie, the recent one. And it's pretty freaking awesome. And you've got incredible special effects, great script, multiple takes, amazing actors, and great sound, and it's very engaging. But if you were to instead say, OK-- even if you had the same script, so at least it's same script. And you said, OK, now that script, instead of having movies, we're going to have that script performed by the local town troop. OK, and so in every small town in America, if movies didn't exist, they'd have to recreate The Dark Night. With like home-sewn costumes and like jumping across the stage. And not really getting their lines quite right. And not really looking like the people in the movie. And no special effects. And I mean that would suck. 39:02 39:03 39:06 39:07 SAL KHAN: So with that-- and I apologize to all of you guys for hogging up all of the time, because, obviously, I could talk for hours about this stuff. But we do have time, probably 5 or 10 minutes for a handful of questions. If none of you all have any, I have about nine more. But, yes. SPEAKER 1: So I noticed-- I picked up two kind of themes from what you were discussing. One was somewhat audacious goals. And the other was I don't think I heard you use the word profit in anything that you spoke about. You seem to be-- each thing is pointed at like re-invigorating an industry or bringing back space missions. How much of your success do you attribute to having really audacious goals or versus just not being focused on the short term, money coming in, or I don't know, investors? ELON MUSK: Unfortunately, one does have to be focused on the short time and money coming in when creating a company, because otherwise the company will die. So I think that a lot of times people think like creating company is going to be fun. I would say it's really not that fun. I mean there are periods of fun. And there are periods where it's just awful. And, particularly, if you're the CEO of the company, you actually have a distillation of all the worst problems in the company. There's no point in spending your time on things that are going right. So you're only spending your time on things that are going wrong. And there are things that are going wrong that other people can't take care of. So you have like the worst-- you have a filter for the crappest problems in the company. The most pernicious and painful problem. 40:51 40:57 40:59 41:01 41:03 41:06 41:10 41:11 41:13 41:17 41:19 41:25 41:28 41:32 41:37 41:41 41:44 41:46 41:48 41:53 41:55 41:59 42:00 42:03 42:05 42:06 ELON MUSK: Well, it's just a very small percentage of mental energy is on the big picture. Like you know where you're generally heading for and the actual path is going to be some sort of zigzaggy thing in that direction. You're trying not to deviate too far from the path that you want to be on, but you're going to have to that to some degree. 42:33 42:35 42:39 42:40 42:42 42:45 42:47 42:50 42:53 42:56 42:58 43:02 43:05 43:09 43:12 43:16 43:17 SAL KHAN: I think we have time for one more question. Joel. JOEL: Yeah, I have an important one. SAL KHAN: OK, very good. Yes, please. SPEAKER 3: No. JOEL: OK, so few months ago, you teased Hyperloop, and we haven't heard anything since. So, first of all, a few of us engineers were talking about it, and I think we have a few ideas, if you need help. But, if you feel comfortable, maybe you could tell us a little bit more. ELON MUSK: I was reading about the California high-speed rail, and it was quite depressing. Because California taxpayers are going to be on the hook to build the most expensive high-speed rail per mile in the world-- and the slowest. 44:03 44:05 And, it's like, damn, we're in California, we make super high-tech stuff. Why are we going to be spending-- now the estimates are around $100 billion-- for something that will take two hours to go from LA to San Francisco? I'm like, OK, well, I can get on a plane and do that it 45 minutes. It doesn't make much sense. And isn't there some better way to do it than that. So if you just say, OK, well what would you ideally want in a transportation system? You'd say, OK, well you'd want something that relative to existing modes of transportation is faster-- let's say twice as fast-- costs half as much per ticket, can't crash, is immune to weather, and is-- you can make the whole thing like self-powering with like solar panels or something like that. That would be pretty-- SAL KHAN: That would be great, yes. ELON MUSK: --a good outcome. And so what would do that? And what's the fastest way short of inventing teleportation that you could do something like that? And some of the elements of that solution are fairly obvious, and some of them are not so obvious. And then the details-- the devil's in the details of actually making something like that work. But I came to the conclusion that there is something like that that could work. And would be practical. SAL KHAN: Is this around the evacuated tubes? The vacuum tubes? Like the old bank-- ELON MUSK: It's something like that. SAL KHAN: But you haven't been more public with what this is? ELON MUSK: No. Although I did say that once Tesla was profitable that I would talk more about it. But, we haven't done our earnings call yet. So I think I'll probably do it after the earnings call. And the thing is I'm kind of strung out on things that I'm already doing. So adding another thing-- it's like doesn't-- it's a lot SAL KHAN: Learning the guitar You could pick up all sorts of things. ELON MUSK: Right. I tried learning the violin. That's, by the way, a hard thing to learn. SAL KHAN: Yeah. Launching rockets, electric cars, revolutionizing transportation. Yeah, it's easy. ELON MUSK: I cannot play the violin at all. Very horrible. If you think about the future, you want a future that's better than the past, and so if we had something like the Hyperloop, I think that would be like cool. You'd look forward to the day that was working. And if something like that, even if it was only in one place-- from LA to San Francisco, or New York to DC or something like that-- then it would be cool enough that it would be like a tourist attraction. It would be like a ride or something. So even if some of the initial assumptions didn't work out, the economics didn't work out quite as one expected, it would be cool enough that like, I want to journey to that place just to ride on that thing. That would be pretty cool. And so that's I think how-- if you come with a new technology, it should feel like that. You should really-- if you told it to an objective person, would they look forward to the day that that thing became available. And it would be pretty exciting to do something like that. Or an aircraft. Like I thought it was really disappointing when the Concorde was taking out of commission, and there was no supersonic transport available. And of course the 787 has had some issues. 47:37 47:40 47:44 47:46 47:49 47:51 47:54 47:57 47:58 48:00 48:04 48:05 48:08 48:09 48:14 48:17 48:20 48:22 48:24 48:25 48:28 48:29 48:31 48:33 48:34 48:34 48:35" + }, + { + "Q": "why is the variable always X?", + "A": "It s not always x. But x is the most common one to use in simple problems.", + "video_name": "Ye13MIPv6n0", + "transcript": "So we have our scale again. And we've got some masses on the left hand side and some masses on the right hand side. And we see that our scale is balanced. We have the same total mass on the left hand side that we have on the right hand side. Instead of labeling the mystery masses as question mark, I've labeled them all x. And since they all have an x on it, we know that each of these have the same mass. But what I'm curious about is, what is that mass? What is the mass of each of these mystery masses, I guess we could say? And so I'll let think about that for a second. How would you figure out what this x value actually is? How many kilograms is the mass of each of these things? What could you do to either one or both sides of this scale? I'll give you a few seconds to think about that. So you might be tempted to say, well if I could end up with just one mystery mass on the left hand side, and if I keep my scale balanced, then that thing's going to be equal to whatever I have on the right hand side. And that part would actually be a true statement. But then to get only one of these mystery masses on the left hand side, you might say, well why don't I just remove two of them? You might just say, well why don't I just remove-- let me do it a good color for removing-- why don't I just remove that one and that one? And then I'll just be left with that right over there. But if you just removed these two, then the left hand side is going to become lighter or it's going to have a lower mass than the right hand side. So it's going to move up and the right hand side is going to move down. And then you might say, OK, I understand. Whatever I have to do to the left hand side, I have to do to the right hand side in order to keep my scale balanced. So you might say, well why don't I remove two of these mystery masses from the right hand side? But that's a problem too because you don't know what this mystery mass is. You could try to remove two from this, but how many of these blocks represent a mystery mass? We actually don't know. But you might then say, well let's see, I've got three of these things here. If I essentially multiply what I have here by 1/3 or if I only leave a 1/3 of the stuff here, and if I only leave a 1/3 of the stuff here, then the scale should be balanced. If this has the total mass as this, then 1/3 of this total mass is going to be the same thing as 1/3 of that total mass. So let's just keep only 1/3 of this here. So that's the equivalent to multiplying by 1/3. So if we're only going to keep 1/3 there, we're going to be left with only one of the masses. And if we only keep 1/3 here, let's see, we have one, two, three, four, five, six, seven, eight, nine masses. If we multiply this by 1/3, or if we only keep 1/3 of it there, 1/3 times 9 is 3. So we're going to remove these . And so we have 1/3 of what we originally had on the right hand side and 1/3 of what we originally had on the left hand side. And they will be balanced because we took 1/3 of the same total masses. And so what you're left with is just one of these mystery masses, this x thing right over here, whatever x might be. And you have three kilograms on the right hand side. And so you can make the conclusion, and the whole time you kept this thing balanced, that x is equal to 3." + }, + { + "Q": "Question:\nwhat happens if the show people pulled a trick and didn't paint anyone blue? How would they figure that out?", + "A": "Because they said that at least one person s forehead was painted blue", + "video_name": "-xYkTJFbuM0", + "transcript": "So we had the hundred logicians. All of their foreheads were painted blue. And before they entered the room, they were told that at least one of you hundred logicians has your forehead painted blue. And then every time that they turned on the lights, so that they could see each other, they said OK, once you've determined that you have a blue forehead, when the lights get turned off again, we want you to leave the room. And then once that's kind of settled down, they'll turn the lights on again. And people will look at each other again. And then they'll turn them off again. And maybe people will leave the room. And so forth and so on. And they're also all told that everyone in the room is a perfect logician. They have infallible logic. So the question was, what happens? And actually maybe an even more interesting question is why does it happen? So I'll answer the first, what happens? And if just take the answer, and you don't know why, it almost seems mystical. That essentially the light gets turned on and off 100 times, and then after the hundredth time that the light gets turned on, and the lights get turned off again, all of them leave. They all leave. So I mean, it's kind of weird, right? Let's say I'm one of them. Or you're one of them. I go into this room. The lights get turned on. And I see 99 people with blue foreheads. And I can't see my own forehead. They see my forehead, of course. But to any other person, I'm one of the 99, right? But I see 99 blue foreheads. So essentially what happens if we were to watch the show is, the lights get turned on. You see 99 blue foreheads. Then the lights get turned off again. And then the lights get turned on again. And everyone's still sitting there. And I still see 99 blue foreheads. And that happens 100 times. And after the hundredth time the light gets turned on, everyone leaves the room. And at first glance, that seems crazy, because nothing changes. Nothing changes between every time we turn on the light. But the way you need to think about this-- and this is what makes it interesting-- is what happens instead of 100, let's say there was one person in the room. So before the show starts-- they never told me that there were going to be 100 people in the room. They just said, at least one of you, at least one of the people in the room, has your forehead painted blue. And as soon as you know that your forehead is painted blue, you leave the room. And that everyone's a perfect logician. So imagine the situation where instead of 100 there's only one perfect logician. Let's say it's me. So that's the room. I walk in. And I sit down. And maybe I should do it with blue. And then they turn the lights on, and say, look around the room. And I look around the room, and I see nobody else, right? And remember, even in the case of one, we've painted everyone's forehead blue. So in this case, this one dude, or me or whoever you want to call him. His forehead is painted blue. So he looks around and he sees no one in the room. But he remembers the statement, and maybe it's even written down on a card for him in case he forgets. That at least one of you has your forehead painted blue. So if he looks around the room and he says, well I'm the only dude in the room. And they told me that at least one of the dudes in the room is going to have their foreheads blue. Well, I'm the only dude in the room. So I must have a blue forehead. So as soon as they turn the lights off, he's going to leave. Fair enough. That's almost trivially simple. And you might say, so how does this apply to 100? Well what happens when there are two people. And once again, both of them have their foreheads painted blue. So let me draw another. I don't want to keep drawing the blue forehead room. Let's say there's two people now. So let's put ourselves in the head of this guy. Right behind the blue forehead. That's where we're sitting. So when he enters the room. He says, I either have a blue forehead. I either have a blue forehead, or I don't have a blue forehead. No blue. Right? This is what this guy's thinking. Let me draw him. And he has a blue forehead. But he doesn't know it. He can't see it. That's the whole point about painting the forehead blue, as opposed to another part of the body. So he says, I either have a blue forehead or I don't have a blue forehead. He walks in. Let's say this is this guy. He walks in. The first time the lights get turned on, he sees this other dude there who has a blue forehead. And he says OK, now let me think about it. How will this guy respond depending on each of these states? So let's say that I don't have a blue forehead. Let's go into this reality. If I don't have a blue forehead, what is this guy going to see? When the lights get turned on, he's going to see that I don't have a blue forehead. And we were both told that at least one of us has a blue forehead. So this guy, because he's a perfect logician, will deduce that he has to have a blue forehead. Remember, this is in a situation, if I assume that I don't have a blue forehead. We're in this world. I'm a perfect logician, so if I can assume, if I'm simulating the reality where I don't have a blue forehead, then this guy will see, I don't have a blue forehead. And then he'll say, I must have a blue forehead. And so when the lights get turned off, this guy will leave. He'll exit the room. And vice versa. The other guy will make the same logic. But since both of them have blue foreheads, what happens? The lights get turned off, then the lights get turned back on. When the light gets turned back on, this guy's still sitting here. And I just determined that if I didn't have a blue forehead-- and this is me-- this guy would have left. Because he could've said, oh I must be the only guy with a blue forehead. So he would have left. But since he didn't leave, I now know that I have a blue forehead. So then the second time that the lights get turned on, I can deduce that I have a blue forehead. And then when the lights get turned off again, I'll leave. And this guy, he's a perfect logician, so he makes the exact same conclusion. Because he also simulated in his head, OK if I didn't have a blue forehead, then this guy will leave as soon as the lights get turned off the first time. If this guy doesn't have a blue forehead, then this guy will say, well I see no-one else in the room with a blue forehead, and since I know that there's at least one with a blue forehead, I'll have to leave. But since he didn't leave, this guy will also know that he must have a blue forehead as well, so they'll actually leave together. Maybe they'll bump into each other on the way out. Fair enough. Now what happens if you extend it to three people? So we already said, if you have one person in the room, he'll come to the conclusion the first time that the lights And then he'll leave right when they're turned off. If you have two people, it takes them essentially two times for the light to get turned on to reach that conclusion. Now if you have three people, and I think you see where this is going. One, two and three. Now remember, no one knows if their foreheads are painted blue, but the producers of the show actually did paint everyone's forehead blue. So so once again, let's get into this guy's head. So this guy says, he's either blue or he's not blue. So in the reality when he's not blue, what's going to happen? Well, this guy-- and this gets a little bit confusing, but if you think about it from the previous example, it makes a lot of sense. A person who has a not-blue forehead actually shouldn't affect the outcome of what the blue people do. Because let's say that this guy says, well what's going to happen if I'm not blue? Well, this guy's going to look at that guy, and say, oh he has a blue forehead. He doesn't have a blue forehead. So if I don't have a blue forehead, this guy's going to see two people without blue foreheads. And he's going to leave the room the first time that the lights are turned on. He'll come to the conclusion. Now the second time that the light's turned on, this guy will say, gee, this guy didn't leave the room. That guy doesn't have a blue forehead. And this guy didn't leave the room because he must have seen someone with a blue forehead. Therefore I must have a blue forehead. And so, if this guy doesn't have a blue forehead, both of these guys would leave the room the second time that the light gets turned on. Now what happens if they don't leave the room the second time that the lights are turned on? Well if I was not blue, they would have left. So if they haven't left by the third showing of the light, then I know that I'm blue. So when you have three people, they're all perfect logicians, they all have their foreheads painted blue. The light will be shown three times. Or the light will be turned on three times. And then when the light gets turned off, they'll all leave together. And so this logic applies for any. You could have 1,000 people. You can keep extending it. The fourth person will have the exact same logic. If he's not blue, then these guys are going to leave after three turnings on of the light. But if they don't leave after three turnings on of the light, then he must be blue. And so they're all going to leave together. All four of them on the fourth showing of everyone's foreheads. Anyway, and you can keep extending this all the way to 100. And 100 is arbitrary. You could do this with a million people, and they would just keep looking at each other a million times. And then on the millionth showing, they would all reach the conclusion that they all have blue foreheads, and they would leave the room. And if you think about it, it's fairly straightforward logic. But it leads to kind of a very almost eerie result. Hopefully that satisfies you. See you in the next video." + }, + { + "Q": "solve the equation by extracting square root (x+6)^2 = 5", + "A": "(x + 6)\u00c2\u00b2 = 5 We take the square root of both sides: x + 6 = \u00c2\u00b1\u00e2\u0088\u009a5 Subtract 6 from both sides: x = (\u00c2\u00b1\u00e2\u0088\u009a5) - 6 So: x = -6 - (\u00e2\u0088\u009a5) AND x = (\u00e2\u0088\u009a5) + 6", + "video_name": "55G8037gsKY", + "transcript": "In this video, I'm going to do several examples of quadratic equations that are really of a special form, and it's really a bit of warm-up for the next video that we're going to do on completing the square. So let me show you what I'm talking about. So let's say I have 4x plus 1 squared, minus 8 is equal to 0. Now, based on everything we've done so far, you might be tempted to multiply this out, then subtract 8 from the constant you get out here, and then try to factor it. And then you're going to have x minus something, times x minus something else is equal to 0. And you're going to say, oh, one of these must be equal to 0, so x could be that or that. We're not going to do that this time, because you might see something interesting here. We can solve this without factoring it. And how do we do that? Well, what happens if we add 8 to both sides of this equation? Then the left-hand side of the equation becomes 4x plus 1 squared, and these 8's cancel out. The right-hand becomes just a positive 8. Now, what can we do to both sides of this equation? And this is just kind of straight, vanilla equation-solving. This isn't any kind of fancy factoring. We can take the square root of both sides of this equation. We could take the square root. So 4x plus 1-- I'm just taking the square root of both sides. You take the square root of both sides, and, of course, you want to take the positive and the negative square root, because 4x plus 1 could be the positive square root of 8, or it could be the negative square root of 8. So 4x plus 1 is equal to the positive or negative square root of 8. Instead of 8, let me write 8 as 4 times 2. We all know that's what 8 is, and obviously the square root of 4x plus 1 squared is 4x plus 1. So we get 4x plus 1 is equal to-- we can factor out the 4, or the square root of 4, which is 2-- is equal to the plus or minus times 2 times the square root of 2, right? Square root of 4 times square root of 2 is the same thing as square root of 4 times the square root of 2, plus or minus the square root of 4 is that 2 right there. Now, it might look like a really bizarro equation, with this plus or minus 2 times the square of 2, but it really isn't. These are actually two numbers here, and we're actually simultaneously solving two equations. We could write this as 4x plus 1 is equal to the positive 2, square root of 2, or 4x plus 1 is equal to negative 2 times the square root of 2. This one statement is equivalent to this right here, because we have this plus or minus here, this or statement. Let me solve all of these simultaneously. So if I subtract 1 from both sides of this equation, what do I have? On the left-hand side, I'm just left with 4x. On the right-hand side, I have-- you can't really mathematically, I mean, you could do them if you had a calculator, but I'll just leave it as negative 1 plus or minus the square root, or 2 times the square root of 2. That's what 4x is equal to. If we did it here, as two separate equations, same idea. Subtract 1 from both sides of this equation, you get 4x is equal to negative 1 plus 2, times the square root of 2. This equation, subtract 1 from both sides. 4x is equal to negative 1 minus 2, times the square root of 2. This statement right here is completely equivalent to these two statements. Now, last step, we just have to divide both sides by 4, so you divide both sides by 4, and you get x is equal to negative 1 plus or minus 2, times the square root of 2, over 4. Now, this statement is completely equivalent to dividing each of these by 4, and you get x is equal to negative 1 plus 2, times the square root 2, over 4. This is one solution. And then the other solution is x is equal to negative 1 minus 2 roots of 2, all of that over 4. That statement and these two statements are equivalent. And if you want, I encourage you to-- let's substitute one of these back in, just so you feel confident that something as bizarro as one of these expressions can be a solution to a nice, vanilla-looking equation like this. So let's substitute it back in. 4 times x, or 4 times negative 1, plus 2 root 2, over 4, plus 1 squared, minus 8 is equal to 0. Now, these 4's cancel out, so you're left with negative 1 plus 2 roots 2, plus 1, squared, minus 8 is equal to 0. This negative 1 and this positive 1 cancel out, so you're left with 2 roots of 2 squared, minus 8 is equal to 0. And then what are you going to have here? So when you square this, you get 4 times 2, minus 8 is equal to 0, which is true. 8 minus 8 is equal to 0. And if you try this one out, you're going to get the exact same answer. Let's do another one like this. And remember, these are special forms where we have squares of binomials in our expression. And we're going to see that the entire quadratic formula is actually derived from a notion like this, because you can actually turn any, you can turn any, quadratic equation into a perfect square equalling something else. We'll see that two videos from now. But let's get a little warmed up just seeing this type of thing. So let's say you have x squared minus 10x, plus 25 is equal to 9. Now, once again your temptation-- and it's not a bad temptation-- would be to subtract 9 from both sides, so you get a 0 on the right-hand side, but before you do that, just inspect this really fast. And say, hey, is this just maybe a perfect square of a binomial? And you see-- well, what two numbers when I multiply them I get positive 25, and when I add them I get negative 10? And hopefully negative 5 jumps out at you. So this expression right here is x minus 5, times x minus 5. So this left-hand side can be written as x minus 5 squared, and the right-hand side is still 9. And I want to really emphasize. I don't want this to ruin all of the training you've gotten on factoring so far. We can only do this when this is a perfect square. If you got, like, x minus 3, times x plus 4, and that would be equal to 9, that would be a dead end. You wouldn't be able to really do anything constructive with that. Only because this is a perfect square, can we now say x minus 5 squared is equal to 9, and now we can take the square root of both sides. So we could say that x minus 5 is equal to plus or minus 3. Add 5 to both sides of this equation, you get x is equal to 5 plus or minus 3, or x is equal to-- what's 5 plus 3? Well, x could be 8 or x could be equal to 5 minus 3, or x is equal to 2. Now, we could have done this equation, this quadratic equation, the traditional way, the way you were tempted to do it. What happens if you subtract 9 from both sides of this equation? You'll get x squared minus 10x. And what's 25 minus 9? 25 minus 9 is 16, and that would be equal to 0. And here, this would be just a traditional factoring problem, the type that we've seen in the last few videos. What two numbers, when you take their product, you get positive 16, and when you sum them you get negative 10? And maybe negative 8 and negative 2 jump into your brain. So we get x minus 8, times x minus 2 is equal to 0. And so x could be equal to 8 or x could be equal to 2. That's the fun thing about algebra: you can do things in two completely different ways, but as long as you do them in algebraically-valid ways, you're not going to get different answers. And on some level, if you recognize this, this is easier because you didn't have to do that little game in your head, in terms of, oh, what two numbers, when you multiply them you get 16, and when you add them you get negative 10? Here, you just said, OK, this is x minus 5-- oh, I guess you did have to do it. You had to say, oh, 5 times 5 is 25, and negative 10 is negative 5 plus negative 5. So I take that back, you still have to do that little game in your head. So let's do another one. Let's do one more of these, just to really get ourselves nice and warmed up here. So, let's say we have x squared plus 18x, plus 81 is equal to 1. So once again, we can do it in two ways. We could subtract 1 from both sides, or we could recognize that this is x plus 9, times x plus 9. This right here, 9 times 9 is 81, 9 plus 9 is 18. So we can write our equation as x plus 9 squared is equal to 1. Take the square root of both sides, you get x plus 9 is equal to plus or minus the square root of 1, which is just 1. So x is equal to-- subtract 9 from both sides-- negative 9 plus or minus 1. And that means that x could be equal to-- negative 9 plus 1 is negative 8, or x could be equal to-- negative 9 minus 1, which is negative 10. And once again, you could have done this the traditional way. You could have subtracted 1 from both sides and you would have gotten x squared plus 18x, plus 80 is equal to 0. And you'd say, hey, gee, 8 times 10 is 80, 8 plus 10 is 18, so you get x plus 8, times x plus 10 is equal to 0. And then you'd get x could be equal to negative 8, or x could be equal to negative 10. That was good warm up. Now, I think we're ready to tackle completing the square." + }, + { + "Q": "What is a model?", + "A": "a diagram or a 2d picture", + "video_name": "Rd4a1X3B61w", + "transcript": "Here some picture of what most people associate when they think of chemistry. They think of scientists working on a bench with the different vials of different chemicals. They might think of a mad scientist. Some of them boiling and changing colors. They might associate chemistry with chemical equations. Thinking about how different things will react together to form other things. They might think about models of the different molecules that can be depicted different ways. They might associate it with the periodic table of elements. And all of these things are a big part of chemistry. But I want you to do in this video is appreciate what at its essence chemistry is all about. And chemistry is one of the sciences that really just helps us understand and make models and make predictions about our reality. And even something like the periodic table of elements, which you'll see at the front of any chemistry classroom, you take it for granted. But this is the product of, frankly, thousands of years of human beings trying to get to an understanding of all of the different complexity in the world. If you look at the world around us, and it doesn't even have to be our planet, it could be the universe around us, you see all these different substances that seem to be different in certain ways. You see things like fire and rock and water. Even in the planets, you see meteorological patterns. In life, you see all of this complexity and all of these different things and it looks like there's just like a infinite spectrum of differentness out of there. Of different substances. Even in things like our human brain. The complexity and the electrochemical interactions. And you could imagine as a species, this is kind of overwhelming. How do you make the sense of all of this? And it was not an easy path, but over thousands of years, we did start to make sense of it. And why it's very lucky for all of us to be born when we are now or to be around when we are now. To be able to learn chemistry where we are now is that we get the answer. And it's a partial answer, which is also exciting, cause we don't want the full answer. But it's a partial answer that takes us a long way. We realize that the periodic table of elements, that all of this complexity that we're seeing before, that at the end of the day, things are made of basic building blocks. Kind of you could imagine the legos that really make up everything. And there aren't an infinite number of legos. There's actually a finite number of them. We're discovering more all of the time, well not all of the time, now new elements are not discovered that frequently, but there's a few of these elements that are disproportionately showing up in a lot of what we see here. These things that seem so different. Well a lot of this is different compositions of elements like carbon and oxygen and hydrogen. And even the elements themselves are made of things like protons and electrons and neutrons that are just rearranged in different ways to give us these elements that have all of these different properties. So when you think about chemistry, yes, it might visually look something like this. These are obviously much older pictures. But at its essence, it's how do we create models and understand the models that describe a lot of the complexity in the universe around us? And just to put chemistry in, I guess you could say, in context with some of the other sciences, many people would say at the purest level, you would have mathematics. That math, you're studying ideas, which could even be independent, you're seeing logical ideas that could be even independent of anything that you've ever observed or experienced. And a lot of folks that say if we ever communicate with another intelligent species that could be completely different than us, math might be that common language. Because even if we perceive the world differently and think differently in certain ways, math might be that common language. But on top of math, we start to say, well how is our reality actually structured? At the most basic level, what are the constituents of matter and what are the mathematical properties that describe how they react together? And then, or interact with each other? Then you go one level above that, you get to the topic of this video, which is chemistry. Which is very closely related to physics. When we talk about these chemical equations and we create these molecular structures, the interactions between these atoms, these are quantum mechanical interactions which we do not fully understand at the deepest level yet. But with chemistry, we can start to make use of the math and they physics to start to think about how all of these different building blocks can interact to explain all sorts of different phenomena. This chemical equation you see right here, this is combustion. This is hydrogen combusting with oxygen to produce a lot of energy. To produce energy. When we imagine combustion, we think of fire. But what even is fire at its most fundamental level? How do we get, why do we perceive this thing here? And chemistry is super important because on top of that, we build biology. We build biology. And as you'll see as you study all of these things, there's points where these things start to bleed together. But the biology in, say, a human being, or really in any species, it's based on molecular interactions. Interactions between molecules, between atoms, which, at the end of the day, is all about chemistry. As I speak, the only reason why I'm able to speak is because of really, hard to imagine the number of chemical interactions happening in me right now to create this soundness. To create this thing that thinks it exists that wants to make a video about how awesome and amazing chemistry is. And then from biology, you can build out on all of everything else. So sciences like psychology and economics, which of course, these things also leverage math and other things. But this gives you kind of a sense of how we build up and how we explain the reality around us. And not one of these is more important than the other. These are all studying incredibly fascinating things that as humans beings first became thoughtful about their environment, said, \"Gee, why are we here? \"What is this place? \"Why do we exist? \"How do we exist?\" And chemistry builds models for us to understand interactions at a scale and a speed that we can't directly observe, but nonetheless, we can to start to make predictions. So that's what's really cool about this. When you study chemistry, you should not view this as some type of a chore that the school system is forcing you through. There are people who would've done anything 100 years ago to get the answers that are in your chemistry book today or that you can learn from your chemistry teacher or that you can learn from a Khan Academy video. There are people in the world in the past and today who'd do anything to be able to understand deeply what this is. That they consider it a privilege to be able to learn at this level. And then to think about where this could go because none of these fields are complete. We have very partial knowledge of all of these fields. Arguably, there's an infinite more that we could learn relative to what we know. But what's exciting is that we have such a strong start. We're starting to make sense of it. To really describe everything in our reality." + }, + { + "Q": "does anybody know how many numbers are in pie?", + "A": "Actually, pie is irrational as well as transcendental, which means that it has a infinite number of digits. However many people are racing to see who can find the most digits of pi.", + "video_name": "ZyOhRgnFmIY", + "transcript": "A candy machine creates small chocolate wafers in the shape of circular discs. The Diameter of each wafer is 16 millimeters. Whats it the area of each candy? So, the candy they say is in the shape of circular disc and they tell us that the diameter is 16 millimeters. If I draw a line across the circle, that goes through the center. The length of the line all the way across the circle through the center is 16 millimeters. The Diameter here is 16 millimeters. And they want us to figure out the area of the surface of the candy. Essentially the area of this circle. When we think about area, we know that the area of the circle is pi times the radius of the circle square. They gave us the diameter, what is the radius? well, you might remember that the radius is half of the diameter. Distance from the center of the circle to the outside, to the boundary of the circle. So, it will be this distance over here, which is exactly half of the diameter. So, would be 8 millimeters. So, where we see the radius, we can put 8 millimeters. So, the Area is going to be equal to pi times 8 millimeters squared, which would be 64 square millimeters. And, typically this is written as pi after 64. So, you might often see it as 64 pi millimeter squared. Now, this is the answer 64 pi millimeters squared. But sometimes it is not satisfying to leave it as 64 pi millimeter squared. You might well say, that what number it is close to. I want a decimal representation of this. And we could start to use the approximate values of pi. So, the most rough approximate value which is tensed to be used is saying that pi, a very rough approximation, is equal to 3.14. So, in that case we can that this will be equal to 64 times 3.14 millimeters squared. We can get a calculator to figure out what this will be in decimal form. So, we have 64 times 3.14, gives us 200.96 So, we can say that the area is approximately equal to 200.96 square millimeters. Now if we want to get a more accurate representation of this, pi just actually keeps going on and on forever, we could use the calculator's internal representation of pi. In which case, we will say 64 times, and than we have to look for the pi on the calculator, it's up here this yellow, so I'll use the 2nd function to get the pi there. Now, we are using the calculator's internal representation of pi which is going to be more precise than what I had in the last one. And you can 201.06 (to the nearest hundred) So, more precise is 201.06 square millimeters. So, this is closer to the actual answer, because the calculator's representation is more precise than this very rough approximation of what pi is." + }, + { + "Q": "Where would fascism fit in on the graph?", + "A": "The Middle of the graph", + "video_name": "MmRgMAZyYN0", + "transcript": "Thought I would do a video on communism just because I've been talking about it a bunch in the history videos, and I haven't given you a good definition of what it means, or a good understanding of what it means. And to understand communism-- let me just draw a spectrum here. So I'm going to start with capitalism. And this is really just going to be an overview. People can do a whole PhD thesis on this type of thing. Capitalism, and then I'll get a little bit more-- and then we could progress to socialism. And then we can go to communism. And the modern versions of communism are really kind of the brainchild of Karl Marx and Vladimir Lenin. Karl Marx was a German philosopher in the 1800s, who, in his Communist Manifesto and other writings, kind of created the philosophical underpinnings for communism. And Vladimir Lenin, who led the Bolshevik Revolution in the-- and created, essentially, the Soviet Union-- he's the first person to make some of Karl Marx's ideas more concrete. And really every nation or every country which we view as communist has really followed the pattern of Vladimir Lenin. And we'll talk about that in a second. But first, let's talk about the philosophical differences between these things, and how you would move. And Karl Marx himself viewed communism as kind of a progression from capitalism through socialism to communism. So what he saw in capitalism-- and at least this part of what he saw was right-- is that you have private property, private ownership of land. That's the main aspect of capitalism. And this is the world that most of us live in today. The problem that he saw with capitalism is he thought, well, look, when you have private property, the people who start accumulating some capital-- and when we talk about capital, we could be talking about land, we could be talking about factories, we could be talking about any type of natural resources-- so the people who start getting a little bit of them-- so let me draw a little diagram here. So let's say someone has a little bit of capital. And that capital could be a factory, or it could be land. So let me write it. Capital. And let's just say it's land. So let's say someone starts to own a little bit of land. And he owns more than everyone else. So then you just have a bunch of other people who don't own land. But they need, essentially-- and since this guy owns all the land, they've got to work on this guy's land. They have to work on this guy's land. And from Karl Marx's point of view, he said, look, you have all of these laborers who don't have as much capital. This guy has this capital. And so he can make these laborers work for a very small wage. And so any excess profits that come out from this arrangement, the owner of the capital will be able to get it. Because these laborers won't be able to get their wages to go up. Because there's so much competition for them to work on this guy's farm or to work on this guy's land. He really didn't think too much about, well, maybe the competition could go the other way. Maybe you could have a reality eventually where you have a bunch of people with reasonable amounts of capital, and you have a bunch of laborers. And the bunch of people would compete for the laborers, and maybe the laborers could make their wages go up, and they could eventually accumulate their own capital. They could eventually start their own small businesses. So he really didn't think about this reality too much over here. He just saw this reality. And to his defense-- and I don't want to get in the habit of defending Karl Marx too much-- to his defense, this is what was happening in the late 1800s, especially-- we have the Industrial Revolution. Even in the United States, you did have kind of-- Mark Twain called it the Gilded Age. You have these industrialists who did accumulate huge amounts of capital. They really did have a lot of the leverage relative to the laborers. And so what Karl Marx says, well, look, if the guy with all the capital has all the leverage, and this whole arrangement makes some profits, he's going to be able to keep the profits. Because he can keep all of these dudes' wages low. And so what's going to happen is that the guy with the capital is just going to end up with more capital. And he's going to have even more leverage. And he'll be able to keep these people on kind of a basic wage, so that they can never acquire capital for themselves. So in Karl Marx's point of view, the natural progression would be for these people to start organizing. So these people maybe start organizing into unions. So they could collectively tell the person who owns the land or the factory, no, we're not going to work, or we're going to go on strike unless you increase our wages, or unless you give us better working conditions. So when you start talking about this unionization stuff, you're starting to move in the direction of socialism. The other element of moving in the direction of socialism is that Karl Marx didn't like this kind of high concentration-- or this is socialists in general, I should say-- didn't like this high concentration of wealth. That you have this reality of not only do you have these people who could accumulate all of this wealth-- and maybe, to some degree, they were able to accumulate it because they were innovative, or they were good managers of land, or whatever, although the Marxists don't give a lot of credit to the owners of capital. They don't really give a lot of credit to saying maybe they did have some skill in managing some type of an operation. But the other problem is is that it gets handed over. It gets handed over to their offspring. So private property, you have this situation where it just goes from maybe father to son, or from parent to a child. And so it's not even based on any type of meritocracy. It's really just based on this inherited wealth. And this is a problem that definitely happened in Europe. When you go back to the French Revolution, you have generation after generation of nobility, regardless of how incompetent each generation would be, they just had so much wealth that they were essentially in control of everything. And you had a bunch of people with no wealth having to work for them. And when you have that type of wealth disparity, it does lead to revolutions. So another principle of moving in the socialist direction is kind of a redistribution of wealth. So let me write it over here. So redistribution. So in socialism, you can still have private property. But the government takes a bigger role. So you have-- let me write this. Larger government. And one of the roles of the government is to redistribute wealth. And the government also starts having control of the major factors of production. So maybe the utilities, maybe some of the large factories that do major things, all of a sudden starts to become in the hands of the government, or in the words of communists, in the hands of the people. And the redistribution is going on, so in theory, you don't have huge amounts of wealth in the hands of a few people. And then you keep-- if you take these ideas to their natural conclusion, you get to the theoretical communist state. And the theoretical communist state is a classless, and maybe even a little bit-- a classless society, and in Karl Marx's point of view-- and this is a little harder to imagine-- a stateless society. So in capitalism, you definitely had classes. You had the class that owns the capital, and then you had the labor class, and you have all of these divisions, and they're different from each other. He didn't really imagine a world that maybe a laborer could get out of this, they could get their own capital, then maybe they could start their own business. So he just saw this tension would eventually to socialism, and eventually a classless society where you have a central-- Well, he didn't even go too much into the details but you have kind of equal, everyone in society has ownership over everything, and society somehow figures out where things should be allocated, and all of the rest. And it's all stateless. And that's even harder to think about in a concrete fashion. So that's Karl Marx's view of things. But it never really became concrete until Vladimir Lenin shows up. And so the current version of communism that we-- The current thing that most of us view as communism is sometimes viewed as a Marxist-Leninist state. These are sometimes used interchangeably. Marxism is kind of the pure, utopian, we're eventually going to get to a world where everyone is equal, everyone is doing exactly what they want, there's an abundance of everything. I guess to some degree, it's kind of describing what happens in Star Trek, where everyone can go to a replicator and get what they want. And if you want to paint part of the day, you can paint part of the day, and you're not just a painter, you can also do whatever you want. So it's this very utopian thing. Let me write that down. So pure Marxism is kind of a utopian society. And just in case you don't know what utopian means, it's kind of a perfect society, where you don't have classes, everyone is equal, everyone is leading these kind of rich, diverse, fulfilling lives. And it's also, utopian is also kind of viewed as unrealistic. It's kind of, if you view it in the more negative light, is like, hey, I don't know how we'll ever be able to get there. Who knows? I don't want to be negative about it. Maybe we will one day get to a utopian society. But Leninist is kind of the more practical element of communism. Because obviously, after the Bolshevik Revolution, 1917, in the Russian Empire, the Soviet Union gets created, they have to actually run a government. They have to actually run a state based on these ideas of communism. And in a Leninist philosophy-- and this is where it starts to become in tension with the ideas of democracy-- in a Leninist philosophy, you need this kind of a party system. So you need this-- and he calls this the Vanguard Party. So the vanguard is kind of the thing that's leading, the one that's leading the march. So this Vanguard Party that kind of creates this constant state of revolution, and its whole job is to guide society, is to kind of almost be the parent of society, and take it from capitalism through socialism to this ideal state of communism. And it's one of those things where the ideal state of communism was never-- it's kind of hard to know when you get there. And so what happens in a Leninist state is it's this Vanguard Party, which is usually called the Communist Party, is in a constant state of revolution, kind of saying, hey, we're shepherding the people to some future state without a real clear definition of what that future state is. And so when you talk about Marxist-Leninist, besides talking about what's happening in the economic sphere, it's also kind of talking about this party system, this party system where you really just have one dominant party that it will hopefully act in the interest of the people. So one dominant communist party that acts in the interest of the people. And obviously, the negative here is that how do you know that they actually are acting in the interest of people? How do you know that they actually are competent? What means are there to do anything if they are misallocating things, if it is corrupt, if you only have a one-party system? And just to make it clear, the largest existing communist state is the People's Republic of China. And although it is controlled by the Communist Party, in economic terms it's really not that communist anymore. And so it can be confusing. And so what I want to do is draw a little bit of a spectrum. On the vertical axis, over here, I want to put democratic. And up here, I'll put authoritarian or totalitarian. Let me put-- well, I'll put authoritarian. I'll do another video on the difference. And they're similar. And totalitarian is more an extreme form of authoritarian, where the government controls everything. And you have a few people controlling everything and it's very non-democratic. But authoritarian is kind of along those directions. And then on this spectrum, we have the capitalism, socialism, and communism. So the United States, I would put-- I would put the United States someplace over here. I would put the United States over here. It has some small elements of socialism. You do have labor unions. They don't control everything. You also have people working outside of labor unions. It does have some elements of redistribution. There are inheritance taxes. There are-- I mean it's not an extreme form of redistribution. You can still inherit private property. You still have safety nets for people, you have Medicare, Medicaid, you have welfare. So there's some elements of socialism. But it also has a very strong capitalist history, private property, deep market, so I'd stick the United States over there. I would put the USSR-- not current Russia, but the Soviet Union when it existed-- I would put the Soviet Union right about there. So this was the-- I would put the USSR right over there. I would put the current state of Russia, actually someplace over here. Because they actually have fewer safety nets, and they kind of have a more-- their economy can kind of go crazier, and they actually have a bigger disparity in wealth than a place like the United States. So this is current Russia. And probably the most interesting one here is the People's Republic of China, the current People's Republic of China, which is at least on the surface, a communist state. But in some ways, it's more capitalist than the United States, in that they don't have strong wealth redistribution. They don't have kind of strong safety nets for people. So you could put some elements of China-- and over here, closer to the left. And they are more-- less democratic than either the US or even current Russia, although some people would call current Russia-- well, I won't go too much into it. But current China, you could throw it here a little bit. So it could be even a little bit more capitalist than the United States. Definitely they don't even have good labor laws, all the rest. But in other ways, you do have state ownership of a lot, and you do have state control of a lot. So in some ways, they're kind of spanning this whole range. So this right over here is China. And even though it is called a communist state, in some ways, it's more capitalist than countries that are very proud of their capitalism. But in a lot of other ways, especially with the government ownership and the government control of things, and this one dominant party, so it's kind of Leninist with less of the Marxist going on. So in that way, it is more in the communist direction. So hopefully that clarifies what can sometimes be a confusing topic." + }, + { + "Q": "This video series jumps straight in to exercise examples. Is there a video somewhere on introducing the concepts of dependent and independent variables? If not, explain it generically...?", + "A": "The difference is that the independent can easily be found without doing much work, but the dependent variable can only be found by using the independent, so the independent can work alone, but the dependent has to be found by using the other which is why its called the dependent variable, because its depending on the other.", + "video_name": "0eWm-LY23W0", + "transcript": "On your math quiz, you earn 5 points for each question that you answer correctly. In the table below, x represents the number of questions that you answer correctly, and y represents the total number of points that you score on your quiz. Fair enough? The relationship between these two variables can be expressed by the following equation-- y is equal to 5x. Graph the equation below. So you could look at a couple of your points. You could say, well, look, if I got 0 questions right I'm going to have 0 points. So you could literally graph that point-- if I got 0 questions right, I get 0 points. And then if I get one question right, and the table tells us that, or we could logically think about it, every question I would get right I'm going to get 5 more points. So if I get one question right, I'm going to get 5 more points, and we saw that in our table as well. We saw that right over here-- one question, 5 points. We could also plot it two questions, 10 points, But you only have to do two points to define a line. So it looks like we are actually done here." + }, + { + "Q": "I don't understand the graph that he made at 2:35. What does it mean that the ions like to keep the cell at certain potentials? Does that mean that their gated pathways open up at these potentials? Are the potentials just for the individual ions or the sum of the positive and negative charge differences between the intracellular ad extra-cellular environment?", + "A": "The way I understand it is that those voltages are Resting Potentials for each type of Ion. In other words, each type will want to come into or out of the cell until the cell reaches a specific voltage. If the voltage happens to be at the Resting Potential voltage of that ion, then that type of ion will stop moving across the membrane (rest). Watching the earlier videos in the series should be very helpful, if you haven t yet watched them", + "video_name": "rIVCuC-Etc0", + "transcript": "Let's figure out how a heart squeezes, exactly. And to do that, we have to actually get down to the cellular level. We have to think about the heart muscle cells. So we call them cardiac myocytes. These are the cells within the heart muscle. And these are the cells that actually do that squeezing. So if you actually were to go with a microscope and look down at one of these cells, it might look a little bit like this, with proteins inside of it. And when it's relaxed, these proteins are all kind of spread apart. And when it's squeezing down, because each cell has to squeeze for the overall heart to squeeze, these proteins look completely different. They're totally overlapped. And that overlapping is really what we call squeezing. So this is a squeezed version of the cell. And the first one was a relaxed version. And the trigger that kind of gets it from squeezing-- and of course actually, I should probably draw that too. The fact that, of course, at some point it has to go back to relaxed to do it again, to beat again. But the trigger for squeezing is calcium. So it's easy to get confused when you're thinking about all this kind of squeezing relaxing all this kind of stuff. But if you just keep your eye on calcium and think about the fact that calcium is the trigger, then you'll never get confused. You'll always be able to kind of find your way in terms of where the heart is in its cycle. So I'm going to draw for you the heart cycle, and specifically the cycle of an individual cell. This is what one cell is going to kind of go through over time. And the heart cycle, or the cycle for a cell, a heart cell, is going to be measured in millivolts. We're going to use millivolts to think about this. And you could use, I guess, a lot of different things. But this is probably one of the simplest things to kind of summarize what's happening with all of the different ions that are moving back and forth across that cell. Now the major ions, the ones that are going to mostly influence our heart cell, are going to be calcium, sodium, and potassium. So I'll put those three on here. And I'm putting them really just as benchmarks just so you can kind of keep track of where things would like to be. So calcium would like to be at 123 millivolts. Sodium at 67. And what that means is that if these were the only ions moving through, then sodium would like to keep things positive. And potassium, on the other hand, would like to make the membrane potential negative. So this scale is actually the scale for the membrane potential. And if we move up the scale, if we go from negative to something positive, this process would be called depolarization. That just means going from some negative number up towards something positive. And if you were to do the reverse, if you were going to from something positive to something negative, you'd call that repolarization. So these are just a couple of terms I wanted to make sure that we're familiar with, because we're going to be able to then get at some of the interesting things that happen. I'm going to make some space here inside of this cell. So let's start with a little picture of the cell. So let's say that this is our cell here. And I'm going to draw in little gap junctions, which are little connections between cells. So maybe a couple there. Maybe one there and maybe one over here. And let me label that. So these are the gap junctions. And also let's draw in some channels. So we have, let's say, a potassium channel right here. We know the potassium likes to leave cells. So this is going to be the way that potassium's going to flow. And it's going to leave behind a negative membrane potential, And let's say potassium is the main ion for this cell, which it is. Then our membrane potential is going to be really, really negative. In fact, if it was the only ion, it would be negative 92. But it's not. It's actually just the dominant ion. So it's over here and our membrane potential is around negative 90. And it continues around negative 90. So let's say nothing changes over a bit of time. So we stay at negative 90. So this is what things look like with the dominant ion that our cell is permeable to being potassium. Now a neighboring cell, let's say now, has a little bit of a depolarization. So it goes positive and through the gap junctions leak a little bit of sodium and some calcium. So this stuff starts leaking through the gap junctions, right? Now what will happened to our membrane potential? Well it was negative 90, but now that we've got some positive ions sitting inside of our cell, our cell becomes a little bit more positive, right? So it goes up to, let's say, here. And it happens pretty quickly. So now it's at negative 70 up from negative 90. So at this point, you actually get-- I'm going to erase gap functions-- but now that you're at negative 70, you actually get new channels opening up. And I haven't drawn them yet, and I'm going to erase sodium and calcium just to make some space. But you get new channels opening up. And these are going to be the sodium channels. So let me draw those in. Sodium channels. And there's so many of them. Lots and lots of these fast sodium channels open up. And I say fast because the sodium can flow through very quickly. So the sodium starts gushing in. And you know that's going to happen because there's a lot more sodium on the outside of a cell than the inside of a cell. And so sodium gushes in, and it's going to drive the membrane potential very quickly up to a very positive range. Now it would go all the way, let's say close to 67, maybe not exactly 67, because you still have those potassium ions leaving. But close to it, if not for the fact that these voltage-gated channels actually close down. So these sodium channels are voltage-gated. And they will actually close down just as quickly as they opened up. To show that, I'm actually going to do a little cut paste. I'm going to just draw this cell here. And I'm going to move it down here. So we've got our cell just as before. And now these voltage-gated channels, they close down. So let me get rid of all these arrow heads. But we're already now in positive range. So at this point, you could say our channels have caused a depolarization. And let me just quickly show these shut downs so that you don't get confused. There's no more sodium flowing through. You still have some potassium leaking out, but that's kind of as it's always been. And in addition to those potassium channels, that little channel I've drawn here, you have new potassium channels that open up down here. And these are actually voltage-gated potassium So you had them before. They existed. But they were actually not open. So let me just draw little x's. And the only reason they flipped open is because the depolarization happened. You had a negative go to a positive. So now that our cell is in positive territory, actually let me write in positive 20 or so, our potassium voltage-gated channels open up. So these voltage-gated channels open up. And you can guess what's going to happen. Like which direction do you think that the membrane potential will go? Well, if the sodium channels aren't gushing the sodium inwards and potassium is leaking outwards, now you're going to have a downwards repolarization. So now potassium is causing the membrane potential to go back down. And let's say it gets to about positive 5. And if it continued, again, it would go all the way back down to negative 90. But an interesting new development occurs. At this point, I'm going to actually cut paste again. And I'll show you what happens next, which is that calcium-- this is the thing I said keep your eye on the whole time, right?-- calcium finally kind of starts leaking in. So let me get rid of this. And this is the key idea, right? I don't want to forget that this is potassium. So you still have potassium in the same over here. But now calcium leaks in. And let's draw that over here. So you have these calcium voltage-gated channel that allow calcium to come it. So you've got calcium coming in, potassium leaving. Now think about what will happen in this situation. So calcium is going to want to rise the membrane potential this way. Potassium leaving is going to want it to continue going down this way. And because both are happening simultaneously, you basically get something like this. You get kind of a flatline. So because both events are happening, both potassium leaving the cell and calcium entering the cell, you get this kind of flatline. And the membrane potential stays kind of around the same. And so it can just write something similar, something like positive 5. Just so we're clear, these are also voltage-gated calcium channels. So to round this out, then what happens after that? So you have so far, so good. We have all these channels coming in to our cells and allowing different ions passage. And now we get to something like this. And I'm going to try to clean this up a little bit. And what happens is that the calcium channels actually close just as suddenly as they opened. So now you don't have any more calcium coming in. And if calcium was the only thing that was keeping this membrane potential going flat-- you know, I said that the potassium makes it want to go down, but the calcium was making it flat-- well, what will happen now? Well, if again you have just those potassium channels open, well then you're going to have the membrane potential go back down. It's going to go back down to negative 90 or so. So this is kind of the last stage, where those potassium channels are going back down. And those voltage-gated potassium channels also close at this point. So finally, they close down as well. And so now that they're closed, you're going to finally get back to just your initial state, which was having a little bit of potassium kind of leaking out of this cell. And those voltage-gated channel have shut down now. So now that you're at negative 90, you stay down there. And this process is ready to begin again. The last thing I want to say is the stages, how they're named. So this is state four, this kind of baseline negative state that the relaxed muscle cell is. And then this action potential, when it finally fires and it hits that negative 70, this is actually considered a threshold. This is our threshold. When it gets to that point, we call that stage 0. And then on the other side of stage 0 you have stage 1, 2, and 3. So stage 1 is that point when just the potassium channels first open up, the voltage-gated ones. And then stage 2 is when they're balanced with the calcium And stage 3 is again when you have just potassium channels, voltage-gated ones that are open. And then you get back to stage 4 again. So this would be stage 4. And because stage 0 is happening so rapidly, because this is so fast, we actually call this a fast action potential. So compare that to how the action potential goes in the pacemaker cells, where it's much slower. This fast action potential is a result of those really, really amazingly quick voltage-gated sodium channels." + }, + { + "Q": "how do you say it actully?", + "A": "For example, 5.5 is 5 and 5 tenths.", + "video_name": "qSPwUDmpnJ4", + "transcript": "- [Voiceover] Let's say that I had the number zero point one seven. How could I say this number? I said it one way, I said zero point one seven, but what are other ways that I could say it, especially if I wanted to express it in terms of tenths or hundredths or other places? And like always, try to pause the video and try think about it on your own. Alright, so there's actually a couple of ways that we could say this number. One is just to say zero point one seven. Other ways are to say look, I have a one in the tenths place, so that's going to be one tenth, one tenth and one tenth and I have a seven in the hundredths place, so this is a seven right over here in the hundredths place, so I can say one tenth and seven hundredths. Hun- Hundredths. And there you go. That's one way to say this number. Now another to think about it is just say the whole thing in terms of hundredths. So a tenth is how many hundredths? Well a tenth is the same thing as 10 hundredths, so you could say, you could say instead of a tenth, you could say this is 10 hundredths, and the way I'm writing it right now, very few people would actually do it this way. 10 hundredths and and seven hundredths. And seven hundredths. Well not I could just add these hundredths, if I have 10 hundredths and I have another seven hundredths, that's going to be 17 hundredths. So I could just write this down as 17 hundredths. Hundredths. And to make that intuition of how we could just call this 17 hundredths instead of just calling it one tenth and seven hundredths, let's actually count by hundredths. So that is one hundredth, that is two hundredth, and actually, let me just go straight to nine hundredths. So I skipped a bunch right over here. And what would be the next, how would I say 10 hundredths? Well 10 hundredths, let me write it this way, 10 hundredths is the same thing as one tenth. So if we go from nine hundredths, the next, if I'm counting by hundredths, the next one's going to be 10 hundredths. Now once again, 10 hundredths is the same thing as one tenth, just the same way that 10 ones is the same thing as one 10. I hope that doesn't confuse you, but we could keep counting. 10 hundredths, 11 hundredths, 12 hundredths, 13 hundredths, 14 hundredths, 15 hundredths, 16 hundredths, and then finally 17 hundredths. So hopefully that gives you a little intuition for why we can call this number, instead of just calling it zero point one seven, or one tenth and seven hundredths, we could call this 17 hundredths. So with that out of the way, let's do a couple of examples going the other way. Let's say we're given a name of a number or the words, and we wanna write it down as a decimal. So this is four tens and three hundredths. Alright, so four tens. So the four tens right over here. So actually let me just put some places over here, so this would be, if this is our tens place, and then this is our ones place, and then you're gonna have your decimal, and then you're gonna have your tenths place, tenths place, and then you're going to have, and I'll do this in a different color, your hundredths place. Hun- Hundredths place. So we're going to have four tens. Not tenths, four tens. So four tens, zero ones, zero ones, we got our decimal, they don't have any tenths over here so zero tenths, and then we have three hundredths. Three hundredths, so three hundredths. So four tens, which is the same thing as 40, and three hundredths right over here, so 40 point zero three. Let's do another one of these, this is kinda fun. So we have 24 hundredths. 24 hundredths. So by the logic that we saw in the first one, in the first one, we could just write this as, remember, this would be nine hundredths, and if we want one more hundredth, this would be 10 hundredths. So if we want to do 24 hundredths, it would just be zero point two four. And if you're saying well wait a minute, this looks like two tenths and four hundredths, you'd be right! But remember, so actually let me, I could re-write this as, I could re-write this as two tenths, two tenths and four hundredths. Let me say and four hundredths. Hundredths. But remember, a tenth is equal to 10 hundredths, so two tenths is the same thing as 20 hundredths. Hundredths. I have trouble saying- So this is the same thing as 20 hundredths and four hundredths. And four, let me write it neatly. Four hundredths. And of course 20 hundredths and four hundredths is the same thing as 24 hundredths. So hopefully that makes sense." + }, + { + "Q": "Is half a circle Pi?", + "A": "Half of the circumference of a circle isn t pi, unless the radius is 1. (2pi*r = 1) The angle formed by going halfway around the unit circle, however, is pi radians, or 180 degrees.", + "video_name": "1jDDfkKKgmc", + "transcript": "What I want to do in this video is revisit a little bit of what we know about pi, and really how we measure angles in radians. And then think about whether pi is necessarily the best number to be paying attention to. So let's think a little bit about what I just said. So pi, we know, is defined-- and I'll write defined as a triple equal sign, I guess you could call it that way-- pi is defined as the ratio of the circumference of a circle to its diameter, which is the same thing as the ratio of the circumference of the circle to two times the radius. And from that, we get all these interesting formulas that you get in geometry class that, hey, if you have the radius and you want to calculate the circumference, multiply both sides of this definition, or this equation, by two times the radius. And you get two times the radius times pi is equal to the circumference, or more familiarly, it would be circumference is equal to 2 pi r. This is one of those fundamental things that you learn early on in your career and you use it to find circumferences, usually, or figure out radiuses if you know circumference. And from that comes how we measure our angles in radians once we get to trigonometry class. And just as a review here, let me draw myself a circle. Let me draw myself a better circle. So there is my-- it'll do the job. And here is the positive x-axis, and let me make some angle here. I'll make the angle kind of obvious just so that it-- so let me make this angle. And the way that we measure angles, when we talk about radians, we're really talking about the angle subtended by something of a certain arc length. And we measure the arc length in-- well, the way I like to think about it, is the angle is in radians and the arc length itself is in radiuses, which isn't really a word. But that's how I think about it. How many radiuses is this arc length that subtends the angle in radians? So let me show you what I'm talking about. So this arc length right here, if the radius is r, what is the length of this arc length? Well, we know from basic geometry the entire circumference over here is going to be 2 pi r, right? This entire circumference, that's really by definition, this entire circumference is going to be 2 pi r. So what is just this arc length here? And I'm assuming this is a fourth of the circle. So it's going to be 2 pi r over 4. So this arc length over here is going to be 2 pi r over 4, which is the same thing as pi over 2 r. Or you could say this is the same thing as pi over 2 radiuses. One of those-- you know, not a real word, but that's how I like to think about it. Or you could say it subtends an angle of pi over 2 radians. So over here, theta is pi over 2 radians. And so really, when you're measuring angles in radians, it's really you're saying, OK, that angle subtended by an arc that has a length of how many radiusi, or I don't even know what the plural of radius is. Actually, I think it's radii but it's fun to try to say radiuses. Radii, actually let me do that just so no one says, Sal, you're teaching people the wrong plural form of radius. Radii. So this arc length is pi over 2 radii and it subtends an angle of pi over 2 radians. We could do another one just for the sake of making the point clear. If you went all the way around the circle and you got back to the positive x-axis here, what is the arc length? Well now, all of a sudden the arc length is the entire circumference of the circle. It would be 2 pi r, which is the same thing as 2 pi radii. And we would say that the angle subtended by this arc length, the angle that we care about going all the way around the circle, is 2 pi radians. And so, out of this comes all of the things that we know about how to graph trigonometric functions or at least how we measure the graph on the x-axis. And I'll also touch on Euler's formula, which is the most beautiful formula, I think, in all of mathematics. And let's visit those right now, just to remind ourselves of how pi fits in to all of that. So if I think about our trigonometric functions. Remember, if this was-- so on trigonometric functions, we assume we have a unit circle here. So in the trig functions, this is the unit circle definition of the trig function. So this is a nice review of all of that. You assume you have a unit circle, a circle of radius 1. And then the trig functions are defined as, for any angle you have here, for any angle, theta, cosine of theta is how far you have to move in-- or the x-coordinate of the point along the arc that subtends this angle. So that's cosine of theta. And then sine of theta is the y value of that point. Let me make that clear. Cosine of theta is the x value, sine of theta is the y value. And so if you were to graph one of these functions, and I'll just do sine of theta for convenience. But you could try it with cosine of theta. So let's graph sine of theta. Let's do one revolution of sine of theta. And we tend to label it-- so let's do sine. When the angle is 0, sine of theta is 0. Let me draw the x- and y-axis just so you remember. This is the y-axis and this is the x-axis. So when the angle is 0, we're right here on the unit circle. The y value there is 0. So sine of theta is going to be right like that. Let me draw it like this. So this is our theta, and this is-- I'm going to graph sine of theta along the y-axis. So we'll say y is equal to sine of theta in this graph that I'm drawing right over here. And then we could do-- well, I'll just do the simple points here. And then if we make the angle go-- if we did it in degrees, 90 degrees, or if we do it in radians, pi over 2 radians. What is sine of theta? Well, now it is 1. This is the unit circle. It has a radius 1. So when theta is equal to pi over 2, then sine of theta is equal to 1. So if this is 1 right here, sine of theta is equal to 1. If theta-- and then if we go 180 degrees or halfway around the circle-- theta is now equal to pi. Let me do this in a color, orange. I have already used orange. Theta is now equal to pi. When theta is equal to pi, the y value of this point right here is once again 0. So we go back to 0. Remember, we're talking about sine of theta. And then we can go all the way down here, where you can see there's 270 degrees. Or you could view this as 3 pi over 2 radians. So this is in radians, this axis. So 3 pi over 2 radians, sine of theta is the y-coordinate on the unit circle right over here. So it's going to be negative 1. So this is negative 1. And then finally, when you go all the way around the circle, you've gone 2 pi radians, and you're back where you began. And the sine of theta, or the y-coordinate, is now 0 once again. And if you connect the dots, or if you'd plotted more points, you would see a sine curve over just the part that we've graphed right over here. So that's another application. You say hey, Sal, where is this going? Well I'm showing you-- I'm reminding you of all of these things because we're going to revisit it with a different number other than pi. And so I want to do one last visit with pi. You say, look, pi is powerful because-- or one of the reasons why pi seems to have some type of mystical power, and we've shown this in the calculus playlist-- is Euler's formula, that e to the i theta is equal to cosine of theta plus i sine of theta. This, by itself, is just a crazy-- it's just one of those mind boggling formulas. But it sometimes looks even more mind boggling when you put pi in for theta because then, from Euler's formula, you would get e to the i pi is equal to-- well, what's cosine of pi? Cosine of pi is negative 1. And then sine of pi is 0. So 0 times i. So you get this formula, which is pretty profound, and then you say, OK, if I want to put all of the fundamental numbers together in one formula, I can add 1 to both sides of this. And you get e to the i pi plus 1 is equal to 0. Sometimes this is called Euler's Identity, the most beautiful formula or equation in all of mathematics. And it is pretty profound. You have all of the fundamental numbers in one equation. e, i, pi, 1, 0, although for my aesthetic taste it would have been even more powerful if this was a 1 right over here. Because then this would have said, look, e to the i pi, this bizarre thing, would have equaled unity. That would have been super duper profound to me. It seems a little bit of a hack to add 1 to both sides and say, oh, look, now I have 0 here. But this is pretty darn good. But with that, I'm going to make-- well, I'm not going to argue for it. I'm going to show an argument for another number, a number different than pi. And I want to make it clear that these ideas are not my own. It comes from-- well, it's inspired by-- many people are on this movement now, the Tau Movement, but these are kind of the people that gave me the thinking on this. And the first is Robert Palais on \"Pi is Wrong.\" And he doesn't argue that pi is calculated wrong. He still agrees that it is the ratio of the circumference to the diameter of a circle that is 3.14159. But what he's saying is that we're paying attention to the wrong number. And also, you have Michael Hartl, \"The Tau Manifesto.\" All of this is available online. And what they argue for is a number called tau, or what they call tau. And they define tau, and it's a very simple change from pi. They define tau not as the ratio of the circumference the diameter, or the ratio of the circumference to 2 times the radius. They say, hey, wouldn't it be natural to define some number, the ratio of the circumference to the radius? And as you see here, this pi is just one half times this over here, right? Circumference over 2 r, this the same thing as one half times circumference over r. So pi is just half of tau. Or another way to think about it is that tau is just 2 times pi. Or, and I'm sure you probably don't have this memorized, because you're like, wait, I spent all my life memorizing pi, but it's 6.283185 and keeps going on and on and on, never repeating just like pi. It's 2 times pi. And so you're saying, hey Sal, pi has been around for millennia, really. Why mess with such a fundamental number, especially when you just spent all this time showing how profound it is? And the argument that they'd make, and it seems like a pretty good argument, is that actually things seem a little bit more elegant when you pay attention to this number instead of half of this number, when you pay attention to tau. And to see that, let's revisit everything that we did here. Now, all of a sudden, if you pay attention to 2 pi, as opposed to pi. Or we should call it-- if you pay attention to tau instead of tau over 2. What is this angle that we did in magenta? Well first of all, let's think about this formula right What is the circumference in terms of the radius? Well, now we could say the circumference is equal to tau times the radius because tau is the same thing as 2 pi. So it makes that formula a little bit neater, although it does make the pi r squared a little bit messier. So you could argue both sides of that. But it makes the measure of radians much more intuitive because you could say that this is pi over 2 radians, or you could say that this is pi over 2 radians is the same thing as to tau over 4 radians. And where did I get that from? Remember, if you go all the way around the circle, that is the circumference. The arc length would be the circumference. It would be tau radii, or it would be to tau radians would be the angle subtended by that arc length. It would be tau radians. All the way around is tau radians. So that by itself is intuitive. One revolution is one tau radians. If you go only one fourth of that, it's going to be tau over four radians. So the reason why tau is more intuitive here is because it immediately-- you don't have to do this weird conversion where you saying, oh, divide by 2, multiply by 2, all that. You're just like, look, however many radians in terms of tau, that's really how many revolutions you've gone around the circle. And so, if you've gone one fourth around, that's tau over four radians. If you've gone halfway around, that'd be tau over two radians. If you go 3/4 around that'd be three tau over four radians. If you go all the way around that would be tau radians. If someone tells you that they have an angle of 10 tau radians, you'd go around exactly 10 times. It would be much more intuitive. You wouldn't have to do this little mental math, converting, saying, do I multiply or divide by 2 when I convert to radians in terms of pi? No, when you do it in terms of tau radians, it's just natural. One revolution is one tau radians. So that makes-- and it makes a sine function over here. Instead of writing pi over 2-- well, when you look at a graph like this you're like, where was this on the unit circle? Was this one fourth around the circle? Was this one half? And this is actually one fourth around the circle, right over here? But now it becomes obvious if you write it in tau. Pi over 2 is the same thing as to tau over 4. Pi is the same thing as to tau over 2. 3 pi over 2 is 3 pi-- oh, sorry, 3 tau over 4, 3/4 tau. And then one revolution is tau. And then immediately, now when you look at it this way, you know exactly where you are in the unit circle. You're one fourth around the unit circle, you're halfway around the unit circle, you're 3/4 of the way around the unit circle. And then you're all the way around the unit circle. And so the last thing that I think the strong pi defenders would say is well, look Sal, you just pointed out one of the most beautiful identities or formulas in mathematics. How does tau hold up to this? Well, let's just try it out and see what happens. So if we take e to the i tau, that will give us cosine of tau plus i sine of tau. And once again, let's just think about what this is. Tau radians means we've gone all the way around the unit circle. So cosine of tau-- remember, we're back at the beginning of the unit circle right over here-- so cosine of tau is going to be equal to 1. And then sine of tau is equal to 0. So e to the i tau is equal to 1. And I'll leave it up to you to decide which one seems to be more aesthetically profound." + }, + { + "Q": "Hi friends :-) I have a question for you-\n1> at 6:27 \"Sal\" proved us that alternate interior angles are equal , right? so, here is my question ?\nwe know that :- b=c,f=g\nnow b and c are vertical angles and we also know that b = f (corresponding angle) so why can't we say that c = f ,right? and why not alternate exterior angles?", + "A": "Ok, so you said b=c,f=g. Which says b=c and f=g. If you look back at Sals blackboard on the video there is no comma, it simply says b=c=f=g which means they are ALL equal, so yes , c does equal f and c equals g, and b equals g etc. etc.. As for alternate exterior angles, the same rule applies. a=d=e=h. They are all equal.", + "video_name": "H-E5rlpCVu4", + "transcript": "Let's say we have two lines over here. Let's call this line right over here line AB. So A and B both sit on this line. And let's say we have this other line over here. We'll call this line CD. So it goes through point C and it goes through point D. And it just keeps on going forever. And let's say that these lines both sit on the same plane. And in this case, the plane is our screen, or this little piece of paper that we're looking at right over here. And they never intersect. So they're on the same plane, but they never intersect each other. If those two things are true, and when they're not the same line, they never intersect and they can be on the same plane, then we say that these lines are parallel. They're moving in the same general direction, in fact, the exact same general direction. If we were looking at it from an algebraic point of view, we would say that they have the same slope, but they have different y-intercepts. They involve different points. If we drew our coordinate axes here, they would intersect that at a different point, but they would have the same exact slope. And what I want to do is think about how angles relate to parallel lines. So right over here, we have these two parallel lines. We can say that line AB is parallel to line CD. Sometimes you'll see it specified on geometric drawings like this. They'll put a little arrow here to show that these two lines are parallel. And if you've already used the single arrow, they might put a double arrow to show that this line is parallel to that line right over there. Now with that out of the way, what I want to do is draw a line that intersects both of these parallel lines. So here's a line that intersects both of them. Let me draw a little bit neater than that. So let me draw that line right over there. Well, actually, I'll do some points over here. Well, I'll just call that line l. And this line that intersects both of these parallel lines, we call that a transversal. This is a transversal line. It is transversing both of these parallel lines. This is a transversal. And what I want to think about is the angles that are formed, and how they relate to each other. The angles that are formed at the intersection between this transversal line and the two parallel lines. So we could, first of all, start off with this angle right over here. And we could call that angle-- well, if we made some labels here, that would be D, this point, and then something else. But I'll just call it this angle right over here. We know that that's going to be equal to its vertical angles. So this angle is vertical with that one. So it's going to be equal to that angle right over there. We also know that this angle, right over here, is going to be equal to its vertical angle, or the angle that is opposite the intersection. So it's going to be equal to that. And sometimes you'll see it specified like this, where you'll see a double angle mark like that. Or sometimes you'll see someone write this to show that these two are equal and these two are equal right over here. Now the other thing we know is we could do the exact same exercise up here, that these two are going to be equal to each other and these two are going to be equal to each other. They're all vertical angles. What's interesting here is thinking about the relationship between that angle right over there, and this angle right up over here. And if you just look at it, it is actually obvious what that relationship is-- that they are going to be the same exact angle, that if you put a protractor here and measured it, you would get the exact same measure up here. And if I drew parallel lines-- maybe I'll draw it straight left and right, it might be a little bit more obvious. So if I assume that these two lines are parallel, and I have a transversal here, what I'm saying is that this angle is going to be the exact same measure as that angle there. And to visualize that, just imagine tilting this line. And as you take different-- so it looks like it's the case over there. If you take the line like this and you look at it over here, it's clear that this is equal to this. And there's actually no proof for this. This is one of those things that a mathematician would say is intuitively obvious, that if you look at it, as you tilt this line, you would say that these angles are the same. Or think about putting a protractor here to actually measure these angles. If you put a protractor here, you'd have one side of the angle at the zero degree, and the other side would specify that point. And if you put the protractor over here, the exact same thing would happen. One side would be on this parallel line, and the other side would point at the exact same point. So given that, we know that not only is this side equivalent to this side, it is also equivalent to this side over here. And that tells us that that's also equivalent to that side over there. So all of these things in green are equivalent. And by the same exact argument, this angle is going to have the same measure as this angle. And that's going to be the same as this angle, because they are opposite, or they're vertical angles. Now the important thing to realize is just what we've deduced here. The vertical angles are equal and the corresponding angles at the same points of intersection are also equal. And so that's a new word that I'm introducing right over here. This angle and this angle are corresponding. They represent kind of the top right corner, in this example, of where we intersected. Here they represent still, I guess, the top or the top right corner of the intersection. This would be the top left corner. They're always going to be equal, corresponding angles. And once again, really, it's, I guess, for lack of a better word, it is a bit obvious. Now on top of that, there are other words that people will see. We've essentially just proven that not only is this angle equivalent to this angle, but it's also equivalent to this angle right over here. And these two angles-- let me label them so that we can make some headway here. So I'm going to use lowercase letters for the angles themselves. So let's call this lowercase a, lowercase b, lowercase c. So lowercase c for the angle, lowercase d, and then let me call this e, f, g, h. So we know from vertical angles that b is equal to c. But we also know that b is equal to f because they are corresponding angles. And that f is equal to g. So vertical angles are equivalent, corresponding angles are equivalent, and so we also know, obviously, that b is equal to g. And so we say that alternate interior angles are equivalent. So you see that they're kind of on the interior of the intersection. They're between the two lines, but they're on all opposite sides of the transversal. Now you don't have to know that fancy word, alternate interior angles, you really just have to deduce what we just saw over here. Know that vertical angles are going to be equal and corresponding angles are going to be equal. And you see it with the other ones, too. We know that a is going to be equal to d, which is going to be equal to h, which is going to be equal to e." + }, + { + "Q": "At 4:14 where did you get the plus sign from? Wouldn't you put a subtraction sign there? (because the previous color coded terms had a subtraction sign at the beginning?)", + "A": "At that point in the video, Sal is combining: -2x^2 + 3x^2. He does this by adding/subtracting the coefficients: -2 + 3 = +1, not a -1. So, once combined, the 2 terms creates +x^2. Hope this helps.", + "video_name": "FNnmseBlvaY", + "transcript": "Now we have a very, very, very hairy expression. And once again, I'm going to see if you can simplify this. And I'll give you little time to do it. So this one is even crazier than the last few we've looked at. We've got y's and xy's, and x squared and x's, well more just xy's and y squared and on and on and on. And there will be a temptation, because you see a y here and a y here to say, oh, maybe I can add this negative 3y plus this 4xy somehow since I see a y and a y. But the important thing to realize here is that a y is different than an xy. Think about it they were numbers. If y was 3 and an x was a 2, then a y would be a 3 while an xy would have been a 6. And a y is very different than a y squared. Once again, if the why it took on the value 3, then the y squared would be the value 9. So even though you see the same letter here, they aren't the same-- I guess you cannot add these two or subtract these two terms. A y is different than a y squared, is different than an xy. Now with that said, let's see if there is anything that we can simplify. So first, let's think about the y terms. So you have a negative 3y there. Do we have any more y term? We have this 2y right over there. So I'll just write it out-- I'll just reorder it. So we have negative 3y plus 2y. Now, let's think about-- and I'm just going in an arbitrary order, but since our next term is an xy term-- let's think about all of the xy terms. So we have plus 4xy right over here. So let me just write it down-- I'm just reordering the whole expression-- plus 4xy. And then I have minus 4xy right over here. Then let's go to the x squared terms. I have negative 2 times x squared, or minus 2x squared. So let's look at this. So I have minus 2x squared. Do I have any other x squared? Yes, I do. I have this 3x squared right over there. So plus 3x squared. And then let's see, I have an x term right over here, and that actually looks like the only x term. So that's plus 2x. And then I only have one y squared term-- I'll circle that in orange-- so plus y squared. So all I have done is I've reordered the statement and I've color coded it based on the type of term we have. And now it should be a little bit simpler. So let's try it out. If I have negative 3 of something plus 2 of that something, what do I have? Or another way to say it, if I have two of something and I subtract 3 of that, what am I left with? Well, I'm left with negative 1 of that something. So I could write negative 1y, or I could just write negative y. And another way you could think about it, but I like to think about it intuitively more, is what's the coefficient here? It is negative 3. What's the coefficient here? It's 2. Where obviously both are dealing-- they're both y terms, not xy terms, not y squared terms, just y. And so negative 3 plus 2 is negative 1, or negative 1y is the same thing as negative y. So those simplify to this right over here. Now let's look at the xy terms. If I have 4 of this, 4 xy's and I were to take away 4 xy's, how many xy's am I left with? Well, I'm left with no xy's. Or you could say add the coefficients, 4 plus negative 4, gives you 0 xy's. Either way, these two cancel out. If I have 4 of something and I take away those 4 of that something, I'm left with none of them. And so I'm left with no xy's. And then I have right over here-- I could have written 0xy, but that seems unnecessary-- then right over here I have my x squared terms. Negative 2 plus 3 is 1. Or another way of saying it, if I have 3x squared and I were to take away 2 of those x squared, so I'm left with the 1x squared. So this right over here simplifies to 1x squared. Or I could literally just write x squared. 1x squared is the same thing as x squared. So plus x squared, and then these there's nothing really left to simplify. So plus 2x plus y squared. And obviously you might have gotten an answer in some other order, but the order in which I write these terms don't matter. It just matters that you were able to simplify it to these four terms." + }, + { + "Q": "At, 8:40 - 8:42, there is repulsion between the lone pairs & the bond pairs but what about the lone pairs? Don't they repel themselves too?", + "A": "Of course they do. And they repel one another the most.", + "video_name": "BM-My1AheLw", + "transcript": "Voiceover: The concept of steric number is very useful, because it tells us the number of hybridized orbitals that we have. So to find the steric number, you add up the number of stigma bonds, or single-bonds, and to that, you add the number of lone pairs of electrons. So, let's go ahead and do it for methane. So, if I wanted to find the steric number, the steric number is equal to the number of sigma bonds, so I look around my carbon here, and I see one, two, three, and four sigma, or single-bonds. So I have four sigma bonds; I have zero lone pairs of electrons around that carbon, so four plus zero gives me a steric number of four. In the last video, we saw that SP three hybridized situation, we get four hybrid orbitals, and that's how many we need, the steric number tells us we need four hybridized orbitals, so we took one S orbital, and three P orbitals, and that gave us four, SP three hybrid orbitals, so this carbon must be SP three hybridized. So let's go ahead, and draw that in here. So this carbon is SP three hybridized, and in the last video, we also drew everything out, so we drew in those four, SP three hybrid orbitals, for that carbon, and we had one valence electron in each of those four, SP three hybrid orbitals, and then hydrogen had one valence electron, in an un-hybridized S orbital, so we drew in our hydrogens, and the one valence electron, like that. This head-on overlap; this is, of course, a sigma bond, so we talked about this in the last video. And so now that we have this picture of the methane molecule, we can think about these electron pairs, so these electron pairs are going to repel each other: like charges repel. And so, the idea of the VSEPR theory, tell us these electron pairs are going to repel, and try to get as far away from each other as they possibly can, in space. And this means that the arrangement of those electron pairs, ends up being tetrahedral. So let's go ahead and write that. So we have a tetrahedral arrangement of electron pairs around our carbon, like that. When we think about the molecular geometry, so that's like electron group geometry, you wanna think about the geometry of the entire molecule. I could think about drawing in those electrons, those bonding electrons, like that. So we have a wedge coming out at us in space, a dash going away from us in space, and then, these lines mean, \"in the plane of the page.\" And so, we can go ahead a draw in our hydrogens, and this is just one way to represent the methane molecule, which attempts to show the geometry of the entire molecule. So the arrangement of the atoms turns out to also be tetrahedral, so let's go ahead and write that. So, tetrahedral. And, let's see if we can see that four-sided figure, so a tetrahedron is a four-sided figure, so we can think about this being one face, and then let's go ahead and draw a second face. And if I draw a line back here, that gives us four faces to our tetrahedron. So our electron group geometry is tetrahedral, the molecular geometry of methane is tetrahedral, and then we also have a bond angle, let me go ahead and draw that in, so a bond angle, this hydrogen-carbon-hydrogen bond angle in here, is approximately 109 point five degrees. All right, let's go ahead and do the same type of analysis for a different molecule, here. So let's do it for ammonia, next. So we have NH three, if I want to find the steric number, the steric number is equal to the number of sigma bonds, so that's one, two, three; so three sigma bonds. Plus number of lone pairs of electrons, so I have one lone pair of electrons here, so three plus one gives me a steric number of four. So I need four hybridized orbitals, and once again, when I need four hybridized orbitals, I know that this nitrogen must be SP three hybridized, because SP three hybridization gives us four hybrid orbitals, and so let's go ahead and draw those four hybrid orbitals. So we would have nitrogen, and let's go ahead and draw in all four of those. So, one, two, three, and four; those are the four hybrid orbitals. When you're drawing the dot structure for nitrogen, you would have one electron, another electron, another electron, and then you'd have two in this one, like that. And then you'd go ahead, and put in your hydrogens, so, once again, each hydrogen has one electron, in a hybridized S orbital, so we go ahead and draw in those hydrogens, so our overlap of orbitals, so here's a sigma bond, here's a sigma bond, and here's a sigma bond; so three sigma bonds in ammonia, and then we have this lone pair up here. So the arrangement of these electron pairs, is just what we talked about before: So we have this tetrahedral arrangement of electron pairs, or electron groups, so the VSEPR theory tells us that's how they're going to repel. However, that's not the shape of the molecule, so if I go ahead and draw in another picture over here, to talk about the molecular geometry, and go ahead and draw in the bonding electrons, like that, and then I'll put in my non-bonding electrons, up here: this lone pair right here, housed in an SP three hybridized orbital. So, the arrangement of the atoms turns out not to be tetrahedral, and that has to do with this lone pair of electrons up here, at the top. So, this lone pair of electrons is going to repel these bonding electrons more strongly than in our previous example, and because it's going to repel those electrons a little bit more strongly, you're not gonna get a bond angle of 109 point five; it's going to decrease the bond angle. So let me go ahead, and use the same color we used before, so this bond angle is not 109 point five; it goes down a bit, because of the extra repulsion, so it turns out to be approximately 107 degrees. And in terms of the shape of the molecule, we don't say \"tetrahedral\"; we say \"trigonal-pyramidal.\" So let me go ahead, and write that here, so the geometry of the ammonia molecule is trigonal-pyramidal, and let's analyze that a little bit. So, \"trigonal\" refers to the fact that nitrogen is bonded to three atoms here, so nitrogen is bonded to three hydrogens, so that takes care of the trigonal part. The \"pyramidal\" part comes in, because when you're doing molecular geometry, you ignore lone pairs of electrons. So if you ignore that lone pair of electrons, and just look at this nitrogen, at the top of this pyramid right here, so that's where the \"pyramidal\" term comes in. So bonded to three other atoms, like this, this, and this; for our pyramid. So trigonal-pyramidal is the geometry of the ammonia molecule, but the nitrogen is SP three hybridized. All right, let's do one more example; let's do water. So, first we calculate the steric number, so the steric number is equal to the number of sigma bonds, so that's one, two; so, two sigma bonds. Plus numbers of lone pairs of electrons, so here's a lone pair, here's a lone pair; so we have two plus two, which is equal to four, so we need four hybridized orbitals. As we've seen in the previous two examples, when you need four hybridized orbitals, that's an SP three hybridization situation; you have four SP three hybridized orbitals. So this oxygen is SP three hybridized, so I'll go ahead and write that in here, so oxygen is SP three hybridized. So we can draw that out, showing oxygen with its four SP three hybrid orbitals; so there's four of them. So I'm gonna go ahead and draw in all four. In terms of electrons, this orbital gets one, this orbital gets one, and these orbitals are going to get two, like that; so that takes care of oxygen's six valence electrons. When you're drawing in your hyrdogens, so let's go ahead and put in the hydrogen here, so, once again, each hydrogen with one electron, in a un-hybridized S orbital, like that. So in terms of overlap of bonds, here's one sigma bond, and here's another sigma bond; so that's our two sigma bonds for water. Once again, the arrangement of these electron pairs is tetrahedral, so VSEPR theory says the electrons repel, and so the electron group geometry, you could say, is tetrahedral, but that's not the geometry of the entire molecule, 'cause I was just thinking about electron groups, and these hybrid orbitals. The geometry of the molecule is different, so we'll go ahead and draw that over here. So we have our water molecule, and draw in our bonding electrons, and now let's put in our non-bonding electrons, like that, so we have a different situation than with ammonia. With ammonia, we had one lone pair of electrons repelling these bonding electrons up here; for water, we have two lone pairs of electrons repelling these bonding electrons, and so that's going to change the bond angle; it's going to short it even more than in the previous example. So the bond angle decreases, so this bond angle in here decreases to approximately 105 degrees, rounded up a little bit. So, thinking about the molecular geometry, or the shape of the water molecule, so we actually call this \"bent,\" or \"angular,\" so this is, \"bent geometry,\" because you ignore the lone pairs of electrons, and that would just give you this oxygen here, and then this angle; so you could also call this, \"angular.\" So we have this bent molecular geometry, like that, or angular, and once again, for molecular geometry, ignore your lone pairs of electrons. So these are examples of three molecules, and the central atom in all three of these molecules is SP three hybridized, and so, this is one way to figure out your overall molecular geometry, and to think about bond angles, and to think about how those hybrid orbitals affect the structure of these molecules." + }, + { + "Q": "When you divide 13 by 93 on a calculator you get .1397. Did I do something wrong? Sal's answer doing it without a calculator is .138R.", + "A": "I think it is because it is a repeating decimal with about 15 repeating numbers. I did not watch the video but it might mean that it is a repeating decimal and it has rounded to 3 decimal places.", + "video_name": "Llt-KkHugRQ", + "transcript": "Welcome to the presentation on ordering numbers. Let's get started with some problems that I think, as you go through the examples hopefully, you'll understand how to do these problems. So let's see. The first set of numbers that we have to order is 35.7%, 108.1% 0.5, 13/93, and 1 and 7/68. So let's do this problem. The important thing to remember whenever you're doing this type of ordering of numbers is to realize that these are all just different ways to represent-- these are all a percent or a decimal or a fraction or a mixed-- are all just different ways of representing numbers. It's very hard to compare when you just look at it like this, so what I like to do is I like to convert them all to decimals. But there could be someone who likes to convert them all to percentages or convert them all to fractions and then compare. But I always find decimals to be the easiest way to compare. So let's start with this 35.7%. Let's turn this into a decimal. Well, the easiest thing to remember is if you have a percent you just get rid of the percent sign and put it over 100. So 35.7% is the same thing as 35.7/100. Like 5%, that's the same thing as 5/100 or 50% is just the same thing as 50/100. So 35.7/100, well, that just equals 0.357. If this got you a little confused another way to think about percentage points is if I write 35.7%, all you have to do is get rid of the percent sign and move the decimal to the left two spaces and it becomes 0.357. Let me give you a couple of more examples down here. Let's say I had 5%. That is the same thing as 5/100. Or if you do the decimal technique, 5%, you could just move the decimal and you get rid of the percent. And you move the decimal over 1 and 2, and you put a 0 here. It's 0.05. And that's the same thing as 0.05. You also know that 0.05 and 5/100 are the same thing. So let's get back to the problem. I hope that distraction didn't distract you too much. Let me scratch out all this. So 35.7% is equal to 0.357. Similarly, 108.1%. Let's to the technique where we just get rid of the percent and move the decimal space over 1, 2 spaces to the left. So then that equals 1.081. See we already know that this is smaller than this. Well the next one is easy, it's already in decimal form. 0.5 is just going to be equal to 0.5. Now 13/93. To convert a fraction into a decimal we just take the denominator and divide it into the numerator. So let's do that. 93 goes into 13? Well, we know it goes into 13 zero times. So let's add a decimal point here. So how many times does 93 go into 130? Well, it goes into it one time. 1 times 93 is 93. Becomes a 10. That becomes a 2. Then we're going to borrow, so get 37. Bring down a 0. So 93 goes into 370? 4 times 93 would be 372, so it actually goes into it only three times. 3 times 3 is 9. 3 times 9 is 27. So this equals? Let's see, this equals-- if we say that this 0 becomes a 10. This become a 16. This becomes a 2. 81. And then we say, how many times does 93 go into 810? It goes roughly 8 times. And we could actually keep going, but for the sake of comparing these numbers, we've already gotten to a pretty good level of accuracy. So let's just stop this problem here because the decimal numbers could keep going on, but for the sake of comparison I think we've already got a good sense of what this decimal looks like. It's 0.138 and then it'll just keep going. So let's write that down. And then finally, we have this mixed number here. And let me erase some of my work because I don't Actually, let me keep it the way it is right now. The easiest way to convert a mixed number into a decimal is to just say, OK, this is 1 and then some fraction that's less than 1. Or we could convert it to a fraction, an improper fraction like-- oh, actually there are no improper fractions here. Actually, let's do it that way. Let's convert to an improper fraction and then convert that into a decimal. Actually, I think I'm going to need more space, so let me clean up this a little bit. We have a little more space to work with now. So 1 and 7/68. So to go from a mixed number to an improper fraction, what you do is you take the 68 times 1 and add it to the numerator here. And why does this make sense? Because this is the same thing as 1 plus 7/68. 1 and 7/68 is the same thing as 1 plus 7/68. And that's the same thing as you know from the fractions module, as 68/68 plus 7/68. And that's the same thing as 68 plus 7-- 75/68. So 1 and 7/68 is equal to 75/68. And now we convert this to a decimal using the technique we did for 13/93. So we say-- let me get some space. We say 68 goes into 75-- suspicion I'm going to run out of space. 68 goes into 75 one time. 1 times 68 is 68. 75 minus 68 is 7. Bring down the 0. Actually, you don't have to write the decimal there. Ignore that decimal. 68 goes into 70 one time. 1 times 68 is 68. 70 minus 68 is 2, bring down another 0. 68 goes into 20 zero times. And the problem's going to keep going on, but I think we've already once again, gotten to enough accuracy that we can compare. So 1 and 7/68 we've now figured out is equal to 1.10-- and if we kept dividing we'll keep getting more decimals of accuracy, but I think we're now ready to compare. So all of these numbers I just rewrote them as decimals. So 35.7% is 0.357. 108.1%-- ignore this for now because we just used It's 108.1% is equal to 1.081. 0.5 is 0.5. 13/93 is 0.138. And 1 and 7/68 is 1.10 and it'll keep going on. So what's the smallest? So the smallest is 0.-- actually, no. The smallest is right here. So I'm going to rank them from smallest to largest. So the smallest is 0.138. Then the next largest is going to be 0.357. Then the next largest is going to be 0.5. Then you're going to have 1.08. And then you're going to have 1 and 7/68. Well, actually, I'm going to do more examples of this, but for this video I think this is the only one I have time for. But hopefully this gives you a sense of doing these problems. I always find it easier to go into the decimal mode to compare. And actually, the hints on the module will be the same for you. But I think you're ready at least now to try the problems. If you're not, if you want to see other examples, you might just want to either re-watch this video and/or I might record some more videos with more examples right now. Anyway, have fun." + }, + { + "Q": "What is the definition of pizzicato?", + "A": "Pizzicato is when string instruments pluck the strings instead of using the bow. Pizzicato is the Latin word for pluck.", + "video_name": "mVFbGjnysP0", + "transcript": "- [Voiceover] Here is an example of 6/4 time from American composer Joseph Schwantner's The Poet's Hour. Six beats in a measure with quarter note getting one beat. (The Poet's Hour) The next meter that we will discuss is 6/8, six beats in a measure with the eighth note getting one beat. This one is a little more complicated because at a slow tempo, or a slow speed, we can think of each measure with six beats in a bar, but if the tempo was fast, we'd divide the bar into two beats with each beat worth three eighth notes, or a dotted quarter. for a slow version of 6/8, let's listen to this beautiful passage for the oboes and English horn from Ravel's Ballet Daphnis et Chloe. (Daphnis et Chloe) For a fast version, let's listen to part of the Firebird's Variation from Stravinsky's Ballet, The Firebird. (Firebird's Variation) 9/8 continues with the same pattern. In a slow speed, or tempo, each eighth note receives one beat with nine beats in a bar. In a faster tempo, the pulse is three with a dotted quarter note receiving one pulse or one beat. 12/8 continues in the same pattern. In a slow tempo, 12 beats in a measure with an eighth note receiving one beat. If we look at the opening of Stravinsky's Firebird, we see 12/8 at the slow tempo with a feeling of 12 beats in a measure. (Firebird) If we look at the same introduction a few bars later, the pulse moves to four beats with a dotted quarter note getting one beat. (Firebird)" + }, + { + "Q": "why did Hitler do all this stuff and are there still survivors from the holocaust.", + "A": "Hitler didn t like the way that Jewish shops were running so much of Germany, that s why he wanted to wipe them out. He just didn t like them, like, at all. Think about it, though, what if you were a terrorist (hypothetically, of course!) and you wanted to wipe out a whole entire race. How easy would that be? Not very easy. We have survivors from the Holocaust because it s next to virtually impossible to wipe out a whole race, even if they are in capitivity.", + "video_name": "QCkn5bu8GgM", + "transcript": "Adolf Hitler got his start in the military during World War I. He was a dispatch runner on the Western Front. He actually gets fairly decorated. And by most accounts, this is where he finds meaning. He finds meaning in being part of the military. He finds meaning in frankly, the war itself. But then in 1918, we, of course, have the end of the war. Well, first you have the abdication of the Kaiser. You have the Republican government, people who want to form a republic, take control. And then they sign an armistice with the Allies in November. And this is not well received by Hitler. And frankly, it's not well received by many in the military. From their point of view, they somewhat delusionally believed that Germany would have won World War I if they weren't stabbed in the back by these November Criminals, by the folks who had taken over after the Kaiser. So you have this whole stab in the back theory by those who had taken over and signed the armistice. And this wasn't just believed by folks like Hitler. This was believed even by very senior people in the military. This right over here is General Ludendorff, one of two people-- the other gentleman, Hindenburg-- who were in charge of the entire German military. He believed in the stab in the back theory. He thought that they would have won if they didn't sign the armistice, if these November Criminals, these people who had taken control of the government, did not sign this with the Allies. And then you go to 1919. From the point of view of people of like Hitler, things only got worse. You have the Treaty of Versailles that applied all the war guilt to the Germans. You have these huge reparations that would even be paid in resources. You have the former German Empire, a significant amount of its territory is given over to the Allies, or to form new states. Then, you also have the formal establishment of the Weimar Republic. It's called the Weimar Republic because the new German constitution is drafted in the town of Weimar. And it sets it up as a parliamentary democracy. That's why it's called a republic. But it's a little bit of a bizarre parliamentary democracy. It actually gave a good bit of power, a directly elected president that had a reasonable amount of power, especially in emergencies. And that would become relevant later on when Hitler actually comes to power over a decade later. But then in 1919, Hitler was still looking-- he was very upset about the war ending. He stays part of the military. And part of the military he's assigned to start infiltrating or spying on the German Workers' Party. And the acronym in German is the DAP. But the English translation would be the German Workers' Party. But he actually gets quite impressed by the German Workers' Party, which is really ultra-nationalist. And when we talk about ultra-nationalist it's all about German race superiority. It's in line with this whole idea that they would have won the war if they weren't stabbed in the back. And it's also anti-communist. It's anti-capitalist. And it's anti ethnic minorities, in particular anti-Jewish. And all of these ideas Hitler found very impressive. And just to be clear, a lot of times when people talk about ultra-nationalist groups they often will call them as ultra-right wing. And this bears some clarification because the right wing is also often viewed as very capitalist. But ultra-nationalists really put the nation, and the race that they view as indicative of the nation, above all other concerns. So yes, they were anti-communist. They were anti-distribution of wealth. Communists believe in no classes, as little private property as possible. The German Workers' party didn't believe in that. They were anti that. But they were also anti unfettered capitalism, especially capitalism that might get in the way of the nation's interests. But he becomes very impressed with them. And he actually joins as the 55th member. So you can imagine, this, at this point, is a very, very small party. But then we fast forward to 1920. By 1920, the party leadership has taken note of Hitler. They actually notice him when he's arguing with people and that other people are listening. He's actually a really great orator. And so they allow him to give more and more talks. He has more and more authority. And in order for the party to have more of an appeal, especially to nationalists in general, they change their name. To German Workers' Party they add the Nationalist Socialist German Workers' Party, DAP, German Workers' Party, or the NSDAP. And if you pronounce nationalist in German, it sounds something like-- and I'm going to butcher it right now-- Nazionalist. And so, if you were to shorten it, they called themselves the Nazis. And Hitler actually designed the logo for the Nazis, which included this symbol right over here, the swastika. And the swastika is worth talking about because it was really this bizarre corruption of a very ancient symbol. Hitler and the Nazis created this entire mythology around the Germans being the descendants of the Aryans, or being the purest example of the Aryans. And the Aryans are the superior race that's responsible for all of civilization's advancement. It was a delusion because frankly, there was an ancient Aryan race. But the most indicative descendants of them are frankly, the Persians or the Indians. And actually, the swastika symbol here, you might actually even see it at a Hindu temple. It does not mean all of what we associate with Nazism now. It actually is an ancient Hindu symbol of auspiciousness, of good luck. But the Nazis usurped it. But for them, this was a very important idea to create this mythology around race superiority and to even have a symbol like this as opposed to say something like a cross that's a religious symbol. Anyone could believe in Christianity and say, hey, I'm a Christian. But the swastika, at least in Hitler's mind and the Nazi's mind, was a racial symbol. So it represented their superior race. And obviously, if their race was superior, a lot of what they consider the ills of Germany were caused by being infiltrated with what they considered less pure races, like Jews, and also infiltrated with less pure ideas, like the ideas of communism. But Hitler gets more and more recognition with the party. The party membership continues to grow. And by 1921, you have some disagreements in the party. Some people threatened to splinter off. And when Hitler says, hey look, if this is going to happen, I'm going to leave the party, they realize that he has so much value to the party that the party would just dissolve if Hitler leaves. And so they make him the chairman. Hitler takes control. Hitler is the chairman of the Nazi party. And by this point, he's becoming more and more well known on the speaking circuit. And we now have several thousand members of the Nazi party. Although, it's still a fairly small group. But then, things start to get a lot worse in Weimar, Germany. You start having hyperinflation. The government keeps printing more and more currency. The economy is weak. It's trying to pay reparations. And so what you have here-- and this is actually one of the most famous cases of hyperinflation in world history-- you see the value of their currency, it devalues from 1919 to 1923 not by a factor of 1,000 or a million or a billion, but nearly a trillion. So the currency becomes, frankly, worthless. The hyperinflation is happening this entire time. And you see, it accelerates through 1922 and then 1923. But then in 1922, you have Mussolini comes to power in Italy. And he comes to power through his March on Rome. And Mussolini is a fascist. That's where the word comes from. He's a member of the Fascist Party. And the Fascists' ideas were very similar to the Nazis. It was all about extreme nationalism, all about racial superiority, a very strong government. And this Hitler finds quite inspiring. The Weimar Republic is having economic difficulty. You have many other groups, including the communists, attempt their own coup d'etats. They fail. But things are getting less and less stable. And then you fast forward to 1923. The inflation is getting super bad, about as bad as inflation can get. The currency is worthless. The economy is going into a tailspin. And on top of that, because Germany can't pay the reparations to France anymore, France occupies the Ruhr. So the Ruhr region is occupied by France. And you might remember from the terms of the Treaty of Versailles, the Saar region was already being occupied. And all that coal was being shipped out to France. The Ruhr region was another significant region of coal and steel production. And now the French are fully occupying this. They're forcing a lot of the civilians out of the region. They're forcing a lot of the workers to work in the mines and the factories. And then they're shipping all of that supply out to France. So this further debilitates the economy, but it's a huge humiliation. The Treaty of Versailles, in the minds of Germans, especially in the minds of nationalists, was bad enough. But now you have this huge humiliation by the French. And this isn't just amongst the nationalists. The general German population is getting very, very, very, very upset about this. And so this gives a lot of fuel to extreme nationalist groups, like the Nazis. So this fuels the Nazis. And based on the estimates I've seen, entering into the year they're starting to have in excess of 10,000 members, starting to be several tens of thousands. And as we get into the later part of the year, we're approaching, I've seen estimates of 40,000 to 55,000 members of the Nazi party. And that's just formal members. And then on top of that, you might have non-members who are growing increasingly sympathetic. And so what we'll see in the next video, at the end of 1923, Hitler sees this as his chance. He's inspired by Mussolini, the economy is in a tailspin, the Germans have been further humiliated by the French, and the Nazis, in particular, are starting to get quite popular." + }, + { + "Q": "what exactly is a \"special product\"?", + "A": "A special product is the product of the binomials of the form (a+b)(a-b). The product is a^2 - b^2. This is useful to remember because it shows up a lot in rational equations, but you can always prove it to yourself by using FOIL (first-outer-inner-last) or the distributive property twice: (a+b)(a-b) = (a^2) + (-ab) + (ab) + (-b^2) = a^2 - b^2", + "video_name": "4fQeHtSdw80", + "transcript": "- Find the area of a square with side (6x-5y). Let me draw our square and all of the sides of a square are going to have the same length, and they're telling us that the length for each of the sides which is the same for all of them is (6x-5y). So the height would be 6x-5y, and so would the width, 6x-5y, and if we wanted to find the area of the square we just have to multiply the width times the height. So the area for this square is just going to be the width, which is (6x-5y) times the height, times the height which is also (6x-5y), so we just have to multiply these two binomials. To do this, you could either do FOIL if you like memorizing things or you could just remember this is just applying the distributive property twice. So what we could do is distribute this entire magenta, (6x-5y), distribute it over each of these terms, in the yellow (6x-5y). If we do that, we will get this 6x times this entire (6x-5y), so (6x-5y), and then we have -5y, - 5y times once again, the entire magenta, (6x-5y). And what does this give us? So we have, we have 6x times 6x, so when I distributed just this, I'm now doing the distributive property for the second time, 6x times 6x is 36x squared, and then when I take 6x times -5y, I get 6 times -5 is -30, and then I have an x times y, -30xy. And then I want to take, I'm trying to introduce many colors here, so I have this -5y times this 6x right over here so -5 times 6 is -30. - 30 and I have a y and an x or an x and a y, and then finally I have my last distribution to do, let me do that maybe in white, I have -5y times another -5y, so the negative times a negative is a positive so it is positive, 5 times 5 is 25, y times y is y squared. And then we are almost done. Right over here, we could say, we can just add these two terms in the middle right over here, - 30xy-30xy is going to be -60xy. So you get 36x squared -60xy +25y squared. Now, there is a faster way to do this if you recognize. If you recognize that if I'm squaring a binomial, which is essentially what we're doing here, this is the exact same thing as 6x-5y squared. So you might recognize a pattern. If I have (a+b) squared, this is the same thing as (a+b) times (a+b) and if you were to multiply it out this exact same way we just did it here, the pattern here is it's a times a which is a squared, plus a times b, +ab, plus b times a, which is also +ab, we just switched the order, plus b squared, +b squared so this is equal to a squared +2ab +b squared. This is kind of the fast way to look, if you're squaring any binomial, it will be a+b squared, it will be a squared +2ab + b squared. And if you knew this ahead of time, then you could have just applied that to this squaring of the binomial right up here so let's do it that way as well. So if we 6x, (6x-5y) squared, we could just say well, this is going to be a squared. It's going to be a squared in which in this case is 6x squared +2ab, so that's +2 times a which is (6x) times b which is (-5y), - 5y +b squared, which is +(-5y), everything is squared. And then this will simplify too, 6x squared is 36x squared plus, actually it's going to be a negative here because it's going to be 2 times 6 is 12 times -5 is -60, we have x and a y, x and a y, and the -5y squared is +25y squared. So hopefully you saw multiple ways to do this, if you saw this pattern immediately, and if you knew this pattern immediately, you could just cut to the chase and go straight here, you wouldn't have to do distributive property twice, although, this will never be wrong." + }, + { + "Q": "If we see light from stars that are 500 million years old. Then how many years old would be the light from the sun?", + "A": "It depends on what you mean by how old the light is. It takes about 8 minutes for light to get to earth from the surface of the sun but the energy for a photon created inside the sun can take anywhere from a few thousand to 100 s of thousands of years to make it to the surface.", + "video_name": "rcLnMe1ELPA", + "transcript": "In the last video, I hinted that things were about to get wacky. And they are. So if we start where we left off in the last video, we started right over here, looking at the distance to the nearest star. And just as a reminder, in this drawing right here, this depiction right here, this circle right here, this solar system circle, it's not the size of the Sun. It's not the size of the orbits of the Earth, or Pluto, or the Kuiper Belt. This is close to the size of the Oort Cloud. And the actual orbit of Earth is about-- well, the diameter of the orbit of Earth is about 1/50,000 of this. So you wouldn't even see it on this. It would not even make up a pixel on this screen right here, much less the actual size of the Sun or something much smaller. And just remember that orbit of the Earth, that was at a huge distance. It takes eight minutes for light to get from the Sun to the Earth, this super long distance. If you shot a bullet at the Sun from Earth, it would take you that 17 years to actually get to the Sun. So one thing, this huge distance wouldn't even show up on this picture. Now, what we saw in the last video is that if you travel at unimaginably fast speeds, if you travel at 60,000 kilometers per hour-- and I picked that speed because that's how fast Voyager 1 actually is traveling. That's I think the fastest object we have out there in space right here. And it's actually kind of leaving the solar system as we speak. But even if you were able to get that fast, it would still take 80,000 years, 75,000 or 80,000 years, to travel the 4.2 light years to the Alpha Centauri cluster of stars. To the nearest star, it would take 80,000 years. And that scale of time is already an amount of time that I have trouble comprehending. As you can imagine, all of modern civilization has occurred, well, definitely in the last 10,000 years. But most of recorded history is in the last 4,000 or 5,000 years. So this is 80,000 years to travel to the nearest star. So it's a huge distance. Another way to think about it is if the Sun were the size of a basketball and you put that basketball in London, if you wanted to do it in scale, the next closest star, which is actually a smaller basketball, right over here, Proxima Centauri, that smaller basketball you would have to put in Kiev, Ukraine in order to have a similar scale. So these are basketballs sitting in these cities. And you would have to travel about 1,200 miles to place the next basketball. And these basketballs are representing these super huge things that we saw in the first video. The Sun, if you actually made the Earth relative to these basketballs, these would be little grains of sand. So there are any little small planets over here, they would have to be grains of sand in Kiev, Ukraine versus the grain of sand in London. So this is a massive, massive distance, already, at least in my mind, unimaginable. And when it gets really wacky is when you start really looking at this. Even this is a super, super small distance relative to the galactic scale. So this whole depiction of kind of our neighborhood of stars, this thing over here is about, give or take-- and we're doing rough estimates right here-- it's about 30 light years. I'll just do LY for short. So that's about 30 light years, And once again, you can take pictures of our galaxy from our point of view. But you actually can't take a picture of the whole galaxy from above it. So these are going to be artists' depictions. But if this is 30 light years, this drawing right here of kind of our local neighborhood of the galaxy, this right here is roughly-- and these are all approximations. Let me do this in a darker color. This is about 1,000 light years. And this is the 1,000 light years of our Sun's neighborhood, if you can even call it a neighborhood anymore. Even this isn't really a neighborhood if it takes you 80,000 years to get to your nearest neighbor. But this whole drawing over here-- now, it would take forever to get anywhere over here-- it would be 1/30 of this. So it would be about that big, this whole drawing. And what's really going to blow your mind is this would be roughly a little bit more than a pixel on this drawing right here, that spans a 1,000 light years. But then when you start to really put it into perspective-- so now, let's zoom out a little bit-- so this drawing right here, this 1,000 light years is now this 1,000 light years over here. So this is the local vicinity of the Sun. And once again, the word \"local\" is used in a very liberal way at this point. So this right here is 1,000 light years. If you're sitting here and you're looking at an object that's sitting-- let me do this in a darker color-- if we're sitting here on Earth and we're looking at an object out here that's 500 light years away, we're looking at it as it was 500 years ago because the light that is reaching our eyeballs right now, or our telescopes right now, left this guy over here 500 years ago. In fact, he's not going to even be there anymore. He probably has moved around a little bit. So just even on this scale, we're talking about these unimaginably huge distances. And then when we zoom out, this is kind of our local part of the galaxy right over here. This piece right here, this is called the Orion Spur. And people are still trying to work out exactly the details of the actual shape the Milky Way Galaxy, the galaxy that we're in. But we're pretty sure-- actually, we're very sure-- we have these spiral arms and we have these spurs off of them. But it's actually very hard to come up with the actual shape, especially because you can't see a lot of the galaxy, because it's kind of on the other side, on the other side of the center. But really just to get a sense of something that at least-- I mean it blows my mind if you really think about what it's saying-- these unbelievable distances show up as a little dot here. This whole drawing shows up as a dot here. Now when we zoom out, over here that dot would no longer even show up. It wouldn't even register a pixel on this drawing right over here. And then this whole drawing, this whole thing right over here, this whole picture is this grid right over here. It is this right over here. So hopefully, that gives you a sense of how small even our local neighborhood is relative to the galaxy as a whole. And the galaxy as a whole, just to give you a sense, has 200 to 400 billion stars. Or maybe I should say solar systems just to give you a sense that when we saw the solar system, it's not just the Sun. There's all this neat, dynamic stuff. And there are planets, and asteroids, and solar winds. And so there's 200 to 400 billion stars and for the most part, 200 and 400 billion solar systems. So it's an unimaginably, I guess, complex or huge place. And just to make it clear, even when we zoom in to this picture right here, and I think it was obvious based on telling you about this, that these little white pockets right here, This isn't two stars. These are thousands of stars here. So when you go over here, each little blotch of white that you see, that's not a star, that's not a thousand stars. We're starting to talk in the millions of stars when you look at certain blotches here and there. I mean, maybe it might be one star that's closer to you or might be a million stars that are far apart and that are just relatively close together. And everything has to be used in kind of loose terms here. And we'll talk more about other galaxies. But even this isn't the upper bound of galaxies. People believe the Andromeda Galaxy has a trillion stars in it, a trillion solar systems. We're talking about these huge, huge, immense distances. And so just to give you a sense of where we fit in the picture, this is a rough location of our Sun. And remember, that little dot I drew just now is including millions of stars, millions of solar systems, already unimaginable the distances. But if you really want to get at the sense relative to the whole galaxy, this is an artist's depiction. Once again, we could never obviously get this perspective on the galaxy. It would take us forever to travel this far so that you could see the galaxy from above. But this is our best guess looking at things from our vantage point. And we actually can't even see this whole area over here because it's on the other side of the center of the galaxy, which is super, super dense and super bright. And so it's very hard to see things on the other side. We think-- or actually there's a super massive black hole at the center of the galaxy. And we think that they're at the center of all or most galaxies. But you know the whole point of this video, actually this whole series of videos, this is just kind of-- I don't know-- to put you in awe a little bit of just how huge Because when you really think about the scale-- I don't know-- no words can really describe it. But just to give you a sense, we're about 25,000 light years from the center of the galaxy. So even when we look at things in the center of the galaxy, that's as they were 25,000 years ago. It took 25,000 years for that light to get to us. I mean when that light left the center of the galaxy, I won't even guess to think what humanity was like at that point in time. So it's these huge distances in the whole galaxy over here. And once again, like solar systems, it's hard to say the edge of the galaxy, because there's always going to be a few more stars and other things orbiting around the galaxy as you go further and further out, but it gets less dense with stars. But the main density, the main disk, is about 100,000 light years. 100,000 light years is the diameter roughly of the main part of the galaxy. And it's about 1,000 light years thick. So you can kind of imagine it as this disk, this thing that's fairly flat. But it's 1,000 light years thick. It's 1,000 light years thick. You would have to do this distance 250 times just to go from the top part of the galaxy to the bottom part, much less going across the galaxy. So it might seem relatively flat. But it's still immensely, immensely thick. And just as another way to visualize it, if this thing right over here that includes the Oort Cloud, roughly a light year in diameter, is a grain of sand, a millimeter in diameter grain of sand, then the universe as a whole is going to be the diameter of a football field. And that might tell you, OK, those are two tractable things. I can imagine a grain of sand, a millimeter wide grain of sand in a football field. But remember, that grain of sand is still 50,000 or 60,000 times the diameter of Earth's orbit. And Earth's orbit, it would still take a bullet or something traveling as fast as a jet plane 15 hours to just go half of that-- or sorry, not-- 15 years or 17 years, I forgot the exact number. But it was 15, 16, 17 years to even cover half of that distance. So 30 years just to cover the diameter of Earth's orbit. That's 1/60,000 of our little grain of sand in the football field. And just to kind of really, I don't know, have an appreciation for how mind-blowing this really is, this is actually a picture of the Milky Way Galaxy, our galaxy, from our vantage point. As you can see, we're in the galaxy and this is looking towards the center. And even this picture, you start to appreciate the complexity of what 100 billion stars are. But what I really want to point out is even in this picture, when you're looking at these things, some of these things that look like stars, those aren't stars. those are thousands of stars or millions of stars. Maybe it could be one star closer up. But when we're starting to approach the center of the galaxy, these are thousands and thousands and millions of stars or solar systems that we're actually looking at. So really, it starts to boggle the mind to imagine what might actually be going on over there." + }, + { + "Q": "I don't understand the claims at 10:37 :( . There is an equation written :\ndelta P * V = P * delta V\nand Sal says if the pressure is constant, these will cancel out.\nPressure being constant means: delta P = 0. But P still has a value.\nSo filling it in gives:\n0 * V = P * delta V\n0 = P * delta V (because P does have a value, even though it's constant)\nBut P has a value, so the right term IS NOT 0 (unless delta V is 0, or P = 0).\nSo i dont know how both terms cancel out if only the pressure is constant T_T, help?", + "A": "It s delta(PV) = P(delta V) not (delta P)*V = P(delta V)", + "video_name": "fucyI7Ouj2c", + "transcript": "Let me draw a good old PV diagram. That's my pressure axis, this is my volume axis. Just like that, I have pressure and volume. I showed several videos ago that if we start at some state here in the PV diagram, right there, and that I change the pressure and the volume to get to another state, and I do it in a quasistatic way, so essentially I'm always close to equilibrium, so my state variables are always defined, I could have some path that takes me to some other state right there. And this is my path. I'm going from this state to that state. And we showed that if I just did this, the work done by the system is the area under this curve. And then if I were to move back to the previous state, and then if I were, you know, by some path, just some random path that I happened to be drawing, the work done to the system would be the area under this light blue curve. So the net work done by the system ended up being the area inside of this path. So this is-- let me do it in different color. The net work done would be the area inside of this path when I go in this clockwise type of direction. So this is the net work done by the system. And now, we also know that if we're at some point on this PV diagram, that our state is the same as it was before. So if we go all the way here, and then go all the way back, all of our state variables will not have changed. Our pressure is the same as it was before. Our volume was the same as it was before because we went all the way back to that same point on the PV diagram. And our internal energy is also the same point as it was before, so our change in internal energy over this path, you're going to have a different internal energy here than you had here, but when you go around the circle and you get back, your change in internal energy is equal to zero. And we know that our change in internal energy is defined as, and this is from the first law of thermodynamics, the heat added to the system minus the work done by the system. Now, if we go on a closed loop on our PV diagram, then what's our change in internal energy? It's zero. So we get zero change in internal energy, because we're at the same state is equal to the heat applied to the system minus the work done or-- and I've done this little exercise multiple times. I think is probably the fourth or fifth time I'm doing it. We get that the heat added to the system, if we just add w to both sides, is equal to the work done by the system. So this area inside of this path, I already said, it's the work done by the system, and if you don't remember even where that came from, it was, remember, pressure times volume times change in volume is a little incremental change in work, and that's why it relates to the area. But we've done that multiple times. I won't go there just yet. So if you have any area here, some heat was added to the system, some net heat, right? Some heat was added here, and some heat was probably taken out here, but you have some net heat that's added to the system. And I use that argument to say why heat isn't a good state variable. Because-- and I had a whole video on this-- that if I define some state variable, let's just say, heat content. Let's say I want to define some state variable heat content. And I would say that the change in heat content would, of course, be equal to the change in heat. That's what I'm defining. If I'm adding heat to the system, my heat content should go up. But the problem with that heat content state variable was that, let's say over here, I say that the heat content is equal to 5. Now, I just showed you that if we go on some path here and we come back, and there's some area in this little path that I took, that some heat was added. So let's say that this area right here, so this is q is equal to the work done by the system, let's say it's equal to 2. So every time, if I start at heat content is equal to 5, that's just an arbitrary number, and I were to do this entire path, when I go back, the heat content would have to be 7. And then when I go back and do it again, my heat content would have to be 9. And it would have to increment by 2 every time I do this It would have to increment by the amount of area that this path goes around. So heat content can't be a state variable, because it's dependent on how you got there. A state variable-- and remember this. In order to be a state variable, if you're at this point, you have to have the same value. If your internal energy was 10 here, when you do the path and you come back, your internal energy will be 10 again. That's why internal energy is a valid state variable. It's dependent only on your state. If your entropy was 50 here, when you soon. go back and you do all sorts of crazy things, and you come back to this point, your entropy is once again 50. If your pressure here is 5 atmospheres, when you come back here, your pressure will be 5 atmospheres. Your state variable cannot change based on what path you took. If you're at a certain state, that's all that matters to the state variable. Now this heat content didn't work, and that's why we actually led into some videos where I divided by t and we got entropy, which was an interesting variation. But that's still not satisfying. What if we really wanted to develop something that could in some way be a state variable, but at the same time measure heat? So obviously, we're going to have to make some compromises, because if we just do a very arbitrary kind of heat content variable, then every time you go around this, it's going to change. That's not a valid state variable. So let's see if we can make up one. So let's just make up a definition. Let's call my new thing that I'm going to try to maybe approximate heat, let's call it h, and just as a little bit of a preview, we're going to call it enthalpy. And let's just define it. I'm just playing around. Let's just define it as the internal energy plus my pressure times my volume. So then what would my change in enthalpy be? So my change in enthalpy will be, of course, the change of these things. But I could just say, that's my change in my internal energy plus my change in pressure times volume. Now, this is interesting. And I want to make a point here. This, by definition, is a valid state variable. Why is it? Because it's the addition of other state variables, right? At any point in my PV diagram, and it's also true if I did diagrams that were entropy in temperature or anything that dealt with state variables, at any point on my diagram, u is going to be the same, no matter how I got there. p is by definition going to be the same. That's why it's at that point. v is definitely going to be the same point. So if I just add them up, this is a valid state variable because it's just the sum of a bunch of other valid state variables. So let's see if we can somehow relate this thing that we've already established as a valid state variable. From the get-go, from our definition, this works because it's just the sum of completely valid state variables. So let's see if we can relate this somehow to heat. So we know what delta u is. If we're dealing with all of the internal energy or the change in internal energy, and I'm not going to deal with all the other chemical potentials and all of that, it's equal to the heat applied to the system minus the work done by the system, right? Let me put everything else there. The change in enthalpy is equal to the heat applied minus the work done-- that's just the change in internal energy-- plus delta PV. This is just from the definition of my enthalpy. Now this is starting to look interesting. What's the work done by a system? So I could write change in h, or enthalpy, is equal to the heat applied to the system minus-- what's the work done If I have some system here, it's got some piston on it, you know, if we're doing it in a quasistatic, I have those classic pebbles that I've talked about in multiple videos. When I apply heat or let's say I remove some of these pebbles, so I'm at a different equilibrium, but what's actually happening? When is the work being done? You have some pressure being applied up here, and this piston is going to be moving up, and your volume is going And we showed multiple videos ago that the work done by the system can be, and you can kind of view this as the volume expansion work, it's equal to pressure times change in volume. And let's add the other part. So this was our change in internal energy. And I had several videos where I show this. And now let me add the other part of the equation. So our enthalpy, our change in enthalpy, can be defined by this. Something interesting is going on. I said that I wanted to define something, because I wanted to somehow measure heat content. My change in enthalpy will be equal to the heat added to the system, if these last two terms cancel out. If I can somehow get these last two terms to cancel out, then my change in enthalpy will be equal to this, if somehow these are equal to each other. So under what conditions are these equal to each other? Or another way, under what conditions is delta pressure times volume equal to pressure times delta volume? When does this happen? When can I make this statement? Because if I can make this statement, then these two terms are equivalent right here, and then you my change in enthalpy will be equal to the heat added. Well, the only way I can make this statement is if pressure is constant. Now why is that? Let's just think about it mathematically. If this is a constant, then if I just change-- you know, if this is just 5, 5 times a change in something is the same thing as the change in 5 times that thing, so it just mathematically works out. Or if you view it another way, if this is a constant, you can just factor it out, right? Well, if I said, the change in 5x, that would be equal to 5 times x final minus 5 times x initial. And you could say, well, that's just equal to 5 times x final minus x initial. Well, that's just equal to 5 times the change in x. It's kind of almost too obvious for me to explain. I think sometimes when you overexplain things, it might become more confusing. So this applies-- and the 5 I'm just doing as the analogy So if pressure is constant, then this equation is true. So if pressure is constant-- so this is a key assumption-- then if heat is being applied in a constant pressure system-- so we could write it this way. I'll write it multiple times, because this is key. If pressure is constant, then our definition, our little thing we made up, this enthalpy thing, which we defined as internal energy plus pressure and volume, then in a constant pressure system, our change in enthalpy we just showed is equal to the heat added to the system because all of these two things become equivalent under constant pressure, so I should write that. This is only true when heat is added in a constant pressure system. So how does this gel with what we did up here on our PV diagram? What's happening in a constant pressure system? Let me draw our PV diagram. That's P, that's V. So what's happening in constant pressure? We're at some pressure right there. So if we're under constant pressure, that means we can only move along this line. So we could go from here to there and back to there, or we could go from there to there, back to there. So we could go there, all the way there, and then go back. But what do we see about this? Is there any area in this curve? I mean, there is no curve to speak of, because we're staying in a constant pressure. We've kind of squeezed out this diagram. We've made the forward path and the return path the same exact path. So because of this, you don't have that state problem because no net heat is being added to the system when you go from this point all the way to this point and then back to this point. So because of that, you can kind of see visually that enthalpy in a constant pressure, when you're not moving up and down in pressure, is the same thing as heat added. So you might say, hey Sal, this was a bit of a compromise, constant pressure, you know, that's a big assumption to make. Why is this useful at all? Well, it's useful, because most chemical reactions, especially ones that occur in an open beaker, or that might occur at sea level, and that should be a big clue, they occur at constant pressure. You know, if I'm sitting at the beach, and I have my chemistry set, and I have some beaker of something, and I'm throwing other stuff into it, and I'm looking for a reaction or something, it's a constant pressure system. This is going to be atmospheric pressure. I'm sitting at sea level. So this is actually a very useful concept for everyday It might not be so useful for engines, because engines always have pressure changing, but it's very useful for actual chemistry, for actually dealing with what's going to happen to a reaction at a constant pressure. So what we're going to see is that this enthalpy, you can kind of view it as the heat content when pressure is constant. In fact, it is the heat content when pressure is constant. So somehow-- well, not somehow, I showed you how-- we were able to make this definition, which by definition was a state variable, because it was the sum of other state variables, and if we just make that one assumption of constant pressure, it all of a sudden reduces to the heat content of that system. So we'll talk more in the future of measuring enthalpy, but you just have to say, if pressure is constant, enthalpy is the same thing as-- and it's really only useful when we're dealing with a constant pressure. But if we have a pressure constant, enthalpy can be imagined as heat content. And it's very useful for understanding whether chemical reactions need heat to occur or whether they release heat, so on and so forth. See" + }, + { + "Q": "Around 5:05 he describes the occupation of Czechslovakia. Didn't they occupy Bohemia+Moravia while creating a puppet state in Slovakia?", + "A": "To start with, yes - but this was quickly enveloped by the greater German mass.", + "video_name": "-kKCjwNvNkQ", + "transcript": "World War II was the largest conflict in all of human history. The largest and bloodiest conflict And so you can imagine it is quite complex My goal in this video is to start giving us a survey, an overview of the war. And I won't even be able to cover it all in this video. It is really just a think about how did things get started. Or what happened in the lead up? And to start I am actually going to focus on Asia and the Pacific. Which probably doesn't get enough attention when we look at it from a western point of view But if we go back even to the early 1900s. Japan is becoming more and more militaristic. More and more nationalistic. In the early 1900s it had already occupied... It had already occupied Korea as of 1910. and in 1931 it invades Manchuria. It invades Manchuria. So this right over here, this is in 1931. And it installs a puppet state, the puppet state of Manchukuo. And when we call something a puppet state, it means that there is a government there. And they kind of pretend to be in charge. But they're really controlled like a puppet by someone else. And in this case it is the Empire of Japan. And we do remember what is happening in China in the 1930s. China is embroiled in a civil war. So there is a civil war going on in China. And that civil war is between the Nationalists, the Kuomintang and the Communists versus the Communists The Communists led by Mao Zedong. The Kuomintang led by general Chiang Kai-shek. And so they're in the midst of the civil war. So you can imagine Imperial Japan is taking advantage of this to take more and more control over parts of China And that continues through the 30s until we get to 1937. And in 1937 the Japanese use some pretext with, you know, kind of a false flag, kind of... well, I won't go into the depths of what started it kind of this Marco-Polo Bridge Incident But it uses that as justifications to kind of have an all-out war with China so 1937...you have all-out war and this is often referred to as the Second Sino-Japanese War ...Sino-Japanese War Many historians actually would even consider this the beginning of World War II. While, some of them say, ok this is the beginning of the Asian Theater of World War II of the all-out war between Japan and China, but it isn't until Germany invades Poland in 1939 that you truly have the formal beginning, so to speak, of World War II. Regardless of whether you consider this the formal beginning or not, the Second Sino-Japanese War, and it's called the second because there was another Sino-Japanese War in the late 1800s that was called the First Sino-Japanese War, this is incredibly, incredibly brutal and incredibly bloody a lot of civilians affected we could do a whole series of videos just on that But at this point it does become all-out war and this causes the civil war to take a back seat to fighting off the aggressor of Japan in 1937. So that lays a foundation for what's happening in The Pacific, in the run-up to World War II. And now let's also remind ourselves what's happening... what's happening in Europe. As we go through the 1930s Hitler's Germany, the Nazi Party, is getting more and more militaristic. So this is Nazi Germany... Nazi Germany right over here. They're allied with Benito Mussolini's Italy. They're both extremely nationalistic; they both do not like the Communists, at all You might remember, that in 1938... 1938, you have the Anschluss, which I'm sure I'm mispronouncing, and you also have the takeover of the Sudetenland in Czechoslovakia. So the Anschluss was the unification with Austria and then you have the Germans taking over the of Sudetenland in Czechoslovakia and this is kind of the famous, you know, the rest of the, what will be called the Allied Powers kind of say, \"Okay, yeah, okay maybe Hitler's just going to just do that... well we don't want to start another war. We still all remember World War I; it was really horrible. And so they kind of appease Hitler and he's able to, kind of, satisfy his aggression. so in 1938 you have Austria, Austria and the Sudetenland ...and the Sudetenland... are taken over, are taken over by Germany and then as you go into 1939, as you go into 1939 in March they're able to take over all of Czechoslovakia they're able to take over all of Czechoslovakia and once again the Allies are kind of, they're feeling very uncomfortable, they kind of, have seen something like this before they would like to push back, but they still are, kind of, are not feeling good about starting another World War so they're hoping that maybe Germany stops there. So let me write this down... So all of Czechoslovakia... ...Czechoslovakia... is taken over by the Germans. This is in March of 1939. And then in August you have the Germans, and this is really in preparation for, what you could guess is about to happen, for the all-out war that's about to happen the Germans don't want to fight the Soviets right out the gate, as we will see, and as you might know, they do eventually take on the Soviet Union, but in 1939 they get into a pact with the Soviet Union. And so this is, they sign the Molotov\u2013Ribbentrop Pact with the Soviet Union, this is in August, which is essentially mutual non-aggression \"Hey, you know, you do what you need to do, we know what we need to do.\" and they secretly started saying \"Okay were gonna, all the countries out here, we're going to create these spheres of influence where Germany can take, uh, control of part of it and the Soviet Union, and Stalin is in charge of the Soviet Union at this point, can take over other parts of it. And then that leads us to the formal start where in September, let me write this in a different color... so September of 1939, on September 1st, Germany invades Poland Germany invades Poland on September 1st, which is generally considered the beginning of World War II. and then you have the Great Britain and France declares war on Germany so let me write this World War II... starts everyone is declaring war on each other, Germany invades Poland, Great Britain and France declare war on Germany, and you have to remember at this point Stalin isn't so concerned about Hitler he's just signed the Molotov-Ribbentrop Pact and so in mid-September, Stalin himself invades Poland as well so they both can kind of carve out... ...their spheres of influence... so you can definitely sense that things are not looking good for the world at this point you already have Asia in the Second Sino-Japanese War, incredibly bloody war, and now you have kind of, a lot of very similar actors that you had in World War I and then they're starting to get into a fairly extensive engagement." + }, + { + "Q": "Is there fusion going on in the core?", + "A": "Nuclear fusion happens in the core of a star once it has reached a certain temperature (around 15 million degrees). The maximum temperature of the inner core is around 6,000 degrees, so definitely not!", + "video_name": "f2BWsPVN7c4", + "transcript": "What I want to do in this video is talk a little bit about plate tectonics. And you've probably heard the word before, and are probably, or you might be somewhat familiar with what it discusses. And it's really just the idea that the surface of the Earth is made up of a bunch of these rigid plates. So it's broken up into a bunch of rigid plates, and these rigid plates move relative to each other. They move relative to each other and take everything that's on them for a ride. And the things that are on them include the continents. So it literally is talking about the movement of these plates. And over here I have a picture I got off of Wikipedia of the actual plates. And over here you have the Pacific Plate. Let me do that in a darker color. You have the Pacific Plate. You have a Nazca Plate. You have a South American Plate. I could keep going on. You have an Antarctic Plate. It's actually, obviously whenever you do a projection onto two dimensions of a surface of a sphere, the stuff at the bottom and the top look much bigger than they actually are. Antarctica isn't this big relative to say North America or South America. It's just that we've had to stretch it out to fill up the rectangle. But that's the Antarctic Plate, North American Plate. And you can see that they're actually moving relative to each other. And that's what these arrows are depicting. You see right over here the Nazca Plate and the Pacific Plate are moving away from each other. New land is forming here. We'll talk more about that in other videos. You see right over here in the middle of the Atlantic Ocean the African Plate and the South American Plate meet each other, and they're moving away from each other, which means that new land, more plate material I guess you could say, is somehow being created right here-- we'll talk about that in future videos-- and pushing these two plates apart. Now, before we go into the evidence for plate tectonics or even some of the more details about how plates are created and some theories as to why the plates might move, what I want to do is get a little bit of the terminology of plate tectonics out of the way. Because sometimes people call them crustal plates, and that's not exactly right. And to show you the difference, what I want to do is show you two different ways of classifying the different layers of the Earth and then think about how they might relate to each other. So what you traditionally see, and actually I've made a video that goes into a lot more detail of this, is a breakdown of the chemical layers of the Earth. And when I talk about chemical layers, I'm talking about what are the constituents of the different layers? So when you talk of it in this term, the top most layer, which is the thinnest layer, is the crust. Then below that is the mantle. Actually, let me show you the whole Earth, although I'm not going to draw it to scale. So if I were to draw the crust, the crust is the thinnest outer layer of the Earth. You can imagine the blue line itself is the crust. Then below that, you have the mantle. So everything between the blue and the orange line, this over here is the mantle. So let me label the crust. The crust you can literally view as the actual blue pixels over here. And then inside of the mantle, you have the core. And when you do this very high level division, these are chemical divisions. This is saying that the crust is made up of different types of elements. Its makeup is different than the stuff that's in the mantle, which is made up of different things than what's inside the core. It's not describing the mechanical properties of it. And when I talk about mechanical properties I'm talking about whether something is solid and rigid. Or maybe it's so hot and melted it's kind of a magma, or kind of a plastic solid. So this would be the most brittle stuff. If it gets warmed up, if rock starts to melt a little bit, then you have something like a magma, or you can view it as like a deformable or a plastic solid. When we talk about plastic, I'm not talking about the stuff that the case of your cellphone is made of. I'm talking about it's deformable. This rock is deformable because it's so hot and it's somewhat melted. It kind of behaves like a fluid. It actually does behave like a fluid, but it's much more viscous. It's much thicker and slower moving than what we would normally associate with a fluid like water. So this a viscous fluid. And then the most fluid would, of course, be the liquid state. This is what we mean when we talk about the mechanical properties. And when you look at this division over here, the crust is solid. The mantle actually has some parts of it that are solid. So the uppermost part of the mantle is solid. Then below that, the rest of the mantle is kind of in this magma, this deformable, somewhat fluid state, and depending on what depth you go into the mantle there are kind of different levels of fluidity. And then the core, the outer level layer of the core, the outer core is liquid, because the temperature is so high. The inner core is made up of the same things, and the temperature is even higher, but since the pressure is so high it's actually solid. So that's why the mantle, crust, and core differentiations don't tell you whether it's solid, whether it's magma, or whether it's really a liquid. It just really tells you what the makeup is. Now, to think about the makeup, and this is important for plate tectonics, because when we talk about these plates we're not talking about just the crust. We're talking about the outer, rigid layer. Let me just zoom in a little bit. Let's say we zoomed in right over there. So now we have the crust zoomed in. This right here is the crust. And then everything below here we're actually talking about the upper mantle. We haven't gotten too deep in the mantle right here. So that's why we call it the upper mantle. Now, right below the crust, the mantle is cool enough that it is also in real solid form. So this right here is solid mantle. And when we talk about the plates were actually talking about the outer solid layer. So that includes both the crust and the solid part of the mantle. And we call that the lithosphere. When people talk about plate tectonics, they shouldn't say crustal plates. They should call these lithospheric plates. And then below the lithosphere you have the least viscous part of the mantle, because the temperature is high enough for the rock to melt, but the pressure isn't so large as what will happen when you go into the lower part of the mantle that the fluid can actually kind of move past each other, although still pretty viscous. This still a magma. So this is still kind of in its magma state. And this fluid part of the mantle, we can't quite call it a liquid yet, but over large periods of time it does have fluid properties. This, that essentially the lithosphere is kind of riding on top of, we call this the asthenosphere. So when we talk about the lithosphere and asthenosphere we're really talking about mechanical layers. The outer layer, the solid layers, the lithosphere sphere. The more fluid layer right below that is the asthenosphere. When we talk about the crust, mantle, and core, we are talking about chemical properties, what are the things actually made up of." + }, + { + "Q": "Why we have 2 different enthalpy (combustion&reaction enthalpy) but not just reaction enthalpy?", + "A": "They are all just enthalpy changes. It is common to refer to an enthalpy change according to the process involved. Thus, we talk about enthalpy of combustion, enthalpy of neutralization, enthalpy of vaporization, etc. But they are all just enthalpy changes.", + "video_name": "8bCL8TQZFKo", + "transcript": "This problem is from chapter five of the Kotz, Treichel, Townsend Chemistry and Chemical Reactivity textbook. So they tell us, suppose you want to know the enthalpy change-- so the change in total energy-- for the formation of methane, CH4, from solid carbon as a graphite-- that's right there-- and hydrogen gas. So we want to figure out the enthalpy change of this reaction. How do we get methane-- how much energy is absorbed or released when methane is formed from the reaction of-- solid carbon as graphite and hydrogen gas? So they tell us the enthalpy change for this reaction cannot to be measured in the laboratory because the reaction is very slow. So normally, if you could measure it you would have this reaction happening and you'd kind of see how much heat, or what's the temperature change, of the surrounding solution. Maybe this is happening so slow that it's very hard to measure that temperature change, or you can't do it in any meaningful way. We can, however, measure enthalpy changes for the combustion of carbon, hydrogen, and methane. So they're giving us the enthalpy changes for these combustion reactions-- combustion of carbon, combustion of hydrogen, combustion of methane. And they say, use this information to calculate the change in enthalpy for the formation of methane from its elements. So any time you see this kind of situation where they're giving you the enthalpies for a bunch of reactions and they say, hey, we don't know the enthalpy for some other reaction, and that other reaction seems to be made up of similar things, your brain should immediately say, hey, maybe this is a Hess's Law problem. Hess's Law. And all Hess's Law says is that if a reaction is the sum of two or more other reactions, then the change in enthalpy of this reaction is going to be the sum of the change in enthalpies of those reactions. Now, when we look at this, and this tends to be the confusing part, how can you construct this reaction out of these reactions over here? And what I like to do is just start with the end product. So I like to start with the end product, which is methane in a gaseous form. And when we look at all these equations over here we have the combustion of methane. So this actually involves methane, so let's start with this. But this one involves methane and as a reactant, not a product. But what we can do is just flip this arrow and write it as methane as a product. So if we just write this reaction, we flip it. So now we have carbon dioxide gas-- let me write it down here-- carbon dioxide gas plus-- I'll do this in another color-- plus two waters-- if we're thinking of these as moles, or two molecules of water, you could even say-- two molecules of water in its liquid state. That can, I guess you can say, this would not happen spontaneously because it would require energy. But if we just put this in the reverse direction, if you go in this direction you're going to get two waters-- or two oxygens, I should say-- I'll do that in this pink color. So two oxygens-- and that's in its gaseous state-- plus a gaseous methane. CH4. CH4 in a gaseous state. And all I did is I wrote this third equation, but I wrote it in reverse order. I'm going from the reactants to the products. When you go from the products to the reactants it will release 890.3 kilojoules per moles of the reaction going on. But if you go the other way it will need 890 kilojoules. So the delta H here-- I'll do this in the neutral color-- so the delta H of this reaction right here is going to be the reverse of this. So it's positive 890.3 kilojoules per mole of the reaction. All I did is I reversed the order of this reaction right there. The good thing about this is I now have something that at least ends up with what we eventually want to end up with. This is where we want to get. This is where we want to get eventually. Now, if we want to get there eventually, we need to at some point have some carbon dioxide, and we have to have at some point some water to deal with. So how can we get carbon dioxide, and how can we get water? Well, these two reactions right here-- this combustion reaction gives us carbon dioxide, this combustion reaction gives us water. So we can just rewrite those. Let me just rewrite them over here, and I will-- let me use So if I start with graphite-- carbon in graphite form-- carbon in its graphite form plus-- I already have a color for oxygen-- plus oxygen in its gaseous state, it will produce carbon dioxide in its gaseous form. It will produce carbon-- that's a different shade of green-- it will produce carbon dioxide in its gaseous form. And this reaction, so when you take the enthalpy of the carbon dioxide and from that you subtract the enthalpy of these reactants you get a negative number. Which means this had a lower enthalpy, which means energy Because there's now less energy in the system right here. So this is essentially how much is released. But our change in enthalpy here, our change in enthalpy of this reaction right here, that's reaction one. I'll just rewrite it. Minus 393.5 kilojoules per mole of the reaction occurring. So the reaction occurs a mole times. This would be the amount of energy that's essentially released. This is our change in enthalpy. So if this happens, we'll get our carbon dioxide. Now we also have-- and so we would release this much energy and we'd have this product to deal with-- but we also now need our water. And this reaction right here gives us our water, the combustion of hydrogen. So we have-- and I haven't done hydrogen yet, so let me do hydrogen in a new color. That's not a new color, so let me do blue. So right here you have hydrogen gas-- I'm just rewriting that reaction-- hydrogen gas plus 1/2 O2-- pink is my color for oxygen-- 1/2 O2 gas will yield, will it give us some water. Will give us H2O, will give us some liquid water. Now, before I just write this number down, let's think about whether we have everything we need. To make this reaction occur, because this gets us to our final product, this gets us to the gaseous methane, we need a mole. Or we can even say a molecule of carbon dioxide, and this reaction gives us exactly one molecule of carbon dioxide. So that's a check. And we need two molecules of water. Now, this reaction only gives us one molecule of water. So let's multiply both sides of the equation to get two molecules of water. So this is a 2, we multiply this by 2, so this essentially You multiply 1/2 by 2, you just get a 1 there. And then you put a 2 over here. So I just multiplied this second equation by 2. So I just multiplied-- this is becomes a 1, this becomes a 2. And if you're doing twice as much of it, because we multiplied by 2, the delta H now, the change enthalpy of the reaction, is now going to be twice this. Let's get the calculator out. It's now going to be negative 285.8 times 2. Because we just multiplied the whole reaction times 2. So negative 571.6. So it's negative 571.6 kilojoules per mole of the reaction. Now, let's see if the combination, if the sum of these reactions, actually is this reaction up here. And to do that-- actually, let me just copy and paste this top one here because that's kind of the order that we're going to go in. You don't have to, but it just makes it hopefully a little bit easier to understand. So let me just copy and paste this. Actually, I could cut and paste it. Cut and then let me paste it down here. That first one. And let's see now what's going to happen. To see whether the some of these reactions really does end up being this top reaction right here, let's see if we can cancel out reactants and products. Let's see what would happen. So this produces carbon dioxide, but then this mole, or this molecule of carbon dioxide, is then used up in this last reaction. So this produces it, this uses it. Let me do it in the same color so it's in the screen. This reaction produces it, this reaction uses it. Now, this reaction right here produces the two molecules of water. And now this reaction down here-- I want to do that same color-- these two molecules of water. Now, this reaction down here uses those two molecules of water. Now, this reaction right here, it requires one molecule of molecular oxygen. This one requires another molecule of molecular oxygen. So these two combined are two molecules of molecular oxygen. So those are the reactants. And in the end, those end up as the products of this last reaction. So those, actually, they go into the system and then they leave out the system, or out of the sum of reactions unchanged. So they cancel out with each other. So we could say that and that we cancel out. And so what are we left with? What are we left with in the reaction? Well, we have some solid carbon as graphite plus two moles, or two molecules of molecular hydrogen yielding-- all we have left on the product side is some methane. So it is true that the sum of these reactions is exactly what we want. All we have left on the product side is the graphite, the solid graphite, plus the molecular hydrogen, plus the gaseous hydrogen-- do it in that color-- plus two hydrogen gas. And all we have left on the product side is the methane. All we have left is the methane in the gaseous form. So it is true that the sum of these reactions-- remember, we have to flip this reaction around and change its sign, and we have to multiply this reaction by 2 so that the sum of these becomes this reaction that we really care about. So this is the sum of these reactions. Its change in enthalpy of this reaction is going to be the sum of these right here. That is Hess's Law. So this is the fun part. So we just add up these values right here. So we have negative 393.-- no, that's not what I wanted to do. Let me just clear it. So I have negative 393.5, so that step is exothermic. And then we have minus 571.6. That is also exothermic. Those were both combustion reactions, which are, as we know, very exothermic. And we have the endothermic step, the reverse of that last combustion reaction. So plus 890.3 gives us negative 74.8. It gives us negative 74.8 kilojoules for every mole of the reaction occurring. Or if the reaction occurs, a mole time. So there you go. We figured out the change in enthalpy. And it is reasonably exothermic. Nowhere near as exothermic as these combustion reactions right here, but it is going to release energy. And we're done." + }, + { + "Q": "the problem asks for the absolute maximum and the |-8 - 2pi| is clearly the largest so should the value be x = -4 and not x = 5/2", + "A": "No. Absolute maximum means the highest point on the graph of g(x), not the point where the absolute value of g(x) is the highest.", + "video_name": "OvMBNVi5bLY", + "transcript": "Part b. \"Determine the x-coordinate of the point at which g has an absolute maximum on the interval negative 4 is less than or equal to x, is less than or equal to 3. And justify your answer.\" So let's just think about it in general terms. If we just think about a general function over an interval, where it could have an absolute maximum. So let me draw some axes over here. And I'm speaking in general terms first, and then we can go back to our function g, which is derived from this function f right over here. So let's say that these are my coordinate axes, and let's say we care about some interval here. So let's say this is the interval that I care about. A function could look something like this. And in this case, its absolute maximum is going to occur at the beginning of the interval. Or a function could look something like this. And then the absolute maximum could occur at the end point of the interval. Or the other possibility is that the function looks something like this. At which point, the maximum would be at this critical point. And I say critical point, as opposed to just a point where the slope is zero, because it's possible to the functions not differentiable there. You could imagine a function that looks like this, and maybe wouldn't be differentiable there. But that point there still would be the absolute maximum. So what we really just have to do is evaluate g at the different endpoints of this interval, to see how high it gets, or how large of a value we get for the g at the end points. And then we have to see if g has any critical points in between. And then evaluate it there to see if that's a candidate for the global maximum. So let's just evaluate g of the different points. So let's start off, let's evaluate g at negative 4-- at kind of the lowest end, or the starting point of our interval. So g of negative 4 is equal to 2 times negative 4 plus the integral from 0 to negative 4 f of t dt. The first part is very easy, 2 times negative 4 is negative 8. Let me do it over here so I have some real estate. So this is equal to negative 8. And instead of leaving this as 0 to negative 4 f of t dt, let's change the bounds of integration here. Especially so that we can get the lower number as the lower bound. And that way, it becomes a little bit more natural to think of it in terms of areas. So this expression right here can be rewritten as the negative of the integral between negative 4 and 0 f of t dt. And now this expression right over here is the area under f of t, or in this case, f of x-- or the area under f between negative 4 and 0. So it's this area right over here. And we have to be careful, because this part over here is below the x-axis. So this we would consider negative area when we think of it in integration terms. And this would be positive area. So the total area here is going to be this positive area minus this area right over here. So let's think about what this is. So this area-- This section over here we did this in part a, actually, This section-- this is a quarter circle, so it's 1/4-- so these are both quarter circles. So we could multiply 1/4 times the area of this entire circle, if we were to draw the entire thing all the way around. It has a radius of 3. So the area of the entire circle would be pi times 3 squared or it would be 9 pi. And of course we're going to divide it by 4-- multiply it by 1/4 to just get this quarter circle right over there. And then this area right over here, the area of the entire circle, we have a radius of 1. So it's going to be pi times 1 squared. So it's going to be pi, and then we're going to divide it by four, because it's only one fourth of that circle. And we're going to subtract that. So we have negative pi and we were multiplying it times 1/4 out here, because it's just a quarter circle in either case. And we're subtracting it, because the area is below the x-axis. And so this simplifies to-- this is equal to 1/4 times 8 pi, which is the same thing as 2 pi. Did I do that right? 1/4 times 8 pi. So this all simplifies to 2 pi. So g of negative 4 is equal to negative 8 minus 2 pi. So clearly it is a negative number here. More negative than negative 8. So let's try the other bounds. So let's see what g of positive 3 is. I'll do it over here so I have some more space. So g of positive 3, when x is equal to 3, that-- we go back to our definition-- that is 2 times 3 plus the integral from 0 to 3 f of t dt. And this is going to be equal to 2 times 3 is 6. And the integral from 0 to 3 f of t dt, that's this entire area. So we have positive area over here. And then we have an equal negative area right over here because it's below the x-axis. So the integral from 0 to 3 is just going to be 0, you're going to have this positive area, and then this negative area right over here is going to completely cancel it out. Because it's symmetric right over here. So this thing is going to evaluate to 0. So g of 3 is 6. So we already know that our starting point, g of negative 4-- that when x is equal to negative 4-- that is not where g hits a global maximum. Because that's a negative number. And we already found the end point, where g hits a positive value. So negative 4 is definitely not a candidate. x is equal to 3 is still in the running for the x-coordinate where g has a global absolute maximum. Now what we have to do is figure out any critical points that g has in between. So points in there where it's either undifferentiable or its derivative is equal to 0. So let's look at this derivative. So g prime of x-- we just take the derivative of this business up here. Derivative of 2x is 2. Derivative of this definition going from 0 to x of f of t dt-- we did that in part a, this is just the fundamental theorem of calculus-- this is just going to be plus f of x right over there. So it actually turns out that g is differentiable over the entire interval. You give any x value over this interval, we have a value for f of x. f of x isn't differentiable everywhere, but definitely f of x is defined everywhere, over the interval. So you'll get a number here, and obviously two is just two, and you add two to it, and you get the derivative of g at that point in the interval. So g is actually differentiable throughout the interval. So the only critical points would be where this derivative is equal to 0. So let's set this thing equal to 0. So we want to solve the equation-- I'll just rewrite it actually-- So we want to solve the equation g prime of x is equal to 0, or 2 plus f of x is equal to 0. You can subtract 2 from both sides, and you get f of x is equal to negative 2. So any x that satisfies f of x is equal to negative 2 is a point where the derivative of g is equal to 0. And let's see if f of x is equal to negative 2 at any point. So let me draw a line over here at negative 2. We have to look at it visually, because there's only given us this visual definition of f of x. Doesn't equal negative 2, doesn't equal to negative 2, only equals negative 2 right over there. And it looks like we're at about 2 and 1/2, but let's get exact. Let's actually figure out the slope of the line, and figure out what x value actually gives f of x equal to negative 2. And we could figure out the slope of this line fairly visually-- or figure out the equation of this line fairly visually, we can figure out its slope. If we run 3-- if our change in x is 3, then our change in y, our rise, is negative 6. Change in y is equal to negative 6. Slope is rise over run, or change in y over change in x. So negative 6 divided by 3 is negative 2. It has a slope of negative 2. And actually, I could have done that easier. Where if we go forward one, we go down by 2. So we could have seen that the slope is negative 2. So this part of f of x, we have y is equal to negative 2x plus-- and then the y-intercept is pretty straightforward. This is at 3-- 1, 2, 3. Negative 2x plus 3. So part of f of x where clearly it equals negative 2 at some point of that-- this part of f of x is defined by this line. Obviously this part of f of x is not defined by that line. But to figure out the exact value, we just have to figure out when this line is equal to negative 2. So we have negative 2x plus 3 is equal to negative 2. And remember, this isn't-- this is what f of x is equal to, over the interval that we care about. If we were talking about f of x over there, we wouldn't be able to put a negative 2x plus 3, we would have to have some form of equation for these circles. But right over here, this is what f of x is, and now we can solve this pretty straightforwardly. So we can subtract 3 from both sides, and we get negative 2x is equal to negative 5. Divide both sides by negative 2, you get x is equal to negative 5 over negative 2, which is equal to 5/2. Which is exactly what we thought it was when we looked at it visually. It looked like we were at about 2 and 1/2, which is the same thing as 5/2. Now we don't know what this is. We don't know if this is an inflection point. Is this a maximum? Is this a minimum? So really we just want to evaluate g at this point to see if it gets higher than when we evaluate g at 3. So let's evaluate g at 5/2. So g at 5/2 is going to be equal to 2 times 15/2 plus the integral from 0 to 5/2 of f of t dt. So this first part right over here, the 2's cancel out. So this is going to be equal to 5. And then plus the integral from 0 to 5/2. Now, you might be able to do it visually, but we know what the value is of f of t over this interval, we already figured out the equation for it over this interval. So it's the integral of negative 2x plus 3 dt. And then let's just evaluate this. Let me get some real estate. So this is-- let me draw a line here., so we don't get confused. So this is going to be equal to 5 plus and then I take the antiderivative. The antiderivative of negative 2x is negative x squared. So we have negative x squared. And the antiderivative of 3 is just going to be 3x. So plus 3x. And we're going to evaluate it from 0 to 5/2. So this is going to be equal to 5 plus-- and I'll do all this stuff right over here. I'll do this stuff in green. So when we evaluate it at 5/2, this is going to be negative 5/2 squared. So it's going to be negative 25 over 4 plus 3 times 5/2, which is 15 over 2. And then from that, we're going to subtract this evaluated at 0. But negative 0 squared plus 3 times 0 is just 0. So this is what it simplifies to. And so what do we have right over here? So let's get our ourselves a common denominator. Looks like a common denominator right over here could be 4 So this is equal to-- 5 is the same thing as 20 over 4 minus 25 over 4 and then plus 30 over 4. So 20 plus 30 is 50 minus 25 is 25. So this is equal to 25 over 4. And 25 over 4 is the same thing as 6 and 1/4. So when we evaluate our function at this critical point, at this thing where the slope, or the derivative, is equal to 0, we got 6 and 1/4, which is higher than six, which is what g was at this end point. And it's definitely higher than what g was a negative 4. So the x-coordinate of the point at which g has an absolute maximum, on the interval negative 4 to three is x is equal to 5/2." + }, + { + "Q": "What happens when you approach the freezing temperature of something? If you cooled it equally, would it all freeze at the same time?", + "A": "Theoretically yes but most elements or compounds won t cool evenly but instead they form crystal lattices well cooling which eventually connect to form the solid that you would see.", + "video_name": "pKvo0XWZtjo", + "transcript": "I think we're all reasonably familiar with the three states of matter in our everyday world. At very high temperatures you get a fourth. But the three ones that we normally deal with are, things could be a solid, a liquid, or it could be a gas. And we have this general notion, and I think water is the example that always comes to at least my mind. Is that solid happens when things are colder, relatively colder. And then as you warm up, you go into a liquid state. And as your warm up even more you go into a gaseous state. So you go from colder to hotter. And in the case of water, when you're a solid, you're ice. When you're a liquid, some people would call ice water, but let's call it liquid water. I think we know what that is. And then when it's in the gas state, you're essentially vapor or steam. So let's think a little bit about what, at least in the case of water, and the analogy will extend to other types of molecules. But what is it about water that makes it solid, and when it's colder, what allows it to be liquid. And I'll be frank, liquids are kind of fascinating because you can never nail them down, I guess is the best way to view them. Or a gas. So let's just draw a water molecule. So you have oxygen there. You have some bonds to hydrogen. And then you have two extra pairs of valence electrons in the oxygen. And a couple of videos ago, we said oxygen is a lot more electronegative than the hydrogen. It likes to hog the electrons. So even though this shows that they're sharing electrons here and here. At both sides of those lines, you can kind of view that hydrogen is contributing an electron and oxygen is contributing an electron on both sides of that line. But we know because of the electronegativity, or the relative electronegativity of oxygen, that it's hogging these electrons. And so the electrons spend a lot more time around the oxygen than they do around the hydrogen. And what that results is that on the oxygen side of the molecule, you end up with a partial negative charge. And we talked about that a little bit. And on the hydrogen side of the molecules, you end up with a slightly positive charge. Now, if these molecules have very little kinetic energy, they're not moving around a whole lot, then the positive sides of the hydrogens are very attracted to the negative sides of oxygen in other molecules. Let me draw some more molecules. When we talk about the whole state of the whole matter, we actually think about how the molecules are interacting with Not just how the atoms are interacting with each other within a molecule. I just drew one oxygen, let me copy and paste that. But I could do multiple oxygens. And let's say that that hydrogen is going to want to be near this oxygen. Because this has partial negative charge, this has a partial positive charge. And then I could do another one right there. And then maybe we'll have, and just to make the point clear, you have two hydrogens here, maybe an oxygen wants to hang out there. So maybe you have an oxygen that wants to be here because it's got its partial negative here. And it's connected to two hydrogens right there that have their partial positives. But you can kind of see a lattice structure. Let me draw these bonds, these polar bonds that start forming between the particles. These bonds, they're called polar bonds because the molecules themselves are polar. And you can see it forms this lattice structure. And if each of these molecules don't have a lot of kinetic energy. Or we could say the average kinetic energy of this matter is fairly low. And what do we know is average kinetic energy? Well, that's temperature. Then this lattice structure will be solid. These molecules will not move relative to each other. I could draw a gazillion more, but I think you get the point that we're forming this kind of fixed structure. And while we're in the solid state, as we add kinetic energy, as we add heat, what it does to molecules is, it just makes them vibrate around a little bit. If I was a cartoonist, they way you'd draw a vibration is to put quotation marks there. That's not very scientific. But they would vibrate around, they would buzz around a little bit. I'm drawing arrows to show that they are vibrating. It doesn't have to be just left-right it could be up-down. But as you add more and more heat in a solid, these molecules are going to keep their structure. So they're not going to move around relative to each other. But they will convert that heat, and heat is just a form of energy, into kinetic energy which is expressed as the vibration of these molecules. Now, if you make these molecules start to vibrate enough, and if you put enough kinetic energy into these molecules, what do you think is going to happen? Well this guy is vibrating pretty hard, and he's vibrating harder and harder as you add more and more heat. This guy is doing the same thing. At some point, these polar bonds that they have to each other are going to start not being strong enough to contain the vibrations. And once that happens, the molecules-- let me draw a couple more. Once that happens, the molecules are going to start moving past each other. So now all of a sudden, the molecule will start shifting. But they're still attracted. Maybe this side is moving here, that's moving there. You have other molecules moving around that way. But they're still attracted to each other. Even though we've gotten the kinetic energy to the point that the vibrations can kind of break the bonds between the polar sides of the molecules. Our vibration, or our kinetic energy for each molecule, still isn't strong enough to completely separate them. They're starting to slide past each other. And this is essentially what happens when you're in a liquid state. You have a lot of atoms that want be touching each other but they're sliding. They have enough kinetic energy to slide past each other and break that solid lattice structure here. And then if you add even more kinetic energy, even more heat, at this point it's a solution now. They're not even going to be able to stay together. They're not going to be able to stay near each other. If you add enough kinetic energy they're going to start looking like this. They're going to completely separate and then kind of bounce around independently. Especially independently if they're an ideal gas. But in general, in gases, they're no longer touching They might bump into each other. But they have so much kinetic energy on their own that they're all doing their own thing and they're not touching. I think that makes intuitive sense if you just think about what a gas is. For example, it's hard to see a gas. Why is it hard to see a gas? Because the molecules are much further apart. So they're not acting on the light in the way that a liquid or a solid would. And if we keep making that extended further, a solid-- well, I probably shouldn't use the example with ice. Because ice or water is one of the few situations where the solid is less dense than the liquid. That's why ice floats. And that's why icebergs don't just all fall to the bottom of the ocean. And ponds don't completely freeze solid. But you can imagine that, because a liquid is in most cases other than water, less dense. That's another reason why you can see through it a little Or it's not diffracting-- well I won't go into that too much, than maybe even a solid. But the gas is the most obvious. And it is true with water. The liquid form is definitely more dense than the gas form. In the gas form, the molecules are going to jump around, not touch each other. And because of that, more light can get through the substance. Now the question is, how do we measure the amount of heat that it takes to do this to water? And to explain that, I'll actually draw a phase change diagram. Which is a fancy way of describing something fairly straightforward. Let me say that this is the amount of heat I'm adding. And this is the temperature. We'll talk about the states of matter in a second. So heat is often denoted by q. Sometimes people will talk about change in heat. They'll use H, lowercase and uppercase H. They'll put a delta in front of the H. Delta just means change in. And sometimes you'll hear the word enthalpy. Let me write that. Because I used to say what is enthalpy? It sounds like empathy, but it's quite a different concept. At least, as far as my neural connections could make it. But enthalpy is closely related to heat. It's heat content. For our purposes, when you hear someone say change in enthalpy, you should really just be thinking of change in heat. I think this word was really just introduced to confuse chemistry students and introduce a non-intuitive word into their vocabulary. The best way to think about it is heat content. Change in enthalpy is really just change in heat. And just remember, all of these things, whether we're talking about heat, kinetic energy, potential energy, enthalpy. You'll hear them in different contexts, and you're like, I thought I should be using heat and they're talking about enthalpy. These are all forms of energy. And these are all measured in joules. And they might be measured in other ways, but the traditional way is in joules. And energy is the ability to do work. And what's the unit for work? Well, it's joules. Force times distance. But anyway, that's a side-note. But it's good to know this word enthalpy. Especially in a chemistry context, because it's used all the time and it can be very confusing and non-intuitive. Because you're like, I don't know what enthalpy is in my everyday life. Just think of it as heat contact, because that's really But anyway, on this axis, I have heat. So this is when I have very little heat and I'm increasing my heat. And this is temperature. Now let's say at low temperatures I'm here and as I add heat my temperature will go up. Temperature is average kinetic energy. Let's say I'm in the solid state here. And I'll do the solid state in purple. No I already was using purple. I'll use magenta. So as I add heat, my temperature will go up. Heat is a form of energy. And when I add it to these molecules, as I did in this example, what did it do? It made them vibrate more. Or it made them have higher kinetic energy, or higher average kinetic engery, and that's what temperature is a measure of; average kinetic energy. So as I add heat in the solid phase, my average kinetic energy will go up. And let me write this down. This is in the solid phase, or the solid state of matter. Now something very interesting happens. Let's say this is water. So what happens at zero degrees? Which is also 273.15 Kelvin. Let's say that's that line. What happens to a solid? Well, it turns into a liquid. Ice melts. Not all solids, we're talking in particular about water, about H2O. So this is ice in our example. All solids aren't ice. Although, you could think of a rock as solid magma. Because that's what it is. I could take that analogy a bunch of different ways. But the interesting thing that happens at zero degrees. Depending on what direction you're going, either the freezing point of water or the melting point of ice, something interesting happens. As I add more heat, the temperature does not to go up. As I add more heat, the temperature does not go up for a little period. Let me draw that. For a little period, the temperature stays constant. And then while the temperature is constant, it stays a solid. We're still a solid. And then, we finally turn into a liquid. Let's say right there. So we added a certain amount of heat and it just stayed a solid. But it got us to the point that the ice turned into a liquid. It was kind of melting the entire time. That's the best way to think about it. And then, once we keep adding more and more heat, then the liquid warms up too. Now, we get to, what temperature becomes interesting again for water? Well, obviously 100 degrees Celsius or 373 degrees Kelvin. I'll do it in Celsius because that's what we're familiar with. That's the temperature at which water will vaporize or which water will boil. But something happens. And they're really getting kinetically active. But just like when you went from solid to liquid, there's a certain amount of energy that you have to contribute to the system. And actually, it's a good amount at this point. Where the water is turning into vapor, but it's not getting any hotter. So we have to keep adding heat, but notice that the temperature didn't go up. We'll talk about it in a second what was happening then. And then finally, after that point, we're completely vaporized, or we're completely steam. Then we can start getting hot, the steam can then get hotter as we add more and more heat to the system. So the interesting question, I think it's intuitive, that as you add heat here, our temperature is going to go up. But the interesting thing is, what was going on here? We were adding heat. So over here we were turning our heat into kinetic energy. Temperature is average kinetic energy. But over here, what was our heat doing? Well, our heat was was not adding kinetic energy to the system. The temperature was not increasing. But the ice was going from ice to water. So what was happening at that state, is that the kinetic energy, the heat, was being used to essentially break these bonds. And essentially bring the molecules into a higher energy state. So you're saying, Sal, what does that mean, higher energy state? Well, if there wasn't all of this heat and all this kinetic energy, these molecules want to be very close to each other. For example, I want to be close to the surface of the earth. When you put me in a plane you have put me in a higher energy state. I have a lot more potential energy. I have the potential to fall towards the earth. Likewise, when you move these molecules apart, and you go from a solid to a liquid, they want to fall towards each other. But because they have so much kinetic energy, they never quite are able to do it. But their energy goes up. Their potential energy is higher because they want to fall towards each other. By falling towards each other, in theory, they could do some work. So what's happening here is, when we're contributing heat-- and this amount of heat we're contributing, it's called the heat of fusion. Because it's the same amount of heat regardless how much direction we go in. When we go from solid to liquid, you view it as the heat of melting. It's the head that you need to put in to melt the ice into liquid. When you're going in this direction, it's the heat you have to take out of the zero degree water to turn it into ice. So you're taking that potential energy and you're bringing the molecules closer and closer to each other. So the way to think about it is, right here this heat is being converted to kinetic energy. Then, when we're at this phase change from solid to liquid, that heat is being used to add potential energy into the system. To pull the molecules apart, to give them more potential energy. If you pull me apart from the earth, you're giving me potential energy. Because gravity wants to pull me back to the earth. And I could do work when I'm falling back to the earth. A waterfall does work. It can move a turbine. You could have a bunch of falling Sals move a turbine as well. And then, once you are fully a liquid, then you just become a warmer and warmer liquid. Now the heat is, once again, being used for kinetic energy. You're making the water molecules move past each other faster, and faster, and faster. To some point where they want to completely disassociate from each other. They want to not even slide past each other, just completely jump away from each other. And that's right here. This is the heat of vaporization. And the same idea is happening. Before we were sliding next to each other, now we're pulling apart altogether. So they could definitely fall closer together. And then once we've added this much heat, now we're just heating up the steam. We're just heating up the gaseous water. And it's just getting hotter and hotter and hotter. But the interesting thing there, and I mean at least the interesting thing to me when I first learned this, whenever I think of zero degrees water I'll say, oh it must be ice. But that's not necessarily the case. If you start with water and you make it colder and colder and colder to zero degrees, you're essentially taking heat out of the water. You can have zero degree water and it hasn't turned into ice yet. And likewise, you could have 100 degree water that hasn't turned into steam yeat. You have to add more energy. You can also have 100 degree steam. You can also have zero degree water. Anyway, hopefully that gives you a little bit of intuition of what the different states of matter are. And in the next problem, we'll talk about how much heat exactly it does take to move along this line. And maybe we can solve some problems on how much ice we might need to make our drink cool." + }, + { + "Q": "How can you find the y-int by only them giving you the x-int?", + "A": "To find the y-intercept, all you have to do is set the x value to 0. To find the x intercept, you just have to set the y value to 0. This makes sense because a y intercept is always (0,something) and on a graph, it doesn t go horizontally in either direction, it just stays on the axis. Hope this helped :)", + "video_name": "uk7gS3cZVp4", + "transcript": "We are asked to graph y is equal to 1/3x minus 2. Now, whenever you see an equation in this form, this is called slope-intercept form. And the general way of writing it is y is equal to mx plus b, where m is the slope. And here in this case, m is equal to 1/3-- so let me write that down-- m is equal to 1/3, and b is the y-intercept. So in this case, b is equal to negative 2. And you know that b is the y-intercept, because we know that the y-intercept occurs when x is equal to 0. So if x is equal to 0 in either of these situations, this term just becomes 0 and y will be equal to b. So that's what we mean by b is the y-intercept. So whenever you look at an equation in this form, it's actually fairly straightforward to graph this line. b is the y-intercept. In this case it is negative 2, so that means that this line must intersect the y-axis at y is equal to negative 2, so it's this point right here. Negative 1, negative 2, this is the point 0, negative 2. If you don't believe me, there's nothing magical about this, try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y-intercept right there. Now, this 1/3 tells us the slope of the line. How much do we change in y for any change in x? So this tells us that 1/3, so that right there, is the slope. So it tells us that 1/3 is equal to the change in y over the change in x. Or another way to think about it, if x changes by 3, then y would change by 1. So let me graph that. So we know that this point is on the graph, that's the y-intercept. The slope tells us that if x changes by 3-- so let me go 3 three to the right, 1, 2, 3-- that y will change by 1. So this must also be a point on the graph. And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2. It's that same ratio, so 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line, and the line is the graph of this equation up here. So let me graph it. So it'll look something like that. And you're done." + }, + { + "Q": "What are negative numbers?", + "A": "negative numbers are numbers below zero", + "video_name": "tJrSILRXOUc", + "transcript": "Voiceover:Which numbers are greater than six? Select all that apply. We see six here on the number line, so the numbers that are greater than six are going to be the ones that are to the right of six on the number line. We see that we're increasing beyond six as we go to the right. Six, seven, eight, nine, ten. Seven is greater than six, eight is greater than six, nine is greater than six, and 10 is greater than six, and we could keep going. 11 is greater than six, and 12, and on and on and on. Which of these are greater than six? Well we see 10 is to the right, is on the right hand side of six, eight is also to the right of six but four is to the left of six. Four is less than six. These are the two numbers that are greater than, the two choices that are greater than six. Which numbers are less than six? Well that's all of these numbers right over here. The numbers to the left of six. Nine is definitely not less than six but four is. Notice four is to the left of six and three is even more to the left of six, so four and three are definitely less than six. When every day life when you're thinking of well if you have four things, or you have three things you have less than someone who has six things. In every day life if you have 10 bananas, you have a greater number of bananas than someone who has six bananas." + }, + { + "Q": "So do you just multiply the denominator of the first fraction and the numerator of the second fraction", + "A": "Unlike adding you multiply the denominator and the numerator and same with division.", + "video_name": "yb7lVnY_VCY", + "transcript": "Tommy is studying for final exams this weekend. He will spend 1/5 of the weekend studying. What fraction of the weekend will he spend studying for each of his 4 subjects if he spends the same amount of time studying for each subject? So the total amount of time he's going to spend studying this weekend is 1/5 of the weekend. And he has to divide that into 4 equal sections. And he's going to spend that much time on each subject. So he's going to divide this by 4. Now, we've already seen that dividing by a number is the same thing as multiplying by its reciprocal. You might say, hey, well, what's the reciprocal of 4? You just have to remind yourself that 4 is the same thing as 4/1. So 1/5 divided by 4/1 is the same thing as 1/5 times 1/4. And you could also view this as 1/4 of 1/5 or 1/5 of 1/4, either way. But here we multiply our numerators to get 1. And then we multiply our denominators, 4 times 5 is 20. So you get 1/20 of the weekend will be spent studying for each subject. Now, let's also try to think about this visually. Let's imagine that this is his entire weekend. And I've divided it into 5 equal sections. And so we already know that the total amount of his weekend spent studying is 1/5. So that's the total amount studying for the weekend is 1/5. Now, he has to divide this into 4 equals section. So let's do that. He's got four subjects, and he's going to spend the same amount of time on each of the 4 subjects. So he's going to divide this into 4 equal sections. So how much time does he spend on one subject? Well, in each subject, that would be this little area that I'm doing in yellow right over here. And what is that? Well, that's 1 over-- how many equal sections are there of that size in the weekend? Well, I've just drawn out the grid. You had 5 rows, and now you have 4 columns. So 5 rows times 4 columns, you have 20 equal sections. So once again, looking at it visually, he's spending 1/20 of his weekend on each of the 4 subjects. And then if you do this for 4 subjects, that means that in this whole weekend, 1/5 will be spent studying. But the question that they're asking, he's spending 1/20 of the weekend on each subject." + }, + { + "Q": "Hi :)\n\nAt 7:24, it is mentioned that the molecules have the same kinetic energy. How so? If the volume of the box was decreased then the molecules kinetic energy would increase, wouldn't it?\n\nOr does it mean that all the molecules inside this smaller box all have the same average kinetic energy? But this kinetic energy is still higher than the kinetic energy of the molecules on the bigger box, isn't it?\n\nThank you!", + "A": "Me thinks the average K.E of molecules remains the same but the extra energy they gained by virtue of reduced volume is transferred as increase in heat of collision which is very minute.", + "video_name": "tQcB9BLUoVI", + "transcript": "After all the work we've been doing with fluids, you probably have a pretty good sense of what pressure is. Now let's think a little bit about what it really means, especially when we think about it in terms of a gas in a volume. Remember, what was the difference between a gas and a liquid? They're both fluids, they both take the shape of their containers, but a gas is compressible, while a liquid is incompressible. Let's start focusing on gases. Let's say I have a container, and I have a bunch of gas in it. What is a gas made of? It's just made up of a whole bunch of the molecules of the gas itself, and I'll draw each of the molecules with a little dot-- it's just going to have a bunch of molecules in it. There's many, many, many more than what I've drawn, but that's indicative, and they'll all be going in random directions-- this one might be going really fast in that direction, and that one might be going a little bit slower in that direction. They all have their own little velocity vectors, and they're always constantly bumping into each other, and bumping into the sides of the container, and ricocheting here and there and changing velocity. In general, especially at this level of physics, we assume that this is an ideal gas, that all of the bumps that occur, there's no loss of energy. Or essentially that they're all elastic bumps between the different molecules. There's no loss of momentum. Let's keep that in mind, and everything you're going to see in high school and on the AP test is going to deal with ideal gases. Let's think about what pressure means in this context. A lot of what we think about pressure is something pushing on an area. If we think about pressure here-- let's pick an arbitrary area. Let's take this side. Let's take this surface of its container. Where's the pressure going to be generated onto this surface? It's going to be generated by just the millions and billions and trillions of little bumps every time-- let me draw a side view. If this is the side view of the container, that same side, every second there's always these little molecules of gas moving around. If we pick an arbitrary period of time, they're always ricocheting off of the side. We're looking at time over a super-small fraction of time. And over that period of time, this one might end up here, this one maybe bumped into it right after it ricocheted and came here, this one changes momentum and goes like that. This one might have already been going in that direction, and that one might ricochet. But what's happening is, at any given moment, since there's so many molecules, there's always going to be some molecules that are bumping into the side of the wall. When they bump, they have a change in momentum. All force is change in momentum over time. What I'm saying is that in any interval of time, over any period or any change in time, there's just going to be a bunch of particles that are changing their momentum on the side of this wall. That is going to generate force, and so if we think about how many on average-- because it's hard to keep track of each particle individually, and when we did kinematics and stuff, we'd keep track of the individual object at play. But when we're dealing with gases and things on a macro level, you can't keep track of any individual one, unless you have some kind of unbelievable supercomputer. We can say, on average, this many particles are changing momentum on this wall in this amount of time. And so the force exerted on this wall or this surface is going to be x. If we know what that force is, and we you know the area of the wall, we can figure out pressure, because pressure is equal to force divided by area. What does this help us with? I wanted to give you that intuition first, and now I'm just going to give you the one formula that you really just need to know in thermodynamics. And then as we go into the next few videos, I'll prove to you why it works, and hopefully give you more of an intuition. Now you understand, hopefully, what pressure means in the context of a gas in a container. With that out of the way, let me give you a formula. I hope by the end of this video you have the intuition for why this formula works. In general, if I have an ideal gas in a container, the pressure exerted on the gas-- on the side of the container, or actually even at any point within the gas, because it will all become homogeneous at some point-- and we'll talk about entropy in future videos-- but the pressure in the container and on its surface, times the volume of the container, is equal to some constant. We'll see in future videos that that constant is actually proportional to the average kinetic energy of the molecules bouncing around. That should make sense to you. If the molecules were moving around a lot faster, then you would have more kinetic energy, and then they would be changing momentum on the sides of the surface a lot more, so you would have more pressure. Let's see if we can get a little bit more intuition onto why pressure times volume is a constant. Let's say I have a container now, and it's got a bunch of molecules of gas in it. Just like I showed you in that last bit right before I erased, these are bouncing off of the sides at a certain rate. Each of the molecules might have a different kinetic energy-- it's always changing, because they're always transferring momentum to each other. But on average, they all have a given kinetic energy, they keep bumping at a certain rate into the wall, and that determines the pressure. What happens if I were able to squeeze the box, and if I were able to decrease the volume of the box? I just take that same box with the same number of molecules in it, but I squeeze. I make the volume of the box smaller-- what's going to happen? I have the same number of molecules in there, with the same kinetic energy, and on average, they're moving with the same velocities. So now what's going to happen? They're going to be hitting the sides more often-- at the same time here that this particle went bam, bam, now it could go bam, bam, bam. They're going to be hitting the sides more often, so you're going to have more changes in momentum, and so you're actually going to have each particle exert more force on each surface. Because it's going to be hitting them more often in a given amount of time. The surfaces themselves are smaller. You have more force on a surface, and on a smaller surface, you're going to have higher pressure. Hopefully, that gives you an intuition that if I had some amount of pressure in this situation-- if I squeeze the volume, the pressure increases. Another intuition-- if I have a balloon, what blows up a balloon? It's the internal air pressure of the helium, or your own exhales that you put into the balloon. The more and more you try to squeeze a balloon-- if you squeeze it from all directions, it gets harder and harder to do it, and that's because the pressure within the balloon increases as you decrease the volume. If volume goes down, pressure goes up, and that makes sense. That follows that when they multiply each other, you have to have a constant. Let's take the same example again, and what happens if you make the volume bigger? Let's say I have-- it's huge like that, and I should have done it more proportionally, but I think you get the idea. You have the same number of particles, and if I had a particle here, in some period of time it could have gone bam, bam, bam-- it could have hit the walls twice. Now, in this situation, with larger walls, it might just go bam, and in that same amount of time, it will maybe get here and won't even hit the other wall. The particles, on average, are going to be colliding with the wall less often, and the walls are going to have a larger area, as well. So in this case, when our volume goes up, the average pressure or the pressure in the container goes down. Hopefully, that gives you a little intuition, and so you'll never forget that pressure times volume is constant. And then we can use that to do some pretty common problems, which I'll do in the next video. See you soon." + }, + { + "Q": "What about when working with decimals in the equation? Do you get rid of the decimals first? Example problem - -2+.3x-.1x+x=6x HELP...we still cannot get this one right...thanks.", + "A": "you can get rid of the decimals by multiplying the decimals by 10", + "video_name": "PL9UYj2awDc", + "transcript": "We have the equation 3/4x plus 2 is equal to 3/8x minus 4. Now, we could just, right from the get go, solve this the way we solved everything else, group the x terms, maybe on the left-hand side, group the constant terms on the right-hand side. But adding and subtracting fractions are messy. So what I'm going to do, right from the start of this video, is to multiply both sides of this equation by some number so I can get rid of the fractions. And the best number to do it by-- what number is the smallest number that if I multiply both of these fractions by it, they won't be fractions anymore, they'll be whole numbers? That smallest number is going to be 8. I'm going to multiply 8 times both sides of this equation. You say, hey, Sal, how did you get 8? And I got 8 because I said, well, what's the least common multiple of 4 and 8? Well, the smallest number that is divisible by 4 and 8 is 8. So when you multiply by 8, it's going to get rid of the fractions. And so let's see what happens. So 8 times 3/4, that's the same thing as 8 times 3 over 4. Let me do it on the side over here. That's the same thing as 8 times 3 over 4, which is equal to 8 divided by 4 is just 2. So it's 2 times 3, which is 6. So the left-hand side becomes 8 times 3/4x is 6x. And then 8 times 2 is 16. You have to remember, when you multiply both sides, or a side, of an equation by a number, you multiply every term by that number. So you have to distribute the 8. So the left-hand side is 6x plus 16 is going to be equal to-- 8 times 3/8, that's pretty easy, the 8's cancel out and you're just left with 3x. And then 8 times negative 4 is negative 32. And now we've cleaned up the equation a good bit. Now the next thing, let's try to get all the x terms on the left-hand side, and all the constant terms on the right. So let's get rid of this 3x from the right. Let's subtract 3x from both sides to do it. That's the best way I can think of of getting rid of the 3x from the right. The left-hand side of this equation, 6x minus 3x is 3x. 6 minus 3 is 3. And then you have a plus 16 is equal to-- 3x minus 3x, that's the whole point of subtracting 3x, is so they cancel out. So those guys cancel out, and we're just left with a negative 32. Now, let's get rid of the 16 from the left-hand side. So to get rid of it, we're going to subtract 16 from both sides of this equation. Subtract 16 from both sides. The left-hand side of the equation just becomes-- you have this 3x here; these 16's cancel out, you don't have to write anything-- is equal to negative 32 minus 16 is negative 48. So we have 3x is equal to negative 48. To isolate the x, we can just divide both sides of this equation by 3. So let's divide both sides of that equation by 3. The left-hand side of the equation, 3x divided by 3 is just an x. That was the whole point behind dividing both sides by 3. And the right-hand side, negative 48 divided by 3 is negative 16. And we are done. x equals negative 16 is our solution. So let's make sure that this actually works by substituting to the original equation up here. And the original equation didn't have those 8's out front. So let's substitute in the original equation. We get 3/4-- 3 over 4-- times negative 16 plus 2 needs to be equal to 3/8 times negative 16 minus 4. So 3/4 of 16 is 12. And you can think of it this way. What's 16 divided by 4? It is 4. And then multiply that by 3, it's 12, just multiplying fractions. So this is going to be a negative 12. So we get negative 12 plus 2 on the left-hand side, negative 12 plus 2 is negative 10. So the left-hand side is a negative 10. Let's see what the right-hand side is. You have 3/8 times negative 16. If you divide negative 16 by 8, you get negative 2, times 3 is a negative 6. So it's a negative 6 minus 4. Negative 6 minus 4 is negative 10. So when x is equal to negative 16, it does satisfy the Both sides of the equation become negative 10. And we are done." + }, + { + "Q": "why didn't he just use sin (3pi/12 and 4pi/12) to solve this instead of sin(pi/4 and pi/3)?", + "A": "Those are actually the same, because 3pi/12=pi/4 and 4pi/12=pi/3 due to the cancellation of common factors(3 in the first, 4 in the second equation).", + "video_name": "2RbKfRfzD-M", + "transcript": "Voiceover:What I want to attempt to do in this video is figure out what the sine of seven pi over 12 is without using a calculator. And so let's just visualize seven pi over 12 in the unit circle. One side of the angle is going along the positive x-axis if we go straight up, that's pi over two, which is the same thing as six pi over 12, so then we essentially just have another pi over 12 to get right over there. This is the angle that we're talking about that is seven pi over 12 radians, by the unit circle definition of sine, it's the y-coordinate of where this ray intersects the unit circle. This is the unit circle, has radius one where it intersects the unit. The y-coordinate is the sine. Another way to think about it, it's the length of this line right over here. I encourage you to pause the video right now and try to think about it on your own. See if you can use your powers of trigonometry to figure out what sine of seven pi over 12 is or essentially the length of this magenta line. I'm assuming you've given a go at it, and if you're like me, your first temptation might have been just to focus on this triangle right over here that I drew for you. The triangle looks like this. It looks like this, where that's what you're trying to figure out, this length right over here, sine of seven pi over 12. We know the length of the hypotenuse is one. It's a radius of the unit circle. It's a right triangle right over there. We also know this angle right over here, which is this angle right over here, this gets us six pi over 12, and then we have another pi over 12, so we know that that is pi over 12, not pi over 16. We know that this angle right over here is pi over 12. Given this information, we can figure out this, or we can at least relate this side to this other side using a trig function relative to this angle. This is the adjacent side. The cosine of pi over 12 is going to be this magenta side over one, or you could just say it's equal to this magenta side. You could say that this is cosine of pi over 12. We just figured out that sine of seven pi over 12 is the same thing as cosine of pi over 12, but that still doesn't help me. I don't know offhand what the cosine of pi over 12 radians is without using a calculator. Instead of thinking about it this way, let's see if we can compose this angle or if we can decompose it into some angles for which we do know the sine and cosine. What angles are those? Those are the angles in special right triangles. For example, we are very familiar with 30-60-90 triangles. 30-60-90 triangles look something like this. This is my best attempt at hand drawing it. Instead of writing 30-degree side, since we're thinking in radians, I'll write that as pi over six radians. The 60-degree side, I'm going to write that as pi over three radians, and of course, this is the right angle. If the hypotenuse here is one, then the side opposite the 30-degree side, or the pi over six radian side, is going to be half the hypotenuse, which, in this case is 1/2. Then the other side that's opposite the 60-degree side or the pi over three radian side, is going to be square root of three times the shorter side. It's going to be square root of three over two. We've used these types of triangles in the past to figure out the sine or cosine of 30 or 60, or in this case, pi over six or pi over three. We know about pi over six and pi over three. We also know about 45-45-90 triangles. We know that they're isosceles right triangles. They look like this, my best attempt at drawing it. That one actually doesn't look that isosceles, so let me make it a little bit more ... I don't know. That looks closer to being an isosceles right triangle. We know if the length of the hypotenuse is one, and this comes straight out of Pythagorean theorem, then the length of each of the other two sides are going to be square root of two over two times the hypotenuse, which, in this case, is the square root of two over two. Instead of describing these as 45-degree angles, we know that's the same thing as pi over four, pi over four radians. If you give me pi over six, pi over three, pi over four, I can use these triangles either using the classic definition, SOHCAHTOA definitions, or I could stick them on the unit circle here to use the unit circle definition of trig functions to figure out what the sine, cosine, or tangent of these angles are. Can I decompose seven pi over 12 into some combination of pi over sixes, pi over threes, or pi over fours? Think about that. Let me rewrite pi over six, pi over three, and pi over four with a denominator over 12. Let me write that. Pi over six is equal to two pi over 12, pi over three is equal to four pi over 12, and pi over four is equal to three pi over 12. Let's see. Two plus four is not seven, two plus three is not seven, but four plus three is seven. So I could use this and this. Four pi over 12 plus three pi over 12 is seven pi over 12. I could rewrite this. This is the same thing as sine of three pi over 12 plus 4 pi over 12, which, of course, is the same thing, sine of pi over 4, I'll do this in another color, sine of pi over 4 plus ... let me do this ... plus pi over three, Now we can use our angle addition formula for sine in order to write this as the sum of products of cosines and sines of these angles. Let's actually do that. This right over here is going to be equal to, this is going to be equal to the sine, the sine of pi over four times the cosine of pi over three plus the other way around, cosine of pi over four times the sine of pi over three, sine of pi over three. Now we just have to figure out these things, and I've already set up the triangles to do it. What is sine of pi over four? Sine of pi over four, well, let's think about ... This is pi over four right over here. Sine is opposite over hypotenuse. That's just going to be square root of two over two. This is square root of two over two, square root of two over two. What is cosine of pi over three? This is a pi over three radian angle right over here. Cosine is adjacent over a hypotenuse. It's adjacent over a hypotenuse, so this is going to be 1/2. What is cosine of pi over four? Go back to pi over four. It's adjacent over a hypotenuse. It's square root of two over two. It is also square root of two over two, square root of two over two. What's sine of pi over three? Sine is opposite over a hypotenuse, so square root of three over two over one. Square root of three over two divided by one, which is square root of three over two. Now we just have to simplify all of this business. This is going to be equal to the sum of this, or the product, I should say, is just square root of two over four, and then plus the product of these. Let's see. We could write that as square root of six over four, square root of six over four, or we could just rewrite this whole thing as, and we deserve a little bit of a drum roll at this point, this is equivalent to, let me just scroll over to the right a little bit. This is equivalent to square root of two plus square root of six, all of that over four. That's what sine of seven pi over 12 is, or cosine of pi over 12, what that is equal to." + }, + { + "Q": "What is the difference b/w oscillation and vibration?", + "A": "Oscillation refers to some sort of back and forth movement. Vibration is not really a word we would use in this context. You would not say a pendulum vibrates, right? but vibration refers vaguely to some sort of jiggling movement.", + "video_name": "ZcZQsj6YAgU", + "transcript": "- [Instructor] Alright, we should talk about oscillators. And what an oscillator is is an object or variable that can move back and forth or increase and decrease, go up and down, left and right, over and over and over. So for instance, a mass on a spring here is an oscillator if we pull this mass back, it's gonna oscillate back and forth, and that's what we mean by an oscillator. Or another common example is a pendulum, and a pendulum is just a mass connected to a string, and you pull the mass back and then it swings back and forth. So you've got something going back and forth, that's an oscillator. These are the two most common types. Masses on springs, pendulum, but there's many other examples and all those examples share one common feature of why they're an oscillator. So you could ask why do these things oscillate in the first place, and it's because they all share this common fact, that they all have a restoring force. And a restoring force, like the name suggests, tries to restore this system, but restore it to what? Restore the system to the equilibrium position. So every oscillator has an equilibrium position, and that would be the point at which there's no net force on the object that's oscillating. So for instance, for this mass, if this mass on the spring was sitting at the equilibrium position, the net force on that mass would be 0 because that's what we mean by the equilibrium position. In other words, if you just sat the mass there it would just stay there because there's no net force on it. However, if I pull this mass to the right, the spring's like uh uh, now I'm gonna try and restore this mass back to the equilibrium position, the spring would pull to the left. If I push this mass to the left, the spring's like uh uh, we're movin' this thing back to the equilibrium position, we're trying to push it back there. So if I push left, the spring pushes right. And if I pull the mass right, the spring pulls left. It tries to restore always, it tries to restore mass back to the equilibrium position. Sam for the pendulum. If I pull the pendulum to the right, gravity is the restoring force trying to bring it back to the left. But if I pull the mass to the left, gravity tries to pull it back to the right, always trying to restore this mass back to the equilibrium position. That's what we mean by a restoring force. Now there's lots of oscillators, but only some of those oscillators are really special, and we give those a special name. We call them Simple Harmonic Oscillators. And you might be thinking, that's a pretty dumb name because that doesn't sound very simple. But they're something called the Simple Harmonic Oscillator. So what makes Simple Harmonic Oscillator's so special is that even though all oscillators have a restoring force, Simple Harmonic Oscillators have a restoring force that's proportional to the amount of displacement. So what that means is if I pull this mass to the right there will be a restoring force, but if it's proportional to the displacement, if I pulled this mass back twice as much, I'd get twice the restoring force. And if I pulled it back three times as much, I'd get three times the restoring force. Same down here. If I pulled this pendulum back with two times the angle, I'd get two times the restoring force. If that's the case, then you've got what we call a Simple Harmonic Oscillator. And you still might not be impressed, you might be like who cares if the restoring force is proportional to the displacement. Why should I care about that? You should care about that because these satisfy some very special rules that I'll show you throughout this video and it turns out that even though this doesn't sound very simple, they are much simpler than the alternative of Non-Simple Harmonic Oscillators. So these are what we typically study in introductory physics classes, and it turns out a mass on a spring is a Simple Harmonic Oscillator, and a pendulum also for small oscillations, here you have to make a caveat, you have to say only for small angles, but for those small angles, the pendulum is a Simple Harmonic Oscillator as well. Now in this video, we're just going to look at the mass on the spring to make it simple. We could look at the pendulum later. So I'm going to get rid of the pendulum so we can focus on this mass on a spring. Now you might not be convinced, you might be like how do we know this mass on a spring is really a Simple Harmonic Oscillator? Well we can prove it because the force that's providing the restoring force in this case is the spring. So the spring is the restoring force in this case, and we know the formula for the force from a spring, that's given by Hooke's Law. And Hooke's Law says that the spring force, the force provided by the spring, is going to be negative. The spring constant times x, the spring displacement, so x is going to be positive if the spring has been displaced to the right because the spring's going to get longer. So this would be a positive x amount. And if you compress the spring, the length of the spring gets smaller, that's going to count as a negative x value. But think about it, if I compress the spring to the left, my x is going to be negative, and that negative combines with this negative to be a positive so I'd get a positive force. That means the spring is there's a force to the right. And that makes sense. Restoring, it means it opposes what you do. If you push the mass to the left, the spring is going to push to the right. And if we did it the other way, if we pulled the mass to the right, now that would be a positive x value. If I have a positive x value in here and combine that with a negative, I'd get a negative spring force. And that means the spring would be pulling to the left, it's restoring this mass back to the equilibrium position. And that's exactly what an oscillator does. And look at it up here, this spring force, this restoring force, is proportional to the displacement. So x is the displacement, this is a force that's proportional to the displacement. And that's the definition. That was what we meant by Simple Harmonic Oscillator. So that's why masses on springs are going to be Simple Harmonic Oscillators, because the restoring force is proportional to the displacement. Now to be completely honest, it has to be negatively proportional to the displacement. If you just had f equals kx with no negative, then if you displaced it to the right, the force would be to the right which would displace it more to the right, which would create a larger force to the right, this would be a runaway solution, this thing would blow up, that wouldn't be good. So it's really forces that have a negative proportionality to the displacement. That way it's going to restore back to the equilibrium position and if this is proportional, you get a Simple Harmonic Oscillator. And so we should talk about this, what the heck do we mean by simple? Like what is simple about this? It turns out that what's simple is that these types of oscillators are going to be described by sin and cosin functions. So Simple Harmonic Oscillators will be described by sin and cosin and that should make sense because think about sin and cosin, what do those look like? Sin and cosin look like this. So here's what sin looks like, it's a function that oscillates back and forth. And cosin looks like this, it starts up here, so it's also a function that oscillates back and forth. And so these are simple, turns out those are very simple functions that oscillate back and forth. And because of that, we like those. In physics, we love things that are described by sin and cosin, it turns out they're pretty easy to deal with mathematically. Maybe you don't feel that way, but they're much easier than the alternatives of other things that could oscillate. So that's what Simple Harmonic Oscillators mean. But let's try to get some intuition, what is really going on for this mass on a spring? So let's imagine we pull the mass back, right? So the mass, if the mass just continues to sit at the equilibrium position, it's a pretty boring problem because the net force right there would be 0 and it would just continue to sit there. So let's say we pull the mass back, we pull it back by a certain amount. Say we pull it back this far, and then we let go. So since we let go of the mass, we've released it at rest. So it started at rest. And that means the speed initially over here is 0. So it starts off with 0 speed, but the spring has been stretched. And so the spring is going to restore, right, the spring is always trying to restore the mass back to the equilibrium position. So the spring pulling the mass to the left, speeding it up, speeds the mass up until it gets to the equilibrium position, and then the spring realizes, oh crud, I messed up. I wanted to get the mass here but I pulled it so much this mass has a huge speed to the left now. And masses don't just stop on their own, They need some force to do that. So this mass has inertia, and according to Newton's First Law, it's going to try and keep moving. So even though the spring got the mass back to the equilibrium position, that was its goal, it got it back there with this huge speed and the mass continues straight through the equilibrium position and the spring starts getting compressed and the spring's like oh no, I've gotta start pushing this thing to the right. I want to get the mass back to the equilibrium position. So now the spring's pushing to the right, slowing the mass down until it stops it, but the spring is compressed, so it's going to keep pushing to the right. Now it's pushing in the direction the mass is moving. Now it's got it going back to the equilibrium position again, which is good, but again, same mistake, the spring gets this mass back to the equilibrium position with a huge speed to the right, and now the spring's like oh great, I did it again, I got this mass back where I wanted it, but this mass had a huge speed and it's got inertia, and so this mass is going to keep moving to the right, past the equilibrium position. And this is why the oscillation happens. It's a constant fight between inertia of the mass wanting to keep moving because it's got mass and it's got velocity, and the restoring force that is desperately trying to get this mass back to the equilibrium position and they can never quite figure it out because they keep overshooting each other and this oscillation happens over and over and over. So just knowing the story, let's you say some really important things about the oscillation. One of them is that at these end points, at these points of maximum compression or extension, the speed is 0. So this mass is moving the slowest, i.e. it's not moving at all at these maximum points of compression or extension because that's where the spring has stopped the mass and started bringing it back in the other direction. Whereas in the middle, at the equilibrium position, you get the most speed. So this is where the mass is moving fastest, when the spring has got it back to the equilibrium position and the spring at that point realizes oh crap, this mass is going really fast, and the mass is coming at it or going away from it too fast for the spring to stop it immediately. So if the equilibrium point this mass has the most speed during the oscillation. So we could also ask where will the magnitude of the restoring force be biggest and where will it be least during this oscillation? And we've got a formula for that. Look at, the spring force is the restoring force. So we could just ask where will the spring force be biggest? That's going to be where this x is biggest or smallest. So if we wanted to know where the magnitude of this f is largest, we could just ask where will the magnitude of the x be largest? If we don't care about which way the force is, we just want to know where we'll get a really big force, we just try to figure out where will I get the biggest x? X is displacement. So the x value at the equilibrium position is 0. So there's no displacement of the spring right here, that's what it means to be the equilibrium position, this is the natural length of the spring. That's the length that the spring wants to be. If the spring has that shape right there, it doesn't push or pull. But if you've displaced it this way, or the other way, this would be positive displacement, and this would be negative displacement, now the spring's going to exert a force. So where will the force be greatest? It's where the spring has been compressed or stretched the most. So at these points here, at the points of maximum extension or compression, you're going to have the greatest amount of force. So greatest magnitude of force, because the spring is really stretched, it's going to pull with a great amount of force back toward the equilibrium position. And we can say which way it points, right? This spring's going to be pulling to the left, so there's going to be a great spring force to the left. Technically that'd be a negative force, so I mean, if you're taking sins into account, you could say that that's the least force because it's really negative. But if you're just worried about magnitude, that would be a great magnitude of force. And then also over here, at the maximum compression, this spring is really pushing the mass to the right, you get a great amount of force this way because your x, even though it's very negative at that point, it's going to give you a large amount of force. And so here you would also have a great amount of magnitude of force which can be confusing because look it. At these end points, you have the least speed, but the greatest force. Sometimes that freaks people out. They're like, how can you have a great force and your speed be so small? Well that's the point where the spring has stopped the mass and started pulling it in the other direction. So even though the speed is 0, the force is greatest. So, be careful, force does not have to be proportional to the speed. The force has to be proportional to the acceleration, right? Because we know net force, we could say that the net force is equal to ma. So wherever you have the largest amount of force, you'll have the largest amount of acceleration. So we could also say at these endpoints, you'll have not only the greatest magnitude of the force, but the greatest magnitude of acceleration as well. Because where you're pulling or pushing on something with the greatest amount of force, you're going to get the greatest amount of acceleration according to Newton's Second Law. So at these endpoints, the force is greatest, the acceleration is also greatest. The magnitude, the acceleration is also greatest even though the speed is 0 at those points. So those are the points where you get the greatest force and greatest acceleration. Where will you get the least amount of magnitude of force and magnitude of acceleration? Well look at up here. The least force will happen where you get the least possible displacement. And the least possible displacement's right here in the middle, this equilibrium position is when x equals 0. That's when the spring is not pushing or pulling. When it's at this point here. So when the mass is passing through the equilibrium position, there is 0 force. Right, that's the point where the mass got back there and the spring was like I'm glad I got it back to the equilibrium position and then the spring quickly realized, oh no, this mass, I got it back there, but the mass was moving really fast, so it shot straight through that point. But right at that moment, the spring had this glorious moment where it thought it had done it and it stopped exerting any force because at that point, the x is 0. And if x is 0, we know from up here, the force is 0. So this would be the least possible force. And I guess I should say it's actually 0 force, it's not just the least, there is 0 force exerted at this point. And if there's 0 force, by the same argument, we could say that there's 0 acceleration at that point. Hopefully that gives you some intuition about why oscillators do what they do and where you might find the largest speed or force at any given point. So recapping, objects with a restoring force that's negatively proportional to the displacement will be a Simple Harmonic Oscillator and for all Simple Harmonic Oscillators, at the equilibrium position you'll get the greatest speed but 0 restoring force and 0 acceleration. Whereas at the points of maximum displacement, you'll get the maximum magnitude of restoring force and acceleration but the least possible speed." + }, + { + "Q": "Is drawing a star with more than 10 and odd number of points possible", + "A": "Maybe. Try it.", + "video_name": "CfJzrmS9UfY", + "transcript": "Let's say you're me, and you're in math class, and you're supposed to be learning about factoring. Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill for the average person to know, with real-world applications ranging from passing your state exams all the way to getting a higher SAT score. And unfortunately, does not have the time to show you why factoring is actually interesting. It's perfectly reasonable for you to get bored in this situation. So like any reasonable person, you start doodling. Maybe it's because your teacher's soporific voice reminds you of a lullaby, but you're drawing stars. And because you're me, you quickly get bored of the usual five-pointed star and get to wondering, why five? So you start exploring. It seems obvious that a five-pointed star is the simplest one, the one that takes the least number of strokes to draw. Sure, you can make a start with four points, but that's not really a star the way you're defining stars. Then there's a six-pointed star, which is also pretty familiar, but totally different from the five-pointed star because it takes two separate lines to make. And then you're thinking about how, much like you can put two triangles together to make a six-pointed star, you can put two squares together to make an eight-pointed star. And any even-numbered star with p points can be made out of two p/2-gons. It is at this point that you realize that if you wanted to avoid thinking about factoring, maybe drawing stars was not the greatest idea. But wait, four would be an even number of points, but that would mean you could make it out of two 2-gons. Maybe you were taught polygons with only two sides But for the purposes of drawing stars, it works out rather well. Sure, the four-pointed star doesn't look too star-like. But then you realize you can make the six-pointed star out of three of these things, and you've got an asterisk, which is definitely a legitimate star. In fact, for any star where the number of points is divisible by 2, you can draw it asterisk style. But that's not quite what you're looking for. What you want is a doodle game, and here it is. Draw p points in a circle, evenly spaced. Pick a number Q. Starting at one point, go around the circle and connect to the point two places over. Repeat. If you get to the starting place before you've covered all the points, jump to a lonely point, and keep going. That's how you draw stars. And it's a successful game, in that previously you were considering running screaming from the room. Or the window was open, so that's an option, too. But now, you're not only entertained but beginning to become curious about the nature of this game. The interesting thing is that the more points you have, the more different ways there is to draw the star. I happen to like seven-pointed stars because there's two really good ways to draw them, but they're still simple. I would like to note here that I've never actually left a math class by the window, not that I can say the same for other subjects. Eight is interesting, too, because not only are there a couple nice ways to draw it, but one's a composite of two polygons, while another can be drawn without picking up the pencil. Then there's nine, which, in addition to a couple of other nice versions, you can make out of three triangles. And because you're me, and you're a nerd, and you like to amuse yourself, you decide to call this kind of star a square star because that's kind of a funny name. So you start drawing other square stars. Four 4-gons, two 2-gons, even the completely degenerate case of one 1-gon. Unfortunately, five pentagons is already difficult to discern. And beyond that, it's very hard to see and appreciate the structure of square stars. So you get bored and move on to 10 dots in a circle, which is interesting because this is the first number where you can make a star as a composite of smaller stars-- that is, two boring old five-pointed stars. Unless you count asterisk stars, in which case 8 was two 4s's or four 2's or two 2's and a 4. But 10 is interesting because you can make it as a composite in more than one way because it's divisible by 5, which itself can be made in two ways. Then there's 11, which can't be made out of separate parts at all because 11 is prime. Though here you start to wonder how to predict how many times around the circle we'll go before getting back to start. But instead of exploring the exciting world of modular arithmetic, you move on to 12, which is a really cool number because it has a whole bunch of factors. And then something starts to bother you. Is a 25-pointed star composite made of five five-pointed stars a square star? You had been thinking only of pentagons because the lower numbers didn't have this question. How could you have missed that? Maybe your teacher said something interesting about prime numbers, and you accidentally lost focus for a moment. I don't know. It gets even worse. 6 squared would be a 36-pointed star made of six hexagons. But if you allow use of six-pointed stars, then it's the same as a composite of 12 triangles. And that doesn't seem in keeping with the spirit of square stars. You'll have to define square stars more strictly. But you do like the idea that there's three ways to make the seventh square star. Anyway, the whole theory of what kind of stars can be made with what numbers is quite interesting. And I encourage you to explore this during your math class." + }, + { + "Q": "why does carbon only bond with hydrogen??", + "A": "It doesn t! It also bonds with oxygen, nitrogen, sulfur, phosphorus and the halogens, along with some other elements. Organic chemistry is a vast subject and this video is concerned solely with alkanes, which are hydrocarbons. You will encounter different types of bonds as you progress but this course starts with the simple stuff and builds up.", + "video_name": "pMoA65Dj-zk", + "transcript": "The one thing that probably causes some of the most pain in chemistry, and in organic chemistry, in particular, is just the notation and the nomenclature or the naming that we use. And what I want to do here in this video and really the next few videos is to just make sure we have a firm grounding in the notation and in the nomenclature or how we name things, and then everything else will hopefully not be too difficult. So just to start off, and this is really a little bit of review of regular chemistry, if I just have a chain of carbons, and organic chemistry is dealing with chains of carbons. Let me just draw a one-carbon chain, so it's really kind of ridiculous to call it a chain, but if we have one carbon over here and it has four valence electrons, it wants to get to eight. That's the magic number we learned in just regular chemistry. For all molecules, that's the stable valence structure, I guess you could say it. A good partner to bond with is hydrogen. So it has four valence electrons and then hydrogen has one valence electron, so they can each share an electron with each other and then they both look pretty happy. I said eight's the magic number for everybody except for hydrogen and helium. Both of them are happy because they're only trying to fill their 1s orbital, so the magic number for those two guys is two. So all of the hydrogens now feel like they have two electrons. The carbon feels like it has eight. Now, there's several ways to write this. You could write it just like this and you can see the electrons explicitly, or you can draw little lines here. So I could also write this exact molecule, which is methane, and we'll talk a little bit more about why it's called methane later in this video. I can write this exact structure like this: a carbon bonded to four hydrogens. And the way that I've written these bonds right here you could imagine that each of these bonds consists of two electrons, one from the carbon and one from the hydrogen. Now let's explore slightly larger chains. So let's say I have a two-carbon chain. Well, let me do a three-carbon chain so it really looks like a chain. So if I were to draw everything explicitly it might So I have a carbon. It has one, two, three, four electrons. Maybe I have another carbon here that has-- let me do the carbons in slightly different shades of yellow. I have another carbon here that has one, two, three, four electrons. And then let me do the other carbon in that first yellow. And then I have another carbon so we're going to have a three-carbon chain. It has one, two, three, four valence electrons. Now, these other guys are unpaired, and if you don't specify it, it's normally going to be hydrogen, so let me draw some hydrogens over here. So you're going to have a hydrogen there, a hydrogen over there, a hydrogen over here, a hydrogen over here, a hydrogen over there, a hydrogen over here, almost done, a hydrogen there, and then a hydrogen there. Now notice, in this molecular structure that I've drawn, I have three carbons. They were each able to form four bonds. This guy has bonds with three hydrogens and another carbon. This guy has a bond with two hydrogens and two carbons. This guy has a bond with three hydrogens and then this carbon And so this is a completely valid molecular structure, but it was kind of a pain to draw all of these valence electrons here. So what we typically would want to do is, at least in this structure, and we're going to see later in this video there's even simpler ways to write it, so if we want at least do it with these lines, we can draw it like this. So you have a carbon, carbon, carbon, and then they are bonded to the hydrogens. So you'll almost never see it written like this because this is just kind of crazy. Hyrdrogen, hydrogen-- at least crazy to write. It takes forever. And it might be messy, like it might not be clear where these electrons belong. I didn't write it as clearly as I could. So they have two electrons there. They share with these two guys. Hopefully, that was reasonably clear. But if we were to draw it with the lines, it looks just like that. So it's a little bit neater, faster to draw, same exact idea here and here. And in general, and we'll go in more detail on it, this three-carbon chain, where everything is a single bond, is propane. Let me write these words down because it's helpful to get. This is methane. And you're going to see the rhyme-- you're going to see the reason to this naming soon enough. This is methane; this is propane. And there's an even simpler way to write propane. You could write it like this. Instead of explicitly drawing these bonds, you could say that this part right here, you could write that that part right there, that is CH3, so you have a CH3, connected to a-- this is a CH2, that is CH2 which is then connected to another CH3. And the important thing is, no matter what the notation, as long as you can figure out the exact molecular structure, as long as you can-- so there's this last CH3. Whether you have this, this, or this, you know what the molecular structure is. You could draw any one of these given any of the others. Now, there's an even simpler way to write this. You could write it just like this. Let me do it in a different color. You literally could write it so we have three carbons. So one, two, three. Now, this seems ridiculously simple and you're like, how can this thing right here give you the same information as all of these more complicated ways to draw it? Well, in chemistry, and in organic chemistry in particular, any of these-- let me call it a line diagram or a line angle diagram. It's the simplest way and it's actually probably the most useful way to show chains of carbons or to show organic molecules. Once they start to get really, really complicated, because then it's a pain to draw all of the H's, but when you see something like this, you assume that the end points of any lines have a carbon on it. So if you see something like that, you assume that there's a carbon at that end point, a carbon at that end point, and a carbon at that end point. And then you know that carbon makes four bonds. There are no charges here. All the carbons are going to make four bonds, and each of the carbons here, this carbon has two bonds, so the other two bonds are implicitly going to be with hydrogens. If they don't draw them, you assume that they're going to be with hydrogens. This guy has one bond, so the other three must be with hydrogen. This guy has one bond, so the other three must be hydrogens. So just drawing that little line angle thing right there, I actually did convey the exact same information as this depiction, this depiction, or this depiction. So you're going to see a lot of this. This really simplifies things. And sometimes you see things that are in between. You might see someone draw it like this, where they'll write CH3, and then they'll draw it like that. So that's kind of combining this way of writing the molecule where you write the CH3's for the end points, but then you implicitly have the CH2 on the inside. You assume that this end point right here is a C and it's bonded to two hydrogens. So these are all completely valid ways of drawing the molecular structures of these carbon chains or of these organic compounds." + }, + { + "Q": "When would you use vectors/vector notation/vector addition & subtraction like this in \"real life\"? My son asked and I had to admit that I have no idea.", + "A": "Pilots use vector addition. To figure out which direction and how fast they are actually traveling they add two vectors. One for the air, and one for the plane. The wind is usually not that strong compared to a jet, but to go straight , you may end up having to point the plane a little to the right or left.", + "video_name": "9ylUcCOTH8Y", + "transcript": "We've already seen that you can visually represent a vector as an arrow, where the length of the arrow is the magnitude of the vector and the direction of the arrow is the direction of the vector. And if we want to represent this mathematically, we could just think about, well, starting from the tail of the vector, how far away is the head of the vector in the horizontal direction? And how far away is it in the vertical direction? So for example, in the horizontal direction, you would have to go this distance. And then in the vertical direction, you would have to go this distance. Let me do that in a different color. You would have to go this distance right over here. And so let's just say that this distance is 2 and that this distance is 3. We could represent this vector-- and let's call this vector v. We could represent vector v as an ordered list or a 2-tuple of-- so we could say we move 2 in the horizontal direction and 3 in the vertical direction. So you could represent it like that. You could represent vector v like this, where it is 2 comma 3, like that. And what I now want to introduce you to-- and we could come up with other ways of representing this 2-tuple-- is another notation. And this really comes out of the idea of what it means to add and scale vectors. And to do that, we're going to define what we call unit vectors. And if we're in two dimensions, we define a unit vector for each of the dimensions we're operating in. If we're in three dimensions, we would define a unit vector for each of the three dimensions that we're operating in. And so let's do that. So let's define a unit vector i. And the way that we denote that is the unit vector is, instead of putting an arrow on top, we put this hat on top of it. So the unit vector i, if we wanted to write it in this notation right over here, we would say it only goes 1 unit in the horizontal direction, and it doesn't go at all in the vertical direction. So it would look something like this. That is the unit vector i. And then we can define another unit vector. And let's call that unit vector-- or it's typically called j, which would go only in the vertical direction and not in the horizontal direction. And not in the horizontal direction, and it goes 1 unit in the vertical direction. So this went 1 unit in the horizontal. And now j is going to go 1 unit in the vertical. So j-- just like that. Now any vector, any two dimensional vector, we can now represent as a sum of scaled up versions of i and j. And you say, well, how do we do that? Well, you could imagine vector v right here is the sum of a vector that moves purely in the horizontal direction that has a length 2, and a vector that moves purely in the vertical direction that has length 3. So we could say that vector v-- let me do it in that same blue color-- is equal to-- so if we want a vector that has length 2 and it moves purely in the horizontal direction, well, we could just scale up the unit vector i. We could just multiply 2 times i. So let's do that-- is equal to 2 times our unit vector i. So 2i is going to be this whole thing right over here or this whole vector. Let me do it in this yellow color. This vector right over here, you could view as 2i. And then to that, we're going to add 3 times j-- so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times j is going to be this vector right over here. And if you add this yellow vector right over here to the magenta vector, you're going to get-- notice, we're putting the tail of the magenta vector at the head of the yellow vector. And if you start at the tail of the yellow vector and you go all the way to the head of the magenta vector, you have now constructed vector v. So vector v, you could represent it as a column vector like this, 2 3. You could represent it as 2 comma 3, or you could represent it as 2 times i with this little hat over it, plus 3 times j, with this little hat over it. i is the unit vector in the horizontal direction, in the positive horizontal direction. If you want to go the other way, you would multiply it by a negative. And j is the unit vector in the vertical direction. As we'll see in future videos, once you go to three dimensions, you'll introduce a k. But it's very natural to translate between these two Notice, 2, 3-- 2, 3. And so with that, let's actually do some vector operations using this notation. So let's say that I define another vector. Let's say it is vector b. I'll just come up with some numbers here. Vector b is equal to negative 1 times i-- times the unit vector i-- plus 4 times the unit vector in the horizontal direction. So given these two vector definitions, what would the would be the vector v plus b be equal to? And I encourage you to pause the video and think about it. Well once again, we just literally have to add corresponding components. We could say, OK, well let's think about what we're doing in the horizontal direction. We're going 2 in the horizontal direction here, and now we're going negative 1. So our horizontal component is going to be 2 plus negative 1-- 2 plus negative 1 in the horizontal direction. And we're going to multiply that times the unit vector i. And this, once again, just goes back to adding the corresponding components of the vector. And then we're going to have plus 4, or plus 3 plus 4-- And let me write it that way-- times the unit vector j in the vertical direction. And so that's going to give us-- I'll do this all in this one color-- 2 plus negative 1 is 1i. And we could literally write that just as i. Actually, let's do that. Let's just write that as i. But we got that from 2 plus negative 1 is 1. 1 times the vector is just going to be that vector, plus 3 plus 4 is 7-- 7j. And you see, this is exactly how we saw vector addition in the past, is that we could also represent vector b like this. We could represent it like this-- negative 1, 4. And so if you were to add v to b, you add the corresponding terms. So if we were to add corresponding terms, looking at them as column vectors, that is going to be equal to 2 plus negative 1, which is 1. 3 plus 4 is 7. So this is the exact same representation as this. This is using unit vector notation, and this is representing it as a column vector." + }, + { + "Q": "how do you solve\n3y-x=-9\n2y+5x=11", + "A": "I did it like this: 3y-x=-9 2y+5x=11 (3y-x=-9) x 5 15y-5x=-45 2y+5x=11 Add the two together. 17y=34 Divide by 17 y=2 Then just go back, substitute for y and solve for x Hope that helped", + "video_name": "uzyd_mIJaoc", + "transcript": "Use substitution to solve for x and y. And we have a system of equations here. The first equation is 2y is equal to x plus 7. And the second equation here is x is equal to y minus 4. So what we want to do, when they say substitution, what we want to do is substitute one of the variables with an expression so that we have an equation and only one variable. And then we can solve for it. Let me show you what I'm talking about. So let me rewrite this first equation. 2y is equal to x plus 7. And we have the second equation over here, that x is equal to y minus 4. So if we're looking for an x and a y that satisfies both constraints, well we could say, well look, at the x and y have to satisfy both constraints, both of these constraints have to be true. So x must be equal to y minus 4. So anywhere in this top equation where we see an x, anywhere we see an x, we say well look, that x by the second constraint has to be equal to y minus 4. So everywhere we see an x, we can substitute it with a y minus 4. So let's do that. So if we substitute y minus 4 for x in this top equation, the top equation becomes 2y is equal to instead of an x, the second constraint tells us that x needs to be equal to y minus 4. So instead of an x, we'll write a y minus 4, and then we have a plus 7. All I did here is I substituted y minus 4 for x. The second constraint tells us that we need to do it. y minus 4 needs to be equal to x or x needs to be equal to y minus 4. The value here is now we have an equation, one equation with one variable. We can just solve for y. So we get 2y is equal to y, and then we have minus 4 plus 7. So y plus 3. We can subtract y from both sides of this equation. The left hand side, 2y minus y is just y. y is equal to-- these cancel out. y is equal to 3. And then we could go back and substitute into either of these equations to solve for x. This is easier right over here, so let's substitute right over here. x needs to be equal to y minus 4. So we could say that x is equal to 3 minus 4 which is equal to negative 1. So the solution to this system is x is equal to negative 1 and y is equal to 3. And you can verify that it works in this top equation 2 times 3 is 6 which is indeed equal to negative 1 plus 7. Now I want to show you that over here we substituted-- we had an expression that, or we had an equation, that explicitly solved for x. So we were able to substitute the x's. What I want to show you is we could have done it the other way around. We could have solved for y and then substituted for the y's. So let's do that. And we could have substituted from one constraint into the other constraint or vice versa. Either way, we would have gotten the same exact answer. So instead of saying x is equal to y minus 4, in that second equation, if we add 4 to both sides of this equation, we get x plus 4 is equal to y. This and this is the exact same constraint. I just added 4 to both sides of this to get this constraint over here. And now since we've solved this equation explicitly for y, we can use the first constraint, the first equation. And everywhere where we see a y, we can substitute it with x plus 4. So it's 2 times-- instead of 2 times y, we can write 2 times x plus 4. 2 times x plus 4 is equal to x plus 7. We can distribute this 2. So we get 2x plus 8 is equal to x plus 7. We can subtract x from both sides of this equation. And then we can subtract 8 from both sides of this equation, subtract 8. The left hand side, that cancels out. We're just left with an x. On the right hand side, that cancels out, and we are left with a negative 1. And then we can substitute back over here we have y is equal to x plus 4, or so y is equal to negative 1 plus 4 which is equal to 3. So once again, we got the same answer even though this time we substituted for y instead of substituting for x. Hopefully you found that interesting." + }, + { + "Q": "Ask a question...can you cancel out the y instead of the x?", + "A": "Yes you can. That would mean that you solve for x first and then substitute the value to get y. Same method.", + "video_name": "u5dPUHjagSI", + "transcript": "We never know when we might have to do a little bit more party planning. So it doesn't hurt to have some practice. And that's what this exercise is doing for us, is generating problems so that we can try solving systems of equations with elimination. And so in this first problem, it says solve for x and y using elimination. And then this is what they have-- 6x minus 6y is equal to negative 24. Negative 5x minus 5y is equal to negative 60. So let me get my scratch pad out to solve this. Let me rewrite it. So they gave us 6x minus 6y is equal to negative 24. And negative 5x minus 5y is equal to negative 60. So what we have to think about, and we saw this in several of the other videos, is when we want to eliminate a variable, we want to manipulate these two equations. And if we were to add the corresponding sides, that variable might disappear. So if we just added a 6x to a negative 5x, that's not going to cancel it out. If this was a negative 6x, that would work out. Or if this was a positive 5x, that would work out. But this isn't exactly right. So if I want to eliminate the x, I have to manipulate these equations so that these two characters might cancel out. And one thing that pops into my brain is it looks like all of this stuff up here is divisible by 6, and all of this stuff down here is divisible by 5. And if we were to divide all this stuff up here by 6, we'd be left with an x over here. And if we were to divide all this bottom stuff by 5, we'd be left with a negative x right over here. And then they just might cancel out. So let's try that out. Let's take this first equation. And we're going to multiply both sides by 1/6. Or another way you could think about it is we're dividing both sides by 6. And as long as we do the same thing to both sides, the equation holds. The equality holds. So if you multiply everything by 1/6, 6x times 1/6 is just going to be x. 6y times 1/6 is just y. So it's negative y. Negative 24 times 1/6 is negative 4. Or you could just view it as negative 24 divided by 6 is negative 4. So this equation, the blue one, we've simplified as x minus y is equal to negative 4. Let's do something similar with the second one. Here we could say we're going to multiply everything times 1/5. Or you could say that we're dividing everything by 5. If we do that, negative 5x divided by 5 is just negative x. Negative 5y divided by 5 is negative y. And then negative 60 divided by 5 is negative 12. And now, this looks pretty interesting. If we add the two left-hand sides-- and remember, we can keep the equality, because we're essentially adding the same thing to both sides. You can imagine we're starting with the blue equation. And on the left-hand side, we're adding negative x minus y. And on the right-hand side, we're adding negative 12. But the second equation tells us that those two things So we're doing the same principle that we saw when we first started looking to algebra, that you can maintain your equality as long as you add the same thing. On the left-hand side, we're going to add this. And on the right-hand side, we're going to add this. But this second equation tells us that those two things are equal. So we can maintain our equality. So let's do that. What do we get on the left-hand side? Well, you have a positive x and a negative x. They cancel out. That was the whole point behind manipulating them in this way. And then you have negative y minus y, which is negative 2y. And then on the right-hand side, you have negative 4 minus 12, which is negative 16. And these are going to be equal to each other. Once again, we're adding the same thing to both sides. To solve for y, we can divide both sides by negative 2. And we are left with y is equal to positive 8. But we are not done yet. We want to go and substitute back into one of the equations. And we can substitute back into this one and to this one, or this one and this one. The solutions need to satisfy all of these essentially. This blue one is another way of expressing this blue equation. This green equation is another way of expressing this green equation. So I'll go for whichever one seems to be the simplest. And this one seems to be pretty simple right over here. So let's take x minus y-- we just solved that y would be positive 8-- is equal to negative 4. And now to solve for x, we just have to add 8 to both sides. And we are left with, on the left-hand side, negative 8 plus 8 cancels out. You're just left with an x. And negative 4 plus 8 is equal to positive 4. So you get x is equal to 4, y is equal to 8. And you can verify that it would work with either one of these equations. 6 times 4 is 24 minus 6 times 8-- so it's 24 minus 48-- is, indeed, negative 24. Negative 5 times 4 is negative 20, minus negative 40, if y is equal to 8, does, indeed, get you negative 60. So it works out for both of these. And we can try it out by inputting our answers. So x is 4, y is 8. So let's do that. So let me type this in. x is going to be equal to 4. y is going to be equal to 8. And let's check our answer. It is correct. Very good." + }, + { + "Q": "Was it already a clear plan for Germans, by the year 1939, to spread over the new countries as a 'superior race'?", + "A": "Not by the Germans- as in their people, nor even the majority of those in the military. But, yes it WAS the plan of the top echelon of Nazi Party Leadership.", + "video_name": "VTdV9JaHiIA", + "transcript": "As we get into the second half of the 1930s, we see an increasingly aggressive Nazi Germany. In 1935, they publicly announce their intent to rearm their military. The reason why this is significant is not that they were all of a sudden building their military. They, in fact, were doing this as soon as they had taken power, in 1933. But now, they felt confident enough to publicly state their intention -- which is another way of publicly stating that they [could n't] care less about the Treaty of Versailles, which had said that Germany was limited to a 100,000-soldier military. Then, we get into 1936. 1936, you might remember -- another term of the Treaty of Versailles was that Germany was not allowed to occupy the Rhineland -- this area in yellow right over here. And then that was actually reaffirmed in 1925 by the Treaties of Locarno, where Germany, itself, agreed to not occupy the Rhineland. But by 1936, Hitler decides to ignore all of those, and occupies the Rhineland. But once again the allies -- The French are not so happy about this. The UK, in particular, once again, [was] not super happy about this. But they decided this is not reason to potentially start another war over. So they really don't push back on Germany. Then, we get into 1938, and German aggression really goes into full gear. In March of 1938, you have a coup d'\u00e9tat, orchestrated by the Nazis in Austria, that really overthrows the Austrian government and allows the Germans to unify the two countries. So, you have the Germans come into Austria -- really a bloodless takeover. And there was already popular support for the Nazis in Austria. There was a Nazi party in Austria. There had been popular sentiment for many years, amongst many Austrians, to possibly be unified with the Germans. Austria [was and] is, fundamentally, a German-speaking nation. And so in March, this actually happens. This 'Anschluss' -- or unification. And, if you remember, that was also another forbidden term of the Treaty of Versailles. So now, the Germans are pretty much ignoring the Treaty of Versailles and the Treaty of St. Germain, which was the equivalent of the Treaty of Versailles -- but with the Austrians. So, you have the unification of Germany and Austria. Then, as we get into late 1938 -- in September in particular -- Hitler and the Nazis are interested in bringing the German-speaking populations of Czechoslovakia under German control. And this region, right over here in magenta, this is where you have large populations of German speakers. These regions are collectively referred to as the 'Sudetenland.' And really, just continuing the policy of not wanting to rock the boat with Germany, you have France, Great Britain, and Italy agreeing -- And Italy was an ally of the Germans. But France and Great Britain, in particular, are not interested in rocking the boat with the Germans. And so, in September of 1938, they sign the Munich Agreement, which did not actually -- where they actually did not consult the Czechoslovakian government -- where they allowed Germany to take over this region right over here -- the Sudetenland. And that, frankly, with the Germans taking over this significant part of the population of Czechoslovakia, a significant part of the industrial capacity of Czechoslovakia, this eventually leads to early 1939, where all of what we would now consider the Czech Republic -- this area right over here, all [of] this -- becomes a protectorate of Germany. So, they call it the 'Protectorate of Bohemia and Moravia.' So, Bohemia and Moravia go to Germany. And, so this is 1939. So, [by] 1939, [we have seen this pattern repeat itself during the previous] four years -- Nazi Germany ignoring the Treaty of Versailles, by rearming, by occupying the Rhineland, by unifying with Austria. Now, they're expanding their territory. They are actively allowed to take over the German-speaking areas of Czechoslovakia, under the Munich agreement. And eventually, they're able to take over Bohemia and Moravia -- all of what we would currently call the 'Czech Republic.' And this general pattern of German aggression, [allowed] by the other powers in Europe essentially allowing it to happen -- and, in particular, Great Britain allowing it to happen -- has been referred to as a 'policy of appeasement.' Obviously, the word 'appeasement' means there is someone who is angry about something, and you just don't want to make them any angrier -- you just let them do whatever they want -- this is, essentially, what was happening over here. And in hindsight, it might be easy to say, \"Hey, look!. They were allowing Germany to take over more and more -- to become more aggressive, which made [Germany] more and more confident. And this would eventually lead to World War II.\" But at the time, you do have to remember [that[ everyone still had a very strong memory of what had happened in World War I. And no one was interested in starting another [pan-European] war. And so, even [though] in hindsight, it's easy to say that the British -- in particular Neville Chamberlain, who was the Prime Minister from 1937 on -- were weak and allowed German[y] -- Hitler -- to gain confidence, which eventually led to the Nazi invasion of Poland in the fall of 1939. But it's easy to say that in hindsight. But what we see, as we get into 1939, is an aggressive Germany, a Germany that's not being checked by the other powers of Europe. And this is what eventually leads to September of 1939, where, actually, the Germans and the Soviets agree to partition Poland into their own spheres of influence, which allows Germany to invade Poland in early September [of] 1939 -- which is, you could kind of say, 'the straw that broke the camel's back,' and is the beginning of -- So, Poland invasion. The invasion of Poland. -- which is the beginning of World War II." + }, + { + "Q": "what is a fibonacci", + "A": "Fibonacci was a person for whom the series is named. The series is F2 = F1 + F0 as in 0 0 + 1 = 1 1 + 1 = 2 2 + 1 = 3 3 + 2 = 5 5 + 3 = 8", + "video_name": "ahXIMUkSXX0", + "transcript": "Voiceover:Say [unintelligible], you're in math class and your teacher's talking about ... Well, who knows what your teacher's talking about. Probably a good time to start doodling. And you're feeling spirally today, so yeah. Oh, and because of overcrowding in your school, your math class is taking place in greenhouse number three. Plants. You've decided there are three basic types of spirals. There's the kind where, as you spiral out, you keep the same distance. Or you could start big but make it tighter and tighter as you go around, in which case the spiral ends. Or you could start tight but make the spiral bigger as you go out. The first kind is good if you really want to fill up a page with lines. Or if you want to draw curled up snakes. You can start with a wonky shape to spiral around but you've noticed that, as you spiral out, it gets rounder and rounder. Probably something to do with how the ratio between two different numbers approaches one as you repeatedly add the same number to both. But you can bring the wonk back by exaggerating the bumps and it gets all optical illusiony. Anyway, you're not sure what the second kind of spiral is good for, but I guess it's a good way to draw snuggled up slug cats, which are a species you've invented just to keep this kind of spiral from feeling useless. This third spiral, however, is good for all sorts of things. You could draw a snail or a nautilus shell. And elephant with a curled up trunk, the horns of a sheep, a fern frond, a cochlea in an inner ear diagram, an ear itself. Those other spirals can't help but be jealous of this clearly superior kind of spiral. But I draw more slug cats. Here's one way to draw a really perfect spiral. Start with one square and draw another next to it that is the same height. Make the next square fit next to both together, that is each side is length two. The next square has length three. The entire outside shape will always be a rectangle. Keep spiraling around, adding bigger and bigger squares. This one has side length one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13. And now 21. Once you do that you can add a curve going through each square, arcing from one corner to the opposite corner. Resist the urge to zip quickly across the diagonal, if you want a nice smooth spiral. Have you ever looked at the spirally pattern on a pine cone and thought, \"Hey, sure are \"spirals on this pine cone?\" I don't know why there's pine cones in your greenhouse. Maybe the greenhouse is in a forest. Anyway, there's spirals and there's not There's one, two, three, four, five, six, seven, eight going this way. Or you could look at the spirals going the other way and there's one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13. Look familiar? Eight and 13 are both numbers in the Fibonacci series. That's the one where you start by adding one and one to get two, then one and two to get three, two and three to get five. Three plus five is eight, five plus eight is 13, and so on. Some people think that instead of starting with one plus one you should start with zero and one. Zero plus one is one, one plus one is two, one plus two and three, and it continues on the same way as starting with one and one. Or, I guess you could start with one plus zero and that would work too. Or why not go back one more to negative one and so on? Anyway, if you're into the Fibonacci series, you probably have a bunch memorized. I mean, you've got to know one, one, two, three, five. Finish off the single digits with eight and, ooh with 13, how spooky. And once you're memorizing double digits, you might as well know 21, 34, 55, 89 so that whenever someone turns a Fibonacci number you can say, \"Happy Fib Birthday.\" And then, isn't it interesting that 144, 233, 377? But 610 breaks that pattern, so you'd better know that one too. And oh my goodness, 987 is a neat number and, well, you see how these things get out of hand. Anyway, 'tis the season for decorative scented pine cones and if you're putting glitter glue spirals on your pine cones during math class, you might notice that the number of spirals are five and eight or three and five or three and five again. Five and eight. This one was eight and thirteen and one Fibonacci pine cone is one thing, but all of them? What is up with that? This pine cone has this wumpy weird part. Maybe that messes it up. Let's count the top. Five and eight. Now let's check out the bottom. Eight and 13. If you wanted to draw a mathematically realistic pine cone, you might start by drawing five spirals one way and eight going the other. I'm going to mark out starting and ending points for my spirals first as a guide and then draw the arms. Eight one way and five the other. Now I can fill in the little pine coney things. So there's Fibonacci numbers in pine cones but are there Fibonacci numbers in other things that start with pine? Let's count the spirals on this thing. One, two, three, four, five, six, seven, eight. And one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13. The leaves are hard to keep track of, but they're in spirals too. Of Fibonacci numbers. What if we looked at these really tight spirals going almost straight up? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21. A Fibonacci number. Can we find a third spiral on this pine cone? Sure, go down like this. And one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13 (muttering) 19, 20, 21. But that's only a couple examples. How about this thing I found on the side of the road? I don't know what it is. It probably starts with pine, though. Five and eight. Let's see how far the conspiracy goes. What else has spirals in it? This artichoke has five and eight. So does this artichoke looking flower thing. And this cactus fruit does too. Here's an orange cauliflower with five and eight and a green one with five and eight. I mean, five and eight. Oh, it's actually five and eight. Maybe plants just like these numbers though. Doesn't mean it has anything to do with Fibonacci, does it? So let's go for some higher numbers. We're going to need some flowers. I think this is a flower. It's got 13 and 21. These daisies are hard to count, but they have 21 and 34. Now let's bring in the big guns. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34. And one, two, three, four, five, six, seven, eight, nine, 10, 11, (muttering) 17, 24, (muttering) 42, 53, 54, 55. I promise, this is a random flower and I didn't pick it out specially to trick you into thinking there's Fibonacci numbers in things, but you should really count for yourself next time you see something spirally. There's even Fibonacci numbers in how the leaves are arranged on this stalk, or this one, or the Brussels sprouts on this stalk are a beautiful delicious three and five. Fibonacci is even in the arrangement of the petals on this rose, and sunflowers have shown Fibonacci numbers as high as 144. It seems pretty cosmic and wondrous, but the cool thing about the Fibonacci series and spiral is not that it's this big complicated mystical magical super math thing beyond the comprehension of our puny human minds that shows up mysteriously everywhere. We'll find that these numbers aren't weird at all. In fact, it would be weird if they weren't there. The cool thing about it is that these incredibly intricate patterns can result from utterly simple beginnings." + }, + { + "Q": "what is 0 times 0", + "A": "The general rule is 0 times anything is 0. So 0 * 0 = 0", + "video_name": "lR_kUUPL8YY", + "transcript": "We have the number 7,346,521.032. And what I want to think about is if I look at the same digit in two different places, in particular, I'm going to look at the digit 3 here and the digit 3 here, how much more value does this left 3 represent than this right 3? In order to think about that, we have to think about place value. So let's write down all the place values. So this right over here, this is the ones place. Now, we could move to the right. And as we move to the right in place values, each place represents 1/10 of the place before it. Or you could divide by 10 as we're moving to the right. So this is the ones place. This is divide by 10. This is the 1/10 place, or the tenths place. Divide by 10 again, this is the hundredths place. Divide by 10 again, this is the thousandths place. And that \"s\" I'm just saying to be plural-- hundredths, thousandths. Now, if we go to the left, now each place represents a factor of 10 more. So if this is ones, multiply by 10, this is the tens place. This is the hundreds place. This is the thousands place. This is the ten thousands place. I'm going to have to write a little bit smaller. This is the hundred thousands place. And then the 7 is in the millions place. So what does this number, what does this 3 represent? Well, it's in the hundred thousands place. It literally represents 3 hundred thousands, or you could say 300,000, 3 followed by five zeroes. Now, what does this 3 represent? It's in the hundredths place. It literally represents 3 hundredths. It represents 3 times 1/100, which is the same thing as 3, which is equal to 3 over-- let me do the 3 in that purple color. Which is the same thing as 3/100, which is the same thing as 0.03. These are all equivalent statements. Now let's try to answer our original question. How much larger is this 3 than that 3 there? Well, one way to think about it is how much would you have to multiply this 3 by to get to this 3 over here? Well, one way to think about is to look directly at place value. So we got to multiply by 10. Every time we multiply by 10, that's equivalent to thinking about shifting it to one place to the left. So we would have to multiply by 10 one, two, three, four, five, six, seven times. So multiplying by 10 seven times. Let me write this down. So this multiplied by 10 seven times should be equal to this. Let me rewrite this. 300,000 should be equal to 3/100-- let me write it the same way. 3/100 multiplied by 10 seven times, so times 10 times 10 times 10 times 10 times 10-- let's see, that's five times-- times 10 times 10. Now, multiplying by 10 seven times is the same thing as multiplying by 1 followed by seven zeroes. Every time you multiply by 10, you're going to get another zero here. So this is the same thing as 3/100 times 1 followed by one, two, three, four, five, six, seven zeroes. So this is literally 3/100 times 10 million. So let's see if this actually is the case. Does this actually equal 300,000? Well, if you divide 10 million by 100, dividing 10 million by 100, or I guess you'd say in the numerator, you have 10 million and the denominator you have 100, if you were to just multiply it like this, if you view this as 3 over 100 times 10 million over 1. Well, you divide the numerator by 100, you're going to get rid of two of these zeroes. Divide the denominator by 100, you're going to get rid of this 100 here. And so you're going to be left with 3 times-- now we got to be careful with the commas here, because since I removed two zeroes, the commas are going to be different. It's going to be 3 times-- we put our commas in the right place, so just like that. So this simplified to 3 times 100,000, which is indeed 300,000. So it did work out. Shifting the 3 one, two, three, four, five, six, seven decimal places makes that 3 worth 1 followed by seven zeroes more, or it essentially makes that 3 worth 10 million more. So this 3 represents 10 million times the value of this 3. Let me write down the numbers. So this 3 is 10 million times the value of that 3." + }, + { + "Q": "So do weight and gravitational force mean the same ?explain?", + "A": "Weight is the force of gravity on an object.", + "video_name": "IuBoeDihLUc", + "transcript": "In this video, I want to clarify two ideas that we talk about on a regular basis, but are really muddled up in our popular language. And these are the ideas of mass and weight. And first, I'll tell you what they are. And then, we'll talk about how they are muddled up. So mass is literally-- there's a couple of ways to view mass. One way to view mass is-- and this is not a technical definition, but it will give you a sense of it-- is how much the stuff there is. So this is similar to saying matter. So if I have more molecules of a given mass, I will have a total of more mass. Or if I have more atoms, I will have more mass. So how much stuff there is of something. And I want to be careful with this definition right here, because there are other things that aren't what we would traditionally associate with matter, once we start going into more fancy physics, that still will exhibit mass. So another way to define mass is, how does something react to a specific force? And we already learned from Newton's second law that if you have a given force and you have more mass, you'll accelerate less. If you have less mass, you'll accelerate more. So how something responds to a given force. Something with lower mass will accelerate more for a given force. Something with higher mass will accelerate less. Now weight is the force of gravity on a mass, or on an object. So this is the force of gravity on an object. And just to think about the difference here, let's think about, I guess, myself sitting on Earth. So if I'm on Earth, my mass is 70 kilograms. My mass-- let me do this in a new color-- so my mass is 70 kilograms. There's 70 kilograms of stuff that constitute Sal. But my weight is not 70 kilograms. I mean, you'll often hear people say, I weighs 70 kilograms. And that's all right in just conversational usage, but that is not technically correct. Because weight is the force that Earth is pulling down-- or I should say-- the force of gravity on my mass. And so my weight-- let me think about the weight for a second-- the weight is going to be equal to the gravitational field at Earth. Hopefully you've watched the video on gravity. Or if you haven't, feel free to watch it. But the gravitational attraction between two objects, so the force of gravity between two objects is going to big G, the universal gravitational constant, times the mass of the first object-- let me actually-- times the mass of the second object, divided by the distance that separates the two objects squared. And if you're on the Earth, and if you take all of this stuff right over here combined-- so if you say that this right here is the mass of Earth. If you say this, right here, is the distance from the center to the surface of Earth, because that's where I'm sitting right now. So distance from center to surface of Earth. Then all of this stuff over here simplifies to what's sometimes called as lower case g. And lower case g is-- and I'm just rounding it here-- 9.8 meters per second squared. So the force of gravity for something near the surface of the Earth is going to be this quantity right over here times the mass. So my weight on the surface of the Earth is this 9.8 meters per second squared times my mass, times 70 kilograms. And so this is going to be-- I won't do, well, I could get my calculator out. Why don't I just get my calculator out and do the math? I was going to round it to 10 and say it's about 700, but let's just actually calculate it. So we have 9.8 times 70 kilograms. So we have 686. So this gives me 686. And then the units are kilogram meters per second squared. And these units, kilogram meters per second squared, are the same thing as a newton. So my weight-- and you'll never hear people say this-- but my weight on the surface of the Earth is 686 newtons. And notice, I just said that is my weight on the surface of the Earth. Because as you could imagine, weight is the force due to gravity on an object, on a mass. So if I go someplace else, if I go to the moon, for example, my weight will change. But my mass will not. So let's write this. This is the weight on Earth. If I were to take my weight on the moon-- and I haven't looked this up before the video. And you can verify this for me if you like. But I've been told that the gravitational force on the moon, or the gravitational attraction at the surface of the moon, is about 1/6 that of the surface of the Earth. So my weight on the moon will be roughly 1/6 of my weight on the Earth. Times 686 newtons. So that gives me a little bit over-- what is that-- 114 maybe? I'll just get the calculator out. My brain operates a little bit slower while I'm recording videos. Yeah. 114. So that gets us 114, approximately 114 newtons. So this is the thing I really want to emphasize then. Weight is a force due to gravity on an object. Your weight changes from planet to planet you go on. Your weight would actually even change if you went to a very high altitude because you're getting slightly further-- it would be immeasurably small-- but you're getting slightly further from the center of the Earth. Your weight would change an imperceptible amount. In fact, because the Earth is not a perfect sphere-- it's often referred to as an oblique spheroid-- your weight is actually slightly different on different parts If you went to the poles verses the equator, you would have a slightly different weight. Your mass does not change. It doesn't matter where you go, assuming that you don't have some type of nuclear reaction going on inside of you. Your mass does not change. So your mass does not change depending on where you are. Now, you might be saying, hey, look, I don't deal with kilograms, and newtons, and all of this. I operate in America. And in America, we talk about pounds. Is pounds appropriate? And yes, pounds is a unit of weight. So if I say that I weigh 160 pounds, this is indeed weight. I'm saying that the force of gravity on me is 160 pounds. But then you might say, well, what is mass then if you're talking about the English system or sometimes called the imperial system? And here, I will introduce you to a concept that very few people know. It's kind of a good trivial concept. The unit of mass-- so let's just be clear here-- the unit of mass in the imperial system, mass is called the slug. So if you wanted to figure out how many slugs you are, so your weight-- the force of gravity on you is 160 pounds. This is going to be equal to-- if you were to calculate all of this stuff-- the force of gravity on the surface of Earth. But if you were to do it in imperial units, instead of getting 9.8 meters per second squared, you would get 32 feet per second squared, which is also the acceleration near the surface of the Earth due to gravity in feet and seconds, as opposed to meters and seconds. And then, this is times your mass in slugs. So to figure it out, you divide both sides by 32 feet per second squared. So let's do that. Let's divide both sides by 32 feet per second squared. It cancels out. And then let me get my calculator out. So I have 160 pounds divided by 32 feet per second squared. And I get exactly 5. I should have been able to do that in my head. So I get 5. And the units here-- in the numerator, I have pounds. And then I'm dividing by feet per second squared. That's the same thing as multiplying by second squared per feet. And these units, 5 pounds second squareds over feet, this is the same thing as a slug. So if I weigh 160 pounds, my mass is going to be equal to 5 slugs. If my mass is 70 kilograms, my weight is 686 newtons. So hopefully that clarifies things a little bit." + }, + { + "Q": "Solve x5 y5=x-y", + "A": "If you re saying x = 5 and y = 5, the answer to x - y (or even y - x) is 0. Hope this helps! :D", + "video_name": "p5e5mf_G3FI", + "transcript": "We have the equation negative 16 is equal to x over 4, plus 2. And we need to solve for x. So we really just need to isolate the x variable on one side of this equation, and the best way to do that is first to isolate it-- isolate this whole x over 4 term from all of the other terms. So in order to do that, let's get rid of this 2. And the best way to get rid of that 2 is to subtract it. But if we want to subtract it from the right-hand side, we also have to subtract it from the left-hand side, because this is an equation. If this is equal to that, anything we do to that, we also have to do to this. So let's subtract 2 from both sides. So you subtract 2 from the right, subtract 2 from the left, and we get, on the left-hand side, negative 16 minus 2 is negative 18. And then that is equal to x over 4. And then we have positive 2 minus 2, which is just going to be 0, so we don't even have to write that. I could write just a plus 0, but I think that's a little unnecessary. And so we have negative 18 is equal to x over 4. And our whole goal here is to isolate the x, to solve for the x. And the best way we can do that, if we have x over 4 here, if we multiply that by 4, we're just going to have an x. So we can multiply that by 4, but once again, this is an equation. Anything you do to the right-hand side, you have to do to the left-hand side, and vice versa. So if we multiply the right-hand side by 4, we also have to multiply the left-hand side by 4. So we get 4 times negative 18 is equal to x over 4, times 4. The x over 4 times 4, that cancels out. You divide something by 4 and multiply by 4, you're just going to be left with an x. And on the other side, 4 times negative 18. Let's see, that's 40. Well, let's just write it out. So 18 times 4. If we were to multiply 18 times 4, 4 times 8 is 32. 4 times 1 is 4, plus 1 is 72. But this is negative 18 times 4, so it's negative 72. So x is equal to negative 72. And if we want to check it, we can just substitute it back into that original equation. So let's do that. Let's substitute this into the original equation. So the original equation was negative 16 is equal to-- instead of writing x, I'm going to write negative 72-- is equal to negative 72 over 4 plus 2. Let's see if this is actually true. So this right-hand side simplifies to negative 72 divided by 4. We already know that that is negative 18. So this is equal to negative 18 plus 2. This is what the equation becomes. And then the right-hand side, negative 18 plus 2, that's negative 16. So it all comes out true. This right-hand side, when x is equal to negative 72, does indeed equal negative 16." + }, + { + "Q": "I think it would be helpful to define what you mean by d.v. and i.v. before launching into the question.", + "A": "Thats easy D.v means Dependent variable and I.v means independent variable", + "video_name": "i9j_VUMq5yg", + "transcript": "On your math quiz, you earn 5 points for each question that you answer correctly. In the table above, q represents the number of questions that you answer correctly on your math quiz, and p represents the total number of points that you score on your quiz. The relationship between these two variables can be expressed by the following equation-- p is equal to 5q, where p is the points you get and q is the number of questions you answered correctly. And you could see that in the table. If q is 0, if you got no questions right, you get 0 points. If you got no questions right, well, 5 times 0 is going to be 0. If you get one question right, well, 1 times 5 is 5. You get 5 points per question. Two questions right, well, 2 times 5 is 10. 3 times 5 is 15. So this all makes sense. So then they ask us, which of the following statements Check all that apply. So let's think about this. They say the dependent variable is the number of points you score. So when you think about what's happening here, is your number of points you score is being driven by how many questions you get right. It's not like somehow the teacher says you got 15 points and now you have to get exactly three questions right. It's the other way around. The number of questions you get right is the independent variable, and that's driving the number of points you score. So the number of points you score is the dependent variable. And typically, the convention is to have the dependent variable be equal to some expression involving the independent variable. And you see that right over here. p is dependent on what happens to q. Depending on the number of questions, you multiply it by 5, and you get p. So the dependent variable is the number of points you score. The dependent variable is the number of questions you answer correctly. No, we've already talked about that. That's the independent variable. The independent variable is the number of points you score. That's the dependent variable. The independent variable is a number of questions you answer correctly. That's what's driving the dependent variable. And we can check our answer." + }, + { + "Q": "does sound travel fast in solid, liquid or gases and why", + "A": "think about the distance beween molecules and the stiffness of the bonds. Then ask again if you are not sure", + "video_name": "yF4cvbAYjwI", + "transcript": "- [Voiceover] To change the speed of sound you have to change the properties of the medium that sound wave is traveling through. There's two main factors about a medium that will determine the speed of the sound wave through that medium. One is the stiffness of the medium or how rigid it is. The stiffer the medium the faster the sound waves will travel through it. This is because in a stiff material, each molecule is more interconnected to the other molecules around it. So any disturbance gets transmitted faster down the line. The other factor that determines the speed of a sound wave is the density of the medium. The more dense the medium, the slower the sound wave will travel through it. This makes sense because if a material is more massive it has more inertia and therefore it's more sluggish to changes in movement or oscillations. These two factors are taken into account with this formula. V is the speed of sound. Capital B is called the bulk modulus of the material. The bulk modulus is the official way physicists measure how stiff a material is. The bulk modulus has units of pascals because it's measuring how much pressure is required to compress the material by a certain amount. Stiff, rigid materials like metal would have a large bulk modulus. More compressible materials like marshmallows would have a smaller bulk modulus. Row is the density of the material since density is the mass per volume, the density gives you an idea of how massive a certain portion of the material would be. For example, let's consider a metal like iron. Iron is definitely more rigid and stiff than air so it has a much larger bulk modulus than air. This would tend to make sound waves travel faster through iron than it does through air. But iron also has a much higher density than air, which would tend to make sound waves travel slower through it. So which is it? Does sound travel faster though iron or slower? Well it turns out that the higher stiffness of iron more than compensates for the increased density and the speed of sound through iron is about 14 times faster than through air. This means that if you were to place one ear on a railroad track and someone far away struck the same railroad track with a hammer, you should hear the noise 14 times faster in the ear placed on the track compared to the ear just listening through the air. In fact, the larger bulk modulus of more rigid materials usually compensates for any larger densities. Because of this fact, the speed of sound is almost always faster through solids than it is through liquids and faster through liquids than it is through gases because solids are more rigid than liquids and liquids are more rigid than gases. Density is important in some aspects too though. For instance, if you heat up the air that a sound wave is travelling through, the density of the air decreases. This explains why sound travels faster through hotter air compared to colder air. The speed of sound at 20 degrees Celsius is about 343 meters per second, but the speed of sound at zero degrees Celsius is only about 331 meters per second. Remember, the only way to change the speed of sound is to change the properties of the medium it's travelling in and the speed of sound is typically faster through solids than it is through liquids and faster through liquids than it is through gases." + }, + { + "Q": "aren't sound waves supposed to be longitudinal or am I missing something", + "A": "yep but when theyre displayed on an oscilloscope, they still come up the same way trasverse waves would- the amplitudes represent rarefaction and compression", + "video_name": "oTjTXS40pqs", + "transcript": "- [Instructor] So imagine you've got a wave source. This could be a little oscillator that's creating a wave on a string, or a little paddle that goes up and down that creates waves on water, or a speaker that creates sound waves. This could be any wave source whatsoever creates this wave, a nice simple harmonic wave. Now let's say you've got a second wave source. If we take this wave source, the second one, and we put it basically right on top of the first one, we're gonna get wave interference because wave interference happens when two waves overlap. And if we want to know what the total wave's gonna look like we add up the contributions from each wave. So if I put a little backdrop in here and I add the contributions, if the equilibrium point is right here, so that's where the wave would be zero, the total wave can be found by adding up the contributions from each wave. So if we add up the contributions from wave one and wave two wave one here has a value of one unit, wave two has a value of one unit. One unit plus one unit is two units. And then zero units and zero units is still zero. Negative one and negative one is negative two, and you keep doing this and you realize wait, you're just gonna get a really big cosine looking wave. I'm just gonna drop down to here. We say that these waves are constructively interfering. We call this constructive interference because the two waves combined to construct a wave that was twice as big as the original wave. So when two waves combine and form a wave bigger than they were before, we call it constructive interference. And because these two waves combined perfectly, sometimes you'll hear this as perfectly constructive or totally constructive interference. You could imagine cases where they don't line up exactly correct, but you still might get a bigger wave. In that case, it's still constructive. It might not be totally constructive. So that was constructive interference. And these waves were constructive? Think about it because this wave source two looked exactly like wave source one did, and we just overlapped them and we got double the wave, which is kinda like alright, duh. That's not that impressive. But check this out. Let's say you had another wave source. A different wave source two. This one is what we call Pi shifted 'cause look at it. Instead of starting at a maximum, this one starts at a minimum compared to what wave source one is at. So it's 1/2 of a cycle ahead of or behind of wave source one. 1/2 of a cycle is Pi because a whole cycle is two Pi. That's why people often call this Pi shifted, or 180 degrees shifted. Either way, it's out of phase from wave source one by 1/2 of a cycle. So what happens if we overlap these two? Now I'm gonna take these two. Let's get rid of that there, let's just overlap these two and see what happens. I'm gonna overlap these two waves. We'll perform the same analysis. I don't even really need the backdrop now because look at. I've got one and negative one. One and negative one, zero. Zero and zero, zero. Negative one and one, zero. Zero and zero, zero and no matter where I'm at, 1/2, a negative 1/2, zero. These two waves are gonna add up to zero. They add up to nothing, so we call this destructive interference because these two waves essentially destroyed each other. This seems crazy. Two waves add up to nothing? How can that be the case? Are there any applications of this? Well yeah. So imagine you're sitting on an airplane and you're listening to the annoying roar of the airplane engine in your ear. It's very loud and it might be annoying. You put on your noise canceling headphones, and what those noise canceling headphones do? They sit on your ear, they listen to the wave coming in. This is what they listen to. This sound wave coming in, and they cancel off that sound by sending in their own sound, but those headphones Pi shift the sound that's going into your ear. So they match that roar of the engine's frequency, but they send in a sound that's Pi shifted so that they cancel and your ear doesn't hear anything. Now it's often now completely silent. They're not perfect, but they work surprisingly well. They're essentially fighting fire with fire. They're fighting sound with more sound, and they rely on this idea of destructive interference. They're not perfectly, totally destructive, but the waves I've drawn here are totally destructive. If they were to perfectly cancel, we'd call that total destructive interference, or perfectly destructive interference. And it happens because this wave we sent in was Pi shifted compared to what the first wave was. So let me show you something interesting if I get rid of all this. Let me clean up this mess. If I've got wave source one, let me get wave source two back. So this was the wave that was identical to wave source one. We overlap 'em, we get constructive interference because the peaks are lining up perfectly with the peaks, and these valleys or troughs are matching up perfectly with the other valleys or troughs. But as I move this wave source too forward, look at what happens. They start getting out of phase. When they're perfectly lined up we say they're in phase. They're starting to get out of phase, and look at when I move it forward enough what was a constructive situation, becomes destructive. Now all the peaks are lining up with the valleys, they would cancel each other out. And if I move it forward a little more, it lines up perfectly again and you get constructive, move it more I'm gonna get destructive. Keep doing this, I go from constructive to destructive over and over. So in other words, one way to get constructive interference is to take two wave sources that start in phase, and just put them right next to each other. And a way to get destructive is to take two wave sources that are Pi shifted out of phase, and put them right next to each other, and that'll give you destructive 'cause all the peaks match the valleys. But another way to get constructive or destructive is to start with two waves that are in phase, and make sure one wave gets moved forward compared to the other, but how far forward should we move these in order to get constructive and destructive? Well let's just test it out. When they're right next to each other we get constructive. If I move this second wave source that was initially in phase all the way to here, I get constructive again. How far did I move it? I moved it this far. The front of that speaker moved this far. So how far was that? Let me get rid of this. That was one wavelength. So look at this picture. From peak to peak is exactly one wavelength. We're assuming these waves have the same wavelength. So notice that essentially what we did, we made it so that the wave from wave source two doesn't have to travel as far to whatever's detecting the sound. Maybe there's an ear here, or some sort of scientific detector detecting the sound. Wave source two is now only traveling this far to get to the detector, whereas wave source one is traveling this far. In other words, we made it so that wave source one has to travel one wavelength further than wave source two does, and that makes it so that they're in phase and you get constructive interference again. But that's not the only option, we can keep moving wave source two forward. We move it all the way to here, we moved it another wavelength forward. We again get constructive interference, and at this point, wave source one is having to make its wave travel two wavelengths further than wave source two does. And you could probably see the pattern. No matter how many wavelengths we move it forward, as long as it's an integer number of wavelengths we again get constructive interference. So something that turns out to be useful is a formula that tells us alright, how much path length difference should there be? So if I'm gonna call this X two, the distance that the wave from wave source two has to travel to get to whatever's detecting that wave. And the distance X one, that wave source one has to travel to get to that detector. So we could write down a formula that relates the difference in path length, I'll call that delta X, which is gonna be the distance that wave one has to travel minus the distance that wave two has to travel. And given what we saw up here, if this path length difference is ever equal to an integer number of wavelengths, so if it was zero that was when they were right next to each other, you got constructive. When this difference is equal to one wavelength, we also got constructive. When it was two wavelengths, we got constructive. It turns out any integer wavelength gives us constructive. So how would we get destructive interference then? Well let's continue with this wave source that originally started in phase, right? So these two wave sources are starting in phase. How far do I have to move it to get destructive? Well let's just see. I have to move it 'til it's right about here. So how far did the front of that speaker move? It moved about this far, which if I get rid of that speaker you could see is about 1/2 of a wavelength. From peak to valley, is 1/2 of a wavelength, I can keep moving it forward. Let's just see, that's constructive. My next destructive happens here which was an extra this far. How far was that? Let's just see. That's one wavelength, so notice at this point, wave source one is having to go one and 1/2 wavelengths further than wave source two does. So let's just keep going. Move wave source two, that's constructive. We get another destructive here which is an extra this far forward, and that's equal to one more wavelength. So if we get rid of this you could see valley to valley is a whole nother wavelength. So in this case, wave source two has to travel two and 1/2 wavelengths farther than wave source two. Any time wave source one has to travel 1/2 integer more wavelengths than wave source two, you get destructive interference. In other words, if this path length difference here is equal to lambda over two, three lambda over two, which is one and 1/2 wavelengths. Five lambda over two, which is two and 1/2 wavelengths, and so on, that leads to destructive interference. So this is how the path length differences between two wave sources can determine whether you're gonna get constructive or destructive interference. But notice we started with two wave sources that were in phase. These started in phase. So this whole analysis down here assumes that the two sources started in phase with each other, i.e. neither of them are Pi shifted. What would this analysis give you if we started with one that was Pi shifted? So let's get rid of this wave two. Let's put this wave two back in here. Remember this one? This one was Pi shifted relative to relative to wave source one. So if we put this one in here, and we'll get rid of this, now when these two wave sources are right next to each other you're getting destructive interference. So this time for a path length difference of zero, right? These are both traveling the same distance to get to the detector. So X one and X two are gonna be equal. You subtract them, you'd get zero. This time the zero's giving us destructive instead of constructive. So let's see what happens if we move this forward, let's see how far we've gotta move this forward to again get destructive. We'd have to move it over to here. How far did we move it? Let's just check. We moved the front of this speaker that far, which is one whole wavelength. So if we get rid of this, we had to move the front of the speaker one whole wavelength, and look at again it's destructive. So again, zero gave us destructive this time, and the lambda's giving us destructive, and you realize oh wait, all of these integer wavelengths. If I move it another integer wavelength forward, I'm again gonna get destructive interference because all these peaks are lining up with valleys. So interestingly, if two sourcese started Pi out of phase, so I'm gonna change this. Started Pi out of phase, then path length differences of zero, lambda, and two lambda aren't gonna give us constructive, they're gonna give us destructive. And so you could probably guess now, what are these path length differences of 1/2 integer wavelengths gonna give us? Well let's just find out. Let's start here, and we'll get rid of these. Let's just check. We'll move this forward 1/2 of a wavelength and what do I get? Yup, I get constructive. So if I move this Pi shifted source 1/2 a wavelength forward instead of giving me destructive, it's giving me constructive now. And if I move it so it goes another wavelength forward over to here, notice this time wave source one has to move one and 1/2 wavelengths further than wave source two. That's 3/2 wavelengths. But instead of giving us destructive, look. These are lining up perfectly. It's giving us constructive, and you realize oh, all these 1/2 integer wavelength path length differences, instead of giving me destructive are giving me constructive now because one of these wave sources was Pi shifted compared to the other. So I can take this here, and I could say that when the two sources start Pi out of phase, instead of leading to destructive this is gonna lead to constructive interference. And these two ideas are the foundation of almost all interference patterns you find in the universe, which is kind of cool. If there's an interference pattern you see out there, it's probably due to this. And if there's an equation you end up using, it's probably fundamentally based on this idea if it's got wave interference in it. So I should say one more thing, that sources don't actually have to start out of phase. Sometimes they travel around. Things happen, it's a crazy universe. Maybe one of the waves get shifted along its travel. Regardless, if any of them get a Pi shift either at the beginning or later on, you would use this second condition over here to figure out whether you get constructive or destructive. If neither of them get a phase shift, or interestingly, if both of them get a phase shift, you could use this one 'cause you could imagine flipping both of them over, and it's the same as not flipping any of them over. So recapping, constructive interference happens when two waves are lined up perfectly. Destructive interference happens when the peaks match the valleys and they cancel perfectly. And you could use the path length difference for two wave sources to determine whether those waves are gonna interfere constructively or destructively. The path length difference is the difference between how far one wave has to travel to get to a detector compared to how far another wave has to travel to get to that same detector, assuming those two sources started in phase and neither of them got a Pi shift along their travels. Path length differences of integer wavelengths are gonna give you constructive interference, and path length differences of 1/2 integer wavelengths are gonna give you destructive interference. Whereas if the two sources started Pi out of phase, or one of the got a Pi phase shift along its travel, integer wavelengths for the path length difference are gonna give you destructive interference. And 1/2 integer wavelengths for the path length difference are gonna give you constructive interference." + }, + { + "Q": "What is a constant?", + "A": "A constant is a specific number, no variable. For example, these are constants: 5; -8; 11.2; -2/5", + "video_name": "bAerID24QJ0", + "transcript": "Welcome to level one linear equations. So let's start doing some problems. So let's say I had the equation 5-- a big fat 5, 5x equals 20. So at first this might look a little unfamiliar for you, but if I were to rephrase this, I think you'll realize this is a pretty easy problem. This is the same thing as saying 5 times question mark equals 20. And the reason we do the notation a little bit-- we write the 5 next to the x, because when you write a number right next to a variable, you assume that you're multiplying them. So this is just saying 5 times x, so instead of a question mark, we're writing an x. So 5 times x is equal to 20. Now, most of you all could do that in your head. You could say, well, what number times 5 is equal to 20? Well, it equals 4. But I'll show you a way to do it systematically just in case that 5 was a more complicated number. So let me make my pen a little thinner, OK. So rewriting it, if I had 5x equals 20, we could do two things and they're essentially the same thing. We could say we just divide both sides of this equation by 5, in which case, the left hand side, those two 5's will cancel out, we'll get x. And the right hand side, 20 divided by 5 is 4, and we would have solved it. Another way to do it, and this is actually the exact same way, we're just phrasing it a little different. If you said 5x equals 20, instead of dividing by 5, we could multiply by 1/5. And if you look at that, you can realize that multiplying by 1/5 is the same thing as dividing by 5, if you know the difference between dividing and multiplying fractions. And then that gets the same thing, 1/5 times 5 is 1, so you're just left with an x equals 4. I tend to focus a little bit more on this because when we start having fractions instead of a 5, it's easier just to think about multiplying by the reciprocal. Actually, let's do one of those right now. So let's say I had negative 3/4 times x equals 10/13. Now, this is a harder problem. I can't do this one in my head. We're saying negative 3/4 times some number x is equal to 10/13. If someone came up to you on the street and asked you that, I think you'd be like me, and you'd be pretty stumped. But let's work it out algebraically. Well, we do the same thing. We multiply both sides by the coefficient on x. So the coefficient, all that is, all that fancy word means, is the number that's being multiplied by x. So what's the reciprocal of minus 3/4. Well, it's minus 4/3 times, and dot is another way to use times, and you're probably wondering why in algebra, there are all these other conventions for doing times as opposed to just the traditional multiplication sign. And the main reason is, I think, just a regular multiplication sign gets confused with the variable x, so they thought of either using a dot if you're multiplying two constants, or just writing it next to a variable to imply you're multiplying a variable. So if we multiply the left hand side by negative 4/3, we also have to do the same thing to the right hand side, minus 4/3. The left hand side, the minus 4/3 and the 3/4, they cancel out. You could work it out on your own to see that they do. They equal 1, so we're just left with x is equal to 10 times minus 4 is minus 40, 13 times 3, well, that's equal to 39. So we get x is equal to minus 40/39. And I like to leave my fractions improper because it's easier to deal with them. But you could also view that-- that's minus-- if you wanted to write it as a mixed number, that's minus 1 and 1/39. I tend to keep it like this. Let's check to make sure that's right. The cool thing about algebra is you can always get your answer and put it back into the original equation to make sure you are right. So the original equation was minus 3/4 times x, and here we'll substitute the x back into the equation. Wherever we saw x, we'll now put our answer. So it's minus 40/39, and our original equation said that equals 10/13. Well, and once again, when I just write the 3/4 right next to the parentheses like that, that's just another way of writing times. So minus 3 times minus 40, it is minus 100-- Actually, we could do something a little bit simpler. This 4 becomes a 1 and this becomes a 10. If you remember when you're multiplying fractions, you can simplify it like that. So it actually becomes minus-- actually, plus 30, because we have a minus times a minus and 3 times 10, over, the 4 is now 1, so all we have left is 39. And 30/39, if we divide the top and the bottom by 3, we get 10 over 13, which is the same thing as what the equation said we would get, so we know that we've got the right answer. Let's do one more problem. Minus 5/6x is equal to 7/8. And if you want to try this problem yourself, now's a good time to pause, and I'm going to start doing the problem right now. So same thing. What's the reciprocal of minus 5/6? Well, it's minus 6/5. We multiply that. If you do that on the left hand side, we have to do it on the right hand side as well. Minus 6/5. The left hand side, the minus 6/5 and the minus 5/6 cancel out. We're just left with x. And the right hand side, we have, well, we can divide both the 6 and the 8 by 2, so this 6 becomes negative 3. This becomes 4. 7 times negative 3 is minus 21/20. And assuming I haven't made any careless mistakes, that should be right. Actually, let's just check that real quick. Minus 5/6 times minus 21/20. Well, that equals 5, make that into 1. Turn this into a 4. Make this into a 2. Make this into a 7. Negative times negative is positive. So you have 7. 2 times 4 is 8. And that's what we said we would get. So we got it right. I think you're ready at this point to try some level one equations. Have fun." + }, + { + "Q": "is there any other way to find cube roots", + "A": "Some other ways to find cube roots are long division, succesive subtraction, etc. but these are the easiest", + "video_name": "DKh16Th8x6o", + "transcript": "We are asked to find the cube root of negative 512. Or another way to think about it is if I have some number, and it is equal to the cube root of negative 512, this just means that if I take that number and I raise it to the third power, then I get negative 512. And if it doesn't jump out at you immediately what this is the cube of, or what we have to raise to the third power to get negative 512, the best thing to do is to just do a prime factorization of it. And before we do a prime factorization of it to see which of these factors show up at least three times, let's at least think about the negative part a little bit. So negative 512, that's the same thing-- so let me rewrite the expression-- this is the same thing as the cube root of negative 1 times 512, which is the same thing as the cube root of negative 1 times the cube root of 512. And this one's pretty straightforward to answer. What number, when I raise it to the third power, do I get negative 1? Well, I get negative 1. This right here is negative 1. Negative 1 to the third power is equal to negative 1 times negative 1 times negative 1, which is equal to negative 1. So the cube root of negative 1 is negative 1. So it becomes negative 1 times this business right here, times the cube root of 512. And let's think what this might be. So let's do the prime factorization. So 512 is 2 times 256. 256 is 2 times 128. 128 is 2 times 64. We already see a 2 three times. 64 is 2 times 32. 32 is 2 times 16. We're getting a lot of twos here. 16 is 2 times 8. 8 is 2 times 4. And 4 is 2 times 2. So we got a lot of twos. So essentially, if you multiply 2 one, two, three, four, five, six, seven, eight, nine times, you're going to get 512, or 2 to the ninth power is 512. And that by itself should give you a clue of what the cube root is. But another way to think about it is, can we find-- there's definitely three twos here. But can we find three groups of twos, or we could also find-- let me look at it this way. We can find three groups of two twos over here. So that's 2 times 2 is 4. 2 times 2 is 4. So definitely 4 multiplied by itself three times is divisible into this. But even better, it looks like we can get three groups of three twos. So one group, two groups, and three groups. So each of these groups, 2 times 2 times 2, that's 8. That is 8. This is 2 times 2 times 2. That's 8. And this is also 2 times 2 times 2. So that's 8. So we could write 512 as being equal to 8 times 8 times 8. And so we can rewrite this expression right over here as the cube root of 8 times 8 times 8. So this is equal to negative 1, or I could just put a negative sign here, negative 1 times the cube root of 8 times 8 times 8. So we're asking our question. What number can we multiply by itself three times, or to the third power, to get 512, which is the same thing as 8 times 8 times 8? Well, clearly this is 8. So the answer, this part right over here, is just going to simplify to 8. And so our answer to this, the cube root of negative 512, is negative 8. And we are done. And you could verify this. Multiply negative 8 times itself three times. Negative 8 times negative 8 times negative 8. Negative 8 times negative 8 is positive 64. You multiply that times negative 8, you get negative 512." + }, + { + "Q": "So i have been confused. Why is pluto a dwarf planet?", + "A": "It is very small and it has an irregular orbit for a planet.", + "video_name": "kJSOqlcFpJw", + "transcript": "In the last video, we had a large cloud of hydrogen atoms eventually condensing into a high pressure, high mass, I guess you could say, ball of hydrogen atoms. And when the pressure and the temperature got high enough-- and so this is what we saw the last video-- when the pressure and temperature got high enough, we were able to get the hydrogen protons, the hydrogen nucleuses close enough to each other, or hydrogen nuclei close enough to each other, for the strong force to take over and fusion to happen and release energy. And then that real energy begins to offset the actual gravitational force. So the whole star-- what's now a star-- does not collapse on itself. And once we're there, we're now in the main sequence of a star. What I want to do in this video is to take off from that starting point and think about what happens in the star next. So in the main sequence, we have the core of the star. So this is the core-- star's core. And you have hydrogen fusing into helium. And it's releasing just a ton of energy. And that energy is what keeps the core from imploding. It's kind of the outward force to offset the gravitational force that wants to implode everything, that wants to crush everything. And so you have the core of a star, a star like the sun, and that energy then heats up all of the other gas on the outside of the core to create that really bright object that we see as a star, or in our case, in our sun's case, the sun. Now, as the hydrogen is fusing into helium, you could imagine that more and more helium is forming in the core. So I'll do the helium as green. So more, more, and more helium forms in the core. It'll especially form-- the closer you get to the center, the higher the pressures will be, and the faster that this fusion, this ignition, will happen. In fact, the bigger the mass of the star, the more the pressure, the faster the fusion occurs. And so you have this helium building up inside of the core as this hydrogen in the core gets fused. Now what's going to happen there? Helium is a more dense atom. It's packing more mass in a smaller space. So as more and more of this hydrogen here turns into helium, what you're going to have is the core itself is going to shrink. So let me draw a smaller core here. So the core itself is going to shrink. And now it has a lot more helium in it. And let's just take it to the extreme point where it's all helium, where it's depleted. But it's much denser. That same amount of mass that was in this sphere is now in a denser sphere, in a helium sphere. So it's going to have just as much attraction to it, gravitational attraction. But things can get even closer to it. And we know that the closer you are to a mass, the stronger the pull of gravity. So then instead of having just the hydrogen fusion occurring at the core, you're now going to have hydrogen fusion in a shell around the core. So now you're going to have hydrogen fusing in a shell around the core. Let me just be clear. This isn't just happens all of a sudden. It is a gradual process. As we have more and more helium in the core, the core gets denser and denser and denser. And so the pressures become even larger and larger near the core because you're able to get closer to a more massive core since it is now more dense. And as that pressure near the core increases even more and more, the fusion reaction happens faster and faster and faster until you get to this point. So here, let me be clear. You have a helium core. All of the hydrogen in the core has been used up. And then you have the hydrogen right outside of the core is now under enormous pressure. It's actually under more pressure than it was when it was just a pure hydrogen core. Because it's-- there's so much mass on the outside here, trying to, I guess you could say, exerting downwards, or gravitational force trying to get to that even denser helium core because everything is able to get closer in. And so now you have fusion occurring even faster. And it's occurring over a larger radius. So this faster fusion over a larger radius, the force is now going to expel-- the energy that's released from this fusion is now going to expel these outer layers of the star even further. So the whole time, this gradual process as the hydrogen turns into helium, or fuses into helium in the core, the hydrogen right outside of the core, right outside that area, starts to burn faster and faster. I shouldn't say burn. It starts to fuse faster and faster and over a larger and larger radius. The unintuitive thing is the fusion is happening faster over a larger radius. And the reason that is is because you have even a denser core that is causing even more gravitational pressure. And as that's happening, the star's getting brighter. And it's also-- the fusion reactions, since they're happening in a more intense way and over a larger radius, are able to expel the material of the star even larger. So the radius of the star itself is getting bigger and bigger and bigger. So if this star looked like this-- maybe let me draw it in white-- That's not white. Now what's happening to my color changer? There you go. OK, this star looked like this right over here. Now, this star over here, since a faster fusion reaction is happening over a larger radius, is going to be far larger. And I'm not even drawing it to scale. In the case of our sun, when it gets to this point, it's going to be 100 times the diameter. And at this point, it is a red giant. And the reason why it's redder than this one over here is that even though the fusion is happening more furiously, that energy is being dissipated over a larger surface area. So the actual surface temperature of the red giant, at this point, is actually going to be cooler. So it's going to emit a light at a larger wavelength, a redder wavelength than this thing over here. This thing, the core, was not burning as furiously as this thing over here. But that energy was being dissipated over a smaller volume. So this has a higher surface temperature. This over here, the core is burning more-- sorry, the core is no longer burning. The core is now helium that's not burning. It's getting denser and denser as the helium packs in on itself. But the hydrogen fusion over here is occurring more intensely. It's occurring in a hotter way. But the surface here is less hot because it's just a larger surface area. So it doesn't make-- the increased heat is more than mitigated by how large the star has become. Now, this is going to keep happening. And this core is keep-- the pressures keep intensifying because more and more helium is getting And this core keeps collapsing. And the temperature here keeps going up. So we said that the first ignition, the first fusion, occurs at around 10 million Kelvin. This thing will keep heating up until it gets to 100 million Kelvin. And now I'm talking about a star that's about as massive as the sun. Some stars will never even be massive enough to condense the core so that its temperature reaches But let's just talk about the case in which it does. So eventually, you'll get to a point-- so we're still sitting in the red giant phase, so we're this huge star over here. We have this helium core. And that helium core keeps getting condensed and condensed and condensed. And then we have a shell of hydrogen that keeps fusing into helium around it. So this is our hydrogen shell. Hydrogen fusion is occurring in this yellow shell over here that's expelling, that's allowed-- that's causing the radius of the star to get bigger and bigger, to expand. But when the temperature get sufficiently hot-- and now I think you're going to get a sense of how heavier and heavier elements form in the universe, and all of the heavy elements that you see around us, including the ones that are in you, were formed it this way from, initially, hydrogen-- when it gets hot enough at 100 million Kelvin, in this core, because of such enormous pressures, then the helium itself will start to fuse. So then we're going to have a core in here where the helium itself will start to fuse. And now we're talking about a situation. You have helium, and you had hydrogen. And all sorts of combinations will form. But in general, the helium is mainly going to fuse into carbon and oxygen. And it'll form into other things. And it becomes much more complicated. But I don't want to go into all of the details. But let me just show you a periodic table. I didn't have this in the last one. I had somehow lost it. But we see hydrogen here has one proton. It actually has no neutrons. It was getting fused in the main sequence into helium, two protons, two neutrons. You need four of these to get one of those. Because this actually has an atomic mass of 4 if we're talking about helium-4. And then the helium, once we get to 100 million Kelvin, can start being fused. If you get roughly three of them-- and there's all of these other things that are coming and leaving the reactions-- you can get to a carbon. You get four of them, four of them at least as the starting raw material. You get to an oxygen. So we're starting to fuse heavier and heavier elements. So what happens here is this helium is fusing into carbon and oxygen. So you start building a carbon and oxygen core. So I'm going to leave you there. I realize I'm already past my self-imposed limit of 10 minutes. But what I want you to think about is what is likely to happen. What is likely to happen here if this star will never have the mass to begin to fuse this carbon and oxygen? If it does have the mass, if it is a super massive star, it eventually will be able to raise even this carbon and oxygen core to 600 million Kelvin and begin to fuse that into even heavier elements. But let's think about what's going to happen for something like the sun, where it'll never have the mass, it'll never have the pressure, to start to fuse carbon and oxygen. And that'll be the topic of the next video." + }, + { + "Q": "is it possible to find the median except the mode?", + "A": "With only two points, the mean and median will both work.", + "video_name": "Ez_-RwV9WVo", + "transcript": "Let's say I have the point 3 comma negative 4. So that would be 1, 2, 3, and then down 4. 1, 2, 3, 4. So that's 3 comma negative 4. And I also had the point 6 comma 1. So 1, 2, 3, 4, 5, 6 comma 1. So just like that. 6 comma 1. In the last video, we figured out that we could just use the Pythagorean theorem if we wanted to figure out the distance between these two points. We just drew a triangle there and realized that this was the In this video, we're going to try to figure out what is the coordinate of the point that is exactly halfway between this point and that point? So this right here is kind of the distance, the line that connects them. Now what is the coordinate of the point that is exactly halfway in between the two? What is this coordinate right here? It's something comma something. And to do that-- let me draw it really big here. Because I think you're going to find out that it's actually pretty straightforward. At first it seems like a really tough problem. Gee, let me use the distance formula with some variables. But you're going to see, it's actually one of the simplest things you'll learn in algebra and geometry. So let's say that this is my triangle right there. This right here is the point 6 comma 1. This down here is the point 3 comma negative 4. And we're looking for the point that is smack dab in between those two points. What are its coordinates? It seems very hard at first. But it's easy when you think about it in terms of just the x and the y coordinates. What's this guy's x-coordinate going to be? This line out here represents x is equal to 6. This over here-- let me do it in a little darker color-- this over here represents x is equal to 6. This over here represents x is equal to 3. What will this guy's x-coordinate be? Well, his x-coordinate is going to be smack dab in between the two x-coordinates. This is x is equal to 3, this is x is equal to 6. He's going to be right in between. This distance is going to be equal to that distance. His x-coordinate is going to be right in between the 3 and the 6. So what do we call the number that's right in between the 3 and the 6? Well we could call that the midpoint, or we could call it the mean, or the average, or however you want to talk about it. We just want to know, what's the average of 3 and 6? So to figure out this point, the point halfway between 3 and 6, you literally just figure out, 3 plus 6 over 2. Which is equal to 4.5. So this x-coordinate is going to be 4.5. Let me draw that on this graph. 1, 2, 3, 4.5. And you see, it's smack dab in between. That's its x-coordinate. Now, by the exact same logic, this guy's y-coordinate is going to be smack dab between y is equal to negative 4 and y is equal to 1. So it's going to be right in between those. So this is the x right there. The y-coordinate is going to be right in between y is equal to negative 4 and y is equal to 1. So you just take the average. 1 plus negative 4 over 2. That's equal to negative 3 over 2 or you could say negative 1.5. So you go down 1.5. It is literally right there. So just like that. You literally take the average of the x's, take the average of the y's, or maybe I should say the mean to be a little bit more specific. A mean of only two points. And you will get the midpoint of those two points. The point that's equidistant from both of them. It's the midpoint of the line that connects them. So the coordinates are 4.5 comma negative 1.5. Let's do a couple more of these. These, actually, you're going to find are very, very straightforward. But just to visualize it, let me graph it. Let's say I have the point 4, negative 5. So 1, 2, 3, 4. And then go down 5. 1, 2, 3, 4, 5. So that's 4, negative 5. And I have the point 8 comma 2. So 1, 2, 3, 4, 5, 6, 7, 8 comma 2. 8 comma 2. So what is the coordinate of the midpoint The point that is smack dab in between them? Well, we just average the x's, average the y's. So the midpoint is going to be-- the x values are 8 and 4. It's going to be 8 plus 4 over 2. And the y value is going to be-- well, we have a 2 and a negative 5. So you get 2 plus negative 5 over 2. And what is this equal to? This is 12 over 2, which is 6 comma 2 minus 5 is negative 3. Negative 3 over 2 is negative 1.5. So that right there is the midpoint. You literally just average the x's and average the y's, or So let's graph it, just to make sure it looks like midpoint. 6, negative 5. 1, 2, 3, 4, 5, 6. Negative 1.5. Negative 1, negative 1.5. Yep, looks pretty good. It looks like it's equidistant from this point and that point up there. Now that's all you have to remember. Average the x, or take the mean of the x, or find the x that's right in between the two. Average the y's. You've got the midpoint. What I'm going to show you now is what's in many textbooks. They'll write, oh, if I have the point x1 y1, and then I have the point-- actually, I'll just stick it in yellow. It's kind of painful to switch colors all the time-- and then I have the point x2 y2, many books will give you something called the midpoint formula. Which once again, I think is kind of silly to memorize. Just remember, you just average. Find the x in between, find the y in between. So midpoint formula. What they'll really say is the midpoint-- so maybe we'll say the midpoint x-- or maybe I'll call it this way. I'm just making up notation. The x midpoint and the y midpoint is going to be equal to-- and they'll give you this formula. x1 plus x2 over 2, and then y1 plus y2 over 2. And it looks like something you have to memorize. But all you have to say is, look. That's just the average, or the mean, of these two numbers. I'm adding the two together, dividing by two, adding these two together, dividing by two. And then I get the midpoint. That's all the midpoint formula is." + }, + { + "Q": "what is the lower quartile and the upper quartile?", + "A": "The lower and upper quartiles are sections of a box and whisker plot. To find the lower quartile, find the middle of the numbers to the left of the median without including the median in this computation. The area in between that middle number and the median is the lower quartile. You can also find the upper quartile in a similar fashion.", + "video_name": "b2C9I8HuCe4", + "transcript": "An ecologist surveys the age of about 100 trees in a local forest. He uses a box-and-whisker plot to map his data shown below. What is the range of tree ages that he surveyed? What is the median age of a tree in the forest? So first of all, let's make sure we understand what this box-and-whisker plot is even about. This is really a way of seeing the spread of all of the different data points, which are the age of the trees, and to also give other information like, what is the median? And where do most of the ages of the trees sit? So this whisker part, so you could see this black part is a whisker, this is the box, and then this is another whisker right over here. The whiskers tell us essentially the spread of all of the data. So it says the lowest to data point in this sample is an eight-year-old tree. I'm assuming that this axis down here is in the years. And it says at the highest-- the oldest tree right over here is 50 years. So if we want the range-- and when we think of range in a statistics point of view we're thinking of the highest data point minus the lowest data point. So it's going to be 50 minus 8. So we have a range of 42. So that's what the whiskers tell us. It tells us that everything falls between 8 and 50 years, including 8 years and 50 years. Now what the box does, the box starts at-- well, let me explain it to you this way. This line right over here, this is the median. And so half of the ages are going to be less than this median. We see right over here the median is 21. So this box-and-whiskers plot tells us that half of the ages of the trees are less than 21 and half are older than 21. And then these endpoints right over here, these are the medians for each of those sections. So this is the median for all the trees that are less than the real median or less than the main median. So this is in the middle of all of the ages of trees that are less than 21. This is the middle age for all the trees that are greater than 21 or older than 21. And so we're actually splitting all of the data into four groups. This we would call the first quartile. So I'll call it Q1 for our first quartile. Maybe I'll do 1Q. This is the first quartile. Roughly a fourth of the tree, because the way you calculate it, sometimes a tree ends up in one point or another, about a fourth of the trees A fourth of the trees are between 14 and 21. A fourth are between 21 and it looks like 33. And then a fourth are in this quartile. So we call this the first quartile, the second quartile, the third quartile, and the fourth quartile. So to answer the question, we already did the range. There's a 42-year spread between the oldest and the youngest And then the median age of a tree in the forest is at 21. So even though you might have trees that are as old as 50, the median of the forest is actually closer to the lower end of our entire spectrum of all of the ages. So if you view median as your central tendency measurement, it's only at 21 years. And you can even see it. It's closer to the left of the box and closer to the end of the left whisker than the end of the right whisker." + }, + { + "Q": "How do u turn 8.84 into a fraction", + "A": "Put 84 over 100, which is 84/100. Then, add 8.", + "video_name": "EGr3KC55sfU", + "transcript": "Let's see if we can write 0.0727 as a fraction. Now, let's just think about what places these are in. This is in the tenths place. This is in the hundredths place. This 2 is in the thousandths place. And this 7 right here, this last 7, is in the ten-thousandths place. So there's a couple of ways we can do this. The way I like to think of this, this term over here is in the ten-thousandths place. We can view this whole thing right over here as 727 ten-thousandths because this is the smallest place right over here. So let's just rewrite it. This is equal to 727 over 10,000. And we've already written it as a fraction. And I think that's about as simplified as we can get. This number up here is not divisible by 2. It's not divisible by 5. In fact, it's not divisible by 3, which means it wouldn't be divisible by 6 or 9. It doesn't even seem to be divisible by 7. It might be a prime number. But I think we are done." + }, + { + "Q": "At 6:02, why does the hydrocarbon chain have no polarity?", + "A": "Carbon and Hydrogen have very similar electronegativity, so the elements don t pull each other to a specific direction. Thus the hydrocarbon chain is not polar.", + "video_name": "Pk4d9lY48GI", + "transcript": "- DNA gets a lot of attention as the store of our genetic information, and it deserves that. If we didn't have DNA, there would be no way of keeping the information that makes us us, and other organisms what those organisms are. And DNA has some neat properties, it can replicate itself, and we go into a lot of depth on that in other videos. So DNA producing more DNA, we call that, we call that replication, but just being able to replicate yourself on its own isn't enough to actually produce an organism. And to produce an organism, you somehow have to take that information in the DNA, and then produce things like a structural molecules, enzymes, transport molecules, signaling molecules, that actually do the work of the organism. And that process, the first step, and this is all a review that we've seen in other videos. The first step is to go from DNA to RNA, and in particular, messenger RNA. \"Messenger RNA,\" and this process right over here, this is called transcription. \"Transcription,\" we go into a lot of detail on this in other videos. And then you wanna go from that messenger RNA, it goes to the ribosomes and then tRNA goes and grabs amino acids, and they form actual proteins. So you go from messenger RNA, and then in conjunction, so this is all, this is in conjunction with tRNA and amino acids, so let me say \"+tRNA,\" and \"amino acids.\" And I'll write \"amino acids\" in, I'll write it in a brighter color, since that's going to be the focus of this video. So tRNA and amino acids, you're able to construct proteins. You are able to construct proteins, which are made up of chains of amino acids, and it's the proteins that do a lot of the work of the organism. Proteins, which are nothing but chains of amino acids, or they're made up of, sometimes multiple chains of amino acids. So you can image, I'm just going to, that's an amino acid. That's another amino acid. This is an amino acid. This is an amino acid, you could keep going. So these chains of amino acids, based on how these different, based on the properties of these different amino acids, and how the protein takes shape and how it might interact with its surrounding, these proteins can serve all sorts of different functions. Anything from part of your immune system, antibodies, they can serve as enzymes, they can serve as signaling hormones, like insulin. They're involved in muscle contraction. Actin and myosin, we actually have a fascinating video on that. Transport of oxygen. Hemoglobin. So proteins, the way at least my brain of it, is they do a lot of the work. DNA says, well, what contains the information, but a lot of the work of organism is actually done, is actually done by the proteins. And as I just said, the building blocks of the proteins are the amino acids. So let's focus on that a little bit. So up here are some examples of amino acids. And there are 20 common amino acids, there are a few more depending on what organism you look at, and theoretically there could be many more. But in most biological systems, there are 20 common amino acids that the DNA is coding for, and these are two of them. So let's just first look at what is common. So, we see that both these, and actually all three of this, this is just a general form, you have an amino group. You have an amino group, and this where, this is why we call it an \"amino,\" an amino acid. So you have an amino group. Amino group right over here. Now you might say, \"well, it's called an amino acid,\" \"so where is the acid?\" And that comes from this carboxyl group right over here. So that's why we call it an acid. This carboxyl group is acidic. It likes to donate this proton. And then in between, we have a carbon, and we call that the alpha carbon. We call that the alpha carbon. Alpha carbon, and that alpha carbon is bonded, it has a covalent bond to the amino group, covalent bond to the carboxyl group, and a covalent bond to a hydrogen. Now, from there, that's where you get the variation in the different amino acids, and actually, there's even some exceptions for how the nitrogen is, but for the most part, the variation between the amino acids is what this fourth covalent bond from the alpha carbon does. So you see in serine, you have this, what you could call it an alcohol. You could have an alcohol side chain. In valine right over here, you have a fairly pure hydrocarbon, hydrocarbon side chain. And so in general, we refer to these side chains as an R group, and it's these R groups that play a big role in defining the shape of the proteins, and how they interact with their environment and the types of things they can do. And you can even see, just from these examples how these different sides chains might behave differently. This one has an alcohol side chain, and we know that oxygen is electronegative, it likes to hog electrons, it's amazing how much of chemistry or even biology you can deduce from just pure electronegativity. So, oxygen likes to hog electrons, so you're gonna have a partially-negative charge there. Hydrogen has a low electronegativity relative to oxygen, so it's gonna have its electrons hogged, so you're gonna have a partially positive charge, just like that, and so this has a polarity to it, and so it's going to be hydrophilic, it's going to, at least this part of the molecule is going to be able to be attracted and interact with water. And that's in comparison to what we have over here, this hydrocarbon side chain, this has no polarity over here, so this is going to be hydrophobic. So this is going to be hydrophobic. And so when we start talking about the structures of proteins, and how the structures of proteins are influenced by its side chains, you could image that parts of proteins that have hydrophobic side chains, those are gonna wanna get onto the inside of the proteins if we're in an aqueous solution, while the ones that are more hydrophilic will wanna go onto the outside, and you might have some side chains that are all big and bulky, and so they might make it hard to tightly pack, and then you might have other side chains that are nice and small that make it very easy to pack, so these things really do help define the shape, and we're gonna talk about that a lot more when we talk about the structure. But how do these things actually connect? And we're gonna go into much more detail in another video, but if you have... If you have serine right over here, and then you have valine right over here, they connect through what we call peptide bonds, and a peptide is the term for two or more amino acids connected together, so this would be a dipeptide, and the bond isn't this big, I just, actually let me just, let me draw it a little bit smaller. So... That's serine. This is valine. They can form a peptide bond, and this would be the smallest peptide, this would be a dipeptide right over here. \"Peptide,\" \"peptide bond,\" or sometimes called a peptide linkage. And as this chain forms, that polypeptide, as you add more and more things to it, as you add more and more amino acids, this is going to be, this can be a protein or can be part of a protein that does all of these things. Now one last thing I wanna talk about, this is the way, the way these amino acids have been drawn is a way you'll often see them in a textbook, but at physiological pH's, the pH's inside of your body, which is in that, you know, that low sevens range, so it's a pH of roughly 7.2 to 7.4. What you have is this, the carboxyl group right over here, is likely to be deprotonated, it's likely to have given away its hydrogen, you're gonna find that more likely than when you have... It's gonna be higher concentrations having been deprotonated than being protonated. So, at physiological conditions, it's more likely that this oxygen has taken both of those electrons, and now has a negative charge, so it's given, it's just given away the hydrogen proton but took that hydrogen's electron. So it might be like this, and then the amino group, the amino group at physiological pH's, it's likely to actually grab a proton. So nitrogen has an extra loan pair, so it might use that loan pair to grab a proton, in fact it's physiological pH's, you'll find a higher concentration of it having grabbed a proton than not grabbing a proton. So, the nitrogen will have grabbed a proton, use its loan pairs to grab a proton, and so it is going to have... So it is going to have a... It is going to have a positive charge. And so sometimes you will see amino acids described this way, and this is actually more accurate for what you're likely to find at physiological conditions, and these molecules have an interesting name, a molecule that is neutral even though parts of it have charge, like this, this is called a zwitterion. That's a fun, fun word. Zwitterion. And \"zwitter\" in German means \"hybrid,\" and \"ion\" obviously means that it's going to have charge, and so this has hybrid charge, even though it has charges at these ends, the charges net out to be neutral." + }, + { + "Q": "How do you do the standard form of a polynomial?", + "A": "Standard form for a polynomial has the terms listed from highest degree to lowest. For example: 2 + 6x^2 - 7x +9x^3 written in standard form becomes: 9x^3 + 6x^2 - 7x + 2", + "video_name": "vN0aL-_vIKM", + "transcript": "simply 3x squared minus 8x plus 7 plus 2x to the third minus x squared plus eight x minus 3 so when we simplify this we're essentially going to add up like terms and just as a reminder we can only add or subtract like terms or simplify like terms and just a reminder and what I mean by that if I had an x squared to an x squared these are like terms they're both x terms raised to the same power the same degree so if I have one x squared and another x squared well then I have 2x squared - this is 2x squared If I have an x to the third - let's say I have 3x to thirds plus another 4x to the thirds Well that means I have 7x to the thirds - 7x to the thirds I can't take an x squared and add it to an x to the third I cannot simplify this in any way, so this you cannot simplify cannot simply - these are not like terms just because they both have Xs The Xs are not to - they are not to the same degree With that in mind let's look at the Xs to the same degree Let's start with the highest degree So the highest degree or the highest exponent on an x here - is actually this x to the third here but it looks like the only one it's the only place where we're raising x to the third power so that can't be merged or added or subtracted to anything else so let's just write that down. so we have 2x to the third and let's look at the x squared terms. We have 3x squared over there and we have a minus or we can view as a negative x squared over there so if we want to simplify we can add these two terms - we can add- so let me just write it down we can add 3x squared to negative x squared so I'm just rearranging it really right now I'm putting the like terms next to each other so it'll be easy to simplify now let's just worry about the x to the first terms or just the x terms you have a negative 8x term right over here so let me write it over here negative 8x and then you have a positive 8x term right over here, so let me write that down so positive 8x and then finally let's look at the constant terms you can view those as times x to the zeroth power and the constant terms are - you have a positive 7 over here - so plus 7 and then you have a negative 3 over here you have a negative 3 so all I'll I've done is I've really just used the communative property of addition to just change the order - or addition and subtraction - to change the order in which I'm doing this I've just rearranged the things so the like terms are next to each other but now we can simplify so we have 2x to the third - nothing to simplify that with but then if we subtract - if we have - let me do that in the same blue color if we have 3x squared and from that we're taking away an x squared well, we're only gonna have 2x squared left so that's gonna be plus 2x squared and then over here if we have negative 8 Xs and then we add 8 Xs to it or you can actually swap these around you could view it as you are subtracting 8 x from positive 8x we'll those are just going to cancel out so it's just going to be zero - I could just write plus zero here, but that'd just be redundant it wouldn't change the value and then finally I have a plus 7 minus 3 well that is just clearly 7 minus 3 is 4 so I have plus 4 And we're done! We've simplified it! 2x to the third plus 2 x squared plus 4" + }, + { + "Q": "is are blood actually blue but with the mix of oxygen it turns red", + "A": "No, blood is never blue. When it is deoxygenated, it is dark red. When it is oxygenated, it is a brighter red.", + "video_name": "7b6LRebCgb4", + "transcript": "- [Instructor] Let's talk a little bit about arteries and veins and the roles they play in the circulatory system. So I want you to pause this video and first think to yourself, Do you have a sense of what arteries and veins are? Well one idea behind arteries and veins are that well, in most of these drawings, arteries are drawn in red, and I even made the artery word here in red. And veins are drawn in blue. And so maybe that represents how much oxygen they have. And so one possible explanation is that arteries carry oxygenated blood, oxygenated, oxygenated blood, while veins carry deoxygenated blood. So blood that has less oxygen now. Now this is actually incorrect. It is, many times, the case that arteries are the ones carrying oxygenated blood and veins are carrying the deoxygenated blood. But as we will see, this is not always the case. And since we're already talking about oxygenated blood and deoxygenated blood and the colors red and blue, it's worth addressing another misconception. Many times it is said that deoxygenated blood looks blue, and the reason why people believe that is if you look at your wrist and you're able to see some of the vessels in there, you will see some blue vessels. And those, or at least they look blue when you're looking from the outside of your skin. And those, indeed, are veins. And so that's where the misconception has come from, that veins, which, in your arm, are carrying deoxygenated blood. That that deoxygenated blood is blue. It turns out that it is not blue. It is just a deeper red. And the reason why the veins look blue is because of the optics of light going through your skin and then seeing the outside of the veins and then reflecting back. That is not the color of the actual blood. So so far I have not given you a clear definition of what arteries versus veins are. A better definition, so let me cross these two out, are that arteries carry blood away from the heart. Away from the heart. And veins carry blood towards the heart. Towards the heart. And I can get a zoomed in image of the heart right here and that will make it a little bit clearer. And you can also see, or we're about to see, why this first definition, or this first distinction between arteries and veins does not always hold. So let's just imagine some blood that is being pumped away from the heart. So right when it gets pumped away from the heart, it'll be right over here. It gets pumped through the aorta, and you can see the aorta branches, so some blood can go up towards your head, and if it didn't, you would pass out and die. And then a lot of the blood goes down towards the rest of your body. And that, indeed, is the most oxygenated blood. And so it'll flow through your body. And these arteries will keep branching and branching into smaller vessels, all the way until they form these very small branches. And it's that place, especially, where they will lose a lot of their oxygen to the fluid and the cells around them. And then the blood is less oxygenated. And then even though deoxygenated blood is not blue, it often gets depicted as blue in a lot of diagrams. So I will do the same. And these vessels start building into your veins. And these really small vessels that really bridge between arteries and veins, where a lot of the gas and nutrient exchange occurs, these are called capillaries. And so after going through the capillaries, the blood will then come back to the heart and now it's coming towards the heart through the veins. It comes into the right atrium, then the right ventricle. Then that gets pumped towards the lungs. And this is the exception to the first incorrect definition of arteries and veins that we looked at. This right over here, is an artery. Even though it's carrying less oxygenated or deoxygenated blood, it's an artery because it's carrying blood away from the heart. But in this case, it's not carrying it to the rest of the body, it is carrying it to the lungs. That is why it is called the pulmonary artery, even though it's carrying less oxygenated blood. So that it goes to the lungs and then, in the lungs, there's more gas exchange that occurs. The blood gets oxygenated and then it comes back to the heart. And so it comes back to the heart in these vessels right over here, and that even though these are carrying highly oxygenated blood, these are considered veins because they're carrying blood towards the heart. So these are pulmonary veins. And then the cycle starts again. The pulmonary veins bring the oxygenated blood into the left atrium and the left ventricle, and then that pumps it to the rest of the body to the aorta, for your systemic circulation. You have your pulmonary circulation, which circulates the blood to, through and from the lungs. And you have your systemic circulation, which takes the blood to and from the rest of the body. So now that we have this main distinction between arteries and veins, what are some other interesting things that we know about it? Well one thing to keep in mind is that since arteries are being pumped directly by the heart towards the rest of the body, they have high pressure. I'll write that in caps. High pressure. And so if you were to have an accident of some type, which you do not want to have, and you were to accidentally cut an artery, because of that high pressure, it would actually spurt blood, a lot more than if you were to cut a vein. And most of the times where you get a cut, you're really just cutting capillaries. Like if you were to prick your finger, it's usually a series of capillaries that get cut, and that's why the blood would come out very very slowly. Now if arteries are high pressure, veins are low pressure. Low, low pressure. And one way to think about it is the arteries, the blood is being pumped directly by the heart. But then once it goes through the capillaries and comes back through the veins, it's kind of sluggishly making its way back to the heart. It's not being directly pumped. And that's why in veins, because you don't have that high pressure to bring everything back to the heart, you have these valves that make sure that for the most part, the blood is going in one direction. I'm going to draw the blood in red in the veins, just so we don't keep going with that misconception, that blood in the veins is blue somehow. Now related to the fact that the blood in the arteries is under higher pressure, in order to transport a fixed volume of blood in a certain amount of time, you need less volume. And so that's why arteries are low volume. And on the other hand, veins are high volume. And to appreciate the difference, the blood volume in arteries are only approximately 15% of the entire blood volume in your body, while the blood volume in veins are closer to 65%, approximately 65%. And if you're wondering where the rest of the blood is, about five percent is in capillaries, five percent is in your heart, and about 10% is in your lungs. So I will leave you there. The big take away: arteries are the vessels that take blood away from the heart. Veins are the vessels that take blood towards the heart." + }, + { + "Q": "Where is the setting of the portrait of the Mona Lisa.", + "A": "Leonardo began the painting in Florence, Italy but the background landscape in the painting is presumed to be imaginary.", + "video_name": "3kQ_p2EZX4Q", + "transcript": "[MUSIC PLAYING] BETH HARRIS: We thought we would start by looking at what is perhaps the most famous painting in the world, and whether we can actually even really still see. SAL KHAN: Right. Because I have seen this before. And I've even visited it at the Louvre-- I know I'm pronouncing it wrong. Yes, you're right. This is probably the most famous painting world. BETH HARRIS: And I just read that most people spend about 15 seconds in the Louvre looking at the painting, which is a funny statistic. SAL KHAN: Well, it's stressful, because there's people behind you. And on top of that, it's actually surprisingly small when you see it in real life. I mean, now that I'm able to take my time, and not worry about the tourists behind me, and I'm looking at it for real, I'm already-- things are jumping out at me that I actually had never noticed before. BETH HARRIS: Like what? SAL KHAN: Well, it looks like the scenery is some kind of like Vulcan territory or something. [LAUGHING] There's this-- it's like mountainous, and well, I guess, there's a little bridge in there. There's a road. I guess I never paid much attention to that before. Yeah, actually, I'd never even noticed this chair she was on before either. You can see hand resting on it. Actually-- and I never noticed that there's a ledge, right behind her, where's there jars. I could probably keep going. BETH HARRIS: I like your analogy to Vulcan territory, as a Star Trek fan myself. That landscape is otherworldly and very mysterious. But it's Interesting, isn't it, how the bottom part of the landscape at her neck and below looks like an inhabited landscape with a winding road and a bridge, but the landscape that's at her neck and head is more mysterious and looks very much like another planet? SAL KHAN: That's right. And actually, when you point that out and how that painting is divided based on where those landscapes and the ledge divide the painting, I don't have my ruler out, but I would guess that it's pretty close to the golden mean. BETH HARRIS: I think you're probably right. Those things that look like jars are actually the bottom of columns cut off on either side of the painting. SAL KHAN: So Leonardo da Vinci actually painted the columns, and it was cropped? BETH HARRIS: That's right. And so the space that she's in would have made a lot more sense as a balcony. SAL KHAN: Well, you know, all of this-- actually, just take a step back. I mean, we started with this presumption that it's-- and it's true-- that it's probably the most famous painting in the world, but I guess, I've never quite gotten why. I mean, is this just a case of marketing? BETH HARRIS: I think it happened in 1911, when the painting was stolen from the Louvre and disappeared for a couple of years and became notorious. At that point in the 19th century, the \"Mona Lisa\" was not the most popular painting at the Louvre. Paintings by other artists, like Titian and Raphael, were much more popular and even valued more highly for insurance purposes. So it really probably is only in the 20th century that she became as important as she is now. SAL KHAN: If you go back 150 years ago, \"Mona Lisa\" was not something that was just ingrained in our culture. BETH HARRIS: She was important. People were interested in her, and people were writing about her and they said some interesting things. But she wasn't as famous as she is now. And also, don't forget that the technology to reproduce her existed only, really, in the 20th century in terms of mass color reproductions. And so her currency has certainly increased, I think, in the last 100 years or so. SAL KHAN: I see. If you go back 150 years, there was probably no such thing as super famous paintings. BETH HARRIS: I think that might be true, actually. There were paintings that were famous, or important, but not celebrities in the way that the \"Mona Lisa\" is. SAL KHAN: Right. Not something that every person on the street would recognize. BETH HARRIS: Yeah. And of course, now I think most people would say that what's so interesting about her is her look and her smile, which have been interpreted in many different ways. SAL KHAN: Yeah, I know. And I know that's kind of, I guess, one of the claims to fame of the painting. And you see that. I mean, people like to look at it-- is she smirking, is she happy, is she sad. All of these things. Is she looking at you. All of these things that people try to-- but, I guess, trying to look at it without all of the social programming that I've had around this painting, it strikes me is an interesting painting. And it seems very technically well done. And there's something very bright, and just kind of an aura around her face. I don't know if I wasn't programmed to really know this painting and if I were to see this in the museum amongst many, many others, that I would-- it really jump out at me. BETH HARRIS: Portraits really took off during the Renaissance beginning in the 1400s in Italy. And Leonardo painted this in Florence. And that's because of humanism. One way that we define humanism is taking an interest in human beings, and the things of this world, and human achievement, and individuality. All of those values becoming more important in the 15th century. And so we begin to see a lot more portraits. Also with the beginnings of a wealthy merchants class in Florence in the 15th century, people could afford portraits and begin to want them. At first, portraits were painted with the figure in profile. But later, especially in northern Europe, artists like Durer or Memling started to put their figures in believable spaces. SAL KHAN: Right. BETH HARRIS: And so, Leonardo's really the first artist in Italy to do those things. To make an oil painting, which is a relatively new medium SAL KHAN: What did people use before oil? BETH HARRIS: They used fresco and tempera painting. Tempera for panel painting. So this is oil on wood, whereas before, artists would paint tempera on wood. Tempera tends to look more flat than oil paint, where you can really get a sense of modeling and light and dark. So Leonardo's making this three-dimensional figure, and he's using another technique called sfumato, which means a kind of smokey haziness. So he obscures the hard outlines around the forms, which tend to flatten them. One of the things that's fun to talk about with the Mona Lisa, too, is all the things that people have said about her over the years. You might not be aware of the fact that Sigmund Freud actually had a particular interpretation of the Mona Lisa. SAL KHAN: Yes, I'm sure he did. [LAUGHTER] I'm somewhat skeptical of him. I would like to interpret his interpretations someday. But yes. BETH HARRIS: Freud said that the \"Mona Lisa's\" smile combined the two ways that we tend to look at women in our culture. In one way, she's very mothering and nurturing. And in the other way, she seems very seductive. SAL KHAN: I think that says more about Freud than about Leonardo. BETH HARRIS: You could be right. [LAUGHTER] And later artists, another artist that you already know, Duchamp-- SAL KHAN: Duchamp, my favorite. BETH HARRIS: Your favorite. He took a reproduction of the \"Mona Lisa\" and drew a mustache on her. SAL KHAN: I could imagine him doing that. [LAUGHTER] BETH HARRIS: I think the moustache is interesting, because there is something not entirely feminine about her. Something a little bit masculine. SAL KHAN: Do you think it's that? Or I mean, I guess there is a certain-- I mean, it's kind of old now, especially because Duchamp did it, I'm guessing, 80, 90 years ago. But there is something hilarious about drawing a mustache on a feminine form. We all remember doing it as school kids-- just getting a kick out of it. And I could see it's especially funny for this painting. BETH HARRIS: Taking something that's so high art and making it silly, you know? SAL KHAN: Exactly. BETH HARRIS: Recently, the Prado in Madrid, found what turns out to be, after some scientific testing, a copy of the \"Mona Lisa,\" which in and of itself is not that unusual, but it turns out that their copy was made by another artist sitting right next to Leonardo copying what he did stroke for stroke. And they can tell this by analyzing the under drawing. SAL KHAN: Yeah, she looks much younger. BETH HARRIS: She has eyebrows. SAL KHAN: Oh, that's right. That's where the creepiness comes from, because the \"Mona Lisa\" we see looks jaundiced-- it's yellow. And so, the painting is a little bit different. The face is a little bit different, but we can assume that the colors might have not been that different. BETH HARRIS: Exactly. And it's a really interesting thing to think about. What she would look like if she was cleaned. And if she would still mean what she means to us. SAL KHAN: Oh, I don't think she would, because when I look at this cleaned painting, it loses a lot of the mystery. BETH HARRIS: Yeah, I agree. And you can then understand the Louvre's decision not to clean her. SAL KHAN: I mean, the cleaned one, she looks better. She looks younger. She loses a lot of the motherly aspects that Freud seems to want to ascribe to her. Yeah, because the colors are brighter, they're more vibrant, it's not as muted as the one that we've learned to like. BETH HARRIS: Yeah. Although, her reputation has grown over the years, who's to say that we won't care so much about her again. SAL KHAN: There might be a post-celebrity world at some point. [MUSIC PLAYING]" + }, + { + "Q": "At one point Jay says \"a positively charged oxygen\" at another he says \"an oxygen with a plus one formal charge\". Are both the terms equivalent?", + "A": "Yes, they mean the same thing.", + "video_name": "dJhxphep_gY", + "transcript": "So in a hydration reaction, water is added across a double bond. And the OH adds in a Markovnikov way. So according to Markovnikov's rule. So let's go ahead and write that down here. So you have to think about Markovnikov when you're doing this reaction. And this is an acid catalyzed reaction. So technically, this reactions is at equilibrium, and we will cover that at the very end of the video here. So let's look at the mechanism for the acid catalyzed addition of water across a double bond. So here I have my alkene, and I have water present with sulfuric acid. So sulfuric acid, being a strong acid, will donate a proton in solution. And let's say the water molecule picks up that proton. So H2O would go to H3O+. 3 So I'm just jumping ahead to the H3O+ ion called the hydronium ion. So here's my H3O+ ion. So put my one pair of electrons on there, and it's positively charged. So what's going to happen is the pi electrons, the electrons in this pi bond here are going to function as a base. They're going to abstract a proton. They're going to accept a proton. They're going to take, let's say, this proton right here, which would cause these two electrons in this bond to kick off onto your oxygen. So acid base equilibrium. So I'll go ahead and make this an equilibrium arrow here. And what are we going to get if that happens? Well, let's say that the proton added to the carbon on the right-- so the proton added to the carbon on the right here. So I'm saying that the blue electrons on the left are going to be these electrons right here, like that. And that would mean that I took a bond away from the carbon on the left. This carbon over here on the left, this carbon right here, used to have four bonds to it. Now it has only three bonds to it. So it ends up with a plus 1 formal charge. So we have a carbocation in our mechanism. So what's left? In this acid base reaction, we took a proton away from H3O+, which leaves us H2O. So here we have H2O over here, so I'll go ahead and put lone pairs of electrons in on our water molecule. And we know that water can act as a nucleophile here. So this lone pair of electrons is going to be attracted to something that's positively charged. So nucleophilic attack on our carbocation. And this is technically at equilibrium as well, depending on the concentrations of your reactants. So let's go ahead and show that water molecule adding on to the carbon on the left. So the carbon on the right already had a hydrogen or proton added onto it, and the carbon on the left is going to have an oxygen now bonded to that carbon. Two hydrogens bonded to that oxygen, and there was a lone pair of electrons on that oxygen that did not participate in any kind of bonding. This gives this oxygen right here a plus 1 formal charge. So our oxygen is now positively charged, like that. And we're almost to our product. So we're almost there. We need one more acid base reaction to get rid of that proton on our oxygen. So water can function as a base this time. So water comes along, and this time it's going to act as a Bronsted Lowry base and accept a proton. So let's get those electrons in there. So this lone pair of electrons, let's say it takes that proton, leaving these electrons behind on my oxygen. Once again, I'll draw my equilibrium arrows here, acid base reaction. And I'm going to end up with an OH on the carbon on the left, and the carbon on the right there is a hydrogen, like that. So I added water. I ended up adding water across my double bond. And to be complete, this would regenerate my hydronium ion. I'd get H2O plus H+ would give me H3O+. So hydronium is regenerated. And so there you go. So remember, a carbocation is present, so you have to think about Markovnikov addition. And since a carbocation is present, you have to think about possible rearrangements. So Markovnikov and rearrangements. Let's take a look at an example where you have a rearrangement here. So let's look at a reaction. So let's look at this as our starting alkene. And let's go ahead and think about the mechanisms. So we know H3O+ is going to be present. So H3O+ right here. So we're adding our alkene to a solution of water and sulfuric acid. And our first step in the mechanism, the pi electrons are going to function as a base and take a proton from our hydronium ion, leaving these electrons in here letting them kick back off onto the oxygen. So let's see what we would have from that acid base reaction. And I realize I didn't draw an equilibrium arrow here. I'm more concerned with getting the right product. So this is our carbon skeleton. And which side do we add the hydrogen? Which side of the double bond-- do we add the proton to the left side, or do we add the proton to the right side? Well, we want to form the most stable carbocation we possibly can. And if we add the proton to the left side of our double bond, we end up with a secondary carbocation. This carbocation right here is secondary, because this carbon that has the positive charge is bonded to two other carbons. So this is a secondary carbocation. If we had added the proton on to the other side of the double bond, would have a primary carbocation. So secondary carbocation is more stable. But can we form something that's even more stable than a secondary carbocation? Of course we can. We can form a tertiary carbocation if we think about the possibility of a hydride shift. So right here there is a hydrogen attached to that carbon. And if the proton and these two electrons are going to move over here and form a new bond with our positively charged carbon, so we get a hydride shift at this point. So let's draw what would result from that hydride shift. We moved a hydrogen over here. That took a bond away from this carbon. So that is the carbon that's going to end up with the positive charge now. We added a bond to what used to be our secondary carbocation carbon. And so that formal charge goes away. The formal charge moves to this carbon right here, which is now a tertiary carbocation. If you look at the carbons connected to that carbon, this is a tertiary carbocation. So we know tertiary carbocations are more stable than secondary. So now we're at the step of the mechanism where a water molecule is going to come along. So we have a water molecule, which is going to function as a nucleophile and attack our positively charged carbon, like that. So let's go ahead and draw what the result of that nucleophilic attack would look like. So we have our carbon skeleton, and we have an oxygen atom now bonded to that carbon. So two hydrogens here, and once again, one lone pair of electrons now participate in that reaction, giving this oxygen a plus 1 formal charge. And then finally, instead of showing the last step, a water molecule comes along, takes one of the protons off of our positively charged oxygen and gives us our major product with the OH adding on to this carbon right here. So this is a major product. This is our major product. And we would get some of the alcohol that forms from the secondary carbocation. So a minor product, that's what we would get if this oxygen had attacked our secondary carbocation. And you will get some of that. But if your test asks for the major product, you should show the product of this rearrangement. Now, we're lucky in this instance because our product here, this carbon right here, ends up not being a chirality center. Because I have two methyl groups attached to that carbon, so I don't have to worry about my stereochemistry here. Let's do a reaction where we do have to worry about stereochemistry. OK. So let's look at this reaction right here. So we take this as our starting alkene. So I'll put the double bond right there. And let's make that a little bit more clear here. So the double bond is between these two carbons right here. And once again, we're going to add water and sulfuric acid. So H2SO4. And when we think about the mechanism, we know that we're going to add a proton to one side of the double bond and the other side of the double bond is going to end up being our carbocation. So the first thing to think about is OK, which one of these two sides is going to get the proton. We want to form the most stable carbocation we can. So the proton's going to add on to the carbon on the left. So if I can go ahead and show the intermediate here. So for an intermediate-- don't need that arrow. We'll just go ahead and show the proton adding on to the carbon on the left. So we get an H here. And then the carbon on the right of the double bond now ends up being our carbocation. So this is now positively charged, like that. And remember, when we have a carbocation, this carbon is bonded to three other atoms. You have to think about what that looks like. So remember, a carbocation-- when something is bonded to three other carbons, you get this situation where everything is in the same plane. Sp2 hybridized carbon exhibits trigonal planar geometry. Also with your unhybridized p orbital, like that. So this is your sp2 hybridized carbocation situation here. So when your water molecule comes along and acts as a nucleophile, your water molecule could end up attacking from the top here. Or it could end up attacking from below here. So that's where the stereochemistry comes in. So let's go ahead and take our carbocation and let's see if we can draw the products that would result from our nucleophilic attack of water. And then we'll just go ahead and think about the proton going away in our heads for the mechanism. So when you have enough practice, you can do steps of the mechanism in your head. So let's see. What would we get for our two possible products? Well, the OH could've added this way, which would push that methyl group there away from us. So the methyl group would be going away from us. Or the OH could've added from the opposite side. The OH could've been the one back here, and that would've pushed the methyl group out like this. OK. And we know that these are chirality centers. We know that this is our chirality center on these guys. So we get enantiomers here. 50% racemic mixture for our products. And we see that the OH adds on in a Markovnikov fashion. It adds on to the side that's the most stable carbocation here. So one more thing about this reaction. So let's just do one that doesn't have any stereochemistry to worry about. And we'll try to make a different point about this reaction here. So this is our reaction. So we're going to add water to this. And we'll put sulfuric acid up here. And we'll make our arrow a little bit different to illustrate the point here. So we could go-- let's go ahead and draw an equilibrium arrow here. So let's say this whole reaction is at equilibrium. So let me get this equilibrium arrow in here. And our product. So if we don't have to worry about stereochemistry, we think OK, really all I have to do is think about which side of the double bond do I put my OH. And again, It's Markovnikov addition. The more substituted carbon is the one that's going to get your OH. So the more substituted carbon would be the one on the left here. So if you were a product, you would say OK, I know all I have to do is really just go ahead and put my OH in there on the more substituted carbon and I'm done. I don't have to worry about stereochemistry for this reaction. I don't have to worry about rearrangement, since it's the tertiary carbocation. So that takes care of it. Now, this reaction is technically at equilibrium. And you could think about water as being one of your reactants. So if water is one of your reactants and you think about general chemistry, Le Chatelier's principle, how do you shift in equilibrium? If you want to make more of your product, if you wanted to make more of this, your product or your alcohol, one way to do it would be to increase the concentration of water. So if you increase the concentration of water, the equilibrium will shift to decrease the stress that was put on the system. So you're going to get a shift to the right, and you're going to form more and more of your product here. But remember, if you have an alcohol for a product, and if you react this alcohol with sulfuric acid, that's an E1 elimination reaction that we saw in earlier videos. So acid catalyzed dehydration, the addition of concentrated sulfuric acid to your alcohol can actually form your alkene. So that's a reaction that we saw earlier, an E1 elimination acid catalyzed dehydration. Which your major product would be your most substituted alkene here. So you could go back the other way. You could go back to the left. Let's say you decreased the concentration of water to shift the equilibrium to the left, and you'd actually form your alkene here. So the way to control your equilibrium is if you want to go to the right, you just dilute your sulfuric acid. You add more water to it, which would increase this concentration. If you want to shift the equilibrium to the left, you decrease your concentration of water, which means using concentrated sulfuric acid. So I could just write concentrated sulfuric acid here. And that would shift your equilibrium to the left and make more of your alkene. So it all depends on what you're trying to make. And so you have to remember all that general chemistry, shifting equilibrium stuff." + }, + { + "Q": "Is it possible that a an issurance company can be insured but another?", + "A": "Yes, this is called reinsurance, and it is common.", + "video_name": "neAFEvNsiqw", + "transcript": "So let's see if we can get a big picture of everything that's happening in this credit default swap market. I'll speak in generalities. Let's say we have Corporation A, Corporation B, Corporation C. And let's say we have a bunch of people who write the credit default swaps, and I'll call them insurers. Because that's essentially what a credit default swap is, it's insurance on debt. If someone doesn't pay the debt, then the insurance company will pay it for you. In exchange, you're essentially giving some of the interest on the debt. So let's say we have Insurer 1, let's say we have Insurer 2. And some of these were insurance companies, some of these were banks. Some of these may have even been hedge funds. So these are the people who write the credit default swaps, and then there are the people who would actually buy the credit default swaps. In the previous example, I had Pension Fund 1, that was my pension fund. Then you could have another pension fund, Pension Fund 2. Let's re-draw some of the connections between the organizations. Let's say Pension Fund 1 were to lend $1 billion to A. A will pay Pension Fund 1 10%. But Pension Fund 1 wants to make sure that they'll definitely get the money, because they can't lend money to people with anything less than stellar credit ratings. So they get some insurance from Insurer 1. So what they do is out of this 10%, they pay them some of the basis points. So let's say they pay them 100 basis points. And in exchange, they get-- I'll call it Insurance On A. This is this new notation that I'm creating. They get Insurance On A. Fair enough. And the reason why this I1, this first insurer was able to do that is because Moody's has given them a very high credit rating. And so when they insure something, you're essentially the total package, right? The loan to this guy, plus the insurance, kind of is like you're lending the money to this guy, but you're just getting more insurance-- I mean you're getting more interest, right? So this bond becomes a Double A bond. Because the odds that you are not going to get your money are not the odds that this guy defaults, but it's now the odds that this guy defaults. And Moody's or the standard is poor, as I've already said. Hey, these guys are good for their money, they're Double A or whatever. So now your risk is really a Double A risk and not a Double B risk, or whatever. But anyway, this happens. This is Corporation B, and maybe Pension Fund 2 wants to lend to Corporation B. Maybe they lend them $2 billion. They get, I don't know, they get 12%, maybe Corporation B is a little bit more dangerous. But once again, they go to this first insurer. And maybe they get some of it-- well let's just say they get Insurance On B. And B is a little bit riskier, so they have to pay 200 basis points. 200 basis points goes from Pension Fund 2 to B. Now this, already, this is a little bit dangerous, right? Because you can think about what's happening. One, as long as this insurer does not get a downgrade from their credit ratings from S&P or Moody's or whoever, they can just keep it issuing this insurance. There's no limit for how much insurance they can issue. There's no law that says, you know what, if you insure a billion dollars of debt, you have to put a billion dollars aside. So that if that debt defaults, you definitely have that billion dollars there. Or if you insure 2 billion here, you don't have to put that 2 billion aside. What you have is a bunch of people who statistically say, oh, you know, what's the probability that all of this debt defaults? So I just have to keep enough capital so that probabilistically, whatever debt defaults, I can pay it. But you don't keep enough capital to pay all of the defaulting debt. So you already see an interesting risk forming. What if all of these corporations, for whatever reason, do start defaulting simultaneously? Then all of a sudden this insurance company has to pay more out in insurance then it might even have. So you have to wonder whether it even deserves this Double A rating, because it actually is taking on a lot of risk. But in the short term, while these companies are-- everyone is doing well and the economy's doing well, it's a great business for these guys. These guys are just collecting premiums essentially on the insurance, without having to pay out anything. Now let's add another twist on it. These pension funds, P1 and P2, it was reasonable for them to get insurance, because they were giving out these loans and then they got the insurance. So they were essentially hedging the default risk by buying these credit default swaps, which was essentially just an insurance policy from this Insurer 1. But you can have another party. This is no less legitimate, really. But you could call them-- I don't know-- let's call it Hedge Fund 1. And they've done a lot of work, and frankly, they often are much more sophisticated than the pension fund-- in fact, they almost always are. And they say, you know what? Company B looks really, really, really, really shady. I think 200 basis points for the chance that Company B defaults is frankly cheap. Because I think there's a huge probability that Company B defaults. So what I'm going to do, I'm not going to lend Company B money, because if anything, I think that they're maybe about to go out of business. But what I can do is I can buy a credit default swap on Company B's debt. Which is, essentially, I'm getting insurance that they fail without actually lending the money. So let's say I do that from Insurer 2. So I can go and I'll pay Insurer 2 200 basis points a year, or 2% on the notional value of the insurance I'm getting. So let's say it's 200 basis points, and let's say that's Insurance On-- I'm making a big bet-- so they're going to give me Insurance On B for-- I don't know-- $10 billion. And something interesting is going on here already. B might not have even borrowed $10 billion, right? So all of a sudden you have this hedge fund that is getting insurance on more debt than B has even borrowed money on, right? And it's essentially, you just kind of have this side bet between these two parties. This party says, you know what? I think it's a good deal. I get 200 basis points on the 10 billion every year, as long as B doesn't default. And this guy says, I think B's going to default. So I think that's a good deal on that insurance. And just so you understand the math, so the notional value is $10 billion. So what's 2% of 10 billion? 2% on a billion is 20 million, so it's $200 million. 200 if I did my math correct. So they'll pay $200 million a year to this insurer. So the 200 basis points on 10 billion is equal to 200 million. These numbers maybe are a little bit on the big side, but who knows? Actually, this could be a huge hedge fund. This could be a $10 billion hedge fund. Or even worse, maybe it's a billion dollar hedge fund, or maybe it's a $20 million hedge fund, but they've taken a $180 million loan to essentially buy this insurance because they think that B's collapse is imminent. So they're willing to take that bet right now. You know, it might be a good bet. If B collapses tomorrow, what's going to happen? They only dished out maybe 200 million for maybe that first year, although you normally pay it on a quarterly basis. So they'll pay 50 million every three months. Let's say they pay the first payment, 50 million, right? And then over the next three months, B just goes bankrupt and people realize that debt was worth nothing. Then these guys get $10 billion. Right? But something else is interesting here. They probably did insurance to a lot of other people too, maybe on B's debt. Or maybe they also insured A's debt. So maybe they gave some insurance on A's debt, as well. So what happens? Let's say B all of a sudden defaults. So a couple of things happen. I1 is going to owe P2 $2 billion, right? I2, the second insurer, is going to owe this hedge fund $10 billion. Now let's just assume I2's good for the money. They have $10 billion they pay to this hedge fund. This hedge fund is great, they get great bonuses for the year and they go buy yachts, et cetera. But this insurer right here, they pay the money they were good for but something interesting might happen. All of a sudden Moody's finally wakes up, these ratings agencies, and says, oh, my God. Well, there's a couple of things that might make them say, oh, my God. First of all, they might say, oh, look. You have to pay out $10 billion. And I doubt that was the only person you have to pay, maybe they have to pay out a lot of money. Now I2, Insurance Company 2, you are undercapitalized. I am now going to downgrade your rating. So, you were Double A, but since you had to give out all of this capital, Moody's is now going to downgrade you to, I don't know, B+. I'm just making these ratings up. But that's the sound of how these ratings happen, right. A is better, B is worse. The more A's you have, the better it is. But all of a sudden, when this guy is B+, and this guy insured, let's say, some other corporation's debt for this pension fund, now all of a sudden this insurance that this pension fund had is no longer Double A Insurance. It's now B+ Insurance, and maybe this pension fund, by its charter, can't hold something that has a B+ credit rating. So they're going to have to unwind the transaction, or maybe they'll have to unload the debt that was insured. So one, just by Company B defaulting, maybe this guy was holding some of Company A's debt, and it was insured by Insurance Company 1. Now they're going to have to unload that debt. So just one default creates this chain reaction, right? This one default happens, this guy has to pay this guy money, then this guy gets undercapitalized since they have to pay out money. Then Moody's says, oh, my God, you're undercapitalized. We're going to reduce your ratings. Maybe this guy was insuring some of A's debt, but now since he was insuring some of A's debt, all of a sudden that insurance is worth less because it has a lower rating. And now A's debt, less people want to hold it, because there are less people to insure it. I know that's very confusing, but this is really the point that Warren Buffett was saying when he said that the credit defaults swap market, or in general, the derivative market, are financial weapons of mass destruction. Because you have so many people who didn't have to set aside a capital, right? This guy could insure $10 billion worth of debt without having to set aside $10 billion. And you have so many people making all of these side bets, but they're all making two core assumptions. One, that these rating agencies's ratings are valid. And two, that the other person is good for the money. But if all of a sudden you have one failure someplace in the system, you could have this cascade where one, there's just a lot of downgrades. And then a lot of the people end up not being good for the money." + }, + { + "Q": "In my lecture notes, it states that the ATP synthase produces 32 ATP and not 34.\nThis gives a total of 32 (from ATP synthase) + 4 (from glycolysis and the Krebs cycle) = 36 ATP\nand the extra 2 ATP comes from something else, (I think it says fermentation)", + "A": "Different text books and notes vary in the information, but most(if not all) have Cellular Respiration making 36-38 ATP. I hope you found this helpful!", + "video_name": "mfgCcFXUZRk", + "transcript": "After being done with glycolysis and the Krebs Cycle, we're left with 10 NADHs and 2 FADH2s. And I told you that these are going to be used in the electron transport chain. And they're all sitting in the matrix of our mitochondria. And I said they're going to be used in the electron transport chain in order to actually generate ATP. So that's what I'm going to focus on in this video. The electron transport chain. And just so you know, a lot of this stuff is known. But some of the details are actually current areas of research. People have models and they're trying to substantiate the models. But things are happening at such a small scale here that people can just look at the evidence, some of which is indirect, and say, this is probably what's happening. Most of this is very well established, but some of the exact mechanisms-- for example, how exactly some of the proteins work-- aren't completely understood. So I think it's very important for you to understand that this is at the cutting edge, that you're already there. So the basic idea here is that the NADHs-- and that's where FADH2 is kind of the same idea. Although its electrons are just at slightly lower energy state. So they won't produce quite as many ATPs. Each NADH is going to be-- as you'll see-- indirectly responsible for the production of three ATPs. And each FADH2, in a very efficient cell, in both of these cases, will be indirectly responsible for the production of two ATPs. And the reason why this guy produces fewer ATPs is because the electrons that he has going into the electron transport chain are at a slightly lower energy level than the ones from NADH. So in general, I just said indirectly. How does this whole business work? Well in general, NADH, when it gets oxidized-- remember, oxidation is the losing of electrons or the losing of hydrogens that happen to have electrons. We can write its half reaction like this. Its oxidation reaction like this. You'll have some NAD plus, which you can then go and use back in the Krebs Cycle and in glycolysis. You have some NAD plus, you'll have a proton, a positive hydrogen ion is just a proton. And then you'll have two electrons. This is the oxidation of NADH. It's losing these two electrons. Oxidation is losing electrons. OIL RIG. Oxidation is losing electrons. Or you can imagine it's losing hydrogens, from which it can hog electrons. Either one of those is the case. Now this is really the first step of the electron transport chain. These electrons are transported out of the NADH. Now, the last step of the electron transport chain is you have two electrons-- and you could view it as the same two electrons if you like-- two electrons plus two hydrogen protons. And obviously if you just add these two together, you're just going to have two hydrogen atoms, which is just a proton and an electron. Plus one oxygen atom, so I could say one half of molecular oxygen. That's the same thing as saying one oxygen atom. And you're going to produce-- if I have one oxygen and two complete hydrogens, I'm left with water. And you could view this, we're adding electron or we're gaining electrons to oxygen. OIL RIG. Reduction is gaining electrons. So this is the reduction of oxygen to water. This is the oxidation of NADH to NAD plus. Now, these electrons that are popping out of-- these electrons right here-- that are popping out of this NADH. And when they're in NADH they're at a very high energy state. And what happens over the course of the electron transport chain is that these electrons get transported to a series of, I guess you could call them transition molecules. But these transition molecules, as the electrons go from one to the other, they go into slightly lower energy states. And I won't even go into the details of these molecules. One is coenzyme Q, and cytochrome C. And then they eventually end up right here and they are used to reduce your oxygen into water. Now every time an electron goes from a higher energy state to a lower energy state-- and that's what it's doing over the course of this electron transport chain-- it's releasing energy. So energy is released when you go from a higher state to a lower state. When these electrons were in NADH, they were at a higher state than they are when they bond to coenzyme Q. So they release energy. Then they go to cytochrome C and release energy. Now that energy is used to pump protons across the cristae across the inner membrane of the mitochondria. And I know this is all very complicated sounding. And this is the cutting edge. So it maybe should sound a little complicated. Let me draw a mitochondria. So let me draw a small mitochondria just so you know where we're operating. That's its outer membrane. And then its inner membrane, or its cristae, would look like that. And let me zoom in on the membrane. So let's say if I were to zoom in right there. So if I were to zoom that out, that box would look like this. You have your crista here. And I'm going to draw it thick. So I'm zooming in. This green line right here, I'm going to draw it really thick. I'm going to color it in with the green, just like that. And then you have your outer membrane. This outer membrane, I can do it up here. And I'll just color it in. You don't even have to see the outside of the outer membrane. Right here, this space right here, this is the outer compartment. And then we learned in the last video, this space right here is the matrix. This is where our Krebs cycle occurred. And where a lot of our NADH, or really all of our NADH, is sitting. So what happens is, every time NADH gets oxidized to NAD plus, and the electrons keep transferring from one molecule to another, it's occurring in these big protein complexes. And I'm not going to go into the details on this. So each of these protein complexes span-- so let's say that's a protein complex where this first oxidation reaction is occurring and releasing energy. And then let's say there's another protein complex here, where the second oxidation reaction is occurring and releasing energy. And these proteins are able to use that energy to essentially pump-- this might all seem very complicated-- to essentially pump hydrogens into the outer membrane. It actually pumps hydrogen protons. Hydrogen protons into the outer membrane. And every one of these reactions pump out a certain number of hydrogen protons. So by the end of the electron transport chain, or if we just followed one set of electrons, by the time that they've gone from their high energy state in NADH to their lower energy state in water, by the time they've done that, they've supplied the energy to these protein complexes that span our cristae to pump hydrogen from the matrix into the outer membrane. So really the only byproduct of the oxidation of NADH into, eventually, water, or the oxidation of NADH and the reduction of oxygen into water, isn't ATPs yet. It's just this gradient where we have a lot higher hydrogen proton concentration in the outer compartment than we do in the matrix. Or you could say that the outer compartment becomes a lot more acidic. Remember acidity is just hydrogen proton concentration, the concentration of hydrogen protons. So the byproduct of all of this energy is used to really just pump these protons into the outer membrane. So you have two things. The outer membrane becomes more acidic than the matrix inside. Maybe we could call that basic. And obviously these are all positively charged particles. So there's actually an electric gradient, an electric potential between the outer membrane and the inner membrane. This becomes slightly negative, that becomes slightly positive. These guys wouldn't naturally do this on their own. If this is already acidic and it's already positive, left to its own devices, these more protons wouldn't be entering. And the energy to do that is supplied by electrons going from high energy state in NADH to going to a lower energy state, eventually, on the oxygen in the water. That's what's happening. But essentially all that's happening is protons being pumped from the matrix into the outer compartment. Now once that gradient forms, these guys want to get back in. These guys want to get back into the matrix. And that is where the ATPs are formed. So there's a protein that also spans this. Let me draw. Remember this is all this inner membrane right here. Let me just draw it a little bit bigger right here. So that's our inner membrane, our cristae right there. There's a special protein called-- and I'll show you actually a better diagram of what looks like in a second-- called ATP synthase. And what happens is, remember because of the electron transport chain, we have all of these hydrogen ions up here, all of these protons really. All they are is a proton. That really want to get back into the matrix down here. This crista is impermeable to them so they have to find a special way to get through. They were able to go the reverse direction through the special protein complexes. Now they're going to go back into the matrix through ATP synthase. So they're going to go back, but something interesting happens. And this is really an area of current research where people think they know how it works but they're not sure. Because you can't just take these proteins apart and watch them operate like you can a regular mechanical engine. These are ultra-small and they have to be in a living system. And they have to have the right conditions. And you can't even-- it's hard to see hydrogen protons. These are ultra-small things that are, pretty much, you can't see them. But what happens, the current model is, as these enter, as these go through my ATP synthase there's actually an axle. So you can kind of view this as a housing. And then there's an axle. And this is all just a big protein. And there's an axle and then there's another part of the synthase down here. So you can imagine, this is kind of mind-blowing. That something this fancy is occurring on the membranes of pretty much all living systems' cells. It's not just eukaryotes. Even prokaryotes do this. They don't do it in their mitochondria; they do it in their cellular membrane. But it's a pretty neat thing. And what happens is, as these go through, you can kind of imagine as water flowing through a turbine. It mechanically causes this structure in the middle to spin. To actually spin. This is the current thinking. And this thing is all uneven. It's not, like, this nice tube. It'll look all crazy like that. And what happens is that an ADP molecule-- let's say that this is the A part of the ADP. And then you have two phosphate groups. It'll attach to one part of this protein. And maybe a phosphate will just randomly attach to another part of this protein. Just like that. So right now it's just ADP and a phosphate. But as this inner axle turns-- because it's not a symmetrical tube, it has different things sticking out that have different amounts of atomic charge and it's going to play with this outer housing right here. And so as this turns, the outer housing, because of just the proteins bumping against each other and electrical charge and whatever else, it's going to squeeze the ADP and the phosphate together to form, actually form, ATP. And actually the current thinking is that it does it on three different sites simultaneously. So as this spins around, ADP and phosphate groups kind of show up on the inside of this housing. You could imagine it like that. And I don't even know if it's on the inside. But they show up on the housing. And as this thing spins around, it stretches and pulls on this outer part and pushes these two things together. So it's using the energy from this proton gradient to drive this axle. And because it's all strange, it does all these distortions on this outer part and actually pushes the two ATPs together. So when you start off with your 10 NADHs, it'll provided just enough energy and just enough protons to put into the outer membrane that when they go back through our ATP synthase-- you could almost view it as an ATP synthase motor-- just based on people's observations they see that this will produce, on a per-NADH level, roughly three ATPs. On a per-FADH2 level, roughly two ATPs. And I've said multiple times in the videos, this is kind of an ideal. That a lot of times, maybe you'll have some protons leak, so their energy can't be captured properly. Or maybe some of these electrons might somehow jump the gun or jump some steps, so some of the energy gets lost. So you don't always have a completely efficient system. And just so you believe that this is actually occurring on our membrane, there's actual visual depictions of these proteins. This is the actual protein structure of ATP synthase right here. That is actually ATP synthase. And as you can see, there's this piece right here that holds this part and that part. You can kind of imagine it relatively stationary. The hydrogen comes through here. The axle gets spun. And as the axle gets spun, ADP and phosphate groups that are lodged inside this F1 part of the protein, get pushed together. You have to put energy into the reaction in order to make them stick together. But they get pushed together by the protein itself as this axle turns around. And this axle turns around from the energy of the hydrogen going. I don't even know what the mechanics would look like. But you could imagine-- in my head I imagine, the simplest thing is a windmill. Or not a windmill, as maybe some type of water turbine or maybe the simplest thing is, if you have something like that. I don't know if that's what the protein If you have any kind of thing passing by, it's going to spin it. It's going to spin it like that. And you could be more creative if you want to change the angle of the spin and whatnot. And that's all, people are really still trying to understand this at a deeper and deeper level. But for your purposes, especially in an introductory biology level, you just have to realize that two things are happening in the electron transport chain. Electrons are moving from the NADHs and the FADH2s to eventually show up and reduce the oxygen. And as they do that, they're releasing energy as they go from one molecule to another. They're going to lower energy states. That energy is used to pump hydrogen protons into the outer compartment of the mitochondria. And then that gradient, those hydrogen protons want to get back into the matrix of the mitochondria. So as they go back in, that drives this ATP synthase engine, which actually produces the ATP. So just like we said in the past, when you have 10 of these, on average-- let me say this way-- on average each NADH is going to produce 3 ATPs. Not directly. It produces enough of a gradient of hydrogen protons to produce 2 ATPs in the ATP synthase. And each FADH2, on average, produces enough of a hydrogen gradient to produce 2 ATPs. So if we come in with 10 NADH, they're going to produce-- in this ideal world-- 30 ATP. And then our 2 FADH2s are going to produce 4 ATP. And then if you remember from glycolysis, we had 2 net ATPs directly produced. And from the Krebs cycle we had 2 ATPs directly produced. So then you have 4 from glycolysis and Krebs, and that gets us, once again, to our magic 38 ATPs from one molecule of glucose. And now, I think you have a pretty good grasp of cellular respiration." + }, + { + "Q": "I have a question how do i find the percent of a fraction ? please help", + "A": "or if it is simple like 40/100 than it will be .40 = 40%", + "video_name": "X2jVap1YgwI", + "transcript": "Let's do some more percentage problems. Let's say that I start this year in my stock portfolio with $95.00. And I say that my portfolio grows by, let's say, 15%. How much do I have now? I think you might be able to figure this out on your own, but of course we'll do some example problems, just in case it's a little confusing. So I'm starting with $95.00, and I'll get rid of the dollar sign. We know we're working with dollars. 95 dollars, right? And I'm going to earn, or I'm going to grow just because I was an excellent stock investor, that 95 dollars is going to grow by 15%. So to that 95 dollars, I'm going to add another 15% of 95. So we know we write 15% as a decimal, as 0.15, so 95 plus 0.15 of 95, so this is times 95-- that dot is just a times sign. It's not a decimal, it's a times, it's a little higher than a decimal-- So 95 plus 0.15 times 95 is what we have now, right? Because we started with 95 dollars, and then we made another 15% times what we started with. Hopefully that make sense. Another way to say it, the 95 dollars has grown by 15%. So let's just work this out. This is the same thing as 95 plus-- what's 0.15 times 95? Let's see. So let me do this, hopefully I'll have enough space here. 95 times 0.15-- I don't want to run out of space. Actually, let me do it up here, I think I'm about to run out of space-- 95 times 0.15. 5 times 5 is 25, 9 times 5 is 45 plus 2 is 47, 1 times 95 is 95, bring down the 5, 12, carry the 1, 15. And how many decimals do we have? 1, 2. 15.25. Actually, is that right? I think I made a mistake here. See 5 times 5 is 25. 5 times 9 is 45, plus 2 is 47. And we bring the 0 here, it's 95, 1 times 5, 1 times 9, then we add 5 plus 0 is 5, 7 plus 5 is 12-- oh. I made a mistake. It's 14.25, not 15.25. So I'll ask you an interesting question? How did I know that 15.25 was a mistake? Well, I did a reality check. I said, well, I know in my head that 15% of 100 is 15, so if 15% of 100 is 15, how can 15% of 95 be more than 15? I think that might have made sense. The bottom line is 95 is less than 100. So 15% of 95 had to be less than 15, so I knew my answer of 15.25 was wrong. And so it turns out that I actually made an addition error, and the answer is 14.25. So the answer is going to be 95 plus 15% of 95, which is the same thing as 95 plus 14.25, well, that equals what? 109.25. Notice how easy I made this for you to read, especially this 2 here. 109.25. So if I start off with $95.00 and my portfolio grows-- or the amount of money I have-- grows by 15%, I'll end up with $109.25. Let's do another problem. Let's say I start off with some amount of money, and after a year, let's says my portfolio grows 25%, and after growing 25%, I now have $100. How much did I originally have? Notice I'm not saying that the $100 is growing by 25%. I'm saying that I start with some amount of money, it grows by 25%, and I end up with $100 after it grew by 25%. To solve this one, we might have to break out a little bit of algebra. So let x equal what I start with. So just like the last problem, I start with x and it grows by 25%, so x plus 25% of x is equal to 100, and we know this 25% of x we can just rewrite as x plus 0.25 of x is equal to 100, and now actually we have a level-- actually this might be level 3 system, level 3 linear equation-- but the bottom line, we can just add the coefficients on the x. x is the same thing as 1x, right? So 1x plus 0.25x, well that's just the same thing as 1 plus 0.25, plus x-- we're just doing the distributive property in reverse-- equals 100. And what's 1 plus 0.25? That's easy, it's 1.25. So we say 1.25x is equal to 100. Not too hard. And after you do a lot of these problems, you're going to intuitively say, oh, if some number grows by 25%, and it becomes 100, that means that 1.25 times that number is equal to 100. And if this doesn't make sense, sit and think about it a little bit, maybe rewatch the video, and hopefully it'll, over time, start to make a lot of sense to you. This type of math is very very useful. I actually work at a hedge fund, and I'm doing this type of math in my head day and night. So 1.25 times x is equal to 100, so x would equal 100 divided by 1.25. I just realized you probably don't know what a hedge fund is. I invest in stocks for a living. Anyway, back to the math. So x is equal to 100 divided by 1.25. So let me make some space here, just because I used up too much space. Let me get rid of my little let x statement. Actually I think we know what x is and we know how we got to there. If you forgot how we got there, you can I guess rewatch the video. Let's see. Let me make the pen thin again, and go back to the orange color, OK. X equals 100 divided by 1.25, so we say 1.25 goes into 100.00-- I'm going to add a couple of 0's, I don't know how many I'm going to need, probably added too many-- if I move this decimal over two to the right, I need to move this one over two to the right. And I say how many times does 100 go into 100-- how many times does 125 go into 100? None. How many times does it go into 1000? It goes into it eight times. I happen to know that in my head, but you could do trial and error and think about it. 8 times-- if you want to think about it, 8 times 100 is 800, and then 8 times 25 is 200, so it becomes 1000. You could work out if you like, but I think I'm running out of time, so I'm going to do this fast. 8 times 125 is 1000. Remember this thing isn't here. 1000, so 1000 minus 1000 is 0, so you can bring down the 0. 125 goes into 0 zero times, and we just keep getting 0's. This is just a decimal division problem. So it turns out that if your portfolio grew by 25% and you ended up with $100.00 you started with $80.00. And that makes sense, because 25% is roughly 1/4, right? So if I started with $80.00 and I grow by 1/4, that means I grew by $20, because 25% of 80 is 20. So if I start with 80 and I grow by 20, that gets me to 100. Makes sense. So remember, all you have to say is, well, some number times 1.25-- because I'm growing it by 25%-- is equal to 100. Don't worry, if you're still confused, I'm going to add at least one more presentation on a couple of more examples like this." + }, + { + "Q": "at 11:41, why is the average velocity in the horizontal direction is 5 square roots of 3 metres per second? I know Sal said it is because it doesn't change, but why does it not change?", + "A": "Gravity only affects the velocity in the vertical direction, and since we are assuming that there is no air resistance, there is nothing to change the horizontal velocity.", + "video_name": "ZZ39o1rAZWY", + "transcript": "- [Voiceover] So I've got a rocket here. And this rocket is going to launch a projectile, maybe it's a rock of some kind, with the velocity of ten meters per second. And the direction of that velocity is going to be be 30 degrees, 30 degrees upwards from the horizontal. Or the angle between the direction of the launch and horizontal is 30 degrees. And what we want to figure out in this video is how far does the rock travel? We want to figure out how, how far does it travel? Does it travel? And to simplify this problem, what we're gonna do is we're gonna break down this velocity vector into its vertical and horizontal components. We're going to use a vertical component, so let me just draw it visually. So this velocity vector can be broken down into its vertical and its horizontal components. And its horizontal components. So we're gonna get some vertical component, some amount of velocity in the upwards direction, and we can figure, we can use that to figure out how long will this rock stay in the air. Because it doesn't matter what its horizontal component is. Its vertical component is gonna determine how quickly it decelerates due to gravity and then re-accelerated, and essentially how long it's going to be the air. And once we figure out how long it's in the air, we can multiply it by, we can multiply it by the horizontal component of the velocity, and that will tell us how far it travels. And, once again, the assumption that were making this videos is that air resistance is negligible. Obviously, if there was significant air resistance, this horizontal velocity would not stay constant while it's traveling through the air. But we're going to assume that it does, that this does not change, that it is negligible. We can assume that were doing this experiment on the moon if we wanted to have a, if we wanted to view it in purer terms. But let's solve the problem. So the first that we want to do is we wanna break down this velocity vector. We want to break down this velocity vector that has a magnitude of ten meters per second. And has an angle of 30 degrees with the horizontal. We want to break it down it with x- and y-components, or its horizontal and vertical components. so that's its horizontal, let me draw a little bit better, that's its horizontal component, and that its vertical component looks like this. This is its vertical component. So let's do the vertical component first. So how do we figure out the vertical component given that we know the hypotenuse of this right triangle and we know this angle right over here. And the angle, and the side, this vertical component, or the length of that vertical component, or the magnitude of it, is opposite the angle. So we want to figure out the opposite. We have to hypotenuse, so once again we write down so-cah, so-ca-toh-ah. Sin is opposite over hypotenuse. So we know that the sin, the sin of 30 degrees, the sin of 30 degrees, is going to be equal to the magnitude of our vertical component. So this is the magnitude of velocity, I'll say the velocity in the y direction. That's the vertical direction, y is the upwards direction. Is equal to the magnitude of our velocity of the velocity in the y direction. Divided by the magnitude of the hypotenuse, or the magnitude of our original vector. Divided by ten meters per second. Ten meters per second. And then, to solve for this quantity right over here, we multiply both sides by 10. And you get 10, sin of 30. 10, sin of 30 degrees. 10 sin of 30 degrees is going to be equal to the magnitude of our, the magnitude of our vertical component. And so what is the sin of 30 degrees? And this, you might have memorized this from your basic trigonometry class. You can get the calculator out if you want, but sin of 30 degrees is pretty straightforward. It is 1/2. So sin of 30 degrees, use a calculator if you don't remember that, or you remember it now so sin of 30 degrees is 1/2. And so 10 times 1/2 is going to be five. So, and I forgot the units there, so it's five meters per second. Is equal to the magnitude, is equal to the magnitude of our vertical component. Let me get that in the right color. It's equal to the magnitude of our vertical component. So what does that do? What we're, this projectile, because vertical component is five meters per second, it will stay in the air the same amount of time as anything that has a vertical component of five meters per second. If you threw a rock or projectile straight up at a velocity five meters per second, that rocket projectile will stay up in the air as long as this one here because they have the same vertical component. So let's think about how long it will stay in the air. Since were dealing with a situation where we're starting in the ground and we're also finishing at the same elevation, and were assuming the air resistance is negligible, we can do a little bit of a simplification here. Although I'll do another version where we're doing the more complicated, but I guess the way that applies to more situations. We could say, we could say \"well what is our \"change in velocity here?\" So if we think about just the vertical velocity, our initial velocity, let me write it this way. Our initial velocity, and we're talking, let me label all of this. So we're talking only in the vertical. Let me do all the vertical stuff that we wrote in blue. So vertical, were dealing with the vertical here. So our initial velocity, in the vertical direction, our initial velocity in the vertical direction is going to be five meters per second. Is going to be five meters per second. And we're going to use a convention, that up, that up is positive and that down is negative. And now what is going to be our final velocity? We're going to be going up and would be decelerated by gravity, We're gonna be stationary at some point. And then were to start accelerating back down. And, if we assume that air resistance is negligible, when we get back to ground level, we will have the same magnitude of velocity but will be going in the opposite direction. So our final velocity, remember, we're just talking about the vertical component right now. We haven't even thought about the horizontal. We're just trying to figure out how long does this thing stay in the air? So its final velocity is going to be negative five. Negative five meters per second. And this is initial velocity, the final velocity is going to be looking like that. Same magnitude, just in the opposite direction. So what's our change in velocity in the vertical direction? Change in velocity, in the vertical direction, or in the y-direction, is going to be our final velocity, negative five meters per second, minus our initial velocity, minus five meters per second, which is equal to negative 10 meters per second. So how do we use this information to figure out how long it's in the air? Well we know! We know that our vertical, our change our change in our, in our vertical velocity, is going to be the same thing or it's equal to our acceleration in the vertical direction times the change in time. Times the amount of time that passes by. What's our acceleration in the vertical direction? What's the acceleration due to gravity, or acceleration that gravity, that the force of gravity has an object in freefall? and so this, right here, is going to be negative 9.8 meters per second squared. So this quantity over here is negative 10 meters per second, we figured that out, that's gonna be the change in velocity. Negative 10 meters per second is going to be equal to negative 9.8, negative 9.8 meters per second squared times our change in time. So to figure out the total amount of time that we are the air, we just divide both sides by negative 9.8 meters per second squared. So we get, lets just do that, I wanna do that in the same color. So I do it in, that's not, well, that close enough. So we get negative 9.8 meters per second squared. Negative 9.8 meters per second squared. That cancels out, and I get my change in time. And I'll just get the calculator. I have a negative divided by a negative so that's a positive, which is good, because we want to go in positive time. We assume that the elapsed time is a positive one. And so what we get? If I get my calculator out, I get my calculator out. I have, this is the same thing as positive 10 divided by 9.8. 10, divided by 9.8. Gives me 1.02. I'll just round to two digits right over there. So that gives me 1.02 seconds So our change in time, so this right over here is 1.02. So our change in time, delta t, I'm using lowercase now but I can make this all lower case. Is equal to 1.02 1.02 seconds. Now how do we use this information to figure out how far this thing travels? Well if we assume that it retains its horizontal component of its velocity the whole time, we just assume we can this multiply that times our change in time and we'll get the total displacement in the horizontal direction. So to do that, we need to figure out this horizontal component, So this is the component of our velocity in the x direction, or the horizontal direction. Once again, we break out a little bit of trigonometry. This side is adjacent to the angle, so the adjacent over hypotenuse is the cosine of the angle. Cosine of an angle is adjacent over hypotenuse. So we get cosine. Cosine of 30 degrees, I just want to make sure I color-code it right, cosine of 30 degrees is equal to the adjacent side. Is equal to the adjacent side, which is the magnitude of our horizontal component, is equal to the adjacent side over the hypotenuse. Over 10 meters per second. multiply both sides by 10 meters per second, you get the magnitude of our adjacent side, color transitioning is difficult, the magnitude of our adjacent side is equal to 10 meters per second. Is equal to 10 meters per second. Times the cosine, times the cosine of 30 degrees. And you might not remember the cosine of 30 degrees, you can use a calculator for this. Or you can just, if you do remember it, you know that it's the square root of three over two. Square root of three over two. So to figure out the actual component, I'll stop to get a calculator out if I want, well I don't have to use it, do it just yet, because I have 10 times the square root of three over two. Which is going to be 10 divided by two is five. So it's going to be five times the square root of three meters per second. So if I wanna figure out the entire horizontal displacement, so let's think about it this way, the horizontal displacement, we're trying to figure out, the horizontal displacement, a S for displacement, is going to be equal to the average velocity in the x direction, or the horizontal direction. And that's just going to be this five square root of three meters per second because it doesn't change. So it's gonna be five, I don't want to do that same color, is going to be the five square roots of 3 meters per second times the change in time, times how long it is in the air. And we figure that out! Its 1.02 seconds. Times 1.02 seconds. The seconds cancel out with seconds, and we'll get that answers in meters, and now we get our calculator out to figure it out. so we have five time the square root of three, times 1.02. It gives us 8.83 meters, So this is going to be equal to, this is going to be equal to, this is going to be oh, sorry. this is going to be equal to 8.8, is that the number I got? 8.83, 8.83 meters. And we're done. And the next video, I'm gonna try to, I'll show you another way of solving for this delta t. To show you, really, that there's multiple ways to solve this. It's a little bit more complicated but it's also a little bit more powerful if we don't start and end at the same elevation." + }, + { + "Q": "So is size a noun", + "A": "Watch this: What is your size? ( Size in that form is a noun) and this: Can I get your size? ( That way it is a verb!) So @volpeana000 there you go!", + "video_name": "ETzngG8N3AU", + "transcript": "- Hello grammarians! Let's talk about singular and plural nouns. Nouns, as we discussed previously, are a type of word. They are a part of speech. A noun is any word that is a person, a place, a thing, or an idea. In English, we can figure out just by looking at a noun whether or not there is one of something, whether it's a singular, or whether or not there is more than one of something. There's an easy way to tell the difference between singular and plural. If you write the words down, singular contains the word single. Single, means there's only one of it. Plural is maybe a little bit less obvious, but it comes to us from Latin. It comes to us from this word plus, which means more, which you might recognize plus, as we call it in English, from mathematics, from arithmetic. We usually says it looks like this little plus symbol. So, whenever you think, whenever you see plural, just think more; just think plus. There is more than one. Singular is one thing. Plural, more than one thing; there is more. Let's go through it. Let's do some examples. I'll show you how you make the plural in English, how you indicate using your language that there is more than one thing. So let's just throw out a couple of words. Dog, cat, dinosaur, and whale. All you need to do to make it plural is very simply just take an s and you add it onto the end like so. Dogs, cats, dinosaurs, whales. If you want to make something plural, think about plus, more. All you have to do is add an s like that: add an s. This is what we call the regular plural. This is the regular plural. What that means is it obeys this one rule. All you have to do to say that there's more than one dog is throw on an s, and we're lucky because most English nouns behave that way. Most nouns are regular. However, here's the bad news. There are some irregular plurals. They are not regular, thus irregular, not. Now we have words like leaf, child, and fungus, which is like a mushroom, mouse, and sheep. How would you, you know you can't just add an s to these? That's unfortunately not how these nouns work in English. You can't say leafs, childs, and funguses, and mouses, and sheeps. This is how you do it. Each one of these words corresponds to a class of words that has its own unique pluralization standards. So, leaf becomes leaves. Child becomes children. Fungus becomes fungi. Mouse becomes mice. And sheep stays sheep, believe it or not. These are the irregular plurals, and we'll be covering each of these in turn in later videos, but for now I just want you to focus on the regular plural, which again we can sum up in this way. All you have to do is add an s. Here's a good example, right? We have one elephant here. Down here we have two elephants. The only difference between this word and this word is that this one has an s on the end of it. So if we wanted to say that this elephant here was not, in fact, one elephant, and was two elephants, all we have to do is add an s, changing it from singular to plural. Remember, plural comes from plus. Add an s. So one elephant becomes two elephants. (humming) World's fastest elephant drawing, go! (humming) It's kind of an elephant monkey, but you get the vague idea. If you're ever in need of more than one thing, for the regular plural, just add an s. You can learn anything. David, out!" + }, + { + "Q": "Is reflection actually refraction on a denser media?", + "A": "no", + "video_name": "jxptCXHLxKQ", + "transcript": "Before doing more examples with Snell's Law which essentially amounts to math problems what I do is give you an intuitive understanding for why this straw looks bent in this picture right over here To do that, let me just do a simplified version of that picture This is the side profile of the cup, or glass right over here The best I can draw it And then let me draw the actual straw. I'll first draw the straw where it actually is coming in off the side of the cup and the straw is actually not bending goes to the bottom of the cup just like that and then it goes up like that and then it goes slightly above. Then it actually does bent up here It's irrelevant to what we want to talk about What I want to do in this video is talk about when we look over here why does it look like the straw got bent? It all comes out of the refraction of the light As the light from the straw down here changes as it go from one medium to another Now we know from refraction indices or just in general the light moves slower in water than it does in air slower in water; faster in air Let's think about what's going to happen Let me draw 2 rays that are coming from this point on the straw right over here I draw one ray right over here. I'm gonna take an arbitrary direction. Like that Now when it goes from the slower medium to the faster medium, what's going to happen to it? Until this light go here so the left side of the ray is going to end up in the air before the right side and I'm using the car example to think about which way this light's going to bend So if you visualize it as a car--sometimes people visualize it as a marching band The left side of the marching band is gonna get out before the right side and start moving faster So this is going to turn to the right Let me do another ray Let the ray come from the same point Right along the straw, so another ray just like that It will also turn to the right Now if someone's eye is right over here-- Draw their nose and all the rest If they're looking down where does it look like this 2 light rays? Let's say his eye's big enough to capture both of these rays Where does it look like this 2 light rays are coming from? So if you trace both of these rays back if you just assume that there's a line here--that's what our eyes and brains do-- if you assume whatever direction this ray is currently going it's the direction it came from and same thing for this magenta ray It would look to this observer that this point on the straw is actually right over there And if you kept doing that for bunch of points on the straw it would look like this point on the straw is actually right over here It would look like this point on the straw is actually right over here So to this observer, the straw would look like this. It would look like bent Even though the light from here is going up and it moves out to because it gets bent, when you convert it back, it would converge to this just like we saw with that first point The light from this point when it goes out and gets bent. If you extrapolate backwards from their new directions, you get to that point So to this observer this point on the straw will look to be right over here even though the light was emitted down here And that's why the straw actually looks bent So this is all really just because of refraction from going from a slower medium to a faster one So hopefully you find that interesting. In next video, I'll do some examples of Snell's law just to get ourselves comfortable with the mathematics" + }, + { + "Q": "he says change in y but the way i learned is rise over run so which one is it", + "A": "Those mean the same thing. Rise corresponds to change in y , and run corresponds to change in x . See, if we denote change in y by \u00e2\u0088\u0086y and change in x by \u00e2\u0088\u0086x, rise over run refers to \u00e2\u0088\u0086y / \u00e2\u0088\u0086x, which is the slope.", + "video_name": "5fkh01mClLU", + "transcript": "In this video I'm going to do a bunch of examples of finding the equations of lines in slope-intercept form. Just as a bit of a review, that means equations of lines in the form of y is equal to mx plus b where m is the slope and b is the y-intercept. So let's just do a bunch of these problems. So here they tell us that a line has a slope of negative 5, so m is equal to negative 5. And it has a y-intercept of 6. So b is equal to 6. So this is pretty straightforward. The equation of this line is y is equal to negative 5x plus 6. That wasn't too bad. Let's do this next one over here. The line has a slope of negative 1 and contains the point 4/5 comma 0. So they're telling us the slope, slope of negative 1. So we know that m is equal to negative 1, but we're not 100% sure about where the y-intercept is just yet. So we know that this equation is going to be of the form y is equal to the slope negative 1x plus b, where b is the y-intercept. Now, we can use this coordinate information, the fact that it contains this point, we can use that information to solve for b. The fact that the line contains this point means that the value x is equal to 4/5, y is equal to 0 must satisfy this equation. So let's substitute those in. y is equal to 0 when x is equal to 4/5. So 0 is equal to negative 1 times 4/5 plus b. I'll scroll down a little bit. So let's see, we get a 0 is equal to negative 4/5 plus b. We can add 4/5 to both sides of this equation. So we get add a 4/5 there. We could add a 4/5 to that side as well. The whole reason I did that is so that cancels out with that. You get b is equal to 4/5. So we now have the equation of the line. y is equal to negative 1 times x, which we write as negative x, plus b, which is 4/5, just like that. Now we have this one. The line contains the point 2 comma 6 and 5 comma 0. So they haven't given us the slope or the y-intercept explicitly. But we could figure out both of them from these So the first thing we can do is figure out the slope. So we know that the slope m is equal to change in y over change in x, which is equal to-- What is the change in y? Let's start with this one right here. So we do 6 minus 0. Let me do it this way. So that's a 6-- I want to make it color-coded-- minus 0. So 6 minus 0, that's our change in y. Our change in x is 2 minus 2 minus 5. The reason why I color-coded it is I wanted to show you when I used this y term first, I used the 6 up here, that I have to use this x term first as well. So I wanted to show you, this is the coordinate 2 comma 6. This is the coordinate 5 comma 0. I couldn't have swapped the 2 and the 5 then. Then I would have gotten the negative of the answer. But what do we get here? This is equal to 6 minus 0 is 6. 2 minus 5 is negative 3. So this becomes negative 6 over 3, which is the same thing as negative 2. So that's our slope. So, so far we know that the line must be, y is equal to the slope-- I'll do that in orange-- negative 2 times x plus our y-intercept. Now we can do exactly what we did in the last problem. We can use one of these points to solve for b. We can use either one. Both of these are on the line, so both of these must satisfy this equation. I'll use the 5 comma 0 because it's always nice when you have a 0 there. The math is a little bit easier. So let's put the 5 comma 0 there. So y is equal to 0 when x is equal to 5. So y is equal to 0 when you have negative 2 times 5, when x is equal to 5 plus b. So you get 0 is equal to -10 plus b. If you add 10 to both sides of this equation, let's add 10 to both sides, these two cancel out. You get b is equal to 10 plus 0 or 10. So you get b is equal to 10. Now we know the equation for the line. The equation is y-- let me do it in a new color-- y is equal to negative 2x plus b plus 10. We are done. Let's do another one of these. All right, the line contains the points 3 comma 5 and negative 3 comma 0. Just like the last problem, we start by figuring out the slope, which we will call m. It's the same thing as the rise over the run, which is the same thing as the change in y over the change in x. If you were doing this for your homework, you wouldn't I just want to make sure that you understand that these are all the same things. Then what is our change in y over our change in x? This is equal to, let's start with the side first. It's just to show you I could pick either of these points. So let's say it's 0 minus 5 just like that. So I'm using this coordinate first. I'm kind of viewing it as the endpoint. Remember when I first learned this, I would always be tempted to do the x in the numerator. No, you use the y's in the numerator. So that's the second of the coordinates. That is going to be over negative 3 minus 3. This is the coordinate negative 3, 0. This is the coordinate 3, 5. We're subtracting that. So what are we going to get? This is going to be equal to-- I'll do it in a neutral color-- this is going to be equal to the numerator is negative 5 over negative 3 minus 3 is negative 6. So the negatives cancel out. You get 5/6. So we know that the equation is going to be of the form y is equal to 5/6 x plus b. Now we can substitute one of these coordinates in for b. So let's do. I always like to use the one that has the 0 in it. So y is a zero when x is negative 3 plus b. So all I did is I substituted negative 3 for x, 0 for y. I know I can do that because this is on the line. This must satisfy the equation of the line. Let's solve for b. So we get zero is equal to, well if we divide negative 3 by 3, that becomes a 1. If you divide 6 by 3, that becomes a 2. So it becomes negative 5/2 plus b. We could add 5/2 to both sides of the equation, plus 5/2, plus 5/2. I like to change my notation just so you get familiar with both. So the equation becomes 5/2 is equal to-- that's a 0-- is equal to b. b is 5/2. So the equation of our line is y is equal to 5/6 x plus b, which we just figured out is 5/2, plus 5/2. We are done. Let's do another one. We have a graph here. Let's figure out the equation of this graph. This is actually, on some level, a little bit easier. What's the slope? Slope is change in y over change it x. So let's see what happens. When we move in x, when our change in x is 1, so that is our change in x. So change in x is 1. I'm just deciding to change my x by 1, increment by 1. What is the change in y? It looks like y changes exactly by 4. It looks like my delta y, my change in y, is equal to 4 when my delta x is equal to 1. So change in y over change in x, change in y is 4 when change in x is 1. So the slope is equal to 4. Now what's its y-intercept? Well here we can just look at the graph. It looks like it intersects y-axis at y is equal to negative 6, or at the point 0, negative 6. So we know that b is equal to negative 6. So we know the equation of the line. The equation of the line is y is equal to the slope times x plus the y-intercept. I should write that. So minus 6, that is plus negative 6 So that is the equation of our line. Let's do one more of these. So they tell us that f of 1.5 is negative 3, f of negative 1 is 2. What is that? Well, all this is just a fancy way of telling you that the point when x is 1.5, when you put 1.5 into the function, the function evaluates as negative 3. So this tells us that the coordinate 1.5, negative 3 is on the line. Then this tells us that the point when x is negative 1, f of x is equal to 2. This is just a fancy way of saying that both of these two points are on the line, nothing unusual. I think the point of this problem is to get you familiar with function notation, for you to not get intimidated if you see something like this. If you evaluate the function at 1.5, you get negative 3. So that's the coordinate if you imagine that y is equal to f of x. It would be equal to negative 3 when x is 1.5. Anyway, I've said it multiple times. Let's figure out the slope of this line. The slope which is change in y over change in x is equal to, let's start with 2 minus this guy, negative 3-- these are the y-values-- over, all of that over, negative 1 minus this guy. Let me write it this way, negative 1 minus that guy, minus 1.5. I do the colors because I want to show you that the negative 1 and the 2 are both coming from this, that's why I use both of them first. If I used these guys first, I would have to use both the x and the y first. If I use the 2 first, I have to use the negative 1 first. That's why I'm color-coding it. So this is going to be equal to 2 minus negative 3. That's the same thing as 2 plus 3. So that is 5. Negative 1 minus 1.5 is negative 2.5. 5 divided by 2.5 is equal to 2. So the slope of this line is negative 2. Actually I'll take a little aside to show you it doesn't matter what order I do this in. If I use this coordinate first, then I have to use that coordinate first. Let's do it the other way. If I did it as negative 3 minus 2 over 1.5 minus negative 1, this should be minus the 2 over 1.5 minus the negative 1. This should give me the same answer. This is equal to what? Negative 3 minus 2 is negative 5 over 1.5 minus negative 1. That's 1.5 plus 1. That's over 2.5. So once again, this is equal the negative 2. So I just wanted to show you, it doesn't matter which one you pick as the starting or the endpoint, as long as If this is the starting y, this is the starting x. If this is the finishing y, this has to be the finishing x. But anyway, we know that the slope is negative 2. So we know the equation is y is equal to negative 2x plus some y-intercept. Let's use one of these coordinates. I'll use this one since it doesn't have a decimal in it. So we know that y is equal to 2. So y is equal to 2 when x is equal to negative 1. Of course you have your plus b. So 2 is equal to negative 2 times negative 1 is 2 plus b. If you subtract 2 from both sides of this equation, minus 2, minus 2, you're subtracting it from both sides of this equation, you're going to get 0 on the left-hand side is equal to b. So b is 0. So the equation of our line is just y is equal to negative 2x. Actually if you wanted to write it in function notation, it would be that f of x is equal to negative 2x. I kind of just assumed that y is equal to f of x. But this is really the equation. They never mentioned y's here. So you could just write f of x is equal to 2x right here. Each of these coordinates are the coordinates of x and f of x. So you could even view the definition of slope as change in f of x over change in x. These are all equivalent ways of viewing the same thing." + }, + { + "Q": "At 0:19, when Sal said 8/2 is not a perfect square, what's the difference between a perfect square and a non-perfect square?", + "A": "A perfect square is when a whole number is the square root of the number. like \u00e2\u0088\u009a49=7 But \u00e2\u0088\u009a74 is a non-perfect square.", + "video_name": "d9pO2z2qvXU", + "transcript": "Which of the following real numbers are irrational? Well, irrational just means it's not rational. It means that you cannot express it as the ratio of two integers. So let's see what we have here. So we have the square root of 8 over 2. If you take the square root of a number that is not a perfect square, it is going to be irrational. And then if you just take that irrational number and you multiply it, and you divide it by any other numbers, you're still going to get an irrational number. So square root of 8 is irrational. You divide that by 2, it is still irrational. So this is not rational. Or in other words, I'm saying it is irrational. Now, you have pi, 3.14159-- it just keeps going on and on and on forever without ever repeating. So this is irrational, probably the most famous of all of the irrational numbers. 5.0-- well, I can represent 5.0 as 5/1. So 5.0 is rational. It is not irrational. 0.325-- well, this is the same thing as 325/1000. So I can clearly represent it as a ratio of integers. So this is rational. Just as I could represent 5.0 as 5/1, both of these are rational. They are not irrational. Here I have 7.777777, and it just keeps going on and on and on forever. And the way we denote that, you could just say these dots that say that the 7's keep going. Or you could say 7.7. And this line shows that the 7 part, the second 7, just keeps repeating on forever. Now, if you have a repeating decimal-- in other videos, we'll actually convert them into fractions-- but a repeating decimal can be represented as a ratio of two integers. Just as 1/3 is equal to 0.333 on and on and on. Or I could say it like this. I could say 3 repeating. We can also do the same thing for that. I won't do it here, but this is rational. So it's not irrational. 8 and 1/2? Well, that's the same thing. 8 and 1/2 is the same thing as 17/2. So it's clearly rational. So the only two irrational numbers are the first two right over here." + }, + { + "Q": "Why can't the answer also be 14 because when 14 is plugged into the inequality it becomes 3(14) - 6 > 8 which is in fact true.", + "A": "It can be, the inequality states that l is greater than or equal to 14/3. 14 is greater than 14/3, so it is included in the solutions.", + "video_name": "EkBUTZe_SiM", + "transcript": "- [Voiceover] Three L minus six is greater than or equal to eight. Which of the following best describes the solutions to the inequality shown above? So it's three times L minus six is greater than or equal to eight. Well all of these choices, these are in terms of L. They've said L on one side is greater than or equal to, actually all of these choices are greater than or equal to something else. So let's see what we can do to get just an L on the left-hand side. So the first thing we might want to do is let's get rid of this subtracting the six, and the best way we can do that is we can add six. Let's add six to both sides. This six and this six are going to add to zero, and then we are going to be left with, we are going to be left with three L on the left hand side is greater than or equal to eight plus six is 14. Now to just get an L on the left-hand side we can divide both sides by three, and if you divide both sides by three, you're not going to change the sign, you're not going to change the inequality. If you're dividing by a negative number, you would swap the inequality. Greater than or equal to would turn into less than or equal to, but we're dividing by a positive number, so this is going to be L is greater than or equal to 14 over three which is that choice right there." + }, + { + "Q": "At the end shouldn't the numbering be 1,2,4 instead of 1,4,6 and why?", + "A": "I m sure Sal did that sort of naming because this numbering would be favoring the double bonds.", + "video_name": "KWv5PaoHwPA", + "transcript": "Everything we've named so far has been an alkane. We've seen all single bonds. Let's see if we can expand our repertoire a little bit and do some alkenes. So let's look at this first carbon chain right here. And actually, here I drew out all of the hydrogens just to remind you that everything we were doing before with just the lines, it really was representing something like this. When you start having the double bonds, and we'll explain it in more detail later on, it actually starts to matter a little bit more to draw the constituents, because there's actually different ways that you can arrange it. Because these double bonds, you can imagine, they're more rigid, you can't rotate around them as much. But don't think about that too much right now. Let's just try to name these things. So like we always do, let's try to find the longest chain of carbons. And there's only one chain of carbons here. There's one, two, three, four, five, six, seven carbons in that chain. So we're going to be dealing with hept, that is seven carbons. But it's not going to be a heptane. Heptane would mean that we have all single bonds. Here we have a double bond, so this is going to be an alkene. So this tells us right here that we're dealing with an alkene, not an alkane. If you have a double bond, it's an alkene. Triple bond, alkyne. We'll talk about that in future videos. This is hept, and we'll put an ene here, but we haven't specified where the double bond is and we haven't numbered our carbons. When you see an alkene like this, you start numbering closest to the double bond, just like as if it was a alkyl group, as if it was a side chain of carbons. So this side is closest to the double bond, so let's start numbering there. One, two, three, four, five, six, seven. The double bond is between two and three, and to specify its location, you start at the lowest of these numbers. So this double bond is at two. This is actually hept-2-ene. So this tells us that we have a seven carbon chain that has a double bond starting-- the ene tells us a double bond. Let me write that down. So this double bond right there, that's what the ene tells us. Double bond between two carbons, it's an alkene. The double bond starts-- if you start at this point-- the double bond starts at number two carbon, and then it will go to the number three carbon. Now you might be asking, well, what if I had more than one double bond here? So let me draw a quick example of that. Let's say I have something like, one, two, three, four, five, six, seven. So this is the same molecule again. One, two, three, four, five, six, seven. The way we drew it up here, it would look something like this. What if I had another double bond sitting right here? How would we specify this? Well, once again we have seven carbons. One, two, three, four, five, six, seven. So we're still going to have a hept here. It's still going to be an alkene, so we put our ene here. But we start numbering it, once again, closest to the closest double bond. So one, two, three, four, five, six, seven. But now we have a double bond starting at two to three, so it would be hept-2. And we also have another double bond starting from four and going to five, so hept-2,4-ene. That's what this molecule right there is. Sometimes, this is the-- I guess-- proper naming, but just so you're familiar with it if you ever see it. Sometimes someone would write hept-2-ene, they'll write that as 2-heptene, probably because it's easier to say 2-heptene. And from this, you would be able to draw this thing over here, so it's giving you the same amount of information. Similarly over here, they might say 2,4-heptene. But this is the specific, this is the correct way to write it. It let's you know the two and the four apply to the ene, which you know applies to double bonds. Let's do a couple more. So over here, I have a double bond, and I also have some side chains. Let's see if we can figure out how to deal with all of these things. So first of all, what is our longest chain of carbons? So we have one, two, three, four, five, six. Now we could go in either direction, it doesn't matter. Seven carbons or seven carbons. Let's start numbering closest to the double bond. The double bond actually will take precedence over any other groups that are attached to it. So let's take precedence-- well, over any other groups in this case. There will be other groups that will take precedence in the future. But the double bond takes precedence over this side chain, this methyl group. But it doesn't matter in this case, we'd want to start numbering at this end. It's one, two carbon, three carbon, four carbon, five carbon, six carbon, seven carbon. So we're dealing with a hept again. We have a double bond starting from the second carbon to the third carbon. So this thing right here, this double bond from the second carbon to the third carbon. So it's hept-2,3-ene-- sorry, not 2,3, 2-ene. You don't write both endpoints. If there was a three, then there would have been another double bond there. It's hept-2-ene. And then we have this methyl group here, which is also sitting on the second carbon. So this methyl group right there on the second carbon. So we would say 2-methyl-hept-2-ene. It's a hept-2-ene, that's all of this part over here, double bonds starting on the two if we're numbering from the right. And then the methyl group is also attached to that second carbon. Let's do one more of these. So we have a cycle here, and once again the root is going to be the largest chain or the largest ring here. Our main ring is the largest one, and we have one, two, three, four, five, six, carbon. So we are dealing with hex as our root for kind of the core of our structure. It's in a cycle, so it's going to be cyclohex. So let me write that. So it's going to be cyclohex. But it has a double bond in it. So it's cyclohex ene, cyclohexene. Let me do this in a different color. So we have this double bond here, and that's why we know it is an ene. Now you're probably saying, Hey Sal, how come we didn't have to number where the ene is? So if you only have one double bond in a ring, it's assumed that one end point of the double bond is your 1-carbon. When you write just cyclohexene, you know-- so cyclohexene would look just like this. Just like that. You don't have to specify where it is. It's just, one of these are going to be the double bond. Now when you have other constituents on it, by definition or I guess the proper naming mechanism, is one of the endpoints of the double bond will be the 1-carbon, and if any of those endpoints have something else on it, that will definitely be the 1-carbon. So these both are kind of the candidates for the 1-carbon, but this point right here also has this methyl group. We will start numbering there, one, and then you want to number in the direction of the other side of the double bond. One, two, three, four, five, six. So we have three methyl groups, one on one. So these are the-- let me circle the methyl groups. That's a methyl group right there. That's a methyl group right there. That's just one carbon. So we have three methyl groups, so this is going to be-- it's at the one, the four, and the six. So it is 1, 4, 6. We have three methyl groups, so it's trimethyl cyclohexene. 1, 4, 6-trimethylcyclohexene. That's what that is, hopefully you found that useful." + }, + { + "Q": "what is the area of a heptagon", + "A": "That depends on the heptagon in question. You need to be more specific.", + "video_name": "qG3HnRccrQU", + "transcript": "We already know that the sum of the interior angles of a triangle add up to 180 degrees. So if the measure of this angle is a, the measure of this angle over here is b, and the measure of this angle is c, we know that a plus b plus c is equal to 180 degrees. But what happens when we have polygons with more than three sides? So let's try the case where we have a four-sided polygon-- a quadrilateral. And I am going to make it irregular just to show that whatever we do here it probably applies to any quadrilateral with four sides. Not just things that have right angles, and parallel lines, Actually, that looks a little bit too close to being parallel. So let me draw it like this. So the way you can think about it with a four sided quadrilateral, is well we already know about this-- the measures of the interior angles of a triangle add up to 180. So maybe we can divide this into two triangles. So from this point right over here, if we draw a line like this, we've divided it into two triangles. And so if the measure this angle is a, measure of this is b, measure of that is c, we know that a plus b plus c is equal to 180 degrees. And then if we call this over here x, this over here y, and that z, those are the measures of those angles. We know that x plus y plus z is equal to 180 degrees. And so if we want the measure of the sum of all of the interior angles, all of the interior angles are going to be b plus z-- that's two of the interior angles of this polygon-- plus this angle, which is just going to be a plus x. a plus x is that whole angle. The whole angle for the quadrilateral. Plus this whole angle, which is going to be c plus y. And we already know a plus b plus c is 180 degrees. And we know that z plus x plus y is equal to 180 degrees. So plus 180 degrees, which is equal to 360 degrees. So I think you see the general idea here. We just have to figure out how many triangles we can divide something into, and then we just multiply by 180 degrees since each of those triangles will have 180 degrees. Let's do one more particular example. And then we'll try to do a general version where we're just trying to figure out how many triangles can we fit into that thing. So let me draw an irregular pentagon. So one, two, three, four, five. So it looks like a little bit of a sideways house there. Once again, we can draw our triangles inside of this pentagon. So that would be one triangle there. That would be another triangle. So I'm able to draw three non-overlapping triangles that perfectly cover this pentagon. This is one triangle, the other triangle, and the other one. And we know each of those will have 180 degrees if we take the sum of their angles. And we also know that the sum of all of those interior angles are equal to the sum of the interior angles of the polygon as a whole. And to see that, clearly, this interior angle is one of the angles of the polygon. This is as well. But when you take the sum of this one and this one, then you're going to get that whole interior angle of the polygon. And when you take the sum of that one and that one, you get that entire one. And then when you take the sum of that one plus that one plus that one, you get that entire interior angle. So if you take the sum of all of the interior angles of all of these triangles, you're actually just finding the sum of all of the interior angles of the polygon. So in this case, you have one, two, three triangles. So three times 180 degrees is equal to what? 300 plus 240 is equal to 540 degrees. Now let's generalize it. And to generalize it, let's realize that just to get our first two triangles, we have to use up four sides. We have to use up all the four sides in this quadrilateral. We had to use up four of the five sides-- right here-- in this pentagon. One, two, and then three, four. So four sides give you two triangles. And it seems like, maybe, every incremental side you have after that, you can get another triangle out of it. Let's experiment with a hexagon. And I'm just going to try to see how many triangles I get out of it. So one, two, three, four, five, six sides. I get one triangle out of these two sides. One, two sides of the actual hexagon. I can get another triangle out of these two sides of the actual hexagon. And it looks like I can get another triangle out of each of the remaining sides. So one out of that one. And then one out of that one, right over there. So in general, it seems like-- let's say. So let's say that I have s sides. s-sided polygon. And I'll just assume-- we already saw the case for four sides, five sides, or six sides. So we can assume that s is greater than 4 sides. Let's say I have an s-sided polygon, and I want to figure out how many non-overlapping triangles will perfectly cover that polygon. How many can I fit inside of it? And then I just have to multiply the number of triangles times 180 degrees to figure out what are the sum of the interior angles of that polygon. So let's figure out the number of triangles as a function of the number of sides. So once again, four of the sides are going to be used to make two triangles. So those two sides right over there. And then we have two sides right over there. I can draw one triangle over-- and I'm not even going to talk about what happens on the rest of the sides of the polygon. You could imagine putting a big black piece of construction paper. There might be other sides here. I'm not going to even worry about them right now. So out of these two sides I can draw one triangle, just like that. Out of these two sides, I can draw another triangle right over there. So four sides used for two triangles. And then, no matter how many sides I have left over-- so I've already used four of the sides, but after that, if I have all sorts of craziness here. I could have all sorts of craziness here. Let me draw it a little bit neater than that. So I could have all sorts of craziness right over here. It looks like every other incremental side I can get another triangle out of it. So that's one triangle out of there, one triangle out of that side, one triangle out of that side, one triangle out of that side, and then one triangle out of this side. So for example, this figure that I've drawn is a very irregular-- one, two, three, four, five, six, seven, eight, nine, 10. Is that right? One, two, three, four, five, six, seven, eight, nine, 10. It is a decagon. And in this decagon, four of the sides were used for two triangles. So I got two triangles out of four of the sides. And out of the other six sides I was able to get a triangle each. These are six. This is one, two, three, four, five. Actually, let me make sure I'm counting the number of sides right. So I have one, two, three, four, five, six, seven, eight, nine, 10. So let me make sure. Did I count-- am I just not seeing something? Oh, I see. I actually didn't-- I have to draw another line right over These are two different sides, and so I have to draw another line right over here. I can get another triangle out of that right over there. And so there you have it. I have these two triangles out of four sides. And out of the other six remaining sides I get a triangle each. So plus six triangles. I got a total of eight triangles. And so we can generally think about it. The first four, sides we're going to get two triangles. So let me write this down. So our number of triangles is going to be equal to 2. And then, I've already used four sides. So the remaining sides I get a triangle each. So the remaining sides are going to be s minus 4. So the number of triangles are going to be 2 plus s minus 4. 2 plus s minus 4 is just s minus 2. So if I have an s-sided polygon, I can get s minus 2 triangles that perfectly cover that polygon and that don't overlap with each other, which tells us that an s-sided polygon, if it has s minus 2 triangles, that the interior angles in it are going to be s minus 2 times 180 degrees. Which is a pretty cool result. So if someone told you that they had a 102-sided polygon-- so s is equal to 102 sides. You can say, OK, the number of interior angles are going to be 102 minus 2. So it's going to be 100 times 180 degrees, which is equal to 180 with two more zeroes behind it. So it'd be 18,000 degrees for the interior angles of a 102-sided polygon." + }, + { + "Q": "What is recursive", + "A": "Repeating, going on in a pattern from the first case (number) on.", + "video_name": "ayRpoJgph0E", + "transcript": "The following table contains the first five terms of the given Pattern A. Generate Pattern B according to this rule. For every term of Pattern A-- so they give us the terms of Pattern A here-- multiply the term by 3 and add 1 to get the corresponding term of Pattern B. Then graph the pairs of corresponding terms. So for every term in Pattern A, we want to multiply by 3 and 1. So if we multiply 0 by 3, we get 0. And you add 1, you get 1. If you multiply 1 by 3, you get 3. And then you add 1, you get 4. 2 times 3 is 6, plus 1 is 7. 3 times 3 is 9, plus 1 is 10. Remember, we're just multiplying by 3 and adding 1. 4 times 3 is 12, plus 1 is 13. So those are the corresponding terms for Pattern B. And then they ask us to graph them. So let's try to graph these points. So when Pattern A is 0, Pattern B is 1. When Pattern A is 0-- so this is Pattern A equaling 0. That's our horizontal axis, the value of Pattern A-- Pattern B is the value of our vertical axis. Pattern B is 1. When Pattern A is 1, Pattern B is 4. So when Pattern A is 1, Pattern B is 4. Pattern B is on the vertical axis. When Pattern A is 2, Pattern B is 7. When Pattern A is 3, Pattern B is 10, so 3 in the horizontal direction. That's our Pattern A value. And our Pattern B value is 10. And then, finally, when Pattern A is 4, Pattern B is 13. Now, let's just look at these patterns. We see Pattern A is increasing by 1 each time, while Pattern B is increasing by it's-- well, Pattern A starts at 0 and increases by 1, while Pattern B starts at 1 and increases by 3, which makes complete sense. It makes sense that it starts at 1, because all of these, you multiply by 3 and add 1. So you start at 1. And then, the fact that we're multiplying by 3, that's what's leading to the distance between these points being 3. So let's check our answer to make sure we got this right, and we did." + }, + { + "Q": "I think \"9:50\" does not need a proof as they're just i j k l unit vectors.", + "A": "9:54 A proof may be simple, but still needed. That is the case here.", + "video_name": "JUgrBkPteTg", + "transcript": "We've seen in several videos that the column space of a matrix is pretty straightforward to find. In this situation the column space of A is just equal to all of the linear combinations of the column vectors of A. Another way of saying all of the linear combinations is just the span of each of these column vectors. So if we call this one right here a1. This is a2, a3, a4. This is a5. Then the column space of A is just equal to the span of a1, a2, a3, a4, and a5. Fair enough. But a more interesting question is whether these guys form a basis for the column space. Or even more interesting, what is the basis for the column space of A? And in this video I'm going to show you a method for determining the basis, and along the way we'll get an intuition for maybe why it works. And if I have time, actually I probably won't have time in In the next video I'll prove to you why it works. So we want to figure out the basis for the column space of A. Remember the basis just means that vectors span, C, A. Clearly these vectors span our column space. I mean the span of these vectors is the column space. But in order to be a basis, the vectors also have to be linearly, let me just write, linearly independent. And we don't know whether these guys or what subset of these guys are linearly independent. So what you do-- and I'm just really going to describe the process here, as opposed to the proof-- is you put this guy in reduced row echelon form. So let's do that. So let me see if we can do that. Let's keep our first row the same. 1, 0. Let me do it actually in the right side right here. So let's keep the first row the same. 1, 0, minus 1, 0, 4. And then let's replace our second row with the second row minus 2 times the first row. So then our second row. 2 minus 2 times 1 is 0. 1 minus 2 times 0 is 1. 0 minus 2 times negative 1, so that's 0 plus 2. 0 minus 2 times 0 is just 0. And then 9 minus 2 times 4 is 1. Fair enough. Now we want to zero out this guy. Well it seems like a pretty straightforward way. Just replace this row with this row plus the first row. So minus 1 plus 1 is 0. 2 plus 0 is 2. 5 minus 1 is 4. 1 plus 0 is 1. Minus 5 plus 4 is minus 1. And then finally we got this guy right here, and in order to zero him out, let's replace him with him minus the first row. So 1 minus 1 is 0. Minus 1 minus 0 is minus 1. Minus 3 minus negative 1, that's minus 3 plus 1, so that's minus 2. Minus 2 minus 0 is minus 2. And then 9 minus 4 is 5. So we did one round. We got our first pivot column going. Now let's do another round of row operations. Well we want to zero all of these guys out. Luckily this is already 0. So we don't have to change our first row or our second row. So we get 1, 0, minus 1, 0, 4. Our second row becomes 0, 1, 2, 0, 1. And now let us see if we can eliminate this guy right here. And let's do it by replacing our blue row, our third row, with the third row minus 2 times the second row. So 0 minus 2 times 0 is 0. 2 minus 2 times 1 is 0. 4 minus 2 times 2 is 0. 1 minus 2 times 0 is 1. Minus 1 minus 2 times 1 is minus 3. All right. Now this last guy we want to eliminate him, and we want turn this into a 0. Let's replace this fourth row with the fourth row plus the second row. So 0 plus 0 is 0. Minus 1 plus minus 1 is 0. Minus 2 plus minus 2 is 0. Minus 2 plus 0 is minus 2. And then 5 plus 1 is 6. We're getting close. So let's look at our pivot entries. We have this is a pivot entry. That's a pivot entry. And this is not a pivot entry, because it's following obviously another. This guy is a pivot entry right here, or will be. Zero this minus 2 out, and I think we'll be done. So let me write my first row just the way it is, because everything above it is 0, so we don't have So my first row I can just write as 1, 0, minus 1, 0, 4. I can write my second row, 0, 1, 2, 0, 1. I can write my third row as 0, 0, 0, 1 minus 3. And now let's replace my fourth row. Let's replace it with it plus 2 times the second row. So 0 plus 2 times 0, 0 plus 2 times 0, 0 plus 2 times 0, minus 2 plus 2 times 1 is just 0. 6 plus 2 times minus 3, that's 6 minus 6, that's just 0. And there we've actually put our matrix in reduced row echelon form. So let me put brackets around it. It's not so bad if you just kind of go and just do the manipulations. And sometimes you kind of get a headache thinking about doing something like this, but this wasn't too bad. So this is let me just say the reduced row echelon form of A. Let me just call that matrix R. So this is matrix R right there. Now what do we see about matrix R? Well it has 3 pivot entries, or 3 pivot columns. Let me square them out, or circle them out. Column 1 is a pivot column, column 2 is a pivot column, and column 3 is a pivot column. And we've done this in previous videos. There's two things that you can see. These three columns are clearly linearly independent. How do we know that? And that's just with respect to each other. If we just took a set of, let's call this r1, r2, and this would be r3, this would be r4 right here. It's clear that the set r1, r2, and r4 is linearly independent. And you say why is that? Well look, our one's got a 1 here, while the other two have a 0 in that entry, right? And this is by definition of pivot entries. Pivot entries have 0's, or pivot columns have 0's everywhere except for where they have a 1. For any pivot column, it will be the only pivot column that has 0's there. Or it'll be the only pivot column that has a 1 there. So there's no way that you can add up combinations of these guys to get a 1. You can say 100 times 0, minus 3, times 0. You're just going to get a bunch of 0's. So no combination of these two guys is going to be equal to that guy. By the same reasoning, no combination of that and that is going to equal this. This is by definition of a pivot entry. When you put it in reduced row echelon form, it's very clear that any pivot column will be the only one to have 1 in that place. So it's very clear that these guys are linearly independent. Now it turns out, and I haven't proven it to you, that the corresponding columns in A-- this is r1, but it's A before we put it in reduced row echelon form-- that these guys right here, so a1, a2, and a4 are also linearly independent. So a1-- let me circle it-- a2, and a4. So if I write it like this, a1, a2, and a4. Let me write it in set notation. These guys are also linearly independant, which I haven't proven. But I think you can kind of get a sense that these row operations really don't change the sense of the matrix. And I'll do a better explanation of this, but I really just wanted you to understand how to develop a basis for the column space. So they're linearly independent. So the next question is do they span our column space? And in order for them to span, obviously all of these 5 vectors, if you have all of them, that's going to span your column space by definition. But if we can show, and I'm not going to show it in this video, but it turns out that you can always represent the non-pivot columns as linear combinations of the pivot columns. And we've kind of touched on that in previous videos where we find a solution for the null space and all that. So these guys can definitely be represented as linear combinations of these guys. I haven't shown you that, but if you take that on faith, then you don't need that column and that column to span. If you did then, or I guess a better way to think it, you don't need them to span, although they are part of the span. Because if you needed this guy, you can just construct him with linear combinations of these guys. So if you wanted to figure out a basis for the column space of A, you literally just take A into reduced row echelon form. You look at the pivot entries in the reduced row echelon form of A, and that's those three. And then you look at the corresponding columns to those pivot columns in your original A. And those form the basis. Because any linear combination of them, or linear combinations of them can be used to construct the non-pivot columns, and they're linearly independant. So I haven't shown you that. But for this case, if you want to know the basis, it's just a1, a2, and a4. And now we can answer another question. So a1, a2, and a4 form a basis for the column space of A, because you can construct the other two guys with linear combinations of our basis vectors, and they're also linearly independent. Now the next question is what is the dimension of the basis? Or what is the dimension-- not the dimension of the basis-- what is the dimension of the column space of A? Well the dimension is just the number of vectors in any basis for the column space. And all bases have the same number of vectors for any given subspace. So we have 1, 2, 3 vectors. So the dimension of our column space is equal to 3. And the dimension of a column space actually has a specific term for it, and that's called the rank. So the rank of A, which is the exact same thing as the dimension of the column space, it is equal to 3. And another way to think about it is, the rank of A is the number of linearly independent column vectors that you have that can span your entire column space. Or the number of linearly independent column vectors that can be used to construct all of the other column vectors. But hopefully this didn't confuse you too much, because the idea is very simple. Take A, put it into reduced row echelon form, see which columns are pivot columns. The corresponding columns are going to be a basis for your column space. If you want to know the rank for your matrix, you can just count them. Or if you don't want to count those, you could literally just count the number of pivot columns you have in your reduced row echelon form. So that's how you do it. In the next video I'll explain why this worked." + }, + { + "Q": "How to represent 13/-5 on a number line ?", + "A": "That would be -2.6 so you would put it at 2.6 to the left of zero. Hope this helped!", + "video_name": "uC09taczvOo", + "transcript": "- [Voiceover] Plot the following numbers on the number line. The first number we have here is five, and so five is five to the right of zero, five is right over there. That's our five. Then we get 1/3. 1/3. So 1/3 is between zero and one. We can actually split this into thirds. So that would be 1/3, 2/3, and then 3/3, which is one, so 1/3 is going to sit right over there. It's 1/3 of the way from zero to one, that's 1/3. Let me write that. That's 1/3 right over there. Then we have negative 1.2. I'll do that in this blue color. Negative 1.2. So, negative one is right over here. This is more negative than negative one. It's negative 1.2. It's negative one, and then another .2, so it's going to be right over here. This is negative 1.2. Zero is pretty straight forward. Zero is right over there. It's even labeled for us at zero. Five was labeled for us too at five. Then we have negative two and 1/4. So let's go to negative two. Negative two is here, and it's going to be more negative than negative two. It's negative two and then another another negative 1/4. So it's negative two, and then we go 1/4 of the way to negative three. So negative 2 and 1/4 is going to be right over here. So negative two and 1/4. And then finally we have 4.1. 4.1. So four is right over here. .1 is another tenth greater than four, another tenth on the way to five. So four and 1/10 is going to be right over here. 4.1. 4.1. And we are done." + }, + { + "Q": "is there any formula to get sum of terms in geometric sequences?", + "A": "Yes, Sum = a(1-r^n)/(1-r)", + "video_name": "Iq7a2vEsT-o", + "transcript": "So I have the function g of x is equal to 9 times 8 to the x minus 1 power. And it's defined for x being a positive-- or if x is a positive-- integer. If x is a positive integer. So we could say the domain of this function, or all the valid inputs here are positive integers. So 1, 2, 3, 4, 5, on and on and on. So this is an explicitly defined function. What I now want to do is to write a recursive definition of this exact same function. That given an x, it'll give the exact same outputs. So let's first just try to understand the inputs and outputs here. So let's make a little table. Let's make a table here. And let's think about what happens when we put in various x's into this function definition. So the domain is positive integers. So let's try a couple of them. 1, 2, 3, 4. And then see what the corresponding g of x is. g of x. So when x is equal to 1, g of x is 9 times 8 to the 1 minus 1 power, 9 times 8 to the 0 power, or 9 times 1. So g of x is going to be just 9. When x is 2, what's going to happen? It'll be 9 times 8 to the 2 minus 1. So that's the same thing as 9 times 8 to the 1st power. And that's just going to be 9 times 8. So that is 72. Actually let me just write it that way. Let me write it as just 9 times 8. 9 times 8. Then when x is equal to 3, what's going on here? Well this is going to be 3 minus 1 is 2. So it's going to be 8 squared. So it's going to be 9 times 8 squared. So we could write that as 9 times 8 times 8. I think you see a little bit of a pattern forming. When x is 4, this is going to be 8 to the 4 minus 1 power, or 9 to the 3rd power. So that's 9 times 8 times 8 times 8. So this gives a good clue about how we would define this recursively. Notice, if our first term, when x equals 1 is 9, every term after that is 8 times the preceding term. Is 8 times the preceding term. 8 times the preceding term. 8 times the preceding term. So let's define that as a recursive function. So first define our base case. So we could say g of x-- and I'll do this is a new color because I'm overusing the red. I like the blue. g of x. Well we can define our base case. It's going to be equal to 9 if x is equal to 1. g of x equals 9 if x equals 1. So that took care of that right over there. And then if it equals anything else it equals the previous g of x. So if we're looking at-- let's go all the way down to x minus 1, and then an x. So if this entry right over here is g of x minus 1, however many times you multiply the 8s and we have a 9 in front, so this is g of x minus 1. We know that g of x-- we know that this one right over here is going to be the previous entry, g of x minus 1. The previous entry times 8. So we could write that right here. Times 8. So for any other x other than 1, g of x is equal to the previous entry-- so it's g of-- I'll do that in a blue color-- g of x minus 1 times 8. If x is greater than 1, or x is integer greater than 1. Now let's verify that this actually works. So let's draw another table here. So once again, we're going to have x and we're going to have g of x. But this time we're going to use this recursive definition And the reason why it's recursive is it's referring to itself. In its own definition, it's saying hey, g of x, well if x doesn't equal 1 it's going to be g of x minus 1. It's using the function itself. But we'll see that it actually does work out. So let's see... When x is equal to 1, so g of 1-- well if x equals 1, it's equal to 9. It's equal to 9. So that was pretty straightforward. What happens when x equals 2? Well when x equals 2, this case doesn't apply anymore. We go down to this case. So when x is equal to 2 it's going to be equivalent to g of 2 minus 1. Let me write this down. It's going to be equivalent to g of 2 minus 1 times 8, which is the same thing as g of 1 times 8. And what's g of 1? Well g of 1 is right over here. g of 1 is 9. So this is going to be equal to 9 times 8. Exactly what we got over here. And of course this was equivalent to g of 2. So let me write this. This is g of 2. Let me scroll over a little bit so I don't get all scrunched up. So now let's go to 3. Let's go to 3. And right now I'll write g of 3 first. So g of 3 is equal to-- we're going to this case-- it's equal to g of 3 minus 1 times 8. So that's equal to g of 2 times 8. Well what's g of 2? Well g of 2, we already figured out is 9 times 8. So it's equal to 9 times 8-- that's g of 2-- times 8 again. And so you see we get the exact same results. So this is the recursive definition of this function." + }, + { + "Q": "Is atomic mass essentially atomic weight then?", + "A": "Atomic mass is the mass of a specific isotope Relative atomic mass ( atomic weight ) is the average mass of an atom of an element taking into account the percentage of each isotope there is on earth", + "video_name": "NG-rrorZcM8", + "transcript": "Let's have a little bit of a primer on weight and mass, especially if we start talking about atomic weight and atomic mass. If we're sitting in a physics class, weight and mass mean something very, very ... well, they mean different things. It might be a discovery, or a new learning, for some of you, because in everyday life, when we say something's mass, we think, \"Well, the more mass it has, the more weight it has.\" Or, if we think something has more or less weight, we think, \"Okay, that relates to its mass.\" But in physics class, we see that these actually represent two different ideas, albeit related ideas. Mass is a notion of how much of something there is, or you could say, how hard is it to accelerate or decelerate it. Or you could view it as a measure of an object's inertia. We typically, it, kind of a human scale, might measure mass in terms of grams or kilograms. What's confusing is, if you go to Europe, and you ask someone their weight, they'll often give you their weight in terms of kilogram, even though that is a unit of mass. Now weight, on the other hand, is not ... it's different than mass. Weight is a force, it's how much the Earth, or whatever planet you happen to be on, is pulling on you. This right over here is a force. And, in the metric system, you measure weight, not with grams or kilograms, but with Newtons. Newtons. Really, when you ask someone their weight in Europe, they should give it to you in Newtons. If you ask them their mass, what they're telling you is actually their mass. They should say, \"My mass is 60 kilograms,\" or, \"70 kilograms,\" or whatever they might be. It's a very important difference in physics. If I go from Earth to the Moon, my mass does not change, but my weight does change because the force with which the Moon is pulling on me, or that we're pulling on each other, is less than it would be on Earth. In fact, even on the surface of the Earth, if you were to even go to the top of a building, you're just so ... Yeah, it would be very hard to measure it, but you're just slightly further from the center of the Earth, so there's a different gravitational force. Your weight will change ever so slightly, but your mass does not change. You go to deep space, and there's very little gravitational influence, you have pretty much, or close to, zero weight. But you're in deep space, and if there's no planets nearby, but your mass is still going to be whatever your mass happens to be. That's a primer on mass and weight. Now, with that out of the way, I might confuse you because, as we go into a chemistry context, it starts getting a little bit more muddled again. Let me go to chemistry, chemistry. And in any science, if people just talk generally about mass or weight, this is what they're talking about. They're talking about a measure of inertia for mass, and they're talking about a force when they're talking about weight. But in chemistry, we start thinking about things on an atomic scale. You'll hear ... You'll hear the term \"Atomic,\" \"Atomic mass.\" Atomic mass is, literally, a measure of mass. It is measured in atomic mass units. Atomic mass units, which is, and we'll talk in the future videos, a very, very, very, very, small fraction. One atomic mass unit is a very, very, very, very, very, very small fraction of a gram. It is actually defined using the most common isotope of carbon. It's defined using carbon-12. The current definition is carbon-12. Carbon-12 has a mass, has a mass, has a mass of exactly, exactly, exactly 12 atomic mass units. So they can then, you or chemists, use that as the benchmark to figure out what the atomic mass, or the mass of any other atom. And you might say, \"Oh, why didn't they just do a hydrogen, \"and just say that's one atomic mass unit, and all that,\" and actually, they had started there. They had been there at an earlier stage, but for a whole set of reasons, carbon-12 is kind of being the benchmark, as having 12 atomic mass units, is what people went with. Now, what is atomic weight, then? Atomic weight. Let me write this in a different color. I'll do it in blue. Atomic weight. So if you draw the same analogy that we did up here, you might say, \"Okay, this must be a ... \"This must be a force. \"It should maybe, you know, \"an atomic weight unit would be a small fraction of, \"very small fraction of a unit.\" But it turns out in chemistry, when we talk about atomic weight, we're still measuring in atomic mass units. This is still a mass. Atomic mass units. But it's not the mass of just one atom or just one molecule. It's a weighted average across many, many ... of how typically, what you would see, or the makeup of what you would see on Earth. What do I mean by that? Well, on Earth, there are two ... The primary isotope of carbon is carbon-12. Carbon-12, which is defined as having a mass of exactly 12 atomic mass units. But there's also some carbon-14. Carbon-14. What do these numbers mean, just as a reminder? Well, carbon-12 has six protons, and the six protons are what make it carbon. Carbon-14 is also going to have six protons. But carbon-12, carbon-12 also has six neutrons. Six neutrons. While carbon-14 has eight neutrons. I know what you're already thinking. You're, like, \"Well, wait. \"Why don't we say that a proton or a neutron \"weighs one atomic mass unit? \"Because it looks like this is 12, \"and I'm guessing that this, \"that this, the mass of this is going to be \"pretty close to 14.\" If you're thinking that way, that's not an unreasonable way to think. In fact, when I'm kind of just working through chemistry, that is how I think about it. But they don't weigh exactly one atomic mass unit by this definition. Remember, the electron is ever so small, it has very small mass, but it is contributing, or the electrons are contributing, something to the mass. So, a proton or a neutron have very, very, very close ... They are close to one atomic mass unit. Let me write this down. One proton, one proton, or one neutron, one neutron, very close to one atomic mass unit, but not exactly. But anyway, going back to what atomic weight is, right over here, the most common isotope of carbon ... Remember, when we're saying \"isotopes,\" we're saying the same element, we have the same number of protons, but we have different number of neutrons. The most common isotope on Earth is carbon-12, but there's also some carbon-14. If you were to take a weighted average, as found on the Earth, of all the carbon-12 and all of the carbon-14, the weighted average of the atomic masses is the atomic weight. And the atomic weight of carbon ... And you'll see this on a periodic table. In fact, I have one right over here. Notice, the six protons, this is what defines it to be carbon. But then they write 12.011, which is the weighted average of the masses of all of the carbons. Now, it's very close to 12, as opposed to being closer to 14, because most of the carbon on Earth is carbon-12. We could write this down. This is the atomic weight. This is the atomic weight of carbon on Earth. This is 12.011. Typically, if people are telling you, \"Hey, I'm talking about the isotope. \"I am talking about the isotope carbon-12,\" you say, \"Okay, if you're talking about a particular atom \"of carbon-12,\" you would say, \"Okay, that's going to be 12 atomic mass units.\" But then if you said, \"Hey, you know, \"I'm randomly ... \"If I have a big bag of carbon, you could say, \"on average, the weighted average of those carbon atoms, \"they're going to have an atomic mass, \"or a weighted average atomic mass,\" which is atomic weight, \"of 12.011 atomic mass units.\" AMU. Hopefully, this clarified it more than it confused. All right, see you in the next video." + }, + { + "Q": "for the first right hand rule> i know your fingers first point in the direction and velocity and curl towards the magnetic field. But which way do you know the magnetic field is flowing? do you curl your fingers towards the south of the magnetic field.", + "A": "like electric field, magnetic field also has a particular direction.by convention its taken to be originating frm north and end on south. but in the above problems since the the source generating the magnetic field is not specified, its would be better to consider the direction only using the vectors drawn", + "video_name": "b1QFKLZC11U", + "transcript": "In the last video we figured out that if we had a proton coming into the right at a velocity of 6 times 10 to the seventh meters per second. So the magnitude of the velocity is 1/5 the speed of light. And if it were to cross this magnetic field, we used this formula to figure out that the magnitude of the force on this proton would be 4.8 times 10 to the negative 12 newtons. And then the direction, we used our right hand rule because it's a cross product. And we figured out that it would be perpendicular-- well, it has to be perpendicular to both, because we're taking the cross product-- and right when it enters, the net force will be downwards. But then think about what happens. If you have a downward force right there, then the particle will be deflected downward a little bit, so its velocity vector will then look something like that. But it's still in the magnetic field, right? And not only is it still in the magnetic field, but since the particle is still moving within the plane of your video screen, it's still completely perpendicular to the magnetic field. And so the magnitude of the force on the moving particle won't change, just the direction will. Because if we do the right hand rule here, but if we just move our hand down a little bit, if we tilt it down, then our thumb's going to be pointing in this direction. And that just keeps happening. It gets deflected that way a little bit. So the magnitude of the velocity doesn't ever change. It always stays perpendicular to the magnetic field because it's always staying in this plane. But the orientation does change within the plane. And because of that, because the orientation of the velocity changes, the orientation of the force changes. So when the velocity is here, the force is perpendicular. So it acts as kind of a centripetal force, and so the particle will start moving in a circle. So let's see if we can break out our toolkit from what we've learned before in classical mechanics, and figure out what the radius of that circle is. And that might seem more daunting than it really is. Well, what do we know about centripetal forces and radiuses of circles, et cetera? So, what is the formula for centripetal force? And we proved it many, many videos ago, early in the physics playlist. Well, centripetal acceleration is the magnitude of the velocity vector squared over the radius of the circle. And since this is acceleration, if we want to know the centripetal force, it's just the mass times acceleration. So it's the mass of the particle, or the object in question, times the magnitude of its velocity squared divided by the radius of the circle. In this case, this is the radius of the circle. And that's what we're going to try to solve for. And what do we know about the centripetal force? What is causing the centripetal force? Well, it's the magnetic field and we've figured that out. This is going to be equal to this, which we figured out is going to be equal to-- at least the magnitudes-- the magnitude of this is equal to the magnitude of this. And that magnitude is 4.8 times 10 to the minus 12 newtons. And so the radius is going to be-- let's see, if we flip both sides of this equation, we get radius over mass velocity squared is equal to 1 over 4.8 times 10 to the minus 12. I could just figure out what that number is, but I won't Then we can multiply both sides times this mv squared. And we get that the radius of the circle is going to be equal to the mass of the proton times the magnitude of its velocity squared divided by the force from the magnetic field. The centripetal force. 4.8 times 10 to the minus 12 newtons. And the radius should be in meters, since everything is kind of in the standard SI units. And let's see if we can figure this out. Get our calculator. And this is where that constant function is useful again, because what is the mass of a proton? Well, that's something that I personally don't have memorized. But if we go into the built-in constants on the TI-85-- let's see more. Mass of a proton. This is mass of an electron. This is mass of a proton. So mass of a proton-- that's what we care about-- times the magnitude of the velocity squared. What was the velocity? It was 6 times 10 to the seventh meters per second. So times 6 times 10 to the seventh meters per second squared. And all of that divided by the magnitude of the centripetal force. Which is the force that's being generated by the That's 4.8 times 10 to the negative 12. Divided by 4.8 E minus 12. Hopefully we don't get something funky. There we go. That's actually a pretty neat number. 1.25 meters. That's actually a number that we can imagine. So if you have a proton going in this direction at 1/5 the speed of light through a-- what'd I say it was? It was a 0.5 tesla magnetic field, where the vectors are pointing out of the video. We have just shown that this proton will go in a circle of radius 1.25 meters. Which is neat because it's a number that I can actually visualize. And so this whole business of magnetic fields making charged particles go into circles, this is one of the few times that I can actually say has a direct application into things that you've seen. Namely, your TV. Or at least the old-school TVs. The non-plasma or LCD TVs, your cathode ray TVs, take advantage of this. Where you essentially have a beam of not protons but electrons. And a magnet-- if you take apart a TV, which I don't think you should do, because you're more likely to hurt yourself because there's a vacuum in there that can implode, and all that-- but essentially, you have a magnet that deflects this electron beam and does it really fast so it scans your entire screen of different intensities, and that's what forms the image. I won't go into that detail. Maybe one day I'll do a whole video on how TVs work. So that's one application of a magnetic field causing a beam of charged particles to curve. And then the other application, and this is actually one where it's actually useful to make the particle go in a circle, is these cyclotrons that you read about, where they take these protons and they make them go in circles really, really fast, and then they smash them together. Well, have you ever wondered, how do they even make a proton go in a circle? It's not like you could hold it and guide it around in a circle. They pass it through an appropriate strength magnetic field, and it curves the path of the proton so that it can keep going through the same field over and over again. And then they can actually use those electric fields. I don't claim to have any expertise in this, but then they can keep speeding it up using the same devices, because it keeps passing through the same part of the collider. And then once it collides, you've probably seen those pictures. You know, that you spend billions of dollars on supercolliders, and you end up with these pictures. And somehow these physicists are able to take these pictures and say, oh, this is some new particle because of the way it moved. Well, what they're actually talking about is these are moving at relativistic speeds. And since they're at relativistic speeds as they move at different velocities, their mass is changing, and all that. But the basic idea is what we just learned. They move in circles. They move in circles because they're going through a magnetic field. But their radiuses are different because their charges and their velocities are going to be different. And actually some will move to the left and some will move to the right. And that might be because they're positive or negative, and then the radius will be dependent on their masses. Anyway, I don't want to confuse you. But I just wanted to show you that we actually are touching on some physics that a physicist would actually care about. Now with that said, what would have happened if this wasn't a proton but if this was an electron moving at this velocity at 6 times 10 to the seventh meters per second through a 0.5 tesla magnetic field popping out of this video. What would have happened? Well, this formula would have still been safe. The magnitude of the force is the charge-- but it wouldn't have been the charge of a proton, it would have been the charge of an electron, times 6 times 10 to the seventh meters per second times 0.5 teslas. So what's the difference between the charge of a proton and the charge of an electron? Well, the charge of an electron is negative. So if this was an electron, then the net force would actually end up being a negative number. So what does that mean? Well, when we used the right hand rule with the proton example, we said that the-- at least when the proton is moving in this direction-- that the net force would be downwards. But now, all of a sudden, if we reverse the charge, if we say we have a negative charge-- the same magnitude but it's negative, because it's an electron-- what happens? The force is now in this direction, using the right hand rule, but it is a negative. So really it's going to be a positive force of the same magnitude in this direction. So if we have a proton, it'll go in a circle in this direction. It'll go like this. But if we have an electron, it'll go in a circle of the other direction. Now let me ask a question. Is that circle going to be a tighter circle or a wider circle? Well, the mass of an electron is a lot smaller than the mass of a proton. And we had the radius is equal to the mass times the velocity squared divided by the centripetal force. So this mass is smaller and the radius is going to be smaller. So the electron's path would actually move up and it would be a smaller radius. Actually proportional to the difference in their radiuses is the difference in their masses, actually. But that would be the path of the electron. Anyway, I thought you'd be interested in that, as well. I have run out of time. I will see you in the next video." + }, + { + "Q": "What is the electron configuration of Cd2+ and how?", + "A": "The electron configuration for Cd is: [Kr] 4d^10 5s^2 In Cd^2+ the highest energy electrons are lost which are the two 5s electrons, leaving: [Kr] 4d^10", + "video_name": "YURReI6OJsg", + "transcript": "Let's figure out the electron configuration for nickel, right there. 28 electrons. We just have to figure out what shells and orbitals they go in. 28 electrons. So the way we've learned to do it is, we defined this as the s-block. And we can just remember that helium actually belongs here when we talk about orbitals in the s-block. This is the d-block. This is the p-block. And so we could start with the lowest energy electrons. We could either work forward or work backwards. If we work forwards, first we fill up the first two electrons going to 1s2. So remember we're doing nickel. So we fill up 1s2 first with two electrons. Then we go to 2s2. And remember this little small superscript 2 just means we're putting two electrons into that subshell or into that orbital. Actually, let me do each shell in a different color. So 2s2. Then we fill out 2p6. We fill out all of these, right there. So 2p6. Let's see, so far we've filled out 10 electrons. We've configured 10. You can do it that way. Now we're on the third shell. The third shell. So now we go to 3s2. Remember, we're dealing with nickel, so we go to 3s2. Then we fill out in the third shell the p orbital. So 3p6. We're in the third period, so that's 3p6, right there. There's six of them. And then we go to the fourth shell. I'll do it in yellow. So we do 4s2. 4s2. And now we're in the d-block. And so we're filling in one, two, three, four, five, six, seven, eight in this d-block. So it's going to say d8. And remember, it's not going to be 4d8. We're going to go and backfill the third shell. So it will be 3d8. So we could write 3d8 here. So this is the order in which we fill, from lowest energy state electrons to highest energy state. But notice the highest energy state electrons, which are these that we filled in, in the end, these eight, these went into the third shell. So when you're filling the d-block, you take the period that you're in minus one. So we were in the fourth period in the periodic table, but we subtracted one, right? This is 4 minus 1. So this is the electron configuration for nickel. And of course if we remember, if we care about the valence electrons, which electrons are in the outermost shell, then you would look at these right here. These are the electrons that will react, although these are in a higher energy state. And these react because they're the furthest. Or at least, the way I visualize them is that they have a higher probability of being further from the nucleus than these right here. Now, another way to figure out the electron configuration for nickel-- and this is covered in some chemistry classes, although I like the way we just did it because you look at the periodic table and you gain a familiarity with it, which is important, because then you'll start having an intuition for how different elements react with each other -- is to just say, oK, nickel has 28 electrons, if it's neutral. It has 28 electrons, because that's the same number of protons, which is the atomic number. Remember, 28 just tells you how many protons there are. This is the number of protons. We're assuming it's neutral. So it has the same number of electrons. That's not always going to be the case. But when you do these electron configurations, that tends to be the case. So if we say nickel has 28, has an atomic number of 28, so it's electron configuration we can do it this way, too. We can write the energy shells. So one, two, three, four. And then on the top we write s, p, d. Well we're not going to get to f. But you could write f and g and h and keep going. What's going to happen is you're going to fill this one first, then you're going to fill this one, then that one, then this one, then this one. Let me actually draw it. So what you do is, these are the shells that exist, period. These are the shells that exist, in green. What I'm drawing now isn't the order that you fill them. This is just, they exist. So there is a 3d subshell. There's not a 3f subshell. There is a 4f subshell. Let me draw a line here, just so it becomes a little bit neater. And the way you fill them is you make these diagonals. So first you fill this s shell like that, then you fill this one like that. Then you do this diagonal down like that. Then you do this diagonal down like that. And then this diagonal down like that. And you just have to know that there's only two can fit in s, six in p, in this case, 10 in d. And we can worry about f in the future, but if you look at the f-block on a periodic table, you know how many there are in f. So you fill it like that. So first you just say, OK. For nickel, 28 electrons. So first I fill this one out. So that's 1s2. 1s2. Then I go, there's no 1p, so then I go to 2s2. Let me do this in a different color. So then I go right here, 2s2. That's that right there. Then I go up to this diagonal, and I come back down. And then there's 2p6. And you have to keep track of how many electrons you're dealing with, in this case. So we're up to 10 now. So we used that one up. Then the arrow tells us to go down here, so now we do the third energy shell. So 3s2. And then where do we go next? 3s2. Then we follow the arrow. We start there, there's nothing there, there's something here. So we go to 3p6. And then the next thing we fill out is 4s2. So then we go to 4s2. And then what's the very next thing we fill out? We have to go back to the top. We come here and then we fill out 3d. And then how many electrons do we have left to fill out? So we're going to be in 3d. So 3d. And how many have we used so far? 2 plus 2 is 4. 4 plus 6 is 10. 10 plus 2 is 12. 18. 20. We've used 20, so we have 8 more electrons to configure. And the 3d subshell can fit the 8 we need, so we have 3d8. And there you go, you've got the exact same answer that we had when we used the first method. Now I like the first method because you're looking at the periodic table the whole time, so you kind of understand an intuition of where all the elements are. And you also don't have to keep remembering, OK, how many have I used up as I filled the shells? Right? Here you have to say, i used two, then I used two more. And you have to draw this kind of elaborate diagram. Here you can just use the periodic table. And the important thing is you can work backwards. Here there's no way of just eyeballing this and saying, OK, our most energetic electrons are going to be and our highest energy shell is going to be 4s2. There's no way you could get that out of this without going through this fairly involved process. But when do you use this method, you can immediately say, OK, if I'm worried about element Zr, right here. If I'm worried about element Zr. I could go through the whole exercise of filling out the entire electron configuration. But usually the highest shell, or the highest energy electrons, are the ones that matter the most. So you immediately say, OK, I'm filling in 2 d there, but remember, d, you go one period below. So this is 4d2. Right? Because the period is five. So you say, 4d2. 4d2. And then, before that, you filled out the 5s2 electrons. The 5s2 electrons. And then you could keep going backwards. And you filled out the 4p6. 4p6. And then, before you filled out the 4p6. then you had 10 in the d here. But what is that? It's in the fourth period, but d you subtract one from it, so this is 3d10. So 3d10. And then you had 4s2. This is getting messy. Let me just write that. So you have 4d2. That's those two there. Then you have 5s2. 5s2. Then we had 4p6. That's over here. Then we had 3d10. Remember, 4 minus 1, so 3d10. And then you had 4s2. And you just keep going backwards like that. But what's nice about going backwards is you immediately know, OK, what electrons are in my highest energy shell? Well I have this five as the highest energy shell I'm at. And these two that I filled right there, those are actually the electrons in the highest energy shell. They're not the highest energy electrons. These are. But these are kind of the ones that have the highest probability of being furthest away from the nucleus. So these are the ones that are going to react. And these are the ones that matter for most chemistry purposes. And just a little touchpoint here, and this isn't covered a lot, but we like to think that electrons are filling these buckets, and they stay in these buckets. But once you fill up an atom with electrons, they're not just staying in this nice, well-behaved way. They're all jumping between orbitals, and doing all sorts of crazy, unpredictable things. But this method is what allows us to at least get a sense of what's happening in the electron. For most purposes, they do tend to react or behave in ways that these orbitals kind of stay to themselves. But anyway, the main point of here is really just to teach you how to do electron configurations, because that's really useful for later on knowing how things will interact. And what's especially useful is to know what electrons are in the outermost shell, or what are the valence electrons." + }, + { + "Q": "why in sigma complex carbon contains +charge\n+charge appear on it at that time when it loses its one electron.Does it lose its electron?\nI am confused over here can you please explain it for me ?", + "A": "The two electrons in the \u00cf\u0080 bond are used to attack the electrophile. The carbon that is bonded to the electrophile doesn t lose any electrons, but the carbon ortho to it does lose an electron, and that gives it a positive charge.", + "video_name": "eQzbpL0uWVA", + "transcript": "Let's look at the general reaction for electrophilic aromatic substitution. So we start with the benzene ring, and we react benzene with a molecule that contains an electrophile in there. And what happens in electrophilic aromatic substitution. We're going to substitute the electrophile for a proton on our benzene ring. And so over here, we can see the electrophile is now in place of that proton. So that's where that that's where the electrophilic part comes in this. And that's where the substitution parts comes in. You're substituting an electrophile for a proton. The aromatic comes in because you are going to reform an aromatic ring in your mechanism. Electrophilic aromatic substitution requires a catalyst. And the point of a catalyst is to generate your electrophile. So down here, you can see that the catalyst is going to react to produce the positively charged electrophile. So remember, electrophile means loving electron. So if something is positively charged, it's going to love electrons. We also formed this catalyst complex over here, which is going to factor into our mechanism. So now that we formed our electrophile, let's look in more detail as to what happens in electrophilic, aromatic substitution. So we start with our benzene ring. And I'm showing one of the hydrogens on the benzene ring. It could be any of the six, since they are all equivalent. And now we formed our electrophile from our catalyst. So the pi electrons in the benzene ring can be attracted to the positively charged electrophile. Because negative charges are attracted to positive charges. And so pi electrons in your benzene ring are going to function as a nucleophile, and those electrons are going to attack the electrophile. So this is a nucleophile, electrophile attack, where those pi electrons are going to bond to that electrophile there. So those pi electrons are going to form a covalent bond with your electrophile. So let's go ahead and show that. So these pi electrons didn't do anything. The hydrogen stays there. Now, I could show the electrophile adding to either of the two carbons on the side of the double bonds. So it could be that carbon. Or it could be this carbon. Since I've drawn this hydrogen up here at the top, I'm going to go ahead and say that the electrophile adds to the top carbon there. So there's my electrophile there. Let me go ahead and highlight the electrons that are forming that covalent bond. So these pi electrons here are the ones that are functioning as a nucleophile. And those pi electrons are going to form this bond right here. Now in forming that bond, we're taking a bond away from this bottom carbon here. And so that bottom carbon is going to be left with a positive one formal charge. Therefore, we can draw a resonance structure for this cation. So let's go ahead and show a possible resonance structure here. So these pi electrons could move over to here. And let's go ahead and draw what would result if that happened. So now, we have these pi electrons up here. We have our hydrogen. We have our electrophile. And the electrons moved over to this position. Let me go ahead and highlight those in magenta. So I'm saying that these pi electrons right here moved over to here. And when those electrons moved over to there, we're taking a bond away from this carbon this time. So that is the carbon that's going to get a plus 1 formal charge like that. So we can draw another resonance structure. So let's go ahead and do that. So we could take these pi electrons and move them into here. So let's go ahead and show what that would look like. So if those pi electrons moved into there, we would now have, again, our hydrogen, our electrophile, these pi electrons, and then these pi electrons right here. So once again, let me go ahead and highlight those. This time I'll use blue. These pi electrons are going to move over to here. And once again, we're taking a bond away from a carbon. This time, it's this top carbon up here. So that's the carbon that's going to get the plus one formal charge like that. So these are all resonance structures. And remember, the actual cation would be a hybrid of these resonance structures. And we call we call that hybrid a sigma complex. So you have a positive one formal charge de-localized over three carbons in your sigma complex. So the next step in the mechanism-- I'm just going to redraw the first resonance structure that we did here. So I'm going to go and redraw that down here. So let's go ahead and show the first resonance structure. So in our first resonance structure, we had our hydrogen here, our electrophile already bonded to our ring. And we had a positive one formal charge on this carbon right here. Well remember, the catalyst had formed a complex. And I represent it like this. So something bonded to your catalyst like that. So let's just go up here and refresh our memory. So right up here, when we generated our electrophile, we also generated this catalyst complex up here. So y bonded to a catalyst, so I have y bonded to a catalyst down here. And you could think about this as functioning as a base. Or it's going to accept a proton. So I could show these electrons in here taking this proton. And if it takes that proton, that leaves these electrons behind. And those electrons are going to move in here to reform your benzene ring and take away that positive one formal charge. So let's go ahead and show that. So we now have our benzene ring back. And our electrophile is now bonded to our ring. And the proton has left. So the electrophile has completely substituted for that proton. Let's follow those electrons again. So the electrons in magenta in here, so those are the ones that are going to move in here to reform your aromatic ring. So deproteination of the sigma complex restores the aromatic ring. And so we have a stable product here. So the other product you could think about this y here is now going to be bonded to that proton. So you could have the y here bonded to that proton. And you could highlight those electrons. You could say that these electrons right here are now these electrons. And, taking those electrons away from the catalyst would of course regenerate your catalyst. And so it's free to then catalyze another reaction. And so this is the general mechanism for electrophilic aromatic substitution, which the reactions that we're going to see are pretty much going to follow this general mechanism." + }, + { + "Q": "Don't you learn about parabolas in algebra 1? Or does it vary depending on the curriculum?", + "A": "I m pretty sure that it varries upon the curriculum and how fast your class moves.", + "video_name": "0A7RR0oy2ho", + "transcript": "Let's see if we can learn a thing or two about conic sections. So first of all, what are they and why are they called conic sections? Actually, you probably recognize a few of them already, and I'll write them out. They're the circle, the ellipse, the parabola, and the hyperbola. Hyperbola. And you know what these are already. When I first learned conic sections, I was like, oh, I know what a circle is. I know what a parabola is. And I even know a little bit about ellipses and hyperbolas. Why on earth are they called conic sections? So to put things simply because they're the intersection of a plane and a cone. And I draw you that in a second. But just before I do that it probably makes sense to just draw them by themselves. And I'll switch colors. Circle, we all know what that is. Actually let me see if I can pick a thicker line for my circles. so a circle looks something like that. It's all the points that are equidistant from some center, and that distance that they all are that's the radius. So if this is r, and this is the center, the circle is all the points that are exactly r away from this center. We learned that early in our education what a circle is; it makes the world go round, literally. Ellipse in layman's terms is kind of a squished circle. It could look something like this. Let me do an ellipse in another color. So an ellipse could be like that. Could be like that. It's harder to draw using the tool I'm drawing, but it could also be tilted and rotated around. But this is a general sense. And actually, circles are a special case of an ellipse. It's an ellipse where it's not stretched in one dimension It's kind of perfectly symmetric in every way. Parabola. You've learned that if you've taken algebra two and you probably have if you care about conic sections. But a parabola-- let me draw a line here to separate things. A parabola looks something like this, kind of a U shape and you know, the classic parabola. I won't go into the equations right now. Well, I will because you're probably familiar with it. y is equal to x squared. And then, you could shift it around and then you can even have a parabola that goes like this. That would be x is equal to y squared. You could rotate these things around, but I think you know the general shape of a parabola. We'll talk more about how do you graph it or how do you know what the interesting points on a parabola actually are. And then the last one, you might have seen this before, is a hyperbola. It almost looks like two parabolas, but not quite, because the curves look a little less U-ish and a little more open. But I'll explain what I mean by that. So a hyperbola usually looks something like this. So if these are the axes, then if I were to draw-- let me draw some asymptotes. I want to go right through the-- that's pretty good. These are asymptotes. Those aren't the actual hyperbola. But a hyperbola would look something like this. They get to be right here and they get really close to the asymptote. They get closer and closer to those blue lines like that and it happened on this side too. The graphs show up here and then they pop over and they show up there. This magenta could be one hyperbola; I haven't done true justice to it. Or another hyperbola could be on, you could kind of call it a vertical hyperbola. That's not the exact word, but it would look something like that where it's below the asymptote here. It's above the asymptote there. So this blue one would be one hyperbola and then the magenta one would be a different hyperbola. So those are the different graphs. So the one thing that I'm sure you're asking is why are they called conic sections? Why are they not called bolas or variations of circles or whatever? And in fact, wasn't even the relationship. It's pretty clear that circles and ellipses are somehow related. That an ellipse is just a squished circle. And maybe it even seems that parabolas and hyperbolas are somewhat related. This is a P once again. They both have bola in their name and they both kind of look like open U's. Although a hyperbola has two of these going and kind of opening in different directions, but they look related. But what is the connection behind all these? And that's frankly where the word conic comes from. So let me see if I can draw a three-dimensional cone. So this is a cone. That's the top. I could've used an ellipse for the top. Looks like that. Actually, it has no top. It would actually keep going on forever in that direction. I'm just kind of slicing it so you see that it's a cone. This could be the bottom part of it. So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. So if we have a plane that goes directly-- I guess if you call this the axis of this three-dimensional cone, so this is the axis. So if we have a plane that's exactly perpendicular to that axis-- let's see if I can draw it in three dimensions. The plane would look something like this. So it would have a line. This is the front line that's closer to you and then they would have another line back here. That's close enough. And of course, you know these are infinite planes, so it goes off in every direction. If this plane is directly perpendicular to the axis of these and this is where the plane goes behind it. The intersection of this plane and this cone is going to look like this. We're looking at it from an angle, but if you were looking straight down, if you were listening here and you look at this plane-- if you were looking at it right above. If I were to just flip this over like this, so we're looking straight down on this plane, that intersection would be a circle. Now, if we take the plane and we tilt it down a little bit, so if instead of that we have a situation like this. Let me see if I can do it justice. We have a situation where it's-- whoops. Let me undo that. Undo. Where it's like this and has another side like this, and I connect them. So that's the plane. Now the intersection of this plane, which is now not orthogonal or it's not perpendicular to the axis of this three-dimensional cone. If you take the intersection of that plane and that cone-- and in future videos, and you don't do this in your But eventually we'll kind of do the three-dimensional intersection and prove that this is definitely the case. You definitely do get the equations, which I'll show you in the not too far future. This intersection would look something like this. I think you can visualize it right now. It would look something like this. And if you were to look straight down on this plane, if you were to look right above the plane, this would look something-- this figure I just drew in purple-- would look something like this. Well, I didn't draw it that well. It'd be an ellipse. You know what an ellipse looks like. And if I tilted it the other way, the ellipse would squeeze the other way. But that just gives you a general sense of why both of these are conic sections. Now something very interesting. If we keep tilting this plane, so if we tilt the plane so it's-- so let's say we're pivoting around that point. So now my plane-- let me see if I can do this. It's a good exercise in three-dimensional drawing. Let's say it looks something like this. I want to go through that point. So this is my three-dimensional plane. I'm drawing it in such a way that it only intersects this bottom cone and the surface of the plane is parallel to the side of this top cone. In this case the intersection of the plane and the cone is going to intersect right at that point. You can almost view that I'm pivoting around this point, at the intersection of this point and the plane and the cone. Well this now, the intersection, would look something like this. It would look like that. And it would keep going down. So if I were to draw it, it would look like this. If I was right above the plane, if I were to just draw the plane. And there you get your parabola. If you keep kind of tilting-- if you start with a circle, tilt a little bit, you get an ellipse. You get kind of a more and more skewed ellipse. And at some point, the ellipse keeps getting more and more skewed like that. It kind of pops right when you become exactly parallel to the side of this top cone. And I'm doing it all very inexact right now, but I think I want to give you the intuition. It pops and it turns into a parabola. So you can kind of view a parabola-- there is this relationship. Parabola is what happens when one side of an ellipse pops open and you get this parabola. And then, if you keep tilting this plane, and I'll do it another color-- so it intersects both sides of the cone. Let me see if I can draw that. So if this is my new plane-- whoops. That's good enough. So if my plane looks like this-- I know it's very hard to read now-- and you wanted the intersection of this plane, this green plane and the cone-- I should probably redraw it all, but hopefully you're not getting overwhelmingly confused-- the intersection would look like this. It would intersect the bottom cone there and it would intersect the top cone over there. And then you would have something like this. This would be intersection of the plane and the bottom cone. And then up here would be the intersection of the plane and the top one. Remember, this plane goes off in every direction infinitely. So that's just a general sense of what the conic sections are and why frankly they're called conic sections. And let me know if this got confusing because maybe I'll do another video while I redraw it a little bit cleaner. Maybe I can find some kind of neat 3D application that can do it better than I can do it. This is kind of just the reason why they all are conic sections, and why they really are related to each other. And will do that a little more in depth mathematically in a few videos. But in the next video, now that you know what they are and why they're all called conic sections, I'll actually talk about the formulas about these and how do you recognize the formulas. And given a formula, how do you actually plot the graphs of these conic sections? See you in the next video." + }, + { + "Q": "At 1:19- 1:24 the video says that the ft would cancel out. I don't see how they cancel out. In class I'm doing this and my teacher says use a chart to do this, but how do i get things to cancel out. i really want to pass the 9th eoct.", + "A": "It is known as simplifying the equation . Suppose you are given 9/3*27/6. You could simplify it and you will get 3/1*9/2, that is 27/2. If you do it the long way you will the same answer. Sal did the same thing.", + "video_name": "F0LLR7bs7Qo", + "transcript": "A squirrel is running across the road at 12 feet per second. It needs to run 9 feet to get across the road. How long will it take the squirrel to run 9 feet? Round to the nearest hundredth of a second. Fair enough. A car is 50 feet away from the squirrel-- OK, this is a high-stakes word problem-- driving toward it at a speed of 100 feet per second. How long will it take the car to drive 50 feet? Round to the nearest hundredth of a second. Will the squirrel make it 9 feet across the road before the car gets there? So this definitely is high stakes, at least for the squirrel. So let's answer the first question. Let's figure out how long will it take the squirrel to run 9 feet. So let's think about it. So the squirrel's got to go 9 feet, and we want to figure out how many seconds it's going to take. So would we divide or multiply this by 12? Well, to think about that, you could think about the units where we want to get an answer in terms of seconds. We want to figure out time, so it'd be great if we could multiply this times seconds per foot. Then the feet will cancel out, and I'll be left with seconds. Now, right over here, we're told that the squirrel can run at 12 feet per second, but we want seconds per foot. So the squirrel, every second, so they go 12 feet per second, then we could also say 1 second per every 12 feet. So let's write it that way. So it's essentially the reciprocal of this because the units are the reciprocal of this. So, it's 1 second for every 12 feet. Notice, all I did is I took this information right over here, 12 feet per second, and I wrote it as second per foot-- 12 feet for every 1 second, 1 second for every 12 feet. What's useful about this is this will now give me the time it takes for the squirrel in seconds. So the feet cancel out with the feet, and I am left with 9 times 1/12, which is 9/12 seconds. And 9/12 seconds is the same thing as 3/4 seconds, which is the same thing as 0.75 seconds for the squirrel to cross the street. Now let's think about the car. So now let's think about the car. And it's the exact same logic. They tell us that the car is 50 feet away. So the squirrel is trying to cross the road like that, and the car is 50 feet away coming in like that, and we want to figure out if the squirrel will survive. So the car is 50 feet away. So it's 50 feet away. We want to figure out the time it'll take to travel that 50 feet. Once again, we would want it in seconds. So we would want seconds per feet. So we would want to multiply by seconds per foot. They give us the speed in feet per second, 100 feet per second. And so we just have to realize that this is 100 feet for every 1 second, or 1/100 seconds per feet. This is once again just this information, but we took the reciprocal of it, because we don't want feet per second, we want seconds per feet. And if we do that, that cancels with that, and we're left with 50/100 seconds. So this is 50/100 is 0.50 seconds. And so now let's answer the question, this life and death situation for the squirrel. Will the squirrel make it 9 feet across the road before the car gets there? Well, it's going to take the squirrel 0.75 seconds to cross, and it's going to take the car only half a second. So the car is going to get to where the squirrel is crossing before the squirrel has a chance to get all the way across the road. So unfortunately for the squirrel, the answer is no." + }, + { + "Q": "How do I solve x+6 divides by 5 is greater than or equal to 10", + "A": "(x + 6) / 5 \u00e2\u0089\u00a5 10 1. Multiply both sides by 5 --> (x + 6) \u00e2\u0089\u00a5 10 x 5 2. Subtract 6 from both sides --> x \u00e2\u0089\u00a5 50 - 6 3. Therefore, x \u00e2\u0089\u00a5 44", + "video_name": "Yh4TXMVq9eg", + "transcript": "- [Voiceover] We have two inequalities here, the first one says that x plus two is less than or equal to two x. This one over here in I guess this light-purple-mauve color, is three x plus four is greater than five x. Over here we have four numbers and what I want to do in this video is test whether any of these four numbers satisfy either of these inequalities. I encourage you to pause this video and try these numbers out, does zero satisfy this inequality? Does it satisfy this one? Does one satisfy this one? Does it satisfy that one? I encourage you to try these four numbers out on these two inequalities. Assuming you have tried that, let's work through this together. Let's say, if we try out zero on this inequality right over here, let's substitute x with zero. So, we'll have zero plus two needs to be less than or equal to two times zero. Is that true? Well, on the left hand side, this is two needs to be less than or equal to zero. Is that true, is two less than or equal to zero? No, two is larger than zero. So this is not going to be true, this does not satisfy the left hand side inequality, let's see if it satisfies this inequality over here. In order to satisfy it, three times zero plus four needs to be greater than five times zero. Well three times zero is just zero, five times zero is zero. So four needs to be greater than zero, which is true. So it does satisfy this inequality right over here so zero does satisfy this inequality. Let's try out one. To satisfy this one, one plus two needs to be less than or equal to two. One plus two is three, is three less than or equal than two? No, three is larger than two. This does not satisfy the left hand inequality. What about the right hand inequality right over here? Three times one plus four needs to be greater than five times one. So three times one is three, plus four. So seven needs to be greater than five, well that's true. Both zero and one satisfy three x plus four is greater than five x, neither of them satisfy x plus two is less than or equal to two x. Now let's go to the two. I know it's getting a little bit unaligned, but I'll just do it all in the same color so you can tell. Let's try out two here, two plus two needs to be less than or equal to two times two. Four needs to be less than or equal to four. Well four is equal to four and it just has to be less than or equal, so this satisfies. This satisfies this inequality. What about this purple inequality? Let's see, three times two plus four needs to be greater than five times two. Three times two is six plus four is ten, needs to be greater than 10. 10 is equal to 10, it's not greater than 10. It does not satisfy this inequality. If this was a greater than or equal to it would have satisfied but it's not. 10 is not greater than 10. It would satisfy greater than or equal to because 10 is equal to 10. So two satisfies the left hand one but not the right hand one. Let's try out five. Five plus two needs to be less than or equal to two times five, once again everywhere we see an x, we replace it with a five. Seven needs to be less than or equal to 10. Which is absolutely true, seven is less than 10. So it satisfies less than or equal to. Five satisfies this inequality and what you're probably noticing now is that an inequality can have many numbers that satisfy. In fact they sometimes will have nothing that satisfies it and sometimes they might have an infinite number of numbers that satisfy it and you see that right over here. We're just testing out a few numbers. For this left one, zero and one didn't work, two and five did work. This right one, zero and one worked, two didn't work. Let's see what five does. In order for five to satisfy it, three times x. Now we're gonna try x being five. Three times five plus four needs to be greater than five times five. Three times five is fifteen, fifteen plus four is nineteen. Nineteen is to be greater than 25, it is not. So five does not satisfy this inequality right over here. Anyway, hopefully you found that fun." + }, + { + "Q": "Is an isoprene ever a functional unit? Would I be correct to describe geraniol as 2 isoprene units and an alcohol?", + "A": "Yes, structurally, the skeleton of geraniol consists of two isoprene units. But an isoprene unit is not a functional group. A functional group would be a double bond or an alcohol group. The functional groups in geraniol are the two C=C double bonds and the alcohol group.", + "video_name": "esJ5MbAHswc", + "transcript": "- [Voiceover] Let's practice identifying functional groups in different compounds. So this molecule on the left is found in perfumes, and let's look for some of the functional groups that we've talked about in the previous videos. Well here is a carbon-carbon double bond, and we know that a carbon-carbon double bond is an alkene. So here is an alkene functional group. Here's another alkene, right, here's another carbon-carbon double bond. What is this functional group? We have an OH and then we have the rest of the molecule, so we have ROH. ROH is an alcohol, so there's also an alcohol present in this compound. Next let's look at aspirin. So what functional groups can we find in aspirin? Well, here is an aromatic ring. So this is an arene, so there is an arene functional group present in aspirin. What about this one up here? We have an OH, and the oxygen is directly bonded to a carbonyl, so let's go ahead and write that out. We have an OH where the oxygen is directly bonded to a carbon double bonded to an oxygen, and then we have the rest of the molecule, so hopefully you recognize this as being a carboxylic acid. So let me go ahead and write that out here. So this is a carboxylic acid. All right, our next functional group. We have an oxygen, and that oxygen is directly bonded to a carbonyl. So here's a carbon double bonded to an oxygen, so let's write this out. We have an oxygen directly bonded to a carbonyl, and then for this oxygen, we have the rest of the molecule so that's all of this stuff over here, and then, on the other side of the carbonyl we have another R group. So I'll go ahead and write that in, so that is an ester. RO, C double bond O, R, is an ester. So there's an ester functional group present in the aspirin molecule. Let's look at some of the common mistakes that students make. One of them is, students will say a carboxylic acid is an alcohol. So let me write out here a carboxylic acid, so we can talk about that. So sometimes the students will look at that and say, oh, well I see an OH, and then I see the rest of the molecule, so isn't that an alcohol? But since this oxygen is right next to this carbonyl, this is a carboxylic acid. So this is an example of a carboxylic acid. If we moved the OH further away, from the carbonyl, let's go ahead and draw one out like that. So here is our carbonyl, and now the OH is moved further away, now we do have an alcohol, now we have an OH and then the rest of the molecule. So this would be, we can go ahead and use a different color here. So now we are talking about an alcohol, so this is an alcohol. And what would this one be? We have a carbonyl and then we have an R group on one side, an R group on the other side. That is a ketone, let me draw this out. So when you have a carbonyl and an R group on one side, an R group on the other side, they could be the same R group, they could be a different R group. Sometimes you'll see R prime drawn for that. So this is a ketone. So now we have a ketone and an alcohol, so two functional groups present in the same compound. So hopefully you can see the difference between this compound and this compound. This one is a carboxylic acid, and this one is a ketone and an alcohol. Another common mistake that I've seen a lot is on this functional group right here, on aspirin, students will look at this oxygen here, and say, okay, I have an oxygen, and then I have an R group on one side, and I have an R group on the other side. So an R group on one side of the oxygen, an R group on the other side of the oxygen, isn't that an ether? Well, this is, ROR would represent an ether, however, we have this carbonyl here. So this carbonyl right next to this oxygen is what makes this an ester. How could we turn that into an ether? Let me go ahead and redraw this molecule here. So I'll first put in our ring, so I drew the double bonds a little bit differently from how I drew it up here but it doesn't really matter, and then I'll put in our carboxylic acid up here, and now, when I draw in this oxygen, I'm gonna take out the carbonyl. So now the carbonyl is gone, and now we do have an ether. So this actually is an ether now, we have an oxygen, we have an R group on one side, and we have the rest of the molecule over here on the other side, so now this is an ether. So hopefully you see the difference there. Look for the carbonyl right next to the oxygen, that makes it an ester. All right, so more common mistakes that students make is they mix up these two functional groups, so let's look at the functional groups in these two molecules here. And we start with benzaldehyde, and the name is a dead giveaway as to the functional group, we're talking about an aldehyde here. So first, we have our aromatic ring, our arene, and then we have an aldehyde. We have a carbonyl and we have a hydrogen that's directly bonded to the carbonyl carbon. So we have an R group, and then we have a carbonyl, and then we have a hydrogen directly bonded to our carbonyl carbon, that is an aldehyde. If we took off that hydrogen, and we put a CH3 instead, that would be the compound on the right so now we have a CH3 directly bonded to this carbonyl carbon. So now we have an R group on one side, a carbonyl, and then another R group, so we have R, C double bond O, R, and that is a ketone. And you can tell by the ending of our name here that we have a ketone present in this compound. So again, this difference is subtle, but it's important, and a lot of students mess this up. An aldehyde has a hydrogen directly bonded to this carbonyl carbon, but if there's no hydrogen, we're talking about a ketone here, so R, C double bond O, R, is a ketone. Finally, let's look at one giant compound with lots of different functional groups, and let's see if we can identify all the functional groups present in this molecule. This molecule, it is called atenolol. This is a beta blocker. So this is a heart medication. Let's look for some functional groups we've seen before. Here is that aromatic ring, so we know that an arene is present in atenolol, so let me go ahead and write this in here. Next, we have an oxygen, and there's an R group on one side of the oxygen, and an R group on the other side of the oxygen, so ROR, we know that's an ether. So there's an ether present in this compound. Next, we have an OH, and then the rest of the molecule. So ROH would be an alcohol. So there's an alcohol present. All right, next we have a nitrogen with a lone pair of electrons. There's an R group on one side, there's an R group on the other side. So this is an amine. So we have an amine, and finally, over here on the left, so this is one that is messed up a lot. We do have a nitrogen with a lone pair of electrons on it, so it's tempting to say we have an amine here. But this nitrogen is right next to a carbonyl, so it's not an amine. It's an amide, or amid. So this is an amide, so a lot of people pronounce this \"amid\", all right, so it's not an amine. So let's talk more about the difference between an amide and an amine. So let me go ahead and draw out another compound here, so we can see we have our NH2, and then we have our carbonyl. So for this one, we have our nitrogen, directly bonded to the carbonyl carbon. And that's what makes this an amide. We can move these electrons into here, and push these electrons off onto the oxygen. So resonance is possible with this compound. So this is an amide, or an \"amid\" If we move the nitrogen further away from the carbonyl, let's go ahead and do that over here. So we have our carbonyl, and now our nitrogen is further away. Now we don't have anymore resonance right? You can't draw a resonance structure showing the delocalization of the lone pair of electrons on the nitrogen. So now, now we do have an amine, so this over here, this would be an amine. Let me change colors, let me do blue. This is an amine. And then, what would this functional group be? We have a carbonyl and then we have an R group on one side, R group on the other side, that is a ketone. So this is a ketone and an amine. And then over here, we have an amide, or an \"amid\", so make sure to know the difference between these. I've see a lot of very smart students mess up the difference between these two functional groups." + }, + { + "Q": "when I typed in 81 for 9 x 8 it said it was incorrect how is it incorrect?", + "A": "9 x 9 actually equals 81. 9 x 8 is 72.", + "video_name": "NehkLV77ITk", + "transcript": "You're just walking down the street and someone comes up to you and says \"Quick! Quick!-- 4792. Is this divisible by 3? This is an emergency! Tell me as quickly as possible! And luckily you have a little tool in your toolkit where you know how to test for divisibility by 3 Well, you say I can just add up the digits If the sum of that is a multiple of 3 then this whole thing is a multiple of 3 So you say 4 plus 7 plus 9 plus 2 That's 11. Plus 9, it's 20. Plus 2 is 22 That's not divisible by 3 If you're unsure, you can even add the digits of that 2 plus 2 is 4. Clearly not divisible by 3 So this thing right over here is not divisible by 3 And so luckily that emergency was saved But then you walk down the street a little bit more and someone comes up to you--- \"Quick! Quick! Quick! 386,802-- Is that divisible by 3?\" Well, you employ the same tactic You say, what's 3 plus 8 plus 6 plus 8 plus 0 plus 2? 3 plus 8 is 11. Plus 6 is 17. Plus 8 is 25. Plus 2 is 27 Well, 27 is divisible by 3 And if you're unsure, you could add these digits right over here 2 plus 7 is equal to 9. Clearly divisible by 3 So this is divisible by 3 as well So now you feel pretty good You've helped two perfect strangers with their emergencies You figured out if these numbers were divisible by 3 very very very very quickly But you have a nagging feeling Because you're not quite sure why that worked You've just kind of always known it And so, let's think about why it worked To think about it, I'll just pick a random number But we could do this really for any number But I don't want to go too puffy on it just so you can see it's pretty common sense here And the number we'll use is 498 I can literally use any number in this situation And to think about why this whole little tool this little system works we just have to rewrite 498 We can rewrite the 4- since it's in the hundred's place we can write that as 4 times 100 Or 4 times 100, that's the same thing as 4 times 1 plus 99 That's all this 4 is 400, which is the same thing as 4 times 100 which is the same thing as 4 times 1 plus 99 And the little trick here is I want to write- instead of writing 100, I want to write this as the sum of 1 plus something that is divisible by 3 And 99 is divisible by 3 If I add more digits here- 999, 9999-- they're all divisible by 3 And this is why you can do the same reasoning for divisibility by 9 Because they are divisible by 9 as well Anyway, that's what the 4 in the hundred's place represents This 9 in the ten's place- well that represents 90 or 9 times 10, or 9 times 1 plus 9 And then finally this 8. That's in the one's place 8 times 1, or we just write plus 8 Now we can distribute this 4 This is 4 times 1 plus 4 times 99. So it's 4 plus 4 times 99 Actually let me write it like this. I'm going to write-- Actually let me write it first like 4 plus 4 times 99 Do the same thing over here This is the same thing as plus 9-- do that magenta color- plus 9 plus 9 times 9 And then finally I have this 8 right over here And I can rearrange everything These terms right over here, the 4 times 99, and the 9 times 9 I can write over here 4 times 99- I'll write what's like a different notation plus the 9 times the 9, that's those two terms and then we have the plus 4 plus 9 plus 8 Well, can we now tell whether this is divisible by 3? These terms, these first two terms are definitely divisible by 3 This's divisible by 3 because 99 is divisible by 3 regardless of what we have already you don't even have to look at this This is divisible by 3, so if you're multiplying it it's still going to be divisible by 3 This is divisible by 3, so if you're multiplying this whole thing it's still going to be divisible by 3 If you add two things that are divisible by 3 the whole thing is going to be divisible by 3 So all of this is divisible by 3 And if you have another digit here, you'd done the same exact thing Instead of having 1 plus 99, you'd had 1 plus 999, 1 plus 9999, etc So the only thing you have to really worry about is this part right over here you have to ask yourself in order for this whole thing to be divisible by 3 this part is- well that part is, then this part in order for the whole thing has to be divisible by 3 that also has to be divisible by 3 But what is this right over here? These are just our original digits 498. 4 and 9 and 8 We just have to make sure that when we take the sum it's divisible by 3" + }, + { + "Q": "I still do not see how the parabola equation helps when trying to solve for a parabola with a given set of foci and a directrix, the previous activity just equated the directrix and foci using the distance formulas. So what are you supposed to use the parabola equation for?", + "A": "He s showing how to convert between the two forms, focus/directrix and vertex , and how these two different ways of thinking about parabolas give the same set of functions.", + "video_name": "w56Vuf9tHfA", + "transcript": "- This right here is an equation for a parabola and the role of this video is to find an alternate or to explore an alternate method for finding the focus and directrix of this parabola from the equation. So the first thing I like to do is solve explicitly for y. I don't know, my brain just processes things better that way. So, let's get this 23 over four to the right hand side. So let's add 23 over four to both sides and then we'll get y is equal to negative one-third times x minus one squared plus 23 over four. Now let's remind ourselves what we've learned about foci and directrixes, I think is how to say it. So, the focus. If the focus of a parabola is at the point a, b and the directrix, the directrix, directrix is the line y equals k. We've shown in other videos with a little bit of hairy algebra that the equation of the parabola in a form like this is going to be y is equal to one over two times b minus k. This b minus k is then the difference between this y coordinate and this y value, I guess you could say. Times x minus one squared plus b plus k. I'm sorry, not x minus one. I'm getting confused with this. x minus a squared. x minus a squred plus b plus k over two. The focus is a,b and the directrix is y equals k and this is gonna be the equation of the parabola. Well, we've already seen the technique where, look, we can see the different parts. We can see that, okay, this x minus one squared. Actually, let me do this in a different color. This x minus one squared corresponds to the x minus a squared and so one corresponds to a, so just like that, we know that a is going to be equal to one and actually let me just write that down. a is equal to one in this example right over here. And then you could see that the negative one-third over here corresponds to the one over two b minus k and you would see that the 23 over four corresponds to the b plus k over two. Now the first technique that we explored, we said, \"Okay, let's set negative one-third \"to this thing right over here. \"Solve for b minus k.\" We're not solving for b or k, we're solving for the expression b minus k. So you got b minus k equals something. And then you could use 23 over four and this to solve for b plus k. So you get b plus k equals something and then you have two equations, two unknowns, you can solve for b and k. What I wanna do in this video is explore a different method that really uses our knowledge of the vertex of a parabola to be able to figure out where the focus and the directrix is going to be. So let's think about the vertex of this parabola right over here. Remember, the vertex, if the parabola is upward opening like this, the vertex is this minimum point. If it is downward opening, it's going to be this maximum point. And so when you look over here, you see that you have a negative one-third in front of the x minus one squared. So this quantity over here is either going to be zero or negative. It's not going to add to 23 over four, it's either gonna add nothing or take away from it. So this thing's going to hit a maximum point, when this thing is zero, when this thing is zero, and that's just gonna go down from there and when this thing is zero, y is going to be equal to 23 over four. So our vertex is going to be that maximum point. Well, when does this equal zero? Well, when x equals one. When x equals one, you get one minus one squared. So zero squared times negative one-third, this is zero. So when x is equal to one, we're at our maximum y value of 23 over four which five and three-fourths. Actually, let me write that as a . Actually, I'll leave just that's our vertex. and it is a downward opening parabola. So actually, let me start to draw this. So we'd get some axis here. So we have to go all the way up to five and three-fourths. So. Let's make this our y, this is our y axis. This is the x axis. That's the x axis. We're gonna see, we're gonna go to one. Let's call that one. Let's call that two. And then I wanna get, let's see, if I go to five and three-fourths, let's go up to, let's see one, two, three, four five, six, seven. We can label 'em. One, two, three, four five, six and seven and so our vertex is right over here. One comma 23 over four, so that's five and three-fourths. So it's gonna be right around right around there and as we said, since we have a negative value in front of this x minus one squared term, I guess we could call it, this is going to be a downward opening parabola. This is going to be a maximum point. So our actual parabola is going to look is going to look something it's gonna look something like this. It's gonna look something like this and we could, obviously, I'm hand drawing it, so it's not going to be exactly perfect, but hopefully you get the general idea of what the parabola is going look like and actually, let me just do part of it, 'cause I actually don't know that much information about the parabola just yet. I'm just gonna draw it like that. So we don't know just yet where the directrix and focus is, but we do know a few things. The focus is going to sit on the same, I guess you could say, the same x value as the vertex. So if we draw, this is x equals one, if x equals one, we know from our experience with focuses, foci, (laughs) I guess, that they're going to sit on the same axis as the vertex. So the focus might be right over here and then the directrix is going to be equidistant on the other side, equidistant on the other side. So the directrix might be something like this. Might be right over here. And once again, I haven't figured it out yet, but what we know is that because this point, the vertex, sits on the parabola, by definition has to be equidistant from the focus and the directrix. So. This distance has to be the same as this distance right over here and what's another way of thinking about this entire distance? Remember, this coordinate right over here is a, b and this is the line y is equal to k. This is y equals k. So what's this distance in yellow? What's this difference in y going to be? Well, you could call that, in this case, the directrix is above the focus, so you could say that this would be k minus b or you could say it's the absolute value of b minus k. This would actually always work. It'll always give you kind of the positive distance. So if we knew what the absolute value of b minus k is, if we knew this distance, then just split it in half with the directrix is gonna be that distance, half the distance above and then the focus is gonna be half the distance below. So let's see if we can figure this out. And we can figure this out because we see in this, I guess you could say, this equation, you can see where b minus k is involved. One over two times b minus k needs to be equal to negative one-third. So let's solve for b minus k. So we get we get one over two times b minus k is going to be equal to negative one-third. Once again, this corresponds to that. It's going to be equal to negative one-third. We could take the reciprocal of both sides and we get two times b minus k is equal to, is equal to three, is equal to three. Now we can divide both sides we can divide both sides by two and so we're gonna get we're gonna get b b minus k is equal to is equal to, what is that, three-halves, three-halves. b minus k is equal to, oh, let me make sure that has to be a negative three, so this has to be negative three-halves. And so if you took the absolute value of b minus k you're gonna get positive three-halves, or if you took k minus b, you're going to get positive three-halves. So just like that, using this part, just actually matching the negative one-third to this part of this equation, we're able to solve for the absolute value of b minus k which is going to be the distance between the y axis in the y direction between the focus and the directrix. So this distance right over here is three-halves. So what is half that distance? And the reason why I care about half that distance is because then I can calculate where the focus is, because it's going to be half that distance below the vertex and I could say, whatever that distance is is going to be that distance also above the directrix. So half that distance, so one half times three-halves is equal to three-fourths. So just like that, we're able to figure out the directrix is going to be three-fourths above this. So I could say the directrix, so let me see, I'm running out of space, the directrix is gonna be y is equal to the y coordinate of the focus. Sorry, the y coordinate of the vertex. I might be careful with my language. It's gonna be equal to the y coordinate of the vertex plus three-fourths, plus three- fourths. So plus three-fourths, which is equal to 26 over four, which is equal to, what is that, that's equal to six and a half. So this right over here, actually I got pretty close when I drew it is actually going to be the directrix. Y is equal to six and a half and the focus, well, we know the x coordinate of the focus, a is going to be equal to one and b is going to be three-fourths less than the y coordinate of the directrix. So 23 over four minus three-fourths. Gonna be 23 over four 23 over four minus three-fourths which is 20 over four, which is just equal to which is just equal to five. And we are done. That's the focus, one comma five. Directrix is y is equal to six and a half." + }, + { + "Q": "what do i do if i dont have an x value for example 4x cubed plus 2x squared minus 8", + "A": "Write 0x as a placeholder. --> 4x^3 + 2x^2 + 0x - 8. Do the long division and treat the zero as any other number (e.g. 0x - 5x = -5x). Does that make sense?", + "video_name": "MwG6QD352yc", + "transcript": "- [Voiceover] So let's introduce ourselves to the Polynomial Remainder Theorem. And as we'll see a little, you'll feel a little magical at first. But in future videos, we will prove it and we will see, well, like many things in Mathematics. When you actually think it through, maybe it's not so much magic. So what is the Polynomial Remainder Theorem? Well it tells us that if we start with some polynomial, f of x. So this right over here is a polynomial. Polynomial. And we divide it by x minus a. Then the remainder from that essentially polynomial long division is going to be f of a. It is going to be f of a. I know this might seem a little bit abstract right now. I'm talking about f of x's and x minus a's. Let's make it a little bit more concrete. So let's say that f of x is equal to, I'm just gonna make up a, let's say a second degree polynomial. This would be true for any polynomial though. So three x squared minus four x plus seven. And let's say that a is, I don't know, a is one. So we're gonna divide that by, we're going to divide by x minus one. So a, in this case, is equal to one. So let's just do the polynomial long division. I encourage you to pause the video. If you're unfamiliar with polynomial long division, I encourage you to watch that before watching this video because I will assume you know how to do a polynomial long division. So divide three x squared minus four x plus seven. Divide it by x minus one. See what you get as the remainder and see if that remainder really is f of one. So assuming you had a go at it. So let's work through it together. So let's divide x minus one into three x squared minus four x plus seven. All right, little bit of polynomial long division is never a bad way to start your morning. It's morning for me. I don't know what it is for you. All right, so I look at the x term here, the highest degree term. And then I'll start with the highest degree term here. So how many times does x going to three x squared? What was three x times? Three x times x is three x squared. So I'll write three x over here. I'll write it in the, I could say the first degree place. Three x times x is three x squared. Three x times negative one is negative three x. And now we want to subtract this thing. It's just the way that you do traditional long division. And so, what do we get? Well, three x squared minus three x squared. That's just going to be a zero. So this just add up to zero. And this negative four x, this is going to be plus three x, right? And negative of a negative. Negative four x plus three x is going to be negative x. I'm gonna do this in a new color. So it's going to be negative x. And then we can bring down seven. Complete analogy to how you first learned long division in maybe, I don't know, third or fourth grade. So all I did is I multiplied three x times this. You get three x squared minus three x and then I subtract to that from three x squared minus four x to get this right over here or you could say I subtract it from this whole polynomial and then I got negative x plus seven. So now, how many times does x minus one go to negative x plus seven? Well x goes into negative x, negative one times x is negative x. Negative one times negative one is positive one. But then we're gonna wanna subtract this thing. We're gonna wanna subtract this thing and this is going to give us our remainder. So negative x minus negative x. Just the same thing as negative x plus x. These are just going to add up to zero. And then you have seven. This is going to be seven plus one. Remember you have this negative out so if you distribute the negative, this is going to be a negative one. Seven minus one is six. So your remainder here is six. One way to think about it, you could say that, well (mumbles). I'll save that for a future video. This right over here is the remainder. And you know when you got to the remainder, this is just all review of polynomial long division, is when you get something that has a lower degree. This is, I guess you could call this a zero degree polynomial. This has a lower degree than what you are actually dividing into or than the x minus one than your divisor. So this a lower degree so this is the remainder. You can't take this into this anymore times. Now, by the Polynomial Remainder Theorem, if it's true and I just picked a random example here. This is by no means a proof but just kinda a way to make it tangible of Polynomial (laughs) Remainder Theorem is telling us. If the Polynomial Remainder Theorem is true, it's telling us that f of a, in this case, one, f of one should be equal to six. It should be equal to this remainder. Now let's verify that. This is going to be equal to three times one squared, which is going to be three, minus four times one, so that's just going to be minus four, plus seven. Three minus four is negative one plus seven is indeed, we deserve a minor drumroll, is indeed equal to six. So this is just kinda, at least for this particular case, looks like okay, it seems like the Polynomial Remainder Theorem worked. But the utility of it is if someone said, \"Hey, what's the remainder if I were to divide \"three x squared minus four x plus seven \"by x minus one if all I care about is the remainder?\" They don't care about the actual quotient. All they care about is the remainder, you could, \"Hey, look, I can just take that, in this case, a is one. \"I can throw that in. \"I can evaluate f of one and I'm gonna get six. \"I don't have to do all of this business. \"All I had, would have to do is this \"to figure out the remainder of three x squared.\" Well you take three x squared minus four plus seven and divide by x minus one." + }, + { + "Q": "can you write that congruency sign (at 1:39) on a keyboard?", + "A": "In Windows, like in Microsoft Word or Microsoft Excel, you can your font to Symbol. Then, that symbol is SHIFT+2 (or the @ key on your keyboard.). As a side note, with the Symbol font you get most of the Greek letters..alpha, beta, delta, pi, etc.", + "video_name": "CJrVOf_3dN0", + "transcript": "Let's talk a little bit about congruence, congruence And one to think about congruence, it's really kind of equivalence for shapes So, when in algebra when something is equal to another thing it means that their quantities are the same But when we're all of the sudden talking about shapes and we say that those shapes are the same, the shapes are the same size and shape then we say that they're congruent And just to see a simple example here: I have this triangle, right over there and let's say I have this triangle right over here And if you are able to shift, you are able to shift this triangle and flip this triangle, you can make it look exactly like this triangle As long as you're not changing the lengths of any of the sides or the angles here But you can flip it, you can shift it, you can rotate it So you can shift, let me write this, you can shift it, you can flip it and you can rotate If you can do those three procedures to make these the exact same triangle, then they are congruent And if you say that a triangle is congruent, let me label this So, let's call this triangle ABC Now let's call this D, let me call it XYZ XY and Z So, if we were to say, if we make the claim that both of these triangles are congruent So, if we say triangle ABC is congruent And the way you specify it, it almost look like an equal sign But it's equal sign with a curly thing on top Let me write it a little bit either So, we would write it like this If we know that triangle ABC is congruent to triangle XYZ That means their corresponding sides have the same length And their corresponding angles have the same measure So, if we make this assumption or someone tells us that this is true then we know, for example, that AB is going to equal to XY The length of segment AB is gonna be equal to the segment of XY And we could do this like this, and I'm assuming this are the corresponding sides And you can see that actually we've defined these triangles A corresponds to X, B corresponds to Y and C corresponds to Z right over there So, side AB is gonna have the same length as XY Then you can sometimes if you don't have the colors you can denote it just like that These two length are- or this two lines segments have the same length And you can actually say this, you don't always see this written this way You could also make the statement that line segment AB is congruent to line segment XY But congruence of line segments really just means that their lengths are equivalent So, these two things mean the same thing If one line segment is congruent to another line segment that just means the measure of one line segment is equal to the measure of the other line segment And so we can go thru all the corresponding sides If these two characters are congruent, we also know that BC, we also know that the length BC is gonna be the length of YZ Assuming those are the corresponding sides And we can put these double hash marks right over here to show that these lengths are the same And when we go the third side, we also know that these are going to be has same length or the line segments are going to be congruent So, we also know that the length of AC is going to be equal to the length of XZ Not only do we know that all of the sides, the corresponding sides are gonna have the same length If someone tells that a triangle is congruent We also know that all the corresponding angles are going to have the same measure So, for example: we also know that this angle's measure is going to be the same as the corresponding angle's measure, and the corresponding angle is right over It's between these orange side and blue side Or orange side and purple side, I should say And between the orange side and this purple side And so it also tells us that the measure of angle is BAC is equal to the measure of angle of YXZ Let me write that angle symbol, a little less like that, measure of angle of YXZ YXZ We can also write that as angle BAC is congruent angle YXZ And once again, like line segment, if one line segment is congruent to another line segment It just means that their lengths are equal And if one angle is congruent to another angle it just means that their measures are equal So, we know that those two corresponding angles have the same measure, they're congruent We also know that these two corresponding angles I'll use a double arch to specify that this has the same measure as that So, we also know the measure of angle ABC is equal to the measure of angle XYZ And then finally we know that this angle, if we know that these two characters are congruent, then this angle is gonna have the same measure as this angle as a corresponding angle So, we know that the measure of angle ACB is gonna be equal to the measure of angle XZY Now what we're gonna concern ourselves a lot with is how do we prove congruence? 'Cause it's cool, 'cause if you can prove congruence of 2 triangles then all of the sudden you can make all of these assumptions And what we're gonna find out, and this is going to be, we're gonna assume it for the sake of introductory geometry course This is an axiom or a postulate or just something you assume So, an axiom, very fancy word Postulate, also a very fancy word It really just means things we are gonna assume are true An axiom is sometimes, there's a little bit of distinction sometimes where someone would say \"an axiom is something that is self-evident\" or it seems like a universal truth that is definitely true and we just take it for granted You can't prove an axiom A postulate kinda has that same role but sometimes let's just assume this is true and see if we assume that it's true what can we derive from it, what we can prove if we assume its true But for the sake of introductory geometry class and really most in mathematics today, these two words are use interchangeably An axiom or a postulate, just very fancy words that things we take as a given Things that we'll just assume, we won't prove them, we will start with this assumptions and then we're just gonna build up from there And one of the core ones that we'll see in geometry is the axiom or the postulate That if all of the sides are congruent, if the length of all the sides of the triangle are congruent, then we are dealing with congruent triangles So, sometimes called side, side, side postulate or axiom We're not gonna prove it here, we're just gonna take it as a given So this literally stands for side, side, side And what it tells is, if we have two triangles and So I say that's another triangle right over there And we know that corresponding sides are equal So, we know that this side right over here is equal into, like, that side right over there Then we know and we're just gonna take this as an assumption and we can build off of this We know that they are congruent, the triangle, that these two triangles are congruent to each other I didn't put any labels there so it's kinda hard for me to refer to them But these two are congruent triangles And what's powerful there is we know that the corresponding sides are equal Then we know they're congruent and we can make all the other assumptions Which means that the corresponding angles are also equal So, that we know, is gonna be congruent to that or have the same measure That's gonna have the same measure as that and then that is gonna have the same measure as that right over there And to see why that is a reasonable axiom or a reasonable assumption or a reasonable postulate to start off with Let's take one, let's start with one triangle So, let's say I have this triangle right over here So, it has this side and then it has this side and then it has this side right over here And what I'm gonna do is see if I have another triangle that has the exact same line, side lengths is there anyway for me to construct a triangle with the same side lengths that is different, that can't be translated to this triangle thru flipping, shifting or rotating So, we assume this other triangle is gonna have the same size, the same length as that one over there So, I'll try to draw it like that Roughly the same length We know that it's going to have a size that's that length So, it's gonna have a side that is that length Let me put it on this side just to make it look a little bit more interesting So, we know that it's gonna have a side like that So, I'm gonna draw roughly the same length but I'm gonna try to do it in a different angle Now we know that's it's gonna have that looks like that So, let me, I'll put it right over here It's about that length right over there And so clearly this isn't a triangle, in order to make it a triangle, I'll have to connect this point to that point right over there And really there's only two ways to do it I can rotate it around that little hinge right over there If I connect them over here then I'm going to get a triangle that looks likes this Which is really a just a flip, am I visualizing it right? Yeah, just a flip version You can rotate it a little back this way, and you'd have a magenta on this side and a yellow one on this side And you can flip it, you could flip it vertically and it'll look exactly like this Our other option to make these two points connect is to rotate them out this way And the yellow side is gonna be here And then the magenta side is gonna be here and that's not magenta The magenta side is gonna be just like that And if we do that, then we actually just have to rotate it We just have to rotate it around to get that exact triangle So, this isn't a proof, and actually we're gonna start assuming that his is an axiom But hopefully you'll see that it's a pretty reasonable starting point that all of the sides, all of the corresponding sides of two different triangles are equal Then we are going to- we know that they are congruent We are just gonna assume that it's an axiom for that we're gonna build off, that they are congruent And we also know that he corresponding angles are going to be equivalent" + }, + { + "Q": "if i = radical(-1), then i^2 = radical(-1) * radical(-1), which means i^2 should equal radical(-1 * -1) which simplies to radical(1). Making i^2 = 1", + "A": "No, because the property\u00e2\u0088\u009a(ab) = (\u00e2\u0088\u009aa)(\u00e2\u0088\u009ab) does not hold for imaginary numbers. Thus, (\u00e2\u0088\u009a-1)(\u00e2\u0088\u009a-1) \u00e2\u0089\u00a0 \u00e2\u0088\u009a(\u00e2\u0088\u00921 \u00c3\u0097 \u00e2\u0088\u00921)", + "video_name": "s03qez-6JMA", + "transcript": "We're asked to simplify the principal square root of negative 52. And we're going to assume, because we have a negative 52 here inside of the radical, that this is the principal branch of the complex square root function. That we can actually put, input, negative numbers in the domain of this function. That we can actually get imaginary, or complex, results. So we can rewrite negative 52 as negative 1 times 52. So this can be rewritten as the principal square root of negative 1 times 52. And then, if we assume that this is the principal branch of the complex square root function, we can rewrite this. This is going to be equal to the square root of negative 1 times-- or I should say, the principal square root of negative 1 times the principal square root of 52. Now, I want to be very, very clear here. You can do what we just did. If we have the principal square root of the product of two things, we can rewrite that as the principal square root of each, and then we take the product. But you can only do this, or I should say, you can only do this if either both of these numbers are positive, or only one of them is negative. You cannot do this if both of these were negative. For example, you could not do this. You could not say the principal square root of 52 is equal to negative 1 times negative 52. So far, I haven't said anything wrong. 52 is definitely negative 1 times negative 52. But then, since these are both negative, you cannot then say that this is equal to the square root of negative 1 times the square root of negative 52. In fact, I invite you to continue on this train of reasoning. You're going to get a nonsensical answer. This is not OK. You cannot do this, right over here. And the reason why you cannot do this is that this property does not work when both of these numbers are negative. Now with that said, we can do it if only one of them are negative or both of them are positive, obviously. Now, the principal square root of negative 1, if we're talking about the principal branch of the complex square root function, is i. So this right over here does simplify to i. And then let's think if we can simplify the square root of 52 any. And to do that, we can think about its prime factorization, see if we have any perfect squares sitting in there. So 52 is 2 times 26, and 26 is 2 times 13. So we have 2 times 2 there, or 4 there, which is a perfect square. So we can rewrite this as equal to-- Well, we have our i, now. The principal square root of negative 1 is i. The other square root of negative 1 is negative i. But the principal square root of negative 1 is i. And then we're going to multiply that times the square root of 4 times 13. And this is going to be equal to i times the square root of 4. i times the square root of 4, or the principal square root of 4 times the principal square root of 13. The principal square root of 4 is 2. So this all simplifies, and we can switch the order, over here. This is equal to 2 times the square root of 13. 2 times the principal square root of 13, I should say, times i. And I just switched around the order. It makes it a little bit easier to read if I put the i after the numbers over here. But I'm just multiplying i times 2 times the square root of 13. That's the same thing as multiplying 2 times the principal square root of 13 times i. And I think this is about as simplified as we can get here." + }, + { + "Q": "Where can I find the \"rigorous proof\" of these properties?", + "A": "A rigorous proof can usually be found in any old calculus text, in the section on limits. A fun exercise might be to write down the epsilon-delta definition of limits then try to figure out exactly how one would prove these statements!", + "video_name": "lSwsAFgWqR8", + "transcript": "What I want to do in this video is give you a bunch of properties of limits. And we're not going to prove it rigorously here. In order to have the rigorous proof of these properties, we need a rigorous definition of what a limit is. And we're not doing that in this tutorial, we'll do that in the tutorial on the epsilon delta definition of limits. But most of these should be fairly intuitive. And they are very helpful for simplifying limit problems in the future. So let's say we know that the limit of some function f of x, as x approaches c, is equal to capital L. And let's say that we also know that the limit of some other function, let's say g of x, as x approaches c, is equal to capital M. Now given that, what would be the limit of f of x plus g of x as x approaches c? Well-- and you could look at this visually, if you look at the graphs of two arbitrary functions, you would essentially just add those two functions-- it'll be pretty clear that this is going to be equal to-- and once again, I'm not doing a rigorous proof, I'm just really giving you the properties here-- this is going to be the limit of f of x as x approaches c, plus the limit of g of x as x approaches c. Which is equal to, well this right over here is-- let me do that in that same color-- this right here is just equal to L. It's going to be equal to L plus M. This right over here is equal to M. Not too difficult. This is often called the sum rule, or the sum property, of limits. And we could come up with a very similar one with differences. The limit as x approaches c of f of x minus g of x, is just going to be L minus M. It's just the limit of f of x as x approaches c, minus the limit of g of x as x approaches c. So it's just going to be L minus M. And we also often call it the difference rule, or the difference property, of limits. And these once again, are very, very, hopefully, reasonably intuitive. Now what happens if you take the product of the functions? The limit of f of x times g of x as x approaches c. Well lucky for us, this is going to be equal to the limit of f of x as x approaches c, times the limit of g of x, as x approaches c. Lucky for us, this is kind of a fairly intuitive property of limits. So in this case, this is just going to be equal to, this is L times M. This is just going to be L times M. Same thing, if instead of having a function here, we had a constant. So if we just had the limit-- let me do it in that same color-- the limit of k times f of x, as x approaches c, where k is just some constant. This is going to be the same thing as k times the limit of f of x as x approaches c. And that is just equal to L. So this whole thing simplifies to k times L. And we can do the same thing with difference. This is often called the constant multiple property. We can do the same thing with differences. So if we have the limit as x approaches c of f of x divided by g of x. This is the exact same thing as the limit of f of x as x approaches c, divided by the limit of g of x as x approaches c. Which is going to be equal to-- I think you get it now-- this is going to be equal to L over M. And finally-- this is sometimes called the quotient property-- finally we'll look at the exponent property. So if I have the limit of-- let me write it this way-- of f of x to some power. And actually, let me even write it as a fractional power, to the r over s power, where both r and s are integers, then the limit of f of x to the r over s power as x approaches c, is going to be the exact same thing as the limit of f of x as x approaches c raised to the r over s power. Once again, when r and s are both integers, and s is not equal to 0. Otherwise this exponent would not make much sense. And this is the same thing as L to the r over s power. So this is equal to L to the r over s power. So using these, we can actually find the limit of many, many, many things. And what's neat about it is the property of limits kind of are the things that you would naturally want to do. And if you graph some of these functions, it actually turns out to be quite intuitive." + }, + { + "Q": "let the objects be A and B and let us assume that we are going to throw the objects from the tower of some height then which object would reach the ground earlier or both the objects reach the ground at same time? in other words the speed would be same or different depending on mass?", + "A": "Acceleration due to gravity will always be the same, so with no air resistance the two objects would hit the ground at the same time. If there was air resistance, the lighter object would be slowed more.", + "video_name": "VYgSXBjEA8I", + "transcript": "So I'm curious about how much acceleration does a pilot, or the pilot and the plane, experience when they need to take off from an aircraft carrier? So I looked up a few statistics on the Internet, this right here is a picture of an F/A-18 Hornet right over here. It has a take-off speed of 260 kilometers per hour. If we want that to be a velocity, 260 km/hour in this direction, if it's taking off from this Nimitz class carrier right over here. And I also looked it up, and I found the runway length, or I should say the catapult length, because these planes don't take off just with their own power. They have their own thrusters going, but they also are catapulted off, so they can be really accelerated quickly off of the flight deck of this carrier. And the runway length of a Nimitz class carrier is about 80 meters. So this is where they take off from. This right over here is where they take off from. And then they come in and they land over here. But I'm curious about the take-off. So to do this, let's figure out, well let's just figure out the acceleration, and from that we can also figure out how long it takes them to be catapulted off the flight deck. So, let me get the numbers in one place, so the take-off velocity, I could say, is 260 km/hour, so let me write this down. So that has to be your final velocity when you're getting off, of the plane, if you want to be flying. So your initial velocity is going to be 0, and once again I'm going to use the convention that the direction of the vector is implicit. Positive means going in the direction of take-off, negative would mean going the other way. My initial velocity is 0, I'll denote it as a vector right here. My final velocity over here has to be 260 km/hour. And I want to convert everything to meters and seconds, just so that I can get my, at least for meters, so that I can use my runway length in meters. So let's just do it in meters per second, I have a feeling it'll be a little bit easier to understand when we talk about acceleration in those units as well. So if we want to convert this into seconds, we have, we'll put hours in the numerator, 1 hour, so it cancels out with this hour, is equal to 3600 seconds. I'll just write 3600 s. And then if we want to convert it to meters, we have 1000 meters is equal to 1 km, and this 1 km will cancel out with those kms right over there. And whenever you're doing any type of this dimensional analysis, you really should see whether it makes sense. If I'm going 260 km in an hour, I should go much fewer km in a second because a second is so much shorter amount of time, and that's why we're dividing by 3600. If I can go a certain number of km in an hour a second, I should be able to go a lot, many many more meters in that same amount of time, and that's why we're multiplying by 1000. When you multiply these out, the hours cancel out, you have km canceling out, and you have 260 times 1000 divided by 3600 meters per second. So let me get my trusty TI-85 out, and actually calculate that. So I have 260 times 1000 divided by 3600 gets me, I'll just round it to 72, because that's about how many significant digits I can assume here. 72 meters per second. So all I did here is I converted the take-off velocity, so this is 72 m/s, this has to be the final velocity after accelerating. So let's think about what that acceleration could be, given that we know the length of the runway, and we're going to assume constant acceleration here, just to simplify things a little bit. But what does that constant acceleration have to be? So let's think a little bit about it. The total displacement, I'll do that in purple, the total displacement is going to be equal to our average velocity while we're accelerating, times the difference in time, or the amount of time it takes us to accelerate. Now, what is the average velocity here? It's going to be our final velocity, plus our initial velocity, over 2. It's just the average of the initial and final. And we can only do that because we are dealing with a constant acceleration. And what is our change in time over here? What is our change in time? Well our change in time is how long does it take us to get to that velocity? Or another way to think about it is: it is our change in velocity divided by our acceleration. If we're trying to get to 10 m/s, or we're trying to get 10 m/s faster, and we're accelerating at 2 m/s squared, it'll take us 5 seconds. Or if you want to see that explicitly written in a formula, we know that acceleration is equal to change in velocity over change in time. You multiply both sides by change in time, and you divide both sides by acceleration, so let's do that, multiply both sides by change in time and divide by acceleration. Multiply by change in time and divide by acceleration. And you get, that cancels out, and then you have that cancels out, and you have change in time is equal to change in velocity divided by acceleration. Change in velocity divided by acceleration. So what's the change in velocity? Change in velocity, so this is going to be change in velocity divided by acceleration. Change in velocity is the same thing as your final velocity minus your initial velocity, all of that divided by acceleration. So this delta t part we can re-write as our final velocity minus our initial velocity, over acceleration. And just doing this simple little derivation here actually gives us a pretty cool result! If we just work through this math, and I'll try to write a little bigger, I see my writing is getting smaller, our displacement can be expressed as the product of these two things. And what's cool about this, well let me just write it this way: so this is our final velocity plus our initial velocity, times our final velocity minus our initial velocity, all of that over 2 times our acceleration. Our assumed constant acceleration. And you probably remember from algebra class this takes the form: a plus b times a minus b. And so this equal to -- and you can multiply it out and you can review in our algebra playlist how to multiply out two binomials like this, but this numerator right over here, I'll write it in blue, is going to be equal to our final velocity squared minus our initial velocity squared. This is a difference of squares, you can factor it out into the sum of the two terms times the difference of the two terms, so that when you multiply these two out you just get that over there, over 2 times the acceleration. Now what's really cool here is we were able to derive a formula that just deals with the displacement, our final velocity, our initial velocity, and the acceleration. And we know all of those things except for the acceleration. We know that our displacement is 80 meters. We know that this is 80 meters. We know that our final velocity, just before we square it, we know that our final velocity is 72 meters per second. And we know that our initial velocity is 0 meters per second. And so we can use all of this information to solve for our acceleration. And you might see this formula, displacement, sometimes called distance, if you're just using the scalar version, and really we are thinking only in the scalar, we're thinking about the magnitudes of all of these things for the sake of this video. We're only dealing in one dimension. But sometimes you'll see it written like this, sometimes you'll multiply both sides times the 2 a, and you'll get something like this, where you have 2 times, really the magnitude of the acceleration, times the magnitude of the displacement, which is the same thing as the distance, is equal to the final velocity, the magnitude of the final velocity, squared, minus the initial velocity squared. Or sometimes, in some books, it'll be written as 2 a d is equal to v f squared minus v i squared. And it seems like a super mysterious thing, but it's not that mysterious. We just very simply derived it from displacement, or if you want to say distance, if you're just thinking about the scalar quantity, is equal to average velocity times the change in time. So, so far we've just derived ourselves a kind of a neat formula that is often not derived in physics class, but let's use it to actually figure out the acceleration that a pilot experiences when they're taking off of a Nimitz class carrier. So we have 2 times the acceleration times the distance, that's 80 meters, times 80 meters, is going to be equal to our final velocity squared. What's our final velocity? 72 meters per second. So 72 meters per second, squared, minus our initial velocity. So our initial velocity in this situation is just 0. So it's just going to be minus 0 squared, which is just going to be 0, so we don't even have to write it down. And so to solve for acceleration, to solve for acceleration, you just divide, so this is the same thing as 160 meters, well, let's just divide both sides by 2 times 80, so we get acceleration is equal to 72 m/s squared over 2 times 80 meters. And what we're gonna get is, I'll just write this in one color, it's going to be 72 divided by 160, times, we have in the numerator, meters squared over seconds squared, we're squaring the units, and then we're going to be dividing by meters. So times, I'll do this in blue, times one over meters. Right? Because we have a meters in the denominator. And so what we're going to get is this meters squared divided by meters, that's going to cancel out, we're going to get meters per second squared. Which is cool because that's what acceleration should be in. And so let's just get the calculator out, to calculate this exact acceleration. So we have to take, oh sorry, this is 72 squared, let me write that down. So this is, this is going to be 72 squared, don't want to forget about this part right over here. 72 squared divided by 160. So we have, and we can just use the original number right over here that we calculated, so let's just square that, and then divide that by 160, divided by 160. And if we go to 2 significant digits, we get 33, we get our acceleration is, our acceleration is equal to 33 meters per second squared. And just to give you an idea of how much acceleration that is, is if you are in free fall over Earth, the force of gravity will be accelerating you, so g is going to be equal to 9.8 meters per second squared. So this is accelerating you 3 times more than what Earth is making you accelerate if you were to jump off of a cliff or something. So another way to think about this is that the force, and we haven't done a lot on force yet, we'll talk about this in more depth, is that this pilot would be experiencing more than 3 times the force of gravity, more than 3 g's. 3 g's would be about 30 meters per second squared, this is more than that. So an analogy for how the pilot would feel is when he's, you know, if this is the chair right here, his pilot's chair, that he's in, so this is the chair, and he's sitting on the chair, let me do my best to draw him sitting on the chair, so this is him sitting on the chair, flying the plane, and this is the pilot, the force he would feel, or while this thing is accelerating him forward at 33 meters per second squared, it would feel very much to him like if he was lying down on the surface of the planet, but he was 3 times heavier, or more than 3 times heavier. Or if he was lying down, or if you were lying down, like this, let's say this is you, this is your feet, and this is your face, this is your hands, let me draw your hands right here, and if you had essentially two more people stacked above you, roughly, I'm just giving you the general sense of it, that's how it would feel, a little bit more than two people, that squeezing sensation. So his entire body is going to feel 3 times heavier than it would if he was just laying down on the beach or something like that. So it's very very very interesting, I guess, idea, at least to me. Now the other question that we can ask ourselves is how long will it take to get catapulted off of this carrier? And if he's accelerating at 33 meters per second squared, how long would it take him to get from 0 to 72 meters per second? So after 1 second, he'll be going 33 meters per second, after 2 seconds, he'll be going 66 meters per second, so it's going to take, and so it's a little bit more than 2 seconds. So it's going to take him a little bit more than 2 seconds. And we can calculate it exactly if you take 72 meters per second, and you divide it by 33, it'll take him 2.18 seconds, roughly, to be catapulted off of that carrier." + }, + { + "Q": "What is the purpose of meiosis II? There are already cells with a diploid number of chromosomes so why not just keep them and not undergo meiosis II?", + "A": "Suppose we just say that meiosis 2 is not necessary than what will the offspring be normal for example we just have 46 chromosome than our child is going to be 92 which quite obviously may have abnormal effects on child.In case of Downs syndrome one chromosome can change the mental and physical state.We know that every cell got 46 chromosome which in each one there are genes for function many genes are involved in one function but addition of one chromosome can have large effect on cell", + "video_name": "IQJ4DBkCnco", + "transcript": "- [Voiceover] Before we go in-depth on meiosis, I want to do a very high level overview comparing mitosis to meiosis. So, in mitosis, this is all a review, if you've watched the mitosis video, in mitosis, we start with a cell, that has a diploid number of chromosomes. I'll just write 2n to show it has a diploid number. For human beings, this would be 46 chromosomes. 46 for humans, you get 23 chromosomes from your mother, 23 chromosomes from your father or you can say you have 23 homologous pairs, which leads to 46 chromosomes. Now after the process of mitosis happens and you have your cytokinesis and all the rest, you end up with two cells that each have the same genetic information as the original. So you now have two cells that each have the diploid number of chromosomes. So, 2n and 2n. And now each of these cells are just like this cell was, it can go through interphase again. It grows and it can replicate its DNA and centrosomes and grow some more then each of these can go through mitosis again. And this is actually how most of the cells in your body grow. This is how you turn from a single cell organism into you, or for the most part, into you. So that is mitosis. It's a cycle. After each of these things go through mitosis, they can then go through the entire cell cycle again. Let me write this a little bit neater. Mitosis, that s was a little bit hard to read. Now what happens in meiosis? What happens in meiosis? I'll do that over here. In meiosis, something slightly different happens and it happens in two phases. You will start with a cell that has a diploid number of chromosomes. So you will start with a cell that has a diploid number of chromosomes. And in it's interphase, it also replicates its DNA. And then it goes through something called Meiosis One. And in Meiosis One, what you end up with is two cells that now have haploid number of chromosomes. So you end up with two cells, You now have two cells that each have a haploid number of chromosomes. So you have n and you have n. So if we're talking about human beings, you have 46 chromosomes here, and now you have 23 chromosomes in this nucleus. And now you have 23 in this nucleus. But you're still not done. Then each of these will go through a phase, which I'll talk about in a second, which is very similar to mitosis, which will duplicate this entire cell into two. So actually, let me do it like this. So now, this one, you're going to have four cells that each have the haploid number that each have the haploid number of chromosomes. And they don't all necessarily have the same genetic informatioin anymore. Because as we go through this first phase, right over here of meiosis, and this first phase here you go from diploid to haploid, right over here, this is called Meiosis One. Meiosis One, you're essentially splitting the homologous pairs and so this one might get some of the ones that you originally got from your father, and some that you originally got from your mother, some that you originally got from your father, some that you originally got from your mother, they split randomly, but each homogolous pair gets split up. And then in this phase, Meiosis Two, so this phase right over here is called Meiosis Two, it's very similar to mitosis, except your now dealing with cells that start off with the haploid number. It's important to realize that meiosis is not a cycle. These cells that you have over here, these are gametes. This are sex cells. These are gametes. This can now be used in fertilization. If we're talking about, if you're male, this is happening in your testes, and these are going to be sperm cells If you are female, this is happening in your ovaries and these are going to be egg cells. If you a tree, this could be pollen or it could be an ovul. But these are used for fertilization. These will fuse together in sexual reproduction to get to a fertilized egg, which then can undergo mitosis to create an entirely new organism. So not a cycle here, although these will find sex cells from another organism and fuse with them and those can turn into another organism. And I guess the whole circle of life starts again. But it's not the case with mitosis where this can keep going and going, going. This cell is just like this cell, while these sex cells are differeent than this one right over here. Now, where does this happen in the body? We've talked about this in previous videos. These are your somatic cells right over here. These are the ones that make up the bulk of your body, somatic cells. And where is this happening? Well, this is happening in germ cells, As we mentioned, if you're male it's in your tesis and if you're female it's in your ovaries. And germ cells actually can undergo mitosis to produce other germ cells that have a diploid number of chromosomes, or they can undergo meiosis in order to produce sperm or egg cells in order to produce gametes." + }, + { + "Q": "At 0:16, How would you turn that into a linear equation?", + "A": "Good Question. first take the change of your y and x points. which is y( 7) and x(4). now we need to find the slope. to find the slope lest divide our y difference by our x difference: 7/4 now we have our slope! now so far we have Y=7/4x+b at 2:30 Sal was confirming about the dotted line. as we know we are now trying to find the y-Intercept. looking at the graph we can see that the y-Intercept is -7 so now we get!( Drum roll!) Y=7/4X+(-7) or: Y=7/4X-7 Hope This Helps! =)", + "video_name": "wl2iQAuQl7Y", + "transcript": "f is a linear function whose table of values is shown below. So they give us different values of x and what the function is for each of those x's. Which graphs show functions which are increasing at the same rate as f? So what is the rate at which f is increasing? When x increases by 4, we have our function increasing by 7. So we could just look for which of these lines are increasing at a rate of 7/4, 7 in the vertical direction every time we move 4 in the horizontal direction. And an easy way to eyeball that would actually be just to plot two points for f, and then see what that rate looks like visually. So if we see here when x is 0, f is negative 1. When x is 0, f is negative 1. So when x is 0, f is negative 1. And when x is 4, f is 6, so 1, 2, 3, 4, 5, 6, so just like that. And two points specify a line. We know that it is a linear function. You can even verify it here. When we increase by 4 again, we increase our function by 7 again. We know that these two points are on f and so we get a sense of the rate of change of f. Now, when you draw it like that, it immediately becomes pretty clear which of these has the same rate of change of f. A is increasing faster than f. C is increasing slower. A is increasing much faster than f. C is increasing slower than f. B is decreasing, so that's not even close. But D seems to have the exact same inclination, the exact same slope, as f. So D is what we would go with. And we could even verify it, even if we didn't draw it in this way. Our change in f for a given change in x is equal to-- when x changed plus 4, our function changed plus 7. It is equal to 7/4. And we can verify that on D, if we increase in the x-direction by 4, so we go from 4 to 8, then in the vertical direction we should increase by 7, so 1, 2, 3, 4, 5, 6, 7. And it, indeed, does increase at the exact same rate." + }, + { + "Q": "Can expressions also be polynomials?", + "A": "Yes, a polynomial is just an expression with certain requirements.", + "video_name": "BXHNzUaIRR0", + "transcript": "We're asked to evaluate the expression a squared plus 10b minus 8 when a is equal to 7 and b is equal to 4. So to evaluate the expression, we really just have to substitute a with 7 and substitute b with negative 4 because they're saying evaluate it when a is equal to 7 and b is equal to negative 4. So let's do that. So a everywhere we see an a in the expression, we should put a 7 there. So instead of a squared, we should write 7 squared plus-- I'll do it in that same color-- plus 10 times b. But instead of a b there, we are now going to substitute it with b is equal to negative 4. So 10 times negative 4 instead of the b right over there. And then we have the minus 8. And now we just have to evaluate this thing. 7 squared is 49. And then 10 times negative 4. Remember, order of operations, multiplication comes before addition. So we have to multiply this. 10 times negative 4 is negative 40. So it's negative 40. And then we have minus 8 back over here. And so we get 49 plus negative 40, which is really the same thing as 49 minus 40 is going to be 9. And then we're going to subtract 8 from that. And so we get 1. 49 minus 40 is 9 minus 8 is 1. And we are done. We've evaluated the expression when a is equal to 7 and b is equal to negative 4." + }, + { + "Q": "how many wars did the romans have", + "A": "Lets just say a lot they had battles from 8th century BC all the way to the 4th century and that s a long time.", + "video_name": "kiMNT18c4Ko", + "transcript": "DR. STEVEN ZUCKER: We're standing in the marvelous new museum that was just done by Richard Meier to hold the Ara Pacis, one of the most important monuments from Augustan Rome. DR. BETH HARRIS: Ara Pacis means altar of peace. Augustus was the first emperor of Rome. DR. STEVEN ZUCKER: And the person who established the Pax Romana, that is, the Roman peace. The event that prompted the building of this altar to peace under Augustus was Augustus' triumphal return from military campaigns in what is now Spain and France. DR. BETH HARRIS: And when he returned, the Senate vowed to create an altar commemorating the peace that he established in the empire. DR. STEVEN ZUCKER: And apparently, on July 4 in the year 13, the sacred precinct was marked out on which the altar itself would be built. It's really kind of wonderful because today, it's July 4, 2012. DR. BETH HARRIS: Now we're talking about the Ara Pacis, but of course, this has been reconstructed from many, many fragments that were discovered, some in the 17th century, mostly in the 20th century. DR. STEVEN ZUCKER: Actually, it's a small miracle that we've been able to reconstruct this at all. It had been lost to memory. DR. BETH HARRIS: The remains of it lay under someone's palace. When it was recognized what these fragments were, it became really important to excavate them and to reconstruct the altar. DR. STEVEN ZUCKER: That was finally done under Mussolini, the fascist leader in the years leading up to the Second World War, and during the Second World War. And that was important to Mussolini, because Mussolini identified himself with Augustus, the first emperor of Rome. Mussolini was very much trying to reestablish a kind of Italian empire. We should talk a little bit about what an altar is. We talk about the altar, really what we're looking at are the walls of the precinct around what is in the middle, the altar where sacrifices would have occurred. is interesting and important when we think about Augustus. Augustus is establishing a centralized power. Rome had been, since its earliest founding years when it was under the rule of kings, it had been controlled by the Senate. It had been a republic. DR. BETH HARRIS: That's right. And the Senate was basically a group of the leading elder citizens of Rome. So Rome was a republic, and it really was a republic until Julius Caesar, who was a dictator and Augustus' uncle. And then Caesar is assassinated, there's civil war, and then peace is established by Augustus. DR. STEVEN ZUCKER: Right. Augustus, whose real name was Octavian, is given the term Augustus as a kind of honorific as a way of representing his power. And it's interesting the kind of politics that Augustus involved himself with. He gave great power back to the Senate, but by doing so, he established real and central authority for himself. DR. BETH HARRIS: He made himself princeps, or first among But of course he controlled everything. DR. STEVEN ZUCKER: He also held the title of pontificus maximus, that is, the head priest of the state religion, and so he held tremendous power. DR. BETH HARRIS: Now don't forget, too, that his uncle Julius Caesar had been made a god, and so he also represented himself as the son of a god. DR. STEVEN ZUCKER: And so the idea of establishing this altar has a political as well as spiritual significance. DR. BETH HARRIS: He's looking back to the golden age of Greece of the fifth century BC, but he's also looking back to the Roman republic. He is reestablishing some of the ancient rituals of traditional Roman religion. He is embracing traditional Roman values. DR. STEVEN ZUCKER: But even as he's doing that, he's remaking Rome radically. He's changing Rome from a city of brick to a city of marble, and the Ara Pacis is a spectacular example of that. DR. BETH HARRIS: And when we look closely at the Ara Pacis, what we're going to see is that this speaks to the sense of a golden age that Augustus brought about in the Roman Empire. DR. STEVEN ZUCKER: One of the most remarkable elements of the Ara Pacis is all of the highly decorative relief carving in the lower frieze. DR. BETH HARRIS: And that goes all the way around. It apparently shows more than 50 different species of plants. They're very natural in that we can identify the species, but they're also highly abstracted, and they form these beautiful symmetrical and linear patterns. DR. STEVEN ZUCKER: There is a real order that's given to the complexity of nature here. Let me just describe quickly what I'm seeing. This massive, elegant acanthus leaf, which is a native plant, which were made famous in Corinthian capitals. And then almost like a candelabra growing up from it, we see these tendrils of all kinds of plants that spiral. DR. BETH HARRIS: And there are also animal forms within these leaves and plants. We find frogs and lizards and birds. DR. STEVEN ZUCKER: And the carving is quite deep, so that there is this sharp contrast between the brilliance of the external marble and then the shadows that are cast as it seems to lift off the surface. DR. BETH HARRIS: And art historians interpret all of this as a symbol of fertility, of the abundance of the golden age that Augustus brought about. DR. STEVEN ZUCKER: You also see that same pattern repeated in the pilasters that frame these panels. And then we also have meander that moves horizontally around the entire exterior. And it's above that meander that we see the narrative phrases. DR. BETH HARRIS: These panels relate again to this golden age that Augustus establishes. These refer back to Aeneas, Rome's founder and Augustus' ancestor. We see other allegorical figures representing Rome and peace. DR. STEVEN ZUCKER: We have to be a little bit careful when we try to characterize what precisely is being represented. There are lots of conflicting interpretations. DR. BETH HARRIS: And these allegorical or mythological scenes appear on the front and back of the altar. And then on the sides of the altar we see a procession. DR. STEVEN ZUCKER: We've walked around the outer wall, and we're now looking at a panel that's actually in quite good condition. But that doesn't mean we really know what's going on. DR. BETH HARRIS: No, there's a lot of argument about what the figure in the center represents. Some art historians think this figure represents Venus, some think it represents a figure of peace, some the figure of Tellus, or Mother Earth. In any case, she is clearly a figure that suggests fertility and abundance. DR. STEVEN ZUCKER: She's beautifully rendered. Look at the way the drapery clings to her torso so closely as to really review the flesh underneath, like the goddesses on the Parthenon on the Acropolis in Greece. DR. BETH HARRIS: And on her lap sit two children, one of whom offers her some fruit. There's fruit on her lap. On either side of her sit two mythological figures who art historians think represent the winds of the earth and the sea. DR. STEVEN ZUCKER: Well, look at the way the drapes that they're holding whip up, creating these beautiful almost halos around their bodies. DR. BETH HARRIS: And at her feet we see an ox and a sheep. So there's a sense of harmony, of peace and fertility. DR. STEVEN ZUCKER: And that must have been such a rare thing in the ancient world. DR. BETH HARRIS: Well, Augustus reigns after decades of civil war after the assassination of Julius Caesar. So I think there is a powerful sense that this was the golden age. DR. STEVEN ZUCKER: So, let's walk to the sides now, and take a look at the procession. The frieze moves from the back wall of the precinct up towards the very front on both sides, and the figures are also facing towards the main staircase. DR. BETH HARRIS: Art historians are not really clear what event is being depicted here. DR. STEVEN ZUCKER: Art historians aren't clear about any of this, are we? DR. BETH HARRIS: No. There are a couple of possibilities that have been raised. One is that what we are seeing is the procession that would have taken place at the time that the altar was inaugurated. The figures that we see here are priests, and we can identify those figures because of the veils on their heads, and there also seem to be members of Augustus' family, although their identities are not quite firmly established. DR. STEVEN ZUCKER: We think we know which figure is Augustus, although the marble itself is not in especially good condition, and we've lost the front of his body. And we also think we can identify one of his most important ministers. DR. BETH HARRIS: And that would be Agrippa. If we think about this as looking back to the frieze on the Parthenon from the golden age of Greece, those figures are all ideally beautiful. They don't represent anyone specific so much as the Athenian people generally. DR. STEVEN ZUCKER: But these are portraits. DR. BETH HARRIS: That's right. And we can't always identify them for certain, but they really are specific individuals on a specific date taking part in a specific event. DR. STEVEN ZUCKER: It's interesting to think about it, because of course throughout the republic, portraiture in stone was something that the Romans were extremely good at. And so it doesn't surprise me that they would not look to the idealized so much as look to the specific. DR. BETH HARRIS: We also notice those differences in the depths of the carving. Some figures are represented in high relief. Other figures that are supposed to be in the background are represented in low relief. So there's a real illusion of space and of a crowd, here at the procession. DR. STEVEN ZUCKER: Another way that the specificity of the Romans is expressed is through the inclusion of children. This is a sacred event, and a formal event. And yet there are children doing what children do. That is to say, they're not always paying attention. DR. BETH HARRIS: There are a couple of interpretations that have been offered about the presence of children here. Augustus was actually worried about the birth rate and passed laws that encouraged marriage and the birth of children. It originally was painted. We would have seen pinks and blues and greens, and it's very difficult to imagine that when we look at the marble today. DR. STEVEN ZUCKER: Well, it's true, especially in Meier's building, which is so stark and modern. It's almost a little garish to imagine how brightly painted this would have been. They were pretty bright. DR. BETH HARRIS: They were. So one of the things that Augustus said of himself was that he found Rome a city of brick, and he left it a city of marble. Augustus created an imperial city. And here we are 2,000 years later in the room that Augustus created." + }, + { + "Q": "so you just mutiply the fractions right because he was just checking it.", + "A": "He was trying to show what your are doing by multiplying the fraction.", + "video_name": "hr_mTd-oJ-M", + "transcript": "Let's think a little bit about what it means to multiply fractions. Say I want to multiply 1/2 times 1/4. Well, one way to think about this is we could view this as 1/2 of a 1/4. And what do I mean there? Well let me take a whole, let me take a whole here, and let me divide it into fourths. So let me divide it into fourths, so I'll divided into 4 equal sections. And so 1/4 would be 1 of these 4 equal sections. But we want to take 1/2 of that. So how do we take half of that? Well, we could divide this into 2 equal sections, and then just take 1 of them. So divide it into 2 equal sections, and then take 1 of them. So we're taking this pink area, this whole pink area is 1/4, and now we're going to take 1/2 of it. We're now going to take 1/2 of it. So that's this yellow square right over here. But what fraction of the whole does this yellow represent? Well, it now represents 1 out of 1, 2, 3, 4, 5, 6, 7, 8 equal sections. So this right over here, this represents 1/8 of the whole. And so we see conceptually that 1/2 times 1/4, it completely makes sense, that 1/2 of 1/4 should be 1/8. And it hopefully makes sense that you get this 8 by multiplying the 2 times the 4. You started with 4 equal sections, but then you divided each of those 4 equal sections into 2 equal sections. So then you have 8 total equal sections that you split your whole into. Let's do another example, but now let's multiply two fractions that don't have 1's in the numerator. So let's multiply, let's multiply 2/3 times 4/5. And I encourage you now to pause the video and do something very similar to what I just did. Try to represent 4/5 of a whole and then try to represent 2/3 of that 4/5 and see what fraction of the whole you actually have. So pause now. So let's think about this. Let's represent 4/5. So if I have a whole like this, let me try to divide it into 5 equal sections. 5 equal sections, so let's say that is 1 equal section, that is 2 equal sections, that is 3, 4, and 5-- I can do a better This is always the hard part. I'm trying my best to make them look, at least, like equal sections-- 2, 3, 4, and 5. I think you get the point here. I'm trying to make them equal sections. And we want 4/5. So we want 4 of these 5 equal sections. So this would be 1 of the 5 equal sections, 2 of them, 3 of them, and then 4 of them. So that right over there is 4/5. Now we can view this as 2/3 of the 4/5. So how can we think about that? Well, we could take this section and divide it into thirds. So let's do that. Divide it into thirds. So we're going divide it into 3 equal sections. So that's 1/3, and then 2/3. So we took each of the 5 equal sections, and we divided them into 3 equal sections. Now what's going to be 2/3 of the 4/5? Well, that's going to be this part right over here. So let me make this clear. This is 1/3 of the 4/5. And then this would be 2/3 of the 4/5. So this right over here, would be 2/3 of the 4/5, or 2/3 times 4/5. But what fraction of the whole does that represent? Well, how many total, how many total equal sections do we now have? Well, we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. So we have 15 equal sections. I'm using a new color. We have 15 equal sections, and that make sense. We started with 5 equal sections, but then we divided each of those into 3 equal sections. So now we have 5 times 3 total equal sections. And then how many of those are now colored in? Well, we see it's 2 times 4. 1, 2, 3, 4, 5, 6, 7, 8. How many of them are in the 2/3 of the 4/5, I should say. And there's 8 of them, 8 of the 15 equals sections. And so there you have it. It should hopefully now make visual sense, or it makes conceptual sense, that 2/3 times 4/5-- you can obviously compute it by just multiplying the numerators, 2 times 4 is 8. And then multiplying the denominators, 3 times 5 is 15-- but hopefully this now makes conceptual sense as 2/3 of 4/5." + }, + { + "Q": "What if he stated \"There were sheep on the hill\". Would the statement still be correct?", + "A": "That is a correct statement. Maybe you re getting confused because the singular and plural form of sheep is the same. It can either mean one sheep or more than one sheep depending on the context. In your sentence, we can tell more than one sheep are on the hill.", + "video_name": "UnJmPywSSvg", + "transcript": "- [Voiceover] Hello grammarians. I wanted to talk today about a different kind of irregular plural. So we've been talking about regular plurals where you take a word and you add an S. So for example, the word dog becomes dogs. You add an S. And this is the regular plural here. But I've been talking about the irregular plural, the plural, the multiple form of a verb that is not regular, irregular. But today I figured we'd talk about something called the base plural. Which I will illustrate for you using our friend the sheep. Now, sheep is a very strange word in that it doesn't matter whether not there's more than one of them. The form of the word always looks the same whether it's one sheep or two sheep. It's an irregular plural you don't add an S to. This is called a base plural 'cause the base sheep, the thing that you would normally add this particle S to doesn't change whether it's singular sheep or plural sheep. So that's you know. There was one sheep... on the hill. There's a sentence. What if we put another little baby sheep on that hill? A little lamb. Well now the sentence looks like this. Two sheep... on the hill. Now the only difference between these two sentences is that there's one sheep and two sheep and therefore that means that the verb changes to a plural conjugation. So there was one sheep. There were two sheep on the hill. But everything else stays exactly the same. One sheep, two sheep. This is very strange, it's a base plural. So in standard English, the form is two sheep... and not two sheeps. Now, there are more words that do behave this way. So let's go investigate. So there are a small number of words that also behave this way, the way sheep does, these weird sheep plurals, these base plurals. One of them is fish. So you could say the fish are plentiful this season, but you could also say, you know, the fish... is delicious. You could say the bison migrate west or you could also say the bison migrates west indicating a single bison, you see. Bison can be singular or plural. Fish can be singular or plural. As is so frequently the case, there is a special exception regarding the word fishes, which you may have heard before, and fishes is a word that we would use when we're talking about individual species of fish. And fish is the word that we would use to refer to individual fish. So let's say your uncle Marty is a prodigious fisherman and he catches, he goes fly fishing one weekend, he comes back, he has 30 fish. Marty caught... 30 fish. But let's say on the other hand your aunt Marta is a prodigious marine biologist and she studies 30 different types of fish. You would say Marta studies... 30 fishes. And that doesn't mean that she studies 30 individual fish. That means she studies 30 types of fish. That's the difference. Fishes is referring to species. Fish refers to individuals. That's how you'd use them in the plural. So to review, there's this entire class of words called base plurals where the word itself, the base, doesn't take an S for the plural, it's just the same. The singular is the same as the plural. So that gives us words like sheep, fish, and bison. There aren't a ton of English words that behave this way where the plural is the same as the singular. I just wanted to make you aware of some of the most common ones. There are also more examples in the exercises. So I just wanted you to be aware of them. You can learn anything. David out." + }, + { + "Q": "If both C's are \"equally\" substituted but one has an ethyl and the other a methyl, which will the H2O attack?", + "A": "You can consider the Br\u00e2\u0081\u00ba in the cyclic bromonium ion to be an enhanced leaving group. So the reaction will be similar to an SN1 reaction, and the carbon that forms the more stable carbocation will be the one that is preferentially attacked. An ethyl group provides a little more electron density and stabilization than a methyl group (for example, propionic acid is weaker than acetic acid). You will get attack on both carbons, but the major product will have the OH on the carbon with the ethyl group.", + "video_name": "FaOOx6IZxV8", + "transcript": "Here's the general reaction to make halohydrins from alkenes. So if I start with my alkene on the left, and I add a halogen to it and some water, you can see that an OH and a halogen are added anti to each other. So anti, or on opposite sides of where the double bond used to be. The mechanism for this reaction starts off the exact same way the halogenation reaction did. And so we have our halogen approaching our alkene. And we saw in that video that the halogen is usually, of course, nonpolar, because those two atoms have the exact same electronegativity. So these blue electrons in here are pulled with equal force to either halogen, so it's a nonpolar molecule. However, if the pi electrons in my alkene here get too close to the electrons in blue, we saw how that could induce a temporary dipole on the halogen molecule. So those electrons in blue are repelled by the electrons in magenta and push closer to the top halogen, which gives the top halogen a partial negative charge and leaves the bottom halogen with a partial positive charge. The bottom halogen is now an electrophile, so it wants electrons. It's going to get electrons from those pi electrons here, which are going to move out and nucleophilic attack that partially positively charged halogen atom. And then this lone pair of electrons is going to form a bond with this carbon at the same time these blue electrons move out onto the halogen. So when we draw the result of all those electrons moving around, we're going to form a bond between the carbon on the right and the halogen, and we use the magenta electrons to show that. So there's now a bond there. And so we used red electrons before to show these electrons in here forming a bond with the carbon on the left. That halogen had two lone pairs of electrons still on it, like that, which gives that halogen a plus 1 formal charge. We called this our cyclic halonium ion in an earlier video. And if I think about that cyclic halonium ion, I think about the halogen being very electronegative. It's going to attract, I'll say the electrons in magenta again just to be consistent, closer towards it. So it's going to take away a little bit of electron density from this carbon right down here. So I'm going to say this carbon is partially positive. It's going to have some partial carbocationic character. So in the next step of the mechanism, water's going to come along. And water's going to function as a nucleophile. So one of the lone pairs of electrons on water is going to nucleophilic attack our electrophile, which is this carbon right here. And so when that lone pair of electrons on oxygen attacks this carbon, that's going to kick the electrons in magenta off onto your halogen. And so let's go ahead and draw the product. We're going to have, on the left carbon, this halogen now used to have two lone pairs of electrons. It picked up the ones in magenta, so now it looks like that. On the right, we still have the carbon on the right bonded to other things, except now it's bonded to what used to be our water molecule. So the oxygen is now bonded to the carbon. And there's still one lone pair of electrons on that oxygen, giving it a plus 1 formal charge. So let's go ahead and highlight these electrons here in blue. Those electrons in blue are the ones that formed this bond between the carbon and the oxygen. So we're almost done. The last step of the mechanism would just be an acid-base reaction. So another water molecule comes along, and one of the lone pairs of electrons on the water molecule is going to function as a base and take this proton, leaving these two electrons behind on the oxygen. And we are finally done. We have formed our halohydrin, right? So I have my halogen on one side. And then I now have my OH on the opposite side, like that. Let's go ahead and do an example so we can examine the stereochemistry a little bit more here. So if I start with an alkene, and to this alkene we are going to add bromine and water. And we're going to think about doing this two different ways here. So we'll start with the way on the left. So Br2 and H2O. And then we'll come back and we'll go ahead and do this on the right. So BR2 and H2O. So on the left side, I'm going to think about the formation of that bromonium ion here. So I'm going to once again look at this molecule a little bit from above, so looking down. And I'm going to say the bromonium ion is going to form this way. So the bromine's going to form on top here. And so there's going to be two lone pairs of electrons It's going to have a plus 1 formal charge. And if I look at this carbon right here, that's this carbon. So I'm going to say that my methyl group is now going down in space. So with the addition of my bromonium ion, that would be my intermediate. And so now, when I think about water coming along and acting as a nucleophile, so here is H2O, and I think about which carbon will the oxygen attack? So I have two options, right? This oxygen could attack the carbon on the left, or it could attack the carbon on the right. It's been proven that the option is going to attack the most substituted carbon. So if I look at the carbon on the left, and if I think about what sort of carbocation would that be, the carbon on the left is bonded to two other carbons. So this would be similar to a secondary carbocation, or a partial carbocation in character. So you could think about it as being like a partial secondary carbocation on the left here, if this was a partial positive. Or on the right, if I think about this carbon right here, the one in red. And if I think about that being a carbocation, that would be bonded to one, two, three other carbons. So it's like a tertiary carbocation. And we know that tertiary carbocations are more stable than secondary. So even though this isn't a full carbocation, this carbon in red exhibits some partial carbocation character, and that is where our water is going to attack. So the nucleophile is going to attack the electrophile. And it's more stable for it to attack this one on the right, since it has partial carbocation character similar to a tertiary carbocation. And if it attacks that carbon on the right, these electrons here we kick off onto the bromine. So let's go ahead and draw the results of that nucleophilic attack. OK, so what would we have here? Let's go ahead and draw our ring. And the bromine is going to swing over to the carbon on the left. It's now going to have three lone pairs of electrons around it, like that. And the methyl group that was down relative to the plane is going to be pushed up when that water nucleophilic attacks. And now the methyl group is up, and this oxygen is now going to be bonded to this carbon. And so we still have our two hydrogens attached to it, like that. And there's a lone pair of electrons on this oxygen, giving it a plus 1 formal charge. So once again, let's go ahead and highlight those electrons. I'll draw them in blue here. These electrons right here, those are the ones that formed this bond. So let's go back and let's think about the formation of that cyclic bromonium ion in a different way here. So on the left I showed the bromine adding from the top. If I think about the alkene portion of my starting material, well, there's a chance that the bromonium ion could form from below that plane as well. So let's go ahead and draw the result of that over here on the right. So I'm showing another bromonium ion that is possible. And this time the bromine is going to add from below the plane, like that. It's going to get two lone pairs of electrons and have a plus 1 formal charge just like usual. And I will say that this is the carbon that has the methyl on it, OK? So let's go ahead and draw that in as well. So let's see, I'll put it like that. So that's my CH3. And this time, when I think about where will water attack-- so let's go ahead and think about water as my nucleophile-- it's the same idea if I compare the carbons on either side. So I compare this carbon with this carbon. It's the carbon on the right that's going to be the most stable partial carbocation, right? So I'll draw a partial carbocation here. It would be the most stable one. So that's where my nucleophile is going to attack. So I can think about this lone pair of electrons on oxygen that are going to attack right here, which would kick these electrons off onto the bromine. So let's go ahead and draw the results of that nucleophilic attack. So let's see what that would look like. So I have my ring like that. And now I'm going to have my oxygen bonded here. It still has two hydrogens attached to it. It has a lone pair of electrons on it, which give it a plus 1 formal charge. And when the oxygen attacked, that is going to push down this methyl group. So this methyl group is going to be pushed down relative to the plane of the ring. So now we have a methyl group down at this carbon. And the bromine is going to swing over to the carbon on the left. And so that's the position of my bromine now. And so, in the last step-- now I have these two guys right here-- they're both going to lose a proton in the next step, right? So it's an acid-base reaction. And we could show water coming along. So for the molecule on the left, water comes along. And water's going to take this proton. These electrons are going to kick off on to that oxygen. And we can draw that product. So if we were to draw that product, we would look down this way. And we would treat this as being the top carbon here. So there's a methyl group going up at that carbon. So I can say that this is going to, after it loses a proton, so that carbon in blue is going to have a methyl group up relative to the plane of the ring. And it's going to have an OH group down. So this OH is going to be down relative to the plane. So I can go ahead and put OH going away from me in space. And then this bromine over here. This bromine is going to be coming out at me. So I can go ahead and show a wedge for that bromine. So that's one of our possible products. Over here on the right, if this is what happens on the right, I do the same thing. I put my eye right here and I stare down, with this carbon being the top carbon. So I can go ahead and draw my cyclohexane ring. And I can see that this time my OH will be coming out at me after it loses a proton. So if I wanted to, I could go ahead and draw water in here and show the last step of the mechanism. Lone pair of electrons taking this proton, leaving these electrons behind, giving me an OH coming out at me in space. And then this methyl group would therefore be going away from me in space. So I can go ahead and show that methyl group as a dash. And then finally, this bromine over here would be going away from me. So that would be a dash on my ring, like that. When I finally get to my two products, I can analyze them. And I can see that they are enantiomers to each other. So they're different molecules. And we can look at the absolute configurations really fast. I can see that I have a carbon coming out at me on a wedge and an oxygen going away from me. And it's been reversed over here on the right. This time the oxygen is coming out at me on a wedge, and the carbon is going away from me on a dash. When I look at the bromine, it's coming out at me on the left, and the bromine's going away from me on the right. So I can see that I have different absolute configurations at both chirality centers. And so these two will be enantiomers to each other." + }, + { + "Q": "What is the relationship between acetyl CoA and the intermediates in the Krebs cycle? What are the effects of depleted intermediates on the oxydation of acetyl CoA?", + "A": "two answer the first question, Acetyl CoA donates an acetyl group to oxaloacetate to make citrate, an intermediate in the Krebs cycle. Since it is a cycle, I presume that depleted intermediates would eventually lead to a lack of oxaloacetate to oxidise acetyl CoA, leading to a buildup of the unoxidised molecule. Hope that helps! Ryan, 2nd biochemistry", + "video_name": "juM2ROSLWfw", + "transcript": "So we already know that if we start off with a glucose molecule, which is a 6-carbon molecule, that this essentially gets split in half by glycolysis and we end up 2 pyruvic acids or two pyruvate molecules. So glycolysis literally splits this in half. It lyses the glucose. We end up with two pyruvates or pyruvic acids. ruby And these are 3-carbon molecules. There's obviously a lot of other stuff going on in the carbons. You saw it in the past. And you could look up their chemical structures on the internet or on Wikipedia and But this is kind of the important thing. Is that it was lysed, it was cut in half. And this is what happened in glycolysis. And this happened in the absence of oxygen. Or not necessarily. It can happen in the presence or in the absence of oxygen. It doesn't need oxygen. And we got a net payoff of two ATPs. And I always say the net there, because remember, it used two ATPs in that investment stage, and then it generated four. So on a net basis, it generated four, used two, it gave us two ATPs. And it also produced two NADHs. That's what we got out of glycolysis. And just so you can visualize this a little bit better, let me draw a cell right here. Maybe I'll draw it down here. Let's say I have a cell. That's its outer membrane. Maybe its nucleus, we're dealing with a eukaryotic cell. That doesn't have to be the case. It has its DNA and its chromatin form all spread around like that. And then you have mitochondria. And there's a reason why people call it the power We'll look at that in a second. So there's a mitochondria. It has an outer membrane and an inner membrane just like that. I'll do more detail on the structure of a mitochondria, maybe later in this video or maybe I'll do a whole video on them. That's another mitochondria right there. And then all of this fluid, this space out here that's between the organelles-- and the organelles, you kind of view them as parts of the cell that do specific things. Kind of like organs do specific things within our own bodies. So this-- so between all of the organelles you have this fluidic space. This is just fluid of the cell. And that's called the cytoplasm. And that's where glycolysis occurs. So glycolysis occurs in the cytoplasm. Now we all know-- in the overview video-- we know what the next step is. The Krebs cycle, or the citric acid cycle. And that actually takes place in the inner membrane, or I should say the inner space of these mitochondria. Let me draw it a little bit bigger. Let me draw a mitochondria here. So this is a mitochondria. It has an outer membrane. It has an inner membrane. If I have just one inner membrane we call it a crista. If we have many, we call them cristae. This little convoluted inner membrane, let me give it a label. So they are cristae, plural. And then it has two compartments. Because it's divided by these two membranes. This compartment right here is called the outer compartment. This whole thing right there, that's the outer compartment. And then this inner compartment in here, is called the matrix. Now you have these pyruvates, they're not quite just ready for the Krebs cycle, but I guess-- well that's a good intro into how do you make them ready for the Krebs cycle? They actually get oxidized. And I'll just focus on one of these pyruvates. We just have to remember that the pyruvate, that this happens twice for every molecule of glucose. So we have this kind of preparation step for the Krebs Cycle. We call that pyruvate oxidation. And essentially what it does is it cleaves one of these carbons off of the pyruvate. And so you end up with a 2-carbon compound. You don't have just two carbons, but its backbone of carbons is just two carbons. Called acetyl-CoA. And if these names are confusing, because what is acetyl coenzyme A? These are very bizarre. You could do a web search on them But I'm just going to use the words right now, because it will keep things simple and we'llget the big picture. So it generates acetyl-CoA, which is this 2-carbon compound. And it also reduces some NAD plus to NADH. And this process right here is often given credit-- or the Krebs cycle or the citric acid cycle gets credit for this step. But it's really a preparation step for the Krebs cycle. Now once you have this 2-carbon chain, acetyl-Co-A right here. you are ready to jump into the Krebs cycle. This long talked-about Krebs cycle. And you'll see in a second why it's called a cycle. Acetyl-CoA, and all of this is catalyzed by enzymes. And enzymes are just proteins that bring together the constituent things that need to react in the right way so that they do react. So catalyzed by enzymes. This acetyl-CoA merges with some oxaloacetic acid. A very fancy word. But this is a 4-carbon molecule. These two guys are kind of reacted together, or merged together, depending on how you want to view it. I'll draw it like that. It's all catalyzed by enzymes. And this is important. Some texts will say, is this an enzyme catalyzed reaction? Everything in the Krebs cycle is an enzyme catalyzed reaction. And they form citrate, or citric acid. Which is the same stuff in your lemonade or your orange juice. And this is a 6-carbon molecule. You have a 2-carbon and a 4-carbon. You get a 6-carbon molecule. And then the citric acid is then oxidized over a bunch of steps. And this is a huge simplification here. But it's just oxidized over a bunch of steps. Again, the carbons are cleaved off. Both 2-carbons are cleaved off of it to get back to oxaloacetic acid. And you might be saying, when these carbons are cleaved off, like when this carbon is cleaved off, what happens to it? It becomes CO2. It gets put onto some oxygen and leaves the system. So this is where the oxygen or the carbons, or the carbon dioxide actually gets formed. And similarly, when these carbons get cleaved off, it forms CO2. And actually, for every molecule of glucose you have six carbons. When you do this whole process once, you are generating three molecules of carbon dioxide. But you're going to do it twice. You're going to have six carbon dioxides produced. Which accounts for all of the carbons. You get rid of three carbons for every turn of this. Well, two for every turn. But really, for the steps after glycolysis you get rid of three carbons. But you're going to do it for each of the pyruvates. You're going to get rid of all six carbons, which will have to exhale eventually. But this cycle, it doesn't just generate carbons. The whole idea is to generate NADHs and FADH2s and ATPs. So we'll write that here. And this is a huge simplification. I'll show you the detailed picture in a second. We'll reduce some NAD plus into NADH. We'll do it again. And of course, these are in separate steps. There's intermediate compounds. I'll show you those in a second. Another NAD plus molecule will be reduced to NADH. It will produce some ATP. Some ADP will turn into ATP. Maybe we have some-- and not maybe, this is what happens-- some FAD gets-- let me write it this way-- some FAD gets oxidized into FADH2. And the whole reason why we even pay attention to these, you might think, hey cellular respiration is all about ATP. Why do we even pay attention to these NADHs and these FADH2s that get produced as part of the process? The reason why we care is that these are the inputs into the electron transport chain. These get oxidized, or they lose their hydrogens in the electron transport chain, and that's where the bulk of the ATP is actually produced. And then maybe we'll have another NAD get reduced, or gain in hydrogen. Reduction is gaining an electron. Or gaining a hydrogen whose electron you can hog. NADH. And then we end up back at oxaloacetic acid. And we can perform the whole citric acid cycle over again. So now that we've written it all out, let's account for what we have. So depending on-- let me draw some dividing lines so we know what's what. So this right here, everything to the left of that line right there is glycolysis. We learned that already. And then most-- especially introductory-- textbooks will give the Krebs cycle credit for this pyruvate oxidation, but that's really a preparatory stage. The Krebs cycle is really formally this part where you start with acetyl-CoA, you merge it with oxaloacetic acid. And then you go and you form citric acid, which essentially gets oxidized and produces all of these things that will need to either directly produce ATP or will do it indirectly in the electron transport chain. But let's account for everything that we have. Let's account for everything that we have so far. We already accounted for the glycolysis right there. Two net ATPs, two NADHs. Now, in the citric acid cycle, or in the Krebs cycle, well first we have our pyruvate oxidation. That produced one NADH. But remember, if we want to say, what are we producing for every glucose? This is what we produced for each of the pyruvates. This NADH was from just this pyruvate. But glycolysis produced two pyruvates. So everything after this, we're going to multiply by two for every molecule of glucose. So I'll say, for the pyruvate oxidation times two means that we got two NADHs. And then when we look at this side, the formal Krebs cycle, what do we get? We have, how many NADHs? One, two, three NADHs. So three NADHs times two, because we're going to perform this cycle for each of the pyruvates produced from glycolysis. So that gives us six NADHs. We have one ATP per turn of the cycle. That's going to happen twice. Once for each pyruvic acid. So we get two ATPs. And then we have one FADH2. But it's good, we're going to do this cycle twice. This is per cycle. So times two. We have two FADHs. Now, sometimes in a lot of books these two NADHs, or per turn of the Krebs cycle, or per pyruvate this one NADH, they'll give credit to the Krebs cycle for that. So sometimes instead of having this intermediate step, they'll just write four NADHs right here. And you'll do it twice. Once for each puruvate. So they'll say eight NADHs get produced from the Krebs cycle. But the reality is, six from the Krebs cycle two from the preparatory stage. Now the interesting thing is we can account whether we get to the 38 ATPs promised by cellular respiration. We've directly already produced, for every molecule of glucose, two ATPs and then two more ATPs. So we have four ATPs. Four ATPs. How many NADHs do we have? 2, 4, and then 4 plus 6 10. We have 10 NADHs. And then we have 2 FADH2s. I think in the first video on cellular respiration I said FADH. It should be FADH2, just to be particular about things. And these, so you might say, hey, where are our 38 ATPs? We only have four ATPs right now. But these are actually the inputs in the electron transport chain. These molecules right here get oxidized in the electron transport chain. Every NADH in the electron transport chain produces three ATPs. So these 10 NADHs are going to produce 30 ATPs in the electron transport chain. And each FADH2, when it gets oxidized and gets turned back into FAD in the electron transport chain, will produce two ATPs. So two of them are going to produce four ATPs in the electron transport chain. So we now see, we get four from just what Glycolysis, the preparatory stage and the Krebs or citric acid cycle. And then eventually, these outputs from glycolysis and the citric acid cycle, when they get into the electron transport chain, are going to produce another 34. So 34 plus 4, it does get us to the promised 38 ATP that you would expect in a super-efficient cell. This is kind of your theoretical maximum. In most cells they really don't get quite there. But this is a good number to know if you're going to take the AP bio test or in most introductory biology courses. There's one other point I want to make here. Everything we've talked about so far, this is carbohydrate metabolism. Or sugar catabolism, we could call it. We're breaking down sugars to produce ATP. Glucose was our starting point. But animals, including us, we can catabolize other things. We can catabolize proteins. We can catabolize fats. If you have any fat on your body, you have energy. In theory, your body should be able to take that fat and you should be able to do things with that. You should be able to generate ATP. And the interesting thing, the reason why I bring it up here, is obviously glycolysis is of no use to these things. Although fats can be turned into glucose in the liver. But the interesting thing is that the Krebs cycle is the entry point for these other catabolic mechanisms. Proteins can be broken down into amino acids, which can be broken down into acetyl-CoA. Fats can be turned into glucose, which actually could then go the whole cellular respiration. But the big picture here is acetyl-CoA is the general catabolic intermediary that can then enter the Krebs cycle and generate ATP regardless of whether our fuel is carbohydrates, sugars, proteins or fats. Now, we have a good sense of how everything works out right now, I think. Now I'm going to show you a diagram that you might see in your biology textbook. Or I'll actually show you the actual diagram from Wikipedia. I just want to show you, this looks very daunting and very confusing. And I think that's why many of us have trouble with cellular respiration initially. Because there's just so much information. It's hard to process what's important. But I want to just highlight the important steps here. Just so you see it's the same thing that we talked about. From glycolysis you produce two pyruvates. That's the pyruvate right there. They actually show its molecular structure. This is the pyruvate oxidation step that I talked about. The preparatory step. And you see we produce a carbon dioxide. And we reduce NAD plus into NADH. Then we're ready to enter the Krebs cycle. The acetyl-CoA and the oxaloacetate or oxaloacetic acid, they are reacted together to create citric acid. They've actually drawn the molecule there. And then the citric acid is oxidized through the Krebs cycle right there. All of these steps, each of these steps are facilitated by enzymes. And it gets oxidized. But I want to highlight the interesting parts. Here we have an NAD get reduced to NADH. We have another NAD get reduced to NADH. And then over here, another NAD gets reduced to NADH. So, so far, if you include the preparatory step, we've had four NADHs formed, three directly from the Krebs cycle. That's just what I told you. Now we have, in this diagram they say GDP. GTP gets formed from GDP. The GTP is just guanosine triphosphate. It's another purine that can be a source of energy. But then that later can be used to form an ATP. So this is just the way they happen to draw it. But this is the actual ATP that I drew in the diagram on the top. And then they have this Q group. And I won't go into it. And then it gets reduced over here. It gets those two hydrogens. But that essentially ends up reducing the FADH2s. So this is where the FADH2 gets produced. So as promised, we produced, for each pyruvate that inputted-- remember, so we're going to do it twice-- for each pyruvate we produced one, two, three, four NADHs. We produced one ATP and one FADH2. That's exactly what we saw up here. I'll see you in the next video." + }, + { + "Q": "Why do you need to have one side of the equation equal 0 to solve the equation?", + "A": "The Zero Factor Principle tells you that at least one of the factors must equal zero in your equation. So if one side equals zero, you can then make both sides zero because that makes it easy and quicker to solve.", + "video_name": "STcsaKuW-24", + "transcript": "The height of a triangle is four inches less than the length of the base. The area of the triangle is 30 inches squared. Find the height and base. Use the formula area equals one half base times height for the area of a triangle. OK. So let's think about it a little bit. We have the-- let me draw a triangle here. So this is our triangle. And let's say that the length of this bottom side, that's the base, let's call that b. And then this is the height. This is the height right over here. And then the area is equal to one half base times height. Now in this first sentence they tell us at the height of a triangle is four inch is less than the length of the base. So the height is equal to the base minus 4. That's what that first sentence tells us. The area of the triangle is 30 inches squared. So if we take one half the base times the height we'll get 30 inches squared. Or we could say that 30 inches squared is equal to one half times the base, times the height. Now instead of putting an h in for height, we know that the height is the same thing as 4 less than the base. So let's put that in there. 4 less than the base. And then let's see what we get here. We get-- let me do this in yellow. We get 30 is equal to one half times-- let's distribute the b-- times b, let me make it clear. So let's do it this way. Times b over 2, times b, minus 4. I just multiplied the one half times the b. Now let's distribute the b over 2. So 30 is equal to b squared over 2, be careful. b over 2 times b is just b squared over 2. And then b over 2, times negative 4 is negative 2b. Now just to get rid of this fraction here let's multiply both sides of this equation by 2. So let's multiply that side by 2. And let's multiply that side by 2. On the left hand side you get 60. On the right hand side 2 times b squared over 2 is just b squared. Negative 2b times 2 is negative 4b. And now we have a quadratic here. And the best way to solve a quadratic-- we have a second degree term right here-- is to get all of the terms on one side of the equation, having them equal 0. So let's subtract 60 from both sides of this equation. And we get 0 equal to b squared, minus 4b, minus 60. And so what we need to do here is just factor this thing right now, or factor it. And then, no-- if I have the product of some things, and that equals 0, that means that either one or both of those things need to be equal to 0. So we need to factor b squared, minus 4b, minus 60. So what we want to do, we want to find two numbers whose sum is negative 4 and whose product is negative 60. Now, given that the product is negative, we know there are different signs. And this tells us that their absolute values are going to be four apart. That one is going to be four less than the others. So you could look at the products of the factors of 60. 1 and 60 are too far apart. Even if you made one of the negative, you would either get positive 59 as the sum or negative 59 as the sum. 2 and 30, still too far apart. 3 and 20, still too far apart. If you had made one negative you'd either get negative 17 or positive 17. Then you could have 4 and 15, still too far apart. If you made one of them negative, their sum would be either negative 11 or positive 11. Then you have 5 and a 12, still seems too far apart One of them is negative, then you either have their sum being positive 7 or negative 7. Then you have 6 and 10. Now this looks interesting. They are four apart. So if we make-- and we want the larger absolute magnitude number to be negative so that their sum is negative. So if we make it 6 and negative 10 their sum will be negative 4, and their product is negative 60. So that works. So you could literally say that this is equal to b plus 6, times b, minus 10. b plus the a, plus b minus the b. And let me be very careful here. This b over here, I want to make it very clear, is different than the b that we're using in the equation. I just used this b here to say, look, we're looking for two numbers that add up to this second term It's a different b. I could have said x plus y is equal to negative 4, and x times y is equal to negative 60. In fact, let me do it that way just so we don't get confused. So we could write x plus y is equal to negative 4. And then we have x times y is equal to negative 60. So we have b plus 6, times b plus y. x is 6, y is negative 10. And that is equal to 0. Let's just solve this right here. And then we'll go back and show you. You could also factor this by grouping. But just from this, we know that either one of these is equal to zero. Either b plus 6 is equal to 0, or b minus 10 is equal to 0. If we subtract 6 from both sides of this equation, we get b is equal to negative 6. Or if you add 10 to both sides of this equation, you get b is equal to 10. And those are our two solutions. You could put them back in and verify that they satisfy our constraints. Now the other way that you could solve this, and we're going to get exact same answer. Is you could just break up this negative 4b into its constituents. So you could have broken this up into 0 is equal to b squared. And then you could have broken it up into plus 6b, minus 10b, minus 60. And then factor it by grouping. Group these first two terms. Group these second two terms. Just going to add them together. The first one you could factor out a b. So you have b times b, plus 6. The second one you can factor out a negative 10. So minus 10 times b, plus 6. All that's equal to 0. And now you can factor out a b plus 6. So if you factor out a b plus 6 here, you get 0 is equal to b minus 10, times b, plus 6. We're literally just factoring out this out of the expression. You're just left with a b minus 10. You get the same thing that we did in one step over here. Whatever works for you. But either way, the solutions are either b is equal to negative 6, or b is equal to 10. And we have to be careful here. Remember, this is a word problem. We can't just state, oh b could be negative 6 or b could be 10. We have to think about whether this makes sense in the context of the actual problem. We're talking about lengths of triangles, or lengths of the sides of triangles. We can't have a negative length. So because of that, the base of a triangle can't have length of negative 6. So we can cross that out. So we actually only have one solution here. Almost made a careless mistake. Forgot that we were dealing with the word problem. The only possible base is 10. And let's see, they say find the height and the base. Once again, done. So the base we're saying is 10. The height is four inches less. It's b minus 4. So the height is 6. And then you can verify. The area is 6 times 10 times one half, which is 30." + }, + { + "Q": "can verocity and speed be used intechangebly", + "A": "Speed is a scalar and tells us how fast we are going. Velocity is a vector and tells us how fast we are going (speed) AND direction. While they are similar, the best practice is not to use them interchangeably.", + "video_name": "MAS6mBRZZXA", + "transcript": "The goal of this video is to explore some of the concept of formula you might see in introductional physics class but more importantly to see they are really just common sense ideas So let's just start with a simple example Let's say that and for the sake of this video keep things that magnitudes and velocities that's the direction of velocity etc. let's just assume that if I have a positive number that it means for example postive velocity that it means I'm going to the right let's say I have a negative number we won't see in this video let's assume we are going to the left In that way I can just write a number down only operating in one dimension you know that by specifying the magnitude and the direction if I say velocity is 5m/s that means 5m/s to the right if I say negative 5m/s that means 5m/s to the left let's say just for simplicitiy, say that we start with initial velocity we start with an initial velocity of 5m/s once again I specify the magnitude and the direction because of this convention here, we know it is to the right let's say we have a constant acceleration we have a constant acceleration 2m/s^2 or 2m per second square and once again since this is positive it is to the right and let's say that we do this for a duration so my change in time, let's say we do this for a duration of 4 I will just use s, second and s different places so s for this video is seconds So I want to do is to think about how far do we travel? and there is two things how fast are we going? after 4 seconds and how far have we travel over the course of those 4 seconds? so let's draw ourselves a little diagram here So this is my velocity axis, and this over here is time axis we have to draw a straighter line than that So that is my time axis, time this is velocity This is my velocity right over there and I'm starting off with 5m/s, so this is 5m/s right over here So vi is equal to 5m/s And every second goes by it goes 2m/s faster that's 2m/s*s every second that goes by So after 1 second when it goes 2m/s faster it will be at 7 another way to think about it is the slope of this velocity line is my constant accleration, my constant slope here so it might look something like that So what has happend after 4s? So 1 2 3 4 this is my delta t So my final velocity is going to be right over there I'm writing it here because this get into the way of veloctiy so this is v this is my final velocity what would it be? Well I'm starting at 5m/s So we are doing this both using the variable and concretes Some starting with some initial velocity I'm starting with some initial velocity Subscript i said i for initial and then each second that goes by I'm getting this much faster so if I gonna see how much faster have I gone I multiply the number of second, I will just multiply the number second it goes by times my acceleration, times my acceleration and once again, this right here, subscript c saying that is a constant acceleration, so that will tell my how fast I have gone If I started at this point and multiply the duration time with slope I will get this high, I will get to my final velocity just to make it clear with the numbers, this number can really be anything I'm just taking this to make it concrete in your mind you have 5m/s plus 4s plus, I wanna do it in yellow plus 4s times our acceleration with 2m per second square and what is this going to be equal to? you have a second that is cancelling out one of the second down here You have 4 times, so you have 5m/s plus 4 times 2 is 8 this second gone, we just have 8m/s or this is the same thing as 13m/s which is going to be our final velocity and I wanna take a pause here, you can pause and think about it yourself this whole should be intuitive, we are starting by going with 5m/s every second goes by we are gonna going 2m/s faster so after 1s it would be 7 m/s, after 2s we will be 9m/s after 3s we will be 11m/s, and after 4s we will be at 13 m/s so you multiply how much time pass times acceleration this is how much faster we are gonna be going, we are already going 5m/s 5 plus how much faster? 13 m/s so this right up here is 13m/s So I will take a little pause here hopefully intuitive and the whole play of that is to show you this formula you will see in many physics book is not something that randomly pop out of there it just make complete common sense Now the next thing I wanna talk about is what is the total distance that we would have travel? and we know from the last video that distance is just the area under this curve right over here, so it's just the area under this curve you see this is kind of a strange shape here how do I caculate this area? and we can use a little symbol of geometry to break it down into two different areas, it's very easy to calculate their areas two simple shapes, you can break it down to two, blue part is the rectangle right over here, easy to figure out the area of a rectangle and we can break it down to this purple part, this triangle right here easy to figure out the area of a triangle and that will be the total distance we travel even this will hopefully make some intuition because this blue area is how far we would have travel if we are not accelerated, we just want 5m/s for 4s so you goes 5m/s 1s 2s 3s 4s so you are going from 0 to 4 you change in time is 4s so if you go 5m/s for 4s you are going to go 20 m this right here is 20m that is the area of this right here 5 times 4 this purple or magentic area tells you how furthur than this are you going because you are accelerating because kept going faster and faster and faster it's pretty easy to calculate this area the base here is still 5(4) because that's 5(4) second that's gone by what's the height here? The height here is my final velocity minus my initial velocity minus my initial velocity or it's the change in velocity due to the accleration 13 minus 5 is 8 or this 8 right over here it is 8m/s so this height right over here is 8m/s the base over here is 4s that's the time that past what's this area of the triangle? the area of this triangle is one half times the base which is 4s 4s times the height which is 8m/s times 8m/s second cancel out one half time 4 is 2 times 8 is equal to 16m So the total distance we travel is 20 plus 16 is 36m that is the total I could say the total displacement and once again is to the right, since it's positive so that is our displacement What I wanna do is to do the exact the same calculation keep it in variable form, that will give another formula many people often memorize You might understand this is completely intuitive formula and that just come out of the logical flow of reasoning that we went through this video what is the area once again if we just think about the variables? well the area of this rectangle right here is our initial velocity times our change in time, times our change in time So that is the blue rectangle right over here, and plus what do we have to do? we have the change in time once again we have the change in time times this height which is our final velocity which is our final velocity minus our initial velocity these are all vectors, they are just positive if going to the right we just multiply the base with the height that will just be the area of the entire rectangle I will take it by half because triangle is just half of that rectangle so times one half, so times one half so this is the area, this is the purple area right over here this is the area of this, this is the area of that and let's simplify this a little bit let's factor out the delta t, so you factor out the delta t you get delta t times a bunch of stuff v sub i your initial velocity we factor this out plus this stuff, plus this thing right over here and we can distribute the one half we factor the one half, we factor the delta t out, taking it out and let's multiply one half by each of these things so it's gonna be plus one half times vf, times our final velocity that's not the right color, I will use the right color so you would understand what I am doing, so this is the one half so plus one half times our final velocity final velocity minus one half, minus one half times our initial velocity I'm gonna do that in blue, sorry I have trouble in changing color today minus one half times our initial velocity, times our initial velocity and what is this simplify do? we have something plus one half times something else minus one half of the original something so what is vi minus one half vi? so anything minus its half is just a positive half left so these two terms, this term and this term will simplify to one half vi one half initial velocity plus one half times the final velocity plus one half times the final velocity and all of that is being multiplied with the change in time the time that has gone by and this tells us the distance, the distance that we travel another way to think about it, let's factor out this one half you get distance that is equal to change in time times factoring out the one half vi plus vf, vi no that's not the right color vi plus vf so this is interesting, the distance we travel is equal to one half of the initial velocity plus the final velocity so this is really if you just took this quantity right over here it's just the arithmetic, I have trouble saying that word it's the arithmetic mean of these two numbers, so I'm gonna define, this is something new, I'm gonna call this the average velocity we have to be very careful with this this right here is the average velocity but the only reason why I can just take the starting velocity and ending velocity and adding them together and divide them by two since you took an average of two thing it's some place over here and I take that as average velocity it's because my acceleration is constant which is usually an assumption in introductory physics class but it's not always the assumption but if you do have a constant acceleration like this you can assume that the average velocity is gonna be the average of the initial velocity and the final velocity if this is a curve and the acceleration is changing you could not do that but what is useful about this is if you wanna figure out the distance that was travelled, you just need to know the initial velocity and the final velocity, average their two and multiply the times it goes by so in this situation our final velocity is 13m/s our initial velocity is 5m/s so you have 13 plus 5 is equal to 18 you divided that by 2, you average velocity is 9m/s if you take the average of 13 and 5 and 9m/s times 4s gives you 36m so hopefully it doesn't confuse you I just wannt show you some of these things you will see in your physics class but you shouldn't memorize they can all be deduced" + }, + { + "Q": "How to write 230078 in expanded form", + "A": "In the number 230078, the 2 is in the hundred thousands place, the 3 is in the ten thousands place, the 0 s are in the thousands and hundreds places, the 7 is in the tens place, and the 8 is in the ones place. So 230078 represents 2 hundred thousands, 3 ten thousands, 7 tens, and 8 ones. So the expanded form of 230078 is 200000 + 30000 + 70 + 8, or (2 x 100000) + (3 x 10000) + (7 x 10) + (8 x 1). Have a blessed, wonderful day!", + "video_name": "iK0y39rjBgQ", + "transcript": "Write 14,897 in expanded form. Let me just rewrite the number, and I'll color code it, and that way, we can keep track of our digits. So we have 14,000. I don't have to write it-- well, let me write it that big. 14,000, 800, and 97-- I already used the blue; maybe I should use yellow-- in expanded form. So let's think about what place each of these digits are in. This right here, the 7, is in the ones place. The 9 is in the tens place. This literally represents 9 tens, and we're going to see this in a second. This literally represents 7 ones. The 8 is in the hundreds place. The 4 is in the thousands place. It literally represents 4,000. And then the 1 is in the ten-thousands place. And you see, every time you move to the left, you move one place to the left, you're multiplying by 10. Ones place, tens place, hundreds place, thousands place, ten-thousands place. Now let's think about what that really means. If this 1 is in the ten-thousands place, that means that it literally represents-- I want to do this in a way that my arrows don't get mixed up. Actually, let me start at the other end. Let me start with what the 7 represents. The 7 literally represents 7 ones. Or another way to think about it, you could say it represents 7 times 1. All of these are equivalent. They represent 7 ones. Now let's think about the 9. That's why I'm doing it from the right, so that the arrows don't have to cross each other. So what does the 9 represent? It represents 9 tens. You could literally imagine you have 9 actual tens. You could have a 10, plus a 10, plus a 10. Do that nine times. That's literally what it represents: 9 actual tens. 9 tens, or you could say it's the same thing as 9 times 10, or 90, either way you want to think about it. So let me write all the different ways to think about it. It represents all of these things: 9 tens, or 9 times 10, or 90. So then we have our 8. Our 8 represents-- we see it's in the hundreds place. It represents 8 hundreds. Or you could view that as being equivalent to 8 times 100-- a hundred, not a thousand-- 8 times 100, or 800. That 8 literally represents 8 hundreds, 800. And then the 4. I think you get the idea here. This represents the thousands place. It represents 4 thousands, which is the same thing as 4 times 1,000, which is the same thing as 4,000. 4,000 is the same thing as 4 thousands. Add it up. And then finally, we have this 1, which is sitting in the ten-thousands place, so it literally represents 1 ten-thousand. You can imagine if these were chips, kind of poker chips, that would represent one of the blue poker chips and each blue poker chip represents 10,000. I don't know if that helps you or not. And 1 ten-thousand is the same thing as 1 times 10,000 which is the same thing as 10,000. So when they ask us to write it in expanded form, we could write 14,897 literally as the sum of these numbers, of its components, or we could write it as the sum of these numbers. Actually, let me write this. This top 7 times 1 is just equal to 7. So 14,897 is the same thing as 10,000 plus 4,000 plus 800 plus 90 plus 7. So you could consider this expanded form, or you could use this version of it, or you could say this the same thing as 1 times 10,000, depending on what people consider to be expanded form-- plus 4 times 1,000 plus 8 times 100 plus 9 times 10 plus 7 times 1. I'll scroll to the right a little bit. So either of these could be considered expanded form." + }, + { + "Q": "What is the fraction thirteen thirds reduced to? Why doesn't the person reduce the fractions?", + "A": "The fraction thirteen thirds is the simplest form we can simplify it to. The reason we don t convert it into a decimal is because we d have the monstrosity 4.33333333333333....... Also, in math, it s preferred not to have decimals in a fraction.", + "video_name": "XMJ72mtMn4Y", + "transcript": "A line goes through the points (-1, 6) and (5, 4). What is the equation of the line? Let's just try to visualize this. So that is my x axis. And you don't have to draw it to do this problem but it always help to visualize That is my y axis. And the first point is (-1,6) So (-1, 6). So negative 1 coma, 1, 2, 3, 4 ,5 6. So it's this point, rigth over there, it's (-1, 6). And the other point is (5, -4). So 1, 2, 3, 4, 5. And we go down 4, So 1, 2, 3, 4 So it's right over there. So the line connects them will looks something like this. Line will draw a rough approximation. I can draw a straighter than that. I will draw a dotted line maybe Easier do dotted line. So the line will looks something like that. So let's find its equation. So good place to start is we can find its slope. Remember, we want, we can find the equation y is equal to mx plus b. This is the slope-intercept form where m is the slope and b is the y-intercept. We can first try to solve for m. We can find the slope of this line. So m, or the slope is the change in y over the change in x. Or, we can view it as the y value of our end point minus the y value of our starting point over the x-value of our end point minus the x-value of our starting point. Let me make that clear. So this is equal to change in y over change in x wich is the same thing as rise over run wich is the same thing as the y-value of your ending point minus the y-value of your starting point. This is the same exact thing as change in y and that over the x value of your ending point minus the x-value of your starting point This is the exact same thing as change in x. And you just have to pick one of these as the starting point and one as the ending point. So let's just make this over here our starting point and make that our ending point. So what is our change in y? So our change in y, to go we started at y is equal to six, we started at y is equal to 6. And we go down all the way to y is equal to negative 4 So this is rigth here, that is our change in y You can look at the graph and say, oh, if I start at 6 and I go to negative 4 I went down 10. or if you just want to use this formula here it will give you the same thing We finished at negative 4, we finished at negative 4 and from that we want to subtract, we want to subtract 6. This right here is y2, our ending y and this is our beginning y This is y1. So y2, negative 4 minus y1, 6. or negative 4 minus 6. That is equal to negative 10. And all it does is tell us the change in y you go from this point to that point We have to go down, our rise is negative we have to go down 10. That's where the negative 10 comes from. Now we just have to find our change in x. So we can look at this graph over here. We started at x is equal to negative 1 and we go all the way to x is equal to 5. So we started at x is equal to negative 1, and we go all the way to x is equal to 5. So it takes us one to go to zero and then five more. So are change in x is 6. You can look at that visually there or you can use this formula same exact idea, our ending x-value, our ending x-value is 5 and our starting x-value is negative 1. 5 minus negative 1. 5 minus negative 1 is the same thing as 5 plus 1. So it is 6. So our slope here is negative 10 over 6. wich is the exact same thing as negative 5 thirds. as negative 5 over 3 I devided the numerator and the denominator by 2. So we now know our equation will be y is equal to negative 5 thirds, that's our slope, x plus b. So we still need to solve for y-intercept to get our equation. And to do that, we can use the information that we know in fact we have several points of information We can use the fact that the line goes through the point (-1,6) you could use the other point as well. We know that when is equal to negative 1, So y is eqaul to 6. So y is equal to six when x is equal to negative 1 So negative 5 thirds times x, when x is equal to negative 1 y is equal to 6. So we literally just substitute this x and y value back into this and know we can solve for b. So let's see, this negative 1 times negative 5 thirds. So we have 6 is equal to positive five thirds plus b. And now we can subtract 5 thirds from both sides of this equation. so we have subtracted the left hand side. From the left handside and subtracted from the rigth handside And then we get, what's 6 minus 5 thirds. So that's going to be, let me do it over here We take a common denominator. So 6 is the same thing as Let's do it over here. So 6 minus 5 over 3 is the same thing as 6 is the same thing as 18 over 3 minus 5 over 3 6 is 18 over 3. And this is just 13 over 3. And this is just 13 over 3. And then of course, these cancel out. So we get b is equal to 13 thirds. So we are done. We know We know the slope and we know the y-intercept. The equation of our line is y is equal to negative 5 thirds x plus our y-intercept which is 13 which is 13 over 3. And we can write these as mixed numbers. if it's easier to visualize. 13 over 3 is four and 1 thirds. So this y-intercept right over here. this y-intercept right over here. That's 0 coma 13 over 3 or 0 coma 4 and 1 thirds. And even with my very roughly drawn diagram it those looks like this. And the slope negative 5 thirds that's the same thing as negative 1 and 2 thirds. You can see here the slope is downward because the slope is negative. It's a little bit steeper than a slope of 1. It's not quite a negative 2. It's negative 1 and 2 thirds. if you write this as a negative, as a mixed number. So, hopefully, you found that entertaining." + }, + { + "Q": "I can not answer this problem ? a^-b = 1/(a^b) (and why a^0 =1)", + "A": "Fill in some values for your variables. Say a = 4, b = 2. 4^-2 = 1/(4^2) = 1/(4 * 4) = 1/16. Anything to the power of zero is one. That s just the rules of math. 51256^0 is 1.", + "video_name": "Tqpcku0hrPU", + "transcript": "I have been asked for some intuition as to why, let's say, a to the minus b is equal to 1 over a to the b. And before I give you the intuition, I want you to just realize that this really is a definition. The inventor of mathematics wasn't one person. It was, you know, a convention that arose. But they defined this, and they defined this for the reasons that I'm going to show you. Well, what I'm going to show you is one of the reasons, and then we'll see that this is a good definition, because once you learned exponent rules, all of the other exponent rules stay consistent for negative exponents and when you raise something to the zero power. So let's take the positive exponents. Those are pretty intuitive, I think. So the positive exponents, so you have a to the 1, a squared, a cubed, a to the fourth. What's a to the 1? a to the 1, we said, is a, and then to get to a squared, what did we do? We multiplied by a, right? a squared is just a times a. And then to get to a cubed, what did we do? We multiplied by a again. And then to get to a to the fourth, what did we do? We multiplied by a again. Or the other way, you could imagine, is when you decrease the exponent, what are we doing? We are multiplying by 1/a, or dividing by a. And similarly, you decrease again, you're dividing by a. And to go from a squared to a to the first, you're dividing by a. So let's use this progression to figure out what a to the 0 is. So this is the first hard one. So a to the 0. So you're the inventor, the founding mother of mathematics, and you need to define what a to the 0 is. And, you know, maybe it's 17, maybe it's pi. I don't know. It's up to you to decide what a to the 0 is. But wouldn't it be nice if a to the 0 retained this pattern? That every time you decrease the exponent, you're dividing by a, right? So if you're going from a to the first to a to the zero, wouldn't it be nice if we just divided by a? So let's do that. So if we go from a to the first, which is just a, and divide by a, right, so we're just going to go-- we're just going to divide it by a, what is a divided by a? Well, it's just 1. So that's where the definition-- or that's one of the intuitions behind why something to the 0-th power is equal to 1. Because when you take that number and you divide it by itself one more time, you just get 1. So that's pretty reasonable, but now let's go into So what should a to the negative 1 equal? Well, once again, it's nice if we can retain this pattern, where every time we decrease the exponent we're dividing by a. So let's divide by a again, so 1/a. So we're going to take a to the 0 and divide it by a. a to the 0 is one, so what's 1 divided by a? It's 1/a. Now, let's do it one more time, and then I think you're going to get the pattern. Well, I think you probably already got the pattern. What's a to the minus 2? Well, we want-- you know, it'd be silly now to Every time we decrease the exponent, we're dividing by a, so to go from a to the minus 1 to a to the minus 2, let's just divide by a again. And what do we get? If you take 1/2 and divide by a, you get 1 over a squared. And you could just keep doing this pattern all the way to the left, and you would get a to the minus b is equal to 1 over a to the b. Hopefully, that gave you a little intuition as to why-- well, first of all, you know, the big mystery is, you know, something to the 0-th power, why does that equal 1? First, keep in mind that that's just a definition. Someone decided it should be equal to 1, but they had a good reason. And their good reason was they wanted to keep this pattern going. And that's the same reason why they defined negative exponents in this way. And what's extra cool about it is not only does it retain this pattern of when you decrease exponents, you're dividing by a, or when you're increasing exponents, you're multiplying by a, but as you'll see in the exponent rules videos, all of the exponent rules hold. All of the exponent rules are consistent with this definition of something to the 0-th power and this definition of something to the negative power. Hopefully, that didn't confuse you and gave you a little bit of intuition and demystified something that, frankly, is quite mystifying the first time you learn it." + }, + { + "Q": "do you now of there is a world war 2? if so, when?", + "A": "Yes World War II happened. It was between 1939 and 1945", + "video_name": "eIfQ4GfSz3U", + "transcript": "Narrator: As we'll see in this video and in others, the roots of a lot of the current disagreements in the Middle East and a lot of the conflict in the Middle East can actually be traced back to World War I. I realize this is an incredibly touchy subject that there are people who have very strong feelings on either side of it and my goal here is to really give my best attempt at what really happened. I encourage you to doubt any of this and look it up yourself and come, frankly, to your own conclusions. Let's rewind back to October of 1915, or 1915 in particular. The British were already at war with the Ottoman's. Just as a reminder of some of what happened in 1915, the Gallipoli campaign, by the end of 1915 it was pretty clear that this was a disaster for the allies. The Ottoman's were able to fend off the allies, they were in retreat. The British were able to fend off the Ottoman's when they tried to attack the Suez canal in 1915. This is the background, you can imagine the British are eager to get any other allies they can in their battle against the Ottoman's. In particular, they are eager to get the help of the Arab's who have been under the rule of the Ottoman's for hundreds of years. That's the backdrop where you have this correspondence between the high commissioner in Egypt, the British high commissioner, Sir Henry McMahon and the Sharif of Mecca, Hussein bin \u02bfAli, who had his own aspirations to essentially be the king of an independent Arab state. They kept going back and forth from mid 1915 to early 1916 talking about what the state could be. Obviously the British want his support, wants him to lead a revolt against the Ottoman's. He's already articulated the boundaries for a state that he would like to see. So, that gives us a context for this correspondence in October of 1915. This is from Sir Henry McMahon to Hussein. \"... it is with great pleasure that I communicate to you \"on their behalf,\" the British government's behalf, \"the following statement, which I am confident \"you will receive with satisfaction. \"The two districts of Mersina and Alexandretta \"and portions of Syria lying to the west \"of the districts of Damascus, Homs, Hama, \"and Aleppo cannot be said to be purely Arab, \"and should be excluded from the limits demanded.\" This is referring to the limits that Hussein bin Ali had demanded in previous correspondence. \"With the above modifications,\" so just that region right over there, this right over here is Mersina, Alexandretta, this is Hama, Homs, Damascus, so really what he's referring to is this region, the west, the west of those cities right over here. He's saying look, you can't really consider this to be purely Arab, I'm going to exclude this out of the boundaries of this potential independent Arab state. \"With the above modification, and without prejudice \"to our existing treaties with Arab chiefs \"we accept those limits,\" we accept those limits. \"As for those regions lying within those frontiers \"wherein Great Britain is free to act \"without determinant to the interest of her ally, \"France,\" so as long as I'm not getting in trouble with France, \"I'm empowered in the name of \"the Government of Great Britain to give the following \"assurances and make the following reply \"to your letter; Subject to the above modifications,\" so taking this part out, \"Great Britain is prepared \"to recognize and support the independence \"of the Arabs in all the regions within the limits \"demanded by the Sharif of Mecca.\" So, essentially it included all of this region and actually much beyond what I'm showing here, kind of present day Syria, Jordan, Iraq, parts of present day Saudi Arabia. All of that is essentially, the British are saying, yeah we're going to allow you to have that, an independent state there. \"Great Britain will guarantee the Holy Places \"against all external aggression \"and will recognize their inviolability. \"... I am convinced that this declaration will assure \"you beyond all possible doubt,\" beyond all possible doubt, \"of the sympathy \"of Great Britain towards the aspiration of her friends \"the Arabs, and will result in a firm \"and lasting alliance, the immediate results \"of which will be the expulsion of the Turks \"from the Arab countries and the freeing \"of the Arab peoples from the Turkish yoke, \"which for so many years has pressed heavily \"upon them.\" This actually does help to convince the Arab's to rise up against the Turks, against the Ottoman Empire, they play a significant role in the Palestine Campaign, they rise up in June of 1916. Now, the video that I did on the Palestine Campaign, I got several comments of people being cynical about Britain's intentions and it does look like the British were, indeed, cynical. T.E. Lawrence famous for Lawrence of Arabia was often depicted as this mystical fellow, this guy who had this kinship with the Arab's. His actual correspondence with the British government actually do show that he did have a kind of ... he was doing, I guess, in the words of George W. Bush, a little bit of strategery, he had a more cynical view of this relationship with the Arab's. This is some correspondence that he wrote in early 1916, so right about the same time that all of this was going on. This says he's referring to a possible Arab revolt, or Hussein's activity. \"Hussein's activity seems beneficial to us, \"because it matches with our immediate aims, \"the break-up of the Islamic 'bloc' \"and the defeat and disruption of the \"Ottoman Empire.\" Assuming he didn't really talk about this, this being one of the ... the British didn't talk about that when they were talking to Hussein. \"If we can arrange that this political change \"shall be a violent one, we will have abolished \"the threat of Islam, by dividing it against itself, \"in its very heart.\" \"There will then be a Khalifa,\" kind of a seat of Islam, \"in Turkey \"and a Khalifa in Arabia, in theological warfare.\" This is T.E. Lawrence, I got this from The Golden Warrior: The Life and Legend of Lawrence of Arabia. Even this, somewhat portrayed as a heroic figure, was doing things in very strategic, strategic terms. To make things worse for the Arab's, while the British were trying to convince them to revolt, they were also in secret negotiations with the French on how they would divide the Middle East if they were able to beat the Ottoman's. At this point in the war the British were already making some progress in Mesopotamia, but they really hadn't really started on the Palestine Campaign right here. So, this was all conjecture. The British representatives was Sykes, the French representative was Picot, this was done with the consent of the Russian's. You didn't have a revolution in Russia as of now, so in early 1916, in May this agreement was concluded, this secret agreement. You have the Sykes-Picot Agreement, it's secret. Let me write that, it is a secret agreement between Britain and France and essentially they are carving up the entire Middle East between them. This blue area right here, this would be occupied by the French, part of eastern Turkey or modern day eastern Turkey would be given to the Russian's. The British would be able to occupy, would occupy southern Mesopotamia essentially insuring protection of the oil that is coming out of Persia. Oil is becoming more and more of a relevant factor in kind of global power. Then you have these two protectorates right over here, which in theory could be independent or an independent Arab state, or two independent Arab states under the protection. Let me put that in quotes, because \"protectorate\" is always not as nice as it sounds, under the protection of the French or the British which means, \"Hey you're an independent state, but we will \"protect you in case anyone wants to invade.\" The reality of protectorate is that it usually involves the people doing the protecting have all the real power and all the real influence. The Sykes-Picot Agreement also give this little carve out to Britain so they would have access to the Mediterranean. Palestine, or the Roman Kingdom of Judea, this is carved out as a separate international property something that would be administered by multiple states and I guess the argument would be, this is where the Holy Land's are, multiple religions have some of their holiest sites within here and so they carved it out like this. Once again, this is all in secret, they obviously don't want the Arab's to find out because they're about to convince the Arab's to join in a revolt against the Ottoman's. Now, to make things ... once again, this was all secret up to this point in 1916 when it was all agreed on. Then you forward to 1917 where we have the famous Balfour Declaration. This right over here is the Balfour Declaration and it was essentially a letter from the Foreign Secretary of the U.K., Balfour, to Lord Rothschild who was a leading [Briticizen] , a leading member of the Jewish community. In it he writes, \"Dear Lord Rothschild, \"I have much pleasure in conveying to you, \"on behalf of His Majesty's Government, \"the following declaration of sympathy \"with Jewish Zionist aspirations which has been \"submitted to, and approved by, the Cabinet. \"His Majesty's Government view with favor \"the establishment in Palestine of a national home,\" of a national home, \"for the Jewish people, \"and will use their best endeavors to facilitate \"the achievement of this objective. \"It being clearly understood that nothing shall be done \"which may prejudice the civil and religious rights \"of existing non-Jewish communities in Palestine, \"or the rights and political status enjoyed by Jews \"in any other country. \"I should be grateful if you would bring \"this declaration to the knowledge of the \"Zionist Federation.\" Signed Artur Balfour. In here, he's not explicitly saying ... and they're being very careful here, he's not saying we're supporting a state for the Jewish people, but he's saying he is supporting the return of national home for the Jewish people, but at the same time, he's saying that it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of the existing non-Jewish communities in Palestine. Needless to say, you can imagine that this is making the Arab's fairly uncomfortable. On one side it seems, based on some of the McMahon-Hussein correspondences that were ... especially in 1915, that they were being promised an independent Arab state which included much of this territory, but at the same time, in the Balfour Declaration the British were promising to, kind of the Jewish diaspora, that they could have a homeland there and it might one day, who knows, it might one day turn in to some type of a state. To make the Arab's even more uncomfortable, this was in November 2, 1917. By the end of November, you have to remember that 1917 you first had a revolution, in Russia the Czar was overthrown in February and in March of 1917, and October the Bolshevik's take over. They want to get out of the war, they don't like all these secret deals, not clear that they would even get what they were entitled to these secret deals, so they actually release all the entire text of the Sykes-Picot Agreement. They released this, so in the same month you have the Arab's and the Ottoman's and the Ottoman's were very happy to see this because it would undermine the Arab's belief in maybe supporting the allies, but in one month you have the Arab's finding out about the Balfour Declaration, which was a pulbic declaration and then later that month because of the Russian release of it, the formally secret Sykes-Picot Agreement, so it makes them very, or at least a little bit more suspicious. So you can imagine the British Empire trying to have it both ways, to kind of have support from the Jewish Diaspora while at the same time have support from the Arab's in their revolt against the Ottoman's would lead to very significant conflicts over the decades to come. Regardless of which side of the issue you fall on, a lot of the seed is happening right around now, right around World War I. This has been admitted by the British government. This is right here, this was the then Secretary, or Foreign Secretary Jack Straw, U.K Foreign Secretary in 2002. This is a statement he made to the News Statesman Magazine in 2002. \"A lot of the problems we are having to deal with now, \"I have to deal with now,\" he's the Foreign Secretary, \"are a consequence \"of our colonial past ...\" Consequence of our colonial past. \"The Balfour Declaration and the contradictory assurances,\" \"and the contradictory assurances \"which were being given to Palestinian's \"in private at the same time as they were \"being given to the Israelis ... \"again, an interesting history for us, \"but not an honorable one.\" This is really just the beginning as we'll see in future videos as we go to the Interwar period, the British kind of go back and forth on this issue over, over, and over again, but needless to say, it's lead to a very messy situation in the modern Middle East." + }, + { + "Q": "what does linear mean", + "A": "it means straight like a line, hence LINEar. a wobbly string isn t linear, but a straight string is. A problem that is linear basically means if you graph it, it will be a straight line.", + "video_name": "AZroE4fJqtQ", + "transcript": "Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value. For example, for any one-step change in x, is the change in y always going to be 3? Is it always going to be 5? If it's always going to be the same value, you're dealing with a linear function. If for each change in x--so over here x is always changing by 1, so since x is always changing by 1, the change in y's have to always be the same. If they're not, then we're dealing with a non-linear function. We can actually show that plotting out. If the changes in x-- we're going by different values, if this went from 1 to 2 and then 2 to 4-- what you'd want to do, then, is divide the change in y by the change in x, and that should always be a constant. In fact, let me write that down. If something is linear, then the change in y over the change in x always constant. Now, in this example, the change in x's are always 1, right? We go from 1 to 2, 2 to 3, 3 to 4, 4 to 5. So in this example, the change in x is always going to be 1. So in order for this function to be linear, our change in y needs to be constant because we're just going to take that and divide it by 1. So let's see if our change in y is constant. When we go from 11 to 14, we go up by 3. When we go from 14 to 19, we go up by 5, so I already see that it is not constant. We didn't go up by 3 this time, we went up by 5. And here, we go up by 7. And here, we're going up by 9. So we're actually going up by increasing amounts, so we're definitely dealing with a non-linear function. And we can see that if we graph it out. So let me draw-- I'll do a rough graph here. So let me make that my vertical axis, my y-axis. And we go all the way up to 35. So I'll just do 10, 20, 30. Actually, I can it do a little bit more granularly than that. I could do 5, 10, 15, 20, 25, 30, and then 35. And then our values go 1 through 5. I'll do it on this axis right here. They're not obviously the exact same scale, so I'll do 1, 2, 3, 4, and 5. So let's plot these points. So the first point is 1, 11, when x is 1, y is 11. This is our x-axis. When x is 1, y is 11, that's right about there. When x is 2, y is 14, that's right about there. When x is 3, y is 19, right about there. When x is 4, y is 26, right about there. And then finally, when x is 5, y is 35, right up there. So you can immediately see that this is not tracing out a line. If this was a linear function, then all the points would be on a line that looks something like that. That's why it's called a linear function. In this case, it's not, it's non-linear. The rate of increase as x changes is going up." + }, + { + "Q": "Why are we allowed to take the natural log of e^x when finding the derivative. Isn't that changing the function?", + "A": "Try looking at it this way, maybe this will answer your question. d/dx [ln(e^x)] = d/dx [ln(e^x)] I don t need to prove that, right? It s obvious, everything is equal to itself. And then he differentiates the right hand side and gets d/dx [x*lne] = dx/dx * 1 = 1 and then differentiates the other side of the equation and gets d/dx[e^x]* 1/(e^x) But because we said that d/dx [ln(e^x)] = d/dx [ln(e^x)] we can set our two answers equal to each other. Did I help? Make sure to watch the HD version of the video.", + "video_name": "sSE6_fK3mu0", + "transcript": "Let's prove with the derivative of e to the x's, and I think that this is one of the most amazing things, depending on how you view it about either calculus or math or the universe. Well we're essentially going to prove-- I've already told you before that the derivative of e to the x is equal to e to the x, which is amazing. The slope at any point of that line is equal to the x value-- is equal to the function at that point, not the x value. The slope at any point is equal to e. That is mind boggling. And that also means that the second derivative at any point is equal to the function of that value or the third derivative, or the infinite derivative, and that never ceases to amaze me. But anyway back to work. So how are we going to prove this? Well we already proved-- I actually just did it right before starting this video-- that the derivative-- and some people actually call this the definition of e. They go the other way around. They say there is some number for which this is true, and we call that number e. So it could almost be viewed as a little bit circular, but be we said that e is equal to the limit as n approaches infinity of 1 over 1 plus n to the end. And then using this we actually proved that derivative of ln of x is equal to 1/x. The derivative of log base e of x is equal to 1/x. So now that we prove this out, let's use this to prove this. Let me keep switching colors to keep it interesting. Let's take the derivative of ln of e to the x. This is almost trivial. This is equal to the logarithm of a to the b is equal to b times the logarithm of a, so this is equal to the derivative of x ln of e. And this is just saying e to what power is equal to e. Well, to the first power, right? So this just equals the derivative of x, which we have shown as equal to 1. I think we have shown it, hopefully we've shown it. If we haven't, that's actually a very easy one to prove. OK fair enough. We did that. But let's do this another way. Let's use the chain rule. So what doe the chain rule say? If we have f of g of x, where we have one function embedded in another one, the chain rule say we take the derivative of the inside function, so d/dx of e to the x. And then we take the derivative of the outside function or the derivative of the outside function with respect to the inner function. You can almost view it that way. So the derivative ln of e to the x with respect to e to the x. I know that's a little confusing. You could have written a d e to the x down over here, but I think you know the chain rule by now. That is equal to 1 over e to the x. And that just comes from this. But instead of an x, we have e to the x. So this is just a chain rule. Well what else do we know? We know that this is equal to this, and we also know that this is equal to this. So this must be equal to this. So this must be equal to 1. Well let's just multiply both sides of this equation by e to the x. We get on the left hand side, we're just left with this expression. The derivative of e to the x times- we're multiplying both sides by e to the x, times e to the x over e to the x. I just chose to put the e to the x on this term, is equal to e to the x. This is 1. Scratch it out. We're done. That might not have been completely satisfying for you, but it works. The derivative of e to the x is equal to e to the x. I think the school or the nation should take a national holiday or something, and people should just ponder this, because it really is fascinating. But then actually this will lead us to I would say even more dramatic results in the not too far off future. Anyway, I'll see in the next video." + }, + { + "Q": "What is an example of a quadratic equation with two imaginary solutions?", + "A": "x^2+2x+2=0 (x+1)^2= -1 x+1 = i or -i x = i-1 or x=-i-1", + "video_name": "dnjK4DPqh0k", + "transcript": "We're asked to solve 2x squared plus 5 is equal to 6x. And so we have a quadratic equation here. But just to put it into a form that we're more familiar with, let's try to put it into standard form. And standard form, of course, is the form ax squared plus bx plus c is equal to 0. And to do that, we essentially have to take the 6x and get rid of it from the right hand side. So we just have a 0 on the right hand side. And to do that, let's just subtract 6x from both sides of this equation. And so our left hand side becomes 2x squared minus 6x plus 5 is equal to-- and then on our right hand side, these two characters cancel out, and we just are left with 0. And there's many ways to solve this. We could try to factor it. And if I was trying to factor it, I would divide both sides by 2. If I divide both sides by 2, I would get integer coefficients on the x squared in the x term, but I would get 5/2 for the constant. So it's not one of these easy things to factor. We could complete the square, or we could apply the quadratic formula, which is really just a formula derived from completing the square. So let's do that in this scenario. And the quadratic formula tells us that if we have something in standard form like this, that the roots of it are going to be negative b plus or minus-- so that gives us two roots right over there-- plus or minus square root of b squared minus 4ac over 2a. So let's apply that to this situation. Negative b-- this right here is b. So negative b is negative negative 6. So that's going to be positive 6, plus or minus the square root of b squared. Negative 6 squared is 36, minus 4 times a-- which is 2-- times 2 times c, which is 5. Times 5. All of that over 2 times a. a is 2. So 2 times 2 is 4. So this is going to be equal to 6 plus or minus the square root of 36-- so let me just figure this out. 36 minus-- so this is 4 times 2 times 5. This is 40 over here. So 36 minus 40. And you already might be wondering what's going to happen here. All of that over 4. Or this is equal to 6 plus or minus the square root of negative 4. 36 minus 40 is negative 4 over 4. And you might say, hey, wait Sal. Negative 4, if I take a square root, I'm going to get an imaginary number. And you would be right. The only two roots of this quadratic equation right here are going to turn out to be complex, because when we evaluate this, we're going to get an imaginary number. So we're essentially going to get two complex numbers when we take the positive and negative version of this root. So let's do that. So the square root of negative 4, that is the same thing as 2i. And we know that's the same thing as 2i, or if you want to think of it this way. Square root of negative 4 is the same thing as the square root of negative 1 times the square root of 4, I could even do it one step-- that's the same thing as negative 1 times 4 under the radical, which is the same thing as the square root of negative 1 times the square root of 4. And the principal square root of negative 1 is i times the principal square root of 4 is 2. So this is 2i, or i times 2. So this right over here is going to be 2i. So we are left with x is equal to 6 plus or minus 2i over 4. And if we were to simplify it, we could divide the numerator and the denominator by 2. And so that would be the same thing as 3 plus or minus i over 2. Or if you want to write them as two distinct complex numbers, you could write this as 3 plus i over 2, or 3/2 plus 1/2i. That's if I take the positive version of the i there. Or we could view this as 3/2 minus 1/2i. This and these two guys right here are equivalent. Those are the two roots. Now what I want to do is a verify that these work. Verify these two roots. So this one I can rewrite as 3 plus i over 2. These are equivalent. All I did-- you can see that this is just dividing both of these by 2. Or if you were to essentially factor out the 1/2, you could go either way on this expression. And this one over here is going to be 3 minus i over 2. Or you could go directly from this. This is 3 plus or minus i over 2. So 3 plus i over 2. Or 3 minus i over 2. This and this or this and this, or this. These are all equal representations of both of the roots. But let's see if they work. So I'm first going to try this character right over here. It's going to get a little bit hairy, because we're going to have to square it and all the rest. But let's see if we can do it. So what we want to do is we want to take 2 times this quantity squared. So 2 times 3 plus i over 2 squared plus 5. And we want to verify that that's the same thing as 6 times this quantity, as 6 times 3 plus i over 2. So what is 3 plus i squared? So this is 2 times-- let me just square this. So 3 plus i, that's going to be 3 squared, which is 9, plus 2 times the product of three and i. So 3 times i is 3i, times 2 is 6i. So plus 6i. And if that doesn't make sense to you, I encourage you to kind of multiply it out either with the distributive property or FOIL it out, and you'll get the middle term. You'll get 3i twice. When you add them, you get 6i. I And then plus i squared, and i squared is negative 1. Minus 1. All of that over 4, plus 5, is equal to-- well, if you divide the numerator and the denominator by 2, you get a 3 here and you get a 1 here. And 3 distributed on 3 plus i is equal to 9 plus 3i. And what we have over here, we can simplify it just to save some screen real estate. 9 minus 1 is 8. So if I get rid of this, this is just 8 plus 6i. We can divide the numerator and the denominator right here by 2. So the numerator would become 4 plus 3i, if we divided it by 2, and the denominator here is just going to be 2. This 2 and this 2 are going to cancel out. So on the left hand side, we're left with 4 plus 3i plus 5. And this needs to be equal to 9 plus 3i. Well, you can see we have a 3i on both sides of this equation. And we have a 4 plus 5, which is exactly equal to 9. So this solution, 3 plus i, definitely works. Now let's try 3 minus i. So once again, just looking at the original equation, 2x squared plus 5 is equal to 6x. Let me write it down over here. Let me rewrite the original equation. We have 2x squared plus 5 is equal to 6x. And now we're going to try this root, verify that it works. So we have 2 times 3 minus i over 2 squared plus 5 needs to be equal to 6 times this business. 6 times 3 minus i over 2. Once again, a little hairy. But as long as we do everything, we put our head down and focus on it, we should be able to get the right result. So 3 minus i squared. 3 minus i times 3 minus i, which is-- and you could get practice taking squares of two termed expressions, or complex numbers in this case actually-- it's going to be 9, that's 3 squared, and then 3 times negative i is negative 3i. And then you're going to have two of those. So negative 6i. So negative i squared is also negative 1. That's negative 1 times negative 1 times i times i. So that's also negative 1. Negative i squared is also equal to negative 1. Negative i is also another square root. Not the principal square root, but one of the square roots of negative 1. So now we're going to have a plus 1, because-- oh, sorry, we're going to have a minus 1. Because this is negative i squared, which is negative 1. And all of that over 4. All of that over-- that's 2 squared is 4. Times 2 over here, plus 5, needs to be equal to-- well, before I even multiply it out, we could divide the numerator and the denominator by 2. So 6 divided by 2 is 3. 2 divided by 2 is 1. So 3 times 3 is 9. 3 times negative i is negative 3i. And if we simplify it a little bit more, 9 minus 1 is going to be-- I'll do this in blue. 9 minus 1 is going to be 8. We have 8 minus 6i. And then if we divide 8 minus 6i by 2 and 4 by 2, in the numerator, we're going to get 4 minus 3i. And in the denominator over here, we're going to get a 2. We divided the numerator and the denominator by 2. Then we have a 2 out here. And we have a 2 in the denominator. Those two characters will cancel out. And so this expression right over here cancels or simplifies to 4 minus 3i. Then we have a plus 5 needs to be equal to 9 minus 3i. I We have a negative 3i on the left, a negative 3i on the right. We have a 4 plus 5. We could evaluate it. This left hand side is 9 minus 3i, which is the exact same complex number as we have on the right hand side, 9 minus 3i. So it also checks out. It is also a root. So we verified that both of these complex roots, satisfy this quadratic equation." + }, + { + "Q": "What is a rational number??", + "A": "Rational numbers is which u convert decimal n fraction into like if a fraction is given to u 1_3 so u will getr it 1:3 understood", + "video_name": "VZOHWaw5dqM", + "transcript": "- We're told to look at the rational numbers below, order them from least to greatest. They really didn't have to tell us this first sentence, I would have known to look at the rational numbers to order them from least to greatest. Well anyway, they tell us 1/2, negative five, three point three, zero, 21 over 12, negative five point five, and two and 1/8ths. So the easiest way to visualize this might just be to make a number line that's long enough that it actually can contain all of these numbers, and then we can think about how we can compare them. So let me just draw a huge number line over here. So, take up almost all the entire screen. I'll stick with, we have some negative numbers here, we go as low as negative five point five, and we have some positive numbers here, looks like we go as high as three point three. This thing is still a little less than two, so we go about as high as three point three, so I can put, I can safely I think put zero right here in the middle, I can go a little bit to the right since we have our negative numbers go more negative. So zero, and let's make this negative one, negative two, negative three, negative four, negative five, and well that should be enough. Negative five, and then in the positive direction we have, one, two, three, in the positive direction. And let's see if we can plot these. So, to start off, to start off let's look at 1/2. Where does 1/2 sit, so it sits, let me actually make the scale a little bit better. So this is one, two and three, and four. Alright, so let's start with 1/2. 1/2 is directly in-between zero and one, it is half of a whole. This right here would be one whole. This would be one whole, let me label that. This over here is one. So 1/2 is directly between zero and one. So 1/2 is gonna sit right over here. So that is, let me write that a little bit bigger, you probably have trouble reading that, Alright, one over two, which is also zero point five. So this is also zero point five, anyway that's where it sits. Then we have negative five. Negative five, well this is negative one, negative two, negative three, negative four, negative five. Negative five sits right over there. And then we have three point three. Positive three point three, I'll do that in blue. Positive three point three. So this is one, two, three, and then we want to do another point three. So point three is about a third of the way, a little less than a third of the way, it would be three point three three three forever, if it was a third of the way. So a third of the way, that looks like about right over here. This is three, this right over here would be three point three, let me label. What I'm gonna do is I'm gonna label the numbers on the number line up here so it's one, two, three, four, this is zero, negative one, negative two, negative three, negative four, negative five, and so on and so forth. And then we get to zero, which is one of the numbers that we've already written down. Zero is obviously right over there on the number line, so I'll just write this zero in orange to make it clear, it's this zero. Then we have 21 over 12, which is an improper fraction, and to think about where we should place that on the number line, to think about where to place it on the number line, let me do this in this blue color. To think about where to place this on the number line let's change it into a mixed number, makes it a little bit easier to visualize, at least for my brain. So 12 goes into 21, well it goes into it one time. One times 12 is 12. If you subtract you get a remainder of, well we could actually regroup here, or borrow, if you don't want to do this in your head, you would get nine, but let's do this. So if we borrow one from the two, the two becomes a one, this becomes 11, or we're really regrouping a 10. Anyway, 11 minus two is nine, one minus one is zero. So we have a remainder of nine. So this thing, written as a mixed number, 21 over 12 written as a mixed number is one and 9/12ths. You get one 12/12ths in there and then you get 9/12ths left over. So one and 9/12ths we can also write that, actually we could've simplified this right from the get go, cause both 21 and 12 are divisible by three, but now we can just divide nine, we can simplify 9/12ths, divide both the numerator and the denominator by three, we then get one and three over four, one and 3/4ths. And just to make it clear, I could have simplified this right from the get go, 21 divided by three, is equal to seven, and 12 divided by three, is equal to four. So this is the same thing as 7/4ths, and if you were to divide four into seven, four goes into seven one time, subtract, one times four is four, subtract to get a remainder of three, one and 3/4ths. So going back to where do we plot this? Well it is, it's one, and then we have 3/4ths, we're going to go three fourths of the way. This is half way, this is one fourths, two fourths, three fourths, would be right over there. So this is our 21 over 12, which is the same thing as 7/4ths, which is the same thing as one and 3/4ths. And then we have negative five point five. Negative five point five, I'll do that in magenta again, running out of colors. Negative five point five, well this is negative five, so negative five point five is going to be between negative five and negative six. So let me add negative six to our number line, right here just to make it clear. So let me go a little bit further, let's say that this is negative six. Negative six, and our number line will keep going to smaller values. Let me scroll to the left a little bit. Negative six, so if we go to negative five point five, it's smack dab in-between negative five and negative six. So this is negative five point five, right over there. And then finally we have two and 1/8ths. I'll do that in orange again, or I'll do it in blue. Two and 1/8ths, so it's two and then 1/8th. And so if we want to find the exact place we could divide this into eighths, this would be 4/8ths, this would be 2/8ths, and that would be 6/8ths, and then 1/8th would sit right over here. So that right over there is two and 1/8th. So we've actually plotted, as best as we could, the exact locations. You didn't have to plot the exact locations if you were just trying to order them, but it doesn't hurt to see exactly where they sit when we order them. So now we've essentially ordered them cause we stuck them all on this number line. The order is negative five point five is the smallest, then negative five, then a zero, and then positive 1/2, then 21 over 12, then two and 1/8th, and then three point three. And we're done." + }, + { + "Q": "Are du and dx able to just be treated like normal variables?\n@5:30 he treats the dx as another variable, to be multiplied with the 7", + "A": "Yes, when using Leibnitz notation, you can think of them as quantifiable numbers. What they really mean is this - an infinitely small change in the given variable. For example, du is an infinitely small change in u. dx is an infinitely small change in x. Then, if we think of it this way, du/dx as giving the slope makes sense! The change in u over the change in x gives slope! Long story short, you can treat them as actual numbers - infinitely small changes in a variable.", + "video_name": "oqCfqIcbE10", + "transcript": "Let's take the indefinite integral of the square root of 7x plus 9 dx. So my first question to you is, is this going to be a good case for u-substitution? Well, when you look here, maybe the natural thing to set to be equal to u is 7x plus 9. But do I see its derivative anywhere over here? If we set u to be equal to 7x plus 9, what is the derivative of u with respect to x going to be? Derivative of u with respect to x is just going to be equal to 7. Derivative of 7x is 7. Derivative of 9 is 0. So do we see a 7 lying around anywhere over here? Well, we don't. But what could we do in order to have a 7 lying around, but not change the value of the integral? Well, the neat thing-- and we've seen this multiple times-- is when you're evaluating integrals, scalars can go in and outside of the integral very easily. Just to remind ourselves, if I have the integral of let's say some scalar a times f of x dx, this is the same thing as a times the integral of f of x dx. The integral of the scalar times a function is equal to the scalar times the integral of the functions. So let me put this aside right over here. So with that in mind, can we multiply and divide by something that will have a 7 showing up? Well, we can multiply and divide by 7. So imagine doing this. Let's rewrite our original integral. So let me draw a little arrow here just to go around that aside. We could rewrite our original integral as being 9 to the integral of times 1/7 times 7 times the square root of 7x plus 9 dx. And if we want to, we could take the 1/7 outside We don't have to, but we can rewrite this as 1/7 times the integral of 7, times the square root of 7x plus 9 dx. So now if we set u equal to 7x plus 9, do we have its derivative laying around? Well, sure. The 7 is right over here. We know that du-- if we want to write it in differential form-- du is equal to 7 times dx. So du is equal to 7 times dx. That part right over there is equal to du. And if we want to care about u, well, that's just going to be the 7x plus 9. That is are u. So let's rewrite this indefinite integral in terms of u. It's going to be equal to 1/7 times the integral of-- and I'll just take the 7 and put it in the back. So we could just write the square root of u du, 7 times dx is du. And we can rewrite this if we want as u to the 1/2 power. It makes it a little bit easier for us to kind of do the reverse power rule here. So we can rewrite this as equal to 1/7 times the integral of u to the 1/2 power du. And let me just make it clear. This u I could have written in white if I want it the same color. And this du is the same du right over here. So what is the antiderivative of u to the 1/2 power? Well, we increment u's power by 1. So this is going to be equal to-- let me not forget this 1/7 out front. So it's going to be 1/7 times-- if we increment the power here, it's going to be u to the 3/2, 1/2 plus 1 is 1 and 1/2 or 3/2. So it's going to be u to the 3/2. And then we're going to multiply this new thing times the reciprocal of 3/2, which is 2/3. And I encourage you to verify the derivative of 2/3 u to the 3/2 is indeed u to the 1/2. And so we have that. And since we're multiplying 1/7 times this entire indefinite integral, we could also throw in a plus c right over here. There might have been a constant. And if we want, we can distribute the 1/7. So it would get 1/7 times 2/3 is 2/21 u to the 3/2. And 1/7 times some constant, well, that's just going to be some constant. And so I could write a constant like that. I could call that c1 and then I could call this c2, but it's really just some arbitrary constant. Oh, actually, no we aren't done. We still just have our entire thing in terms of u. So now let's unsubstitute it. So this is going to be equal to 2/21 times u to the 3/2. And we already know what u is equal to. u is equal to 7x plus 9. Let me put a new color here just to ease the monotony. So it's going to be 2/21 times 7x plus 9 to the 3/2 power plus c. And we are done. We were able to take a kind of hairy looking integral and realize that even though it wasn't completely obvious at first, that u-substitution is applicable." + }, + { + "Q": "What about the boiling point of ethers? Are they generally low or are they high as compared to the others?", + "A": "The boiling point of ethers is generally low, the most common ether, diethyl ether (C2H5-O-C2H5), having a bp of 35\u00c2\u00b0C.", + "video_name": "pILGRZ0nT4o", + "transcript": "- [Voiceover] A liquid boils when its molecules have enough energy to break free of the attractions that exist between those molecules. And those attractions between the molecules are called the intermolecular forces. Let's compare two molecules, pentane on the left and hexane on the right. These are both hydrocarbons, which means they contain only hydrogen and carbon. Pentane has five carbons, one, two, three, four, five, so five carbons for pentane. And pentane has a boiling point of 36 degrees Celsius. Hexane has six carbons, one, two, three, four, five, and six. So six carbons, and a higher boiling point, of 69 degrees C. Let's draw in another molecule of pentane right here. So there's five carbons. Let's think about the intermolecular forces that exist between those two molecules of pentane. Pentane is a non-polar molecule. And we know the only intermolecular force that exists between two non-polar molecules, that would of course be the London dispersion forces, so London dispersion forces exist between these two molecules of pentane. London dispersion forces are the weakest of our intermolecular forces. They are attractions between molecules that only exist for a short period of time. So I could represent the London dispersion forces like this. So I'm showing the brief, the transient attractive forces between these two molecules of pentane. If I draw in another molecule of hexane, so over here, I'll draw in another one, hexane is a larger hydrocarbon, with more surface area. And more surface area means we have more opportunity for London dispersion forces. So I can show even more attraction between these two molecules of hexane. So the two molecules of hexane attract each other more than the two molecules of pentane. That increased attraction means it takes more energy for those molecules to pull apart from each other. More energy means an increased boiling point. So hexane has a higher boiling point than pentane. So as you increase the number of carbons in your carbon chain, you get an increase in the boiling point of your compound. So this is an example comparing two molecules that have straight chains. Let's compare, let's compare a straight chain to a branched hydrocarbon. So on the left down here, once again we have pentane, all right, with a boiling point of 36 degrees C. Let's write down its molecular formula. We already know there are five carbons. And if we count up our hydrogens, one, two, three, four, five, six, seven, eight, nine, 10, 11 and 12. So there are 12 hydrogens, so H12. C5 H12 is the molecular formula for pentane. What about neopentane on the right? Well, there's one, two, three, four, five carbons, so five carbons, and one, two, three, four, five, six, seven, eight, nine, 10, 11 and 12 hydrogens. So C5 H12. So these two compounds have the same molecular formula. All right? So the same molecular formula, C5 H12. The difference is, neopentane has some branching, right? So neopentane has branching, whereas pentane doesn't. It's a straight chain. All right. Let's think about the boiling points. Pentane's boiling point is 36 degrees C. Neopentane's drops down to 10 degrees C. Now, let's try to figure out why. If I draw in another molecule of pentane, all right, we just talk about the fact that London dispersion forces exist between these two molecules of pentane. So let me draw in those transient attractive forces between those two molecules. Neopentane is also a hydrocarbon. It's non-polar. So if I draw in another molecule of neopentane, all right, and I think about the attractive forces between these two molecules of neopentane, it must once again be London dispersion forces. Because of this branching, the shape of neopentane in three dimensions resembles a sphere. So it's just an approximation, but if you could imagine this molecule of neopentane on the left as being a sphere, so spherical, and just try to imagine this molecule of neopentane on the right as being roughly spherical. And if you think about the surface area, all right, for an attraction between these two molecules, it's a much smaller surface area than for the two molecules of pentane, right? We can kind of stack these two molecules of pentane on top of each other and get increased surface area and increased attractive forces. But these two neopentane molecules, because of their shape, because of this branching, right, we don't get as much surface area. And that means that there's decreased attractive forces between molecules of neopentane. And because there's decreased attractive forces, right, that lowers the boiling point. So the boiling point is down to 10 degrees C. All right. I always think of room temperature as being pretty close to 25 degrees C. So most of the time, you see it listed as being between 20 and 25. But if room temperature is pretty close to 25 degrees C, think about the state of matter of neopentane. Right? We are already higher than the boiling point of neopentane. So at room temperature and room pressure, neopentane is a gas, right? The molecules have enough energy already to break free of each other. And so neopentane is a gas at room temperature and pressure. Whereas, if you look at pentane, pentane has a boiling point of 36 degrees C, which is higher than room temperature. So we haven't reached the boiling point of pentane, which means at room temperature and pressure, pentane is still a liquid. So pentane is a liquid. And let's think about the trend for branching here. So we have the same number of carbons, right? Same number of carbons, same number of hydrogens, but we have different boiling points. Neopentane has more branching and a decreased boiling point. So we can say for our trend here, as you increase the branching, right? So not talk about number of carbons here. We're just talking about branching. As you increase the branching, you decrease the boiling points because you decrease the surface area for the attractive forces. Let's compare three more molecules here, to finish this off. Let's look at these three molecules. Let's see if we can explain these different boiling points. So once again, we've talked about hexane already, with a boiling point of 69 degrees C. If we draw in another molecule of hexane, our only intermolecular force, our only internal molecular force is, of course, the London dispersion forces. So I'll just write \"London\" here. So London dispersion forces, which exist between these two non-polar hexane molecules. Next, let's look at 3-hexanone, right? Hexane has six carbons, and so does 3-hexanone. One, two, three, four, five and six. So don't worry about the names of these molecules at this point if you're just getting started with organic chemistry. Just try to think about what intermolecular forces are present in this video. So 3-hexanone also has six carbons. And let me draw another molecule of 3-hexanone. So there's our other molecule. Let's think about electronegativity, and we'll compare this oxygen to this carbon right here. Oxygen is more electronegative than carbon, so oxygen withdraws some electron density and oxygen becomes partially negative. This carbon here, this carbon would therefore become partially positive. And so this is a dipole, right? So we have a dipole for this molecule, and we have the same dipole for this molecule of 3-hexanone down here. Partially negative oxygen, partially positive carbon. And since opposites attract, the partially negative oxygen is attracted to the partially positive carbon on the other molecule of 3-hexanone. And so, what intermolecular force is that? We have dipoles interacting with dipoles. So this would be a dipole-dipole interaction. So let me write that down here. So we're talk about a dipole-dipole interaction. Obviously, London dispersion forces would also be present, right? So if we think about this area over here, you could think about London dispersion forces. But dipole-dipole is a stronger intermolecular force compared to London dispersion forces. And therefore, the two molecules here of 3-hexanone are attracted to each other more than the two molecules of hexane. And so therefore, it would take more energy for these molecules to pull apart from each other. And that's why you see the higher temperature for the boiling point. 3-hexanone has a much higher boiling point than hexane. And that's because dipole-dipole interactions, right, are a stronger intermolecular force compared to London dispersion forces. And finally, we have 3-hexanol over here on the right, which also has six carbons. One, two, three, four, five, six. So we're still dealing with six carbons. If I draw in another molecule of 3-hexanol, let me do that up here. So we sketch in the six carbons, and then have our oxygen here, and then the hydrogen, like that. We know that there's opportunity for hydrogen bonding. Oxygen is more electronegative than hydrogen, so the oxygen is partially negative and the hydrogen is partially positive. The same setup over here on this other molecule of 3-hexanol. So partially negative oxygen, partially positive hydrogen. And so hydrogen bonding is possible. Let me draw that in. So we have a hydrogen bond right here. So there's opportunities for hydrogen bonding between two molecules of 3-hexanol. So let me use, let me use deep blue for that. So now we're talking about hydrogen bonding. And we know that hydrogen bonding, we know the hydrogen bonding is really just a stronger dipole-dipole interaction. So hydrogen bonding is our strongest intermolecular force. And so we have an increased attractive force holding these two molecules of 3-hexanol together. And so therefore, it takes even more energy for these molecules to pull apart from each other. And that's reflected in the higher boiling point for 3-hexanol, right? 3-hexanol has a higher boiling point than 3-hexanone and also more than hexane. So when you're trying to figure out boiling points, think about the intermolecular forces that are present between two molecules. And that will allow you to figure out which compound has the higher boiling point." + }, + { + "Q": "So, it doesn't matter if I add or subtract either 2 equation ?", + "A": "That is right the main aim is to cancel out one of the variables and you add and subtract to get there.", + "video_name": "vA-55wZtLeE", + "transcript": "Let's explore a few more methods for solving systems of equations. Let's say I have the equation, 3x plus 4y is equal to 2.5. And I have another equation, 5x minus 4y is equal to 25.5. And we want to find an x and y value that satisfies both of these equations. If you think of it graphically, this would be the intersection of the lines that represent the solution sets to both of these equations. So how can we proceed? We saw in substitution, we like to eliminate one of the variables. We did it through substitution last time. But is there anything we can add or subtract-- let's focus on this yellow, on this top equation right here-- is there anything that we can add or subtract to both sides of this equation? Remember, any time you deal with an equation you have to add or subtract the same thing to both sides. But is there anything that we could add or subtract to both sides of this equation that might eliminate one of the variables? And then we would have one equation in one variable, and we can solve for it. And it's probably not obvious, even though it's sitting right in front of your face. Well, what if we just added this equation to that equation? What I mean by that is, what if we were to add 5x minus 4y to the left-hand side, and add 25.5 to the right-hand side? So if I were to literally add this to the left-hand side, and add that to the right-hand side. And you're probably saying, Sal, hold on, how can you just add two equations like that? And remember, when you're doing any equation, if I have any equation of the form-- well, really, any equation-- Ax plus By is equal to C, if I want to do something to this equation, I just have to add the same thing to both sides of the equation. So I could, for example, I could add D to both sides of the equation. Because D is equal to D, so I won't be changing the equation. You would get Ax plus By, plus D is equal to C plus D. And we've seen that multiple, multiple times. Anything you do to one side of the equation, you have to do to the other side. But you're saying, hey, Sal, wait, on the left-hand side, you're adding 5x minus 4y to the equation. On the right-hand side, you're adding 25.5 to the equation. Aren't you adding two different things to both sides of the equation? And my answer would be no. We know that 5x minus 4y is 25.5. This quantity and this quantity are the same. They're both 25.5. This second equation is telling me that explicitly. So I can add this to the left-hand side. I'm essentially adding 25.5 to it. And I could add 25.5 to the right-hand side. So let's do that. If we were to add the left-hand side, 3x plus 5x is 8x. And then what is 4y minus 4y? And this was the whole point. When I looked at these two equations, I said, oh, I have a 4y, I have a negative 4y. If you just add these two together, they are going to cancel out. They're going to be plus 0y. Or that whole term is just going to go away. And that's going to be equal to 2.5 plus 25.5 is 28. So you divide both sides. So you get 8x is equal to 28. And you divide both sides by 8, and we get x is equal to 28 over 8, or you divide the numerator and the denominator by 4. That's equal to 7 over 2. That's our x value. Now we want to solve for our y value. And we could substitute this back into either of these two equations. Let's use the top one. You could do it with the bottom one as well. So we know that 3 times x, 3 times 7 over 2-- I'm just substituting the x value we figured out into this top equation-- 3 times 7 over 2, plus 4y is equal to 2.5. Let me just write that as 5/2. We're going to stay in the fraction world. So this is going to be 21 over 2 plus 4y is equal to 5/2. Subtract 21 over 2 from both sides. So minus 21 over 2, minus 21 over 2. The left-hand side-- you're just left with a 4y, because these two guys cancel out-- is equal to-- this is 5 minus 21 over 2. That's negative 16 over 2. So that's negative 16 over 2, which is the same thing-- well, I'll write it out as negative 16 over 2. Or we could write that-- let's continue up here-- 4y-- I'm just continuing this train of thought up here-- 4y is equal to negative 8. Divide both sides by 4, and you get y is equal to negative 2. So the solution to this equation is x is equal to 7/2, y is equal to negative 2. This would be the coordinate of their intersection. And you could try it out on both of these equations right here. So let's verify that it also satisfies this bottom equation. 5 times 7/2 is 35 over 2 minus 4 times negative 2, so minus negative 8. That's equivalent to-- let's see, this is 17.5 plus 8. And that indeed does equal 25.5. So this satisfies both equations. Now let's see if we can use our newly found skills to tackle a word problem, our newly found skills in elimination. So here it says, Nadia and Peter visit the candy store. Nadia buys 3 candy bars and 4 Fruit Roll-Ups for $2.84. Peter also buys 3 candy bars, but can only afford 1 additional Fruit Roll-Up. His purchase costs $1.79. What is the cost of each candy bar and each Fruit Roll-Up? So let's define some variables. Let's just use x and y. Let's let x equal cost of candy bar-- I was going to do a c and a f for Fruit Roll-Up, but I'll just stick with x and y-- cost of candy bar. And let y equal the cost of a Fruit Roll-Up. All right. So what does this first statement tell us? Nadia buys 3 candy bars, so the cost of 3 candy bars is going to be 3x. And 4 Fruit Roll-Ups. Plus 4 times y, the cost of a Fruit Roll-Up. This is how much Nadia spends. 3 candy bars, 4 Fruit Roll-Ups. And it's going to cost $2.84. That's what this first statement tells us. It translates into that equation. The second statement. Peter also buys 3 candy bars, but could only afford 1 additional Fruit Roll-Up. So plus 1 additional Fruit Roll-Up. His purchase cost is equal to $1.79. What is the cost of each candy bar and each Fruit Roll-Up? And we're going to solve this using elimination. You could solve this using any of the techniques we've seen so far-- substitution, elimination, even graphing, although it's kind of hard to eyeball things with the graphing. So how can we do this? Remember, with elimination, you're going to add-- let's focus on this top equation right here. Is there something we could add to both sides of this equation that'll help us eliminate one of the variables? Or let me put it this way, is there something we could add or subtract to both sides of this equation that will help us eliminate one of the variables? Well, like in the problem we did a little bit earlier in the video, what if we were to subtract this equation, or what if we were to subtract 3x plus y from 3x plus 4y on the left-hand side, and subtract $1.79 from the right-hand side? And remember, by doing that, I would be subtracting the same thing from both sides of the equation. This is $1.79. Because it says this is equal to $1.79. So if we did that we would be subtracting the same thing from both sides of the equation. So let's subtract 3x plus y from the left-hand side of the equation. And let me just do this over on the right. If I subtract 3x plus y, that is the same thing as negative 3x minus y, if you just distribute the negative sign. So let's subtract it. So you get negative 3x minus y-- maybe I should make it very clear this is not a plus sign; you could imagine I'm multiplying the second equation by negative 1-- is equal to negative $1.79. I'm just taking the second equation. You could imagine I'm multiplying it by negative 1, and now I'm going to add the left-hand side to the left-hand side of this equation, and the right-hand side to the right-hand side of that equation. And what do we get? When you add 3x plus 4y, minus 3x, minus y, the 3x's cancel out. 3x minus 3x is 0x. I won't even write it down. You get 4x minus-- sorry, 4y minus y. That is 3y. And that is going to be equal to $2.84 minus $1.79. What is that? That's $1.05. So 3y is equal to $1.05. Divide both sides by 3. y is equal to-- what's $1.05 divided by 3? So 3 goes into $1.05. It goes into 1 zero times. 0 times 3 is 0. 1 minus 0 is 1. Bring down a 0. 3 goes into 10 three times. 3 times 3 is 9. Subtract. 10 minus 9 is 1. Bring down the 5. 3 goes into 15 five times. 5 times 3 is 15. Subtract. We have no remainder. So y is equal to $0.35. So the cost of a Fruit Roll-Up is $0.35. Now we can substitute back into either of these equations to figure out the cost of a candy bar. So let's use this bottom equation right here. Which was originally, if you remember before I multiplied it by negative 1, it was 3x plus y is equal to $1.79. So that means that 3x plus the cost of a Fruit Roll-Up, 0.35 is equal to $1.79. If we subtract 0.35 from both sides, what do we get? The left-hand side-- you're just left with the 3x; these cancel out-- is equal to-- let's see, this is $1.79 minus $0.35. That's $1.44. And 3 goes into $1.44, I think it goes-- well, 3 goes into $1.44, it goes into 1 zero times. 1 times 3 is 0. Bring down the 1. Subtract. Bring down the 4. 3 goes into 14 four times. 4 times 3 is 12. I'm making this messy. 14 minus 12 is 2. Bring down the 4. 3 goes into 24 eight times. 8 times 3 is 24. No remainder. So x is equal to 0.48. So there you have it. We figured out, using elimination, that the cost of a candy bar is equal to $0.48, and that the cost of a Fruit Roll-Up is equal to $0.35." + }, + { + "Q": "im a fourth grader it dosent make any sense i need help", + "A": "Don t worry about this now. College will come after 12th grade in high school, which is 8 years in the future. But if you want to look into college, please fully understand this topic before you do so you don t waste any of your time. Even if you don t, you will most probably have to start thinking about college in the 8th or 9th grade.", + "video_name": "cGg1j1ZCCOs", + "transcript": "- Hi, I'm Sal Khan, founder of the Khan Academy. I just wanted to welcome you to this resource on navigating the college of, I guess, application, admissions, and paying-for process. You know, for me this is a super-important thing. College was a big part of my life. Obviously, we all know you learn a lot at college, but it also opens up your mind. You have, it's important to be in a community that really challenges you, between your peers, your professors, I met my wife in college, many of the same people that I worked with in college, I remember I was in a entrepreneurship competition my junior year, and many of the people who were on my team then, I'm still working with now, 20 years later, on Khan Academy. And I think it would be fair to say if I hadn't had that experience, Khan Academy might not exist. And because college can be such a mind-expanding, an eye-opening experience, it's really important that everyone realizes that it's actually more accessible than most people think. When I was a kid, my mom raised us as a single mother and I just assumed that we would never be able to afford a selective, private four-year university. But my sister, who was three years older, when she applied to college, she got into actually several really good schools, and her first choice, when she applied, you know, when I looked at the actual dollar amount for the tuition, there's no way. That was actually, the tuition at that university was more than my mom made in a year. But they gave my sister a financial aid package, some was contributed by my mother, but my sister had to take on a loan, do some work study, but they made it so that she could go. And that opened the possibilities for me. When it was my time, I applied where I thought that I could thrive the most, and I applied for financial aid, and like my sister, they made it possible for me to have that very rich and important experience in my own life. So my biggest hope is that, as you go through these resources, you look at it from your own lens, and it opens up the same type of possibilities that I was lucky enough to have in my own life." + }, + { + "Q": "How to weite 770,070 in expanded form", + "A": "if you cannot solve this problem, you need to re-learn the skill. You should be able to find this topic right in Khan Academy, watch the videos, work on the practice problems until you master the skills. That will help you for the long run. That is what I have been doing for myself too. Sincerely.", + "video_name": "iK0y39rjBgQ", + "transcript": "Write 14,897 in expanded form. Let me just rewrite the number, and I'll color code it, and that way, we can keep track of our digits. So we have 14,000. I don't have to write it-- well, let me write it that big. 14,000, 800, and 97-- I already used the blue; maybe I should use yellow-- in expanded form. So let's think about what place each of these digits are in. This right here, the 7, is in the ones place. The 9 is in the tens place. This literally represents 9 tens, and we're going to see this in a second. This literally represents 7 ones. The 8 is in the hundreds place. The 4 is in the thousands place. It literally represents 4,000. And then the 1 is in the ten-thousands place. And you see, every time you move to the left, you move one place to the left, you're multiplying by 10. Ones place, tens place, hundreds place, thousands place, ten-thousands place. Now let's think about what that really means. If this 1 is in the ten-thousands place, that means that it literally represents-- I want to do this in a way that my arrows don't get mixed up. Actually, let me start at the other end. Let me start with what the 7 represents. The 7 literally represents 7 ones. Or another way to think about it, you could say it represents 7 times 1. All of these are equivalent. They represent 7 ones. Now let's think about the 9. That's why I'm doing it from the right, so that the arrows don't have to cross each other. So what does the 9 represent? It represents 9 tens. You could literally imagine you have 9 actual tens. You could have a 10, plus a 10, plus a 10. Do that nine times. That's literally what it represents: 9 actual tens. 9 tens, or you could say it's the same thing as 9 times 10, or 90, either way you want to think about it. So let me write all the different ways to think about it. It represents all of these things: 9 tens, or 9 times 10, or 90. So then we have our 8. Our 8 represents-- we see it's in the hundreds place. It represents 8 hundreds. Or you could view that as being equivalent to 8 times 100-- a hundred, not a thousand-- 8 times 100, or 800. That 8 literally represents 8 hundreds, 800. And then the 4. I think you get the idea here. This represents the thousands place. It represents 4 thousands, which is the same thing as 4 times 1,000, which is the same thing as 4,000. 4,000 is the same thing as 4 thousands. Add it up. And then finally, we have this 1, which is sitting in the ten-thousands place, so it literally represents 1 ten-thousand. You can imagine if these were chips, kind of poker chips, that would represent one of the blue poker chips and each blue poker chip represents 10,000. I don't know if that helps you or not. And 1 ten-thousand is the same thing as 1 times 10,000 which is the same thing as 10,000. So when they ask us to write it in expanded form, we could write 14,897 literally as the sum of these numbers, of its components, or we could write it as the sum of these numbers. Actually, let me write this. This top 7 times 1 is just equal to 7. So 14,897 is the same thing as 10,000 plus 4,000 plus 800 plus 90 plus 7. So you could consider this expanded form, or you could use this version of it, or you could say this the same thing as 1 times 10,000, depending on what people consider to be expanded form-- plus 4 times 1,000 plus 8 times 100 plus 9 times 10 plus 7 times 1. I'll scroll to the right a little bit. So either of these could be considered expanded form." + }, + { + "Q": "how do input cot ,csc and sec on my calculator (fx82) ?", + "A": "Cot= 1/tan CSC = 1/Sin SEC = 1/Cos", + "video_name": "Q7htxHDN8LE", + "transcript": "Determine the six trigonometric ratios for angle A in the right triangle below. So this right here is angle A, its in vertex A and help me remember the definitions of the trig ratios, these are human constructed definitions that have ended up being very, very useful for analyzing a whole series of things in the world and to help me remember them, I use the word, SOH-CAH-TOA let me write that down, so SOH-CAH-TOA sometimes you can think of it as one word, but its really the three parts that define atleast three of the trig functions for you, then we can get the other three by looking at the first three, so SOH tells us that sin of an angle, in this case its sin of A, so sin of A is equal to the opposite opposite, thats the o over the hypotenuse so opposite over the hypotenuse, well in this context what is the opposite sin to angle A? so we go across the triangle it opens up onto side BC, it has length 12, so that is the opposite side. So this is going to be equal to 12, and whats the hypotenuse? well, the hypotenuse is the longest side of the triangle; it's opposite to the 90 degree angle, and so we go opposite the 90 degree angle the longest side is side AB, it has length 13, so this side right over here is the hypotenuse, and so the sin of A is 12/13ths. now lets go to CAH CAH defines, cosine for us. it tells us that cosine of an angle, in this case, cosine of A is equal to the adjacent side, the adjacent side to the angle over the hypotenuse, over the hypotenuse. so whats the adjacent side to angle A? well if we look at angle A, there is 2 sides next to it. One of them is the hypotenuse the other one has length 5, the adjacent one is side CA so its 5, and what is the hypotenuse, well we've already figured that out, the hypotenuse right over here its opposite the 90 degree angle, it the longest side of the right triangle, it has length 13, so the cosine of A is 5/13ths and let me label this, this right over here is the adjacent side and this is all specific to angle A, the hypotenuse would be the same regardless of what angle you pick, but the opposite and adjacent is dependent on the angle we choose in the right triangle, now lets go to TOA. TOA defines tangent for us, it tells us that the tangent the tangent of an angle is equal to the opposite, equal to the opposite side over the adjacent side, so given this definition what is the tangent of A? well the opposite we already figured out has length 12, and the adjacent side we already figured out has length 5 so the tangent of A, which is opposite over adjacent is 12/5ths now we'll go to the other three trig ratios which you can think of as the reciprocals of these right over here, but I'll define them so first you have cosecant, and cosecant; its always a little bit unintuitive, why cosecant is the reciprocal of sine of A even though it starts with a co like cosine but cosecant is the reciprocal of the sin of A, so sin of A is opposite over hypotenuse. Cosecant of A is hypotenuse over opposite and so whats the hypotenuse over the opposite, well hypotenuse is 13 and the opposite side is 12 and notice 13/12ths is the reciprocal of 12/13ths now, secant of A is the reciprocal so instead of it being adjacent over hypotenuse, which we got out of the CAH part of SOH CAH TOA, its hypotenuse over adjacent so what is the secant of A? well the hypotenuse, we figured out multiple times is 13 and what is the adjacent side? it is 5, so 13 13/5ths, which is once again the reciprocal of cosine of A, 5/13ths finally lets get the cotangent, and the cotangent is the reciprocal of the tangent of A, instead of being opposite over adjacent it is adjacent over opposite, so what is the cotangent of A? well we figured out the adjacent side multiple times for angle A, its 5 and the opposite side to angle A is 12, so 5/12ths which is once again the reciprocal of the tangent of A which is 12/5ths" + }, + { + "Q": "what is the point of the scientific method", + "A": "It is a method of reasoning that tries to ensure that its conclusions are logical and accurate descriptions of nature.", + "video_name": "N6IAzlugWw0", + "transcript": "Let's explore the scientific method. Which at first might seem a bit intimidating, but when we walk through it, you'll see that it's actually almost a common-sense way of looking at the world and making progress in our understanding of the world and feeling good about that progress of our understanding of the world. So, let's just use a tangible example here, and we'll walk through what we could consider the steps of the scientific method, and you'll see different steps articulated in different ways, but they all boil down to the same thing. You observe something about reality, and you say, well, let me try to come up with a reason for why that observation happens, and then you try to test that explanation. It's very important that you come up with explanations that you can test, and then you can see if they're true, and then based on whether they're true, you keep iterating. If it's not true, you come up with another explanation. If it is true, but it doesn't explain everything, well once again, you try to explain more of it. So, as a tangible example, let's say that you live in, in I don't know, northern Canada or something, and let's say that you live near the beach, but there's also a pond near your house, and you notice that the pond, it tends to freeze over sooner in the Winter than the ocean does. It does that faster and even does it at higher temperatures than when the ocean seems to freeze over. So, you could view that as your observation. So, the first step is you're making an observation. Observation. In our particular case is that the pond freezes over at higher temperatures than the ocean does, and it freezes over sooner in the Winter. Well, the next question that you might wanna, or the next step you could view as a scientific method. It doesn't have to be this regimented, but this is a structured way of thinking about it. Well, ask yourself a question. Ask a question. Why does, so in this particular question, or in this particular scenario, why does the pond tend to freeze over faster and at higher temperatures than the ocean does? Well, you then try to answer that question, and this is a key part of the scientific method, is what you do in this third step, is that you try to create an explanation, but what's key is that it is a testable explanation. So, you try to create a testable explanation. Testable explanation, and this is kind of the core, one of the core pillars of the scientific method, and this testable explanation is called your hypotypothesis. Your hypothesis. And so, in this particular case, a testable explanation could be that, well the ocean is made up of salt water, and this pond is fresh water, so your testable explanation could be salt water, salt water has lower freezing point. Has lower freezing, freezing point. Lower freezing point, so it takes colder temperatures to freeze it than fresh water. Than fresh water. So, this, right over here, this would be a good hypothesis. It doesn't matter whether the hypothesis is actually true or not. We haven't actually run the experiment, but it's a good one, because we can construct an experiment that tests this very well. Now, what would be an example of a bad hypothesis or of something that you couldn't even necessarily consider as part of the scientific method? Well, you could say that there is a fairy that blesses that blesses, let's say that performs magic, performs magic on the pond to freeze it faster. Freeze it faster. And, the reason why this isn't so good is that this is not so testable, because it's depending on this fairy, and you don't know how to convince the fairy to try to do it again. You haven't seen the fairy. You haven't observed the fairy. It's not based on any observation, and so this one right over here, this would not be a good hypothesis for the scientific method, so we would wanna rule that one out. So, let's go back to our testable explanation, our hypothesis. Salt water has a lower freezing point than fresh water. Well, the next step would be to make a prediction based on that, and this is the part where we're really designing an experiment. So, you could just view all of this as designing. Let me do this in a different color. Where we wanna design an experiment. Design an experiment. And in that experiments lets say, and let's see, the next two steps I will put as part of this experimental. Whoops. I messed up. Let me, I did my undo step. So, the next part that I will do is the experiment. Experiment. And there you go. So, the first thing is, we'll say I take, you know, there's all sorts of things that are going on outside. The ocean has waves. You know, maybe there are boats going by that might potentially break up the ice. So, I just wanna isolate that one variable that I care about, whether something is salt water or not, and I want a control for everything else. So, I want a control for whether there's waves or not or whether there's wind or any other possible explanation for why the pond freezes over faster. So, what I do, in a very controlled environment I take two cups. I take two cups. That's one cup and two cups, and I put water in those cups. I put water in those cups. Now, let's say I start with distilled water, but then this one stays, the first one right over here stays distilled, and distilled means that through evaporation I've taken out all of the impurities of that water, and in the second one I take that distilled water, and I throw a bunch of salt in it. So, this one is fresh, very fresh, and in fact, far fresher than you would find in a pond. It's distilled water. And then this is over here, this is salt water. So, you wouldn't see the salt, but just for our visuals, you depict it. Then we would make a prediction, and we could even view this as step 4, our prediction. We predict that the fresh water will freeze at a higher temperature than the salt water. So, our prediction, let's say the fresh freezes at zero degrees Celsius, but salt doesn't. Salt water doesn't. Salt water doesn't. So, what you then do is that you test your prediction. So, then you test it. And how would you test it? Well, you could have a very accurate freezer that is exactly at zero degrees Celsius, and you put both of these cups into it, and you wanna make sure that they're identical and everything where you control for everything else. You control for the surface area. You control for the material of the glass. You control for how much water there is. But, then you test it. Then you see what happened from your test. Leave it in overnight, and if you see that the fresh water has frozen over, so it's frozen over, but the salt water hasn't, well then that seems to validate your testable explanation. That salt water has a lower freezing point than fresh water, and if it didn't freeze, well it's like, okay, well maybe that, or if there isn't a difference, maybe either both of them didn't freeze or both of them did freeze, then you might say, well, okay, that wasn't a good explanation. I have to find another explanation for why the ocean seems to freeze at a lower temperature. Or, you might say, well that's part of the explanation, but that by itself doesn't explain it, or you might now wanna ask even further questions about, well, when does salt water freeze, and what else is it dependent on? Do the waves have an impact? Does the wind have an impact? So, then you can go into the process of iterating and refining. So, you then refine, refine, refine and iterate on the process. When I'm talking about iterate, you're doing it over again, but then, based on the things that you've learned. So, you might come up with a more refined testable explanation, or you might come up with more experiments that could get you a better understanding of the difference between fresh and salt water, or you might try to come up with experiments for why exactly, what is it about the salt that makes this water harder to freeze? So, that's essentially the essence of the scientific method, and I wanna emphasize this isn't some, you know, bizarre thing. This is logical reasoning. Make a testable explanation for something that you're observing in the world, and then you test it, and you see if your explanation seems to hold up based on the data from your test. And then whether or not it holds up, you then keep going, and you keep refining. And you keep learning more about the world, and the reason why this is better than just saying, oh well, look, okay, I see the pond has frozen over and the ocean hasn't, it must be the salt water, and you know, I just feel good about that, is that you can't feel good about that. There's a million different reasons, and you shouldn't just go on your gut, 'cause at some point, your gut might be right 90% of the time, but that 10% that it's wrong, you're going to be passing on knowledge or assumptions about the world that aren't true, and then other people are going to build on that, and then all of our knowledge is going to be built on kind of a shaky foundation, and so the scientific method ensures that our foundation is strong. And I'll leave you with the gentleman who's often considered to be the father, or one of the fathers of the scientific method. He lived in Cairo, and in what is now Egypt, nearly 1,000 or roughly 1,000 years ago. And he was a famous astronomer and phycisist and mathematician. And his quote is a pretty powerful one, 'cause I think it even stands today: \"The duty of the man who investigates the writings of scientists, if learning the truth is his goal, ...\" Let me start over, just so I can get the dramatic effect right. \"The duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and ... attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.\" Hasan Ibn al-Haytham, and his Latinized name is Alhazen. So, he's saying be skeptical, and not just skeptical of what other people write and read, but even of yourself. And another aspect of the scientific method which is super important is, if someone says they made a hypothesis and they tested and they got a result, in order for that to be a good test and in order for that to be a good hypothesis, that experiment has to be reproducible. Someone can't say, oh it's only, you know, a certain time that only happens once every 100 years and not, that that's why it happened that day. It has to be reproducible, and reproducible is key, because then another skeptical scientist like yourself can say, let me see if I can Let me not just believe it, because that person looks like they're smart, and they said that it is true." + }, + { + "Q": "what does co planar mean?", + "A": "4 or more points are coplanar if they lie on the same plane.", + "video_name": "J2Qz-7ZWDAE", + "transcript": "We've already been exposed to points and lines. Now let's think about planes. And you can view planes as really a flat surface that exists in three dimensions, that goes off in every direction. So for example, if I have a flat surface like this, and it's not curved, and it just keeps going on and on and on in every direction. Now the question is, how do you specify a plane? Well, you might say, well, let's see. Let's think about it a little bit. Could I specify a plane with a one point, right over here? Let's call that point, A. Would that, alone, be able to specify a plane? Well, there's an infinite number of planes that could go through that point. I could have a plane that goes like this, where that point, A, sits on that plane. I could have a plane like that. Or, I could have a plane like this. I could have a plane like this where point A sits on it, as well. So I could have a plane like that. And I could just keep rotating around A. So one point by itself does not seem to be sufficient to define a plane. Well, what about two points? Let's say I had a point, B, right over here. Well, notice the way I drew this, point A and B, they would define a line. For example, they would define this line right over here. So they would define, they could define, this line right over here. But both of these points and in fact, this entire line, exists on both of these planes that I just drew. And I could keep rotating these planes. I could have a plane that looks like this. I could have a plane that looks like this, that both of these points actually sit on. I'm essentially just rotating around this line that is defined by both of these points. So two points does not seem to be sufficient. Let's try three. So there's no way that I could put-- Well, let's be careful here. So I could put a third point right over here, point C. And C sits on that line, and C sits on all of these planes. So it doesn't seem like just a random third point is sufficient to define, to pick out any one of these planes. But what if we make the constraint that the three points are not all on the same line. Obviously, two points will always define a line. But what if the three points are not collinear. So instead of picking C as a point, what if we pick-- Is there any way to pick a point, D, that is not on this line, that is on more than one of these planes? We'll, no. If I say, well, let's see, the point D-- Let's say point D is right over here. So it sits on this plane right over here, one of the first ones that I drew. So point D sits on that plane. Between point D, A, and B, there's only one plane that all three of those points sit on. So a plane is defined by three non-colinear points. So D, A, and B, you see, do not sit on the same line. A and B can sit on the same line. D and A can sit on the same line. D and B can sit on the same line. But A, B, and D does not sit on-- They are non-colinear. So for example, right over here in this diagram, we have a plane. This plane is labeled, S. But another way that we can specify plane S is we could say, plane-- And we just have to find three non-collinear points on that plane. So we could call this plane AJB. We could call it plane JBW. We could call it plane-- and I could keep going-- plane WJA. But I could not specify this plane, uniquely, by saying plane ABW. And the reason why I can't do this is because ABW are all on the same line. And this line sits on an infinite number of planes. I could keep rotating around the line, just as we did over here. It does not specify only one plane." + }, + { + "Q": "what about negative c , can the area be negative? integration of -f(x) is equal to the negative of integration of f(x) . So definite integrals are area or something else ?please help.", + "A": "Good question, this is a tricky topic. A negative c value will produce a negative answer. The results of integrals are allowed to be negative. However, you also mention area, which is not allowed to be negative. I teach my students that the area is the absolute value of the integral and that speed is the absolute value of velocity.", + "video_name": "JyArK4jw3XU", + "transcript": "- [Voiceover] We've already seen and you're probably getting tired of me pointing it out repeatedly, that this yellow area right over here, this area under the curve y is equal to f of x and above the positive x-axis or I guess I can say just above the x-axis between x equals a and x equals b, that we can denote this area right over here as the definite integral of from a to b of f of x dx. Now what I want to explore in this video and it'll come up with kind of an answer that you probably could have guessed on your own, but at least get an intuition for it, is that I want to start thinking about the area under the curve that's a scaled version of f of x. Let's say it's y is equal to c times f of x. Y is equal to some number times f of x, so it's scaling f of x. And so I want this to be kind of some arbitrary number, but just to help me visualize, you know I have to draw something so I'm just gonna kind of in my head let's just pretend the c is a three for visualization purposes. So it's going to be three times, so instead of one, instead of this far right over here it's going to be about this far. For right over here, instead of this far right over here it's going to be that and another right over there. And then instead of it's going to be about there. And then instead of it being like that it's going to be one, two and then three, right around there. So I'm starting to get a sense of what this curve is going to look like, a scaled version of f of x. And at least what I'm drawing is pretty close to three times f of x, but just to give you an idea is going to look something like, and let's see over here if this distance, do a second one, a third one, is gonna be up here. It's gonna look something like this. It's gonna look something like that. So this is a scaled version and the scale I did right here I assumed a positive c greater than zero, but this is just for visualization purposes. Now what do we think the area under this curve is going to be between a and b? So what do we think this area right over here is going to be? Now we already know how we can denote it. That area right over there is equal to the definite integral from a to b of the function we're integrating is c f of x dx. I guess to make the question a little bit clearer, how does this relate to this? How does this green area relate to this yellow area? Well one way to think about it is we just scaled the vertical dimension up by c, so one way that you could reason it is if I'm finding the area of something, if I have the area of a rectangle and I have the vertical dimension is let's say I don't want to use those same letters over and over again. Well let's say the vertical dimension is alpha and the horizontal dimension is beta. We know that the area is going to be alpha times beta. Now if I scale up the vertical dimension by c, so instead of alpha this is c times alpha and this is, the width is beta, if I scale up the vertical dimension by c so this is now c times alpha, what's the area going to be? Well it's going to be c alpha times beta, or another way to think of it, when I scale one of the dimensions by c I take my old area and I scale up my old area up by c. And that's what we're doing, we're scaling up the vertical dimension by c. When you multiply c times f of x, f of x is giving us the vertical height. Now obviously that changes as our x changes, but when you think back to the Reimann sums the f of x was what gave us the height of our rectangles. We're now scaling up the height or scaling I should say because we might be scaling down depending on the c. We're scaling it, we're scaling one dimension by c. If you scale one dimension by c you're gonna scale the area by c. So this right over here, the integral, let me just rewrite it. The integral from a to b of c f of x dx, that's just going to be the scaled, we're just going to take the area of f of x, so let me do that in the same color. We're going to take the area under the curve f of x from a to b f of x dx and we're just going to scale it up by this c. So you might say, \"Okay maybe I could have felt \"that was, you know, if I have a c inside the integral \"now I can take the c out of the integral\", and once again this is not a rigorous proof based on the definition of the definite integral, but it hopefully gives you a little bit of intuition why you can do this. If you scale up the function, you're essentially scaling up the vertical dimension, so the area under this is going to just be a scaled up version of the area under the original function f of x. And once again really, really, really useful property of definite integrals that's going to help us solve a bunch of definite integrals. And kind of clarify what we're even doing with them." + }, + { + "Q": "how did sal reduce the original number/ 8:20 to 2:5", + "A": "common factor of 8 and 20 is 4 and 8/4 = 2 and 20/4 = 5 so 2:5. does this help you?", + "video_name": "UK-_qEDtvYo", + "transcript": "Voiceover:Let's think about another scenario involving ratios. In this case, let's think about the ratio of the number of apples. Number of apples to ... Instead of taking the ratio of the number of apples to the number of oranges, let's take the number of apples to the number of fruit. The number of fruit that we have over here. And I encourage you to pause the video and think about that on your own. Well, how many total apples do we have? We have 2, 4, 6, 8 apples. So we're going to have 8 apples. And then how much total fruit do we have? Well we have 8 apples and we have 3, 6, 9, 12 oranges. So our total fruit is 8 plus 12. We have 20 pieces of fruit. So this ratio is going to be 8 to ... 8 to 20. Or, if we want to write this in a more reduced form, we can divide both of these by 4. 4 is their greatest common divisor. And so this is the same thing as a ratio. 8 divided by 4 is 2 and 20 divided by 4 is 5. So 2 to 5. Now, does this make sense? Well, if we divide ... If we divide everything into groups of 4. So ... Or if we divide into 4 groups, I should say. So 1 group, 2 groups, 3 groups, and 4 groups. That's the largest number of groups that we can divide these into so that we don't have to cut up the apples or the oranges. We see that in each group, for every 2 apples we have 1, 2, 3, 4, 5 pieces of fruit. For every 2 apples we have 5 pieces of fruit. This is actually a good opportunity for us to introduce another way of representing ... Another way of representing ratios, and that's using fraction notation. So we could also represent this ratio as 2 over 5. As the fraction 2 over 5. Whenever we put it in the fraction it's very important to recognize what this represents. This is telling us the fraction of fruit that are apples. So we could say 2/5 of the fruit ... Of the fruit ... Of the number of fruit, I guess I could say. Of number of fruit ... Of fruit is equal to the number of apples. Right, I'm just going to say 2/5 of fruit if we're just speaking in more typical terms. 2/5 of fruit are apples. Are, are apples. So, once again, this is introducing another way of representing ratios. We could say that the ratio of apples to fruit, once again, it could be 2 to 5 like that. It could be 2, instead of putting this little colon there we could literally write out the word to. 2 to 5. Or we could say it's 2/5, the fraction 2/5, which would sometimes be read as 2 to 5. This is also, when it's written this way, you could also read that as a ratio, depending on the context. In a sentence like this I would read this as 2/5 of the fruit are apples." + }, + { + "Q": "At 3:38,sal says atooms are joined by covalent bonds ......so my first question is what are covalent bonds and how can we find radius if atom is not joined to anyone??", + "A": "Covalent bonds are bonds between two atoms sharing electrons.", + "video_name": "q--2WP8wXtk", + "transcript": "Voiceover: Let's think a little bit about the notion of atomic size or atomic radius in this video. At first thought, you might think well this might be a fairly straight-forward thing. If I'm trying to calculate the radius of some type of circular object I'm just thinking about well what's the distance between the center of that circular object and the edge of it. So the length of this line right over here. That would be the radius. And so a lot of people when they conceptualize an atom they imagine a positive nucleus with the protons in the center right over here then they imagine the electrons on these fixed orbits around that nucleus so they might imagine some electrons in this orbit right over here, just kind of orbiting around and then there might be a few more on this orbit out here orbiting around, orbiting around out here. And you might say, \"well okay, that's easy to figure out the atomic radius. I just figure out the distance between the nucleus and the outermost electron and we could call that the radius.\" That would work except for the fact that this is not the right way to conceptualize how electrons or how they move or how they are distributed around a nucleus. Electrons are not in orbits the way that planets are in orbit around the sun and we've talked about this in previous videos. They are in orbitals which are really just probability distributions of where the electrons can be, but they're not that well defined. So, you might have an orbital, and I'm just showing you in 2 dimensions. It would actually be in 3 dimensions, where maybe there's a high probability that the electrons where I'm drawing it in kind of this more shaded in green. But there's some probability that the electrons are in this area right over here and some probability that the electrons are in this area over here, and let's say even a lower probability that the electrons are over this, like this over here. And so you might say, well at a moment the electron's there. The outermost electron we'd say is there. You might say well that's the radius. But in the next moment, there's some probability it might be likely that it ends up here. But there's some probability that it's going to be over there. Then the radius could be there. So electrons, these orbitals, these diffuse probability distributions, they don't have a hard edge, so how can you say what the size of an atom actually is? There's several techniques for thinking about this. One technique for thinking about this is saying, okay, if you have 2 of the same atom, that are- 2 atoms of the same element that are not connected to each other, that are not bonded to each other, that are not part of the same molecule, and you were able to determine somehow the closest that you could get them to each other without them bonding. So, you would kind of see, what's the closest that they can, they can kind of get to each other? So let's say that's one of them and then this is the other one right over here. And if you could figure out that distance, that closest, that minimum distance, without some type of, you know, really, I guess, strong influence happening here, but just the minimum distance that you might see between these 2 and then you could take half of that. So that's one notion. That's actually called the Van der Waals radius. Another way is well what about if you have 2 atoms, 2 atoms of the same element that are bonded to each other? They're bonded to each other through a covalent bond. So a covalent bond, we've already- we've seen this in the past. The most famous of covalent bonds is well, a covalent bond you essentially have 2 atoms. So that's the nucleus of one. That's the nucleus of the other. And they're sharing electrons. So their electron clouds actually, their electron clouds actually overlap with each other, actually overlap with each other so the covalent bond, there the electrons in that bond could spend some of their time on this atom and some of their time on this atom right over here. And so when you have a covalent bond like this, you can then find the distance between the 2 nuclei and take half of that and call that call that the atomic radius. So these are all different ways of thinking about it. Now, with that out of the way, let's think about what the trends for atomic size or atomic radii would be in the periodic table. So the first thing to think about is what do you think will be the trend for atomic radii as we move through a period. So let's say we're in the fourth period and we were to go from potassium to krypton. What do you think is going to be the trend here? And if you want to think about the extremes, how do you think potassium is going to compare to krypton in terms of atomic radius. I encourage you to pause this video and think about that on your own. Well, when you're in the fourth period, the outermost electrons are going to be in your fourth shell. Here, you're filling out 4S1, 4S2. Then you start back filling into the 3D subshell and then you start filling again in 4P1 and so forth. You start filling out the P subshell. So as you go from potassium to krypton, you're filling out that outermost fourth shell. Now what's going on there? Well, when you're at potassium, you have 19- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. You have 19 protons and you have 19 electrons. Well I'll just draw those. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, but you only have 1 electron in that outermost, in that fourth shell, so let's just say that's that electron at a moment, just for visually. It doesn't necessarily have to be there but just to visualize that. So that 1 electron right over there, you have 19, yeah, you have 19 protons. So, you have some, I guess you could say Coulom force that is attracting it, that is keeping it there. But if you go to krypton, all of a sudden you have much more positive charge in the nucleus. So you have 1, 2, 3, 4, 5, 6, 7, 8- I don't have to do them all. You have 36. You have a positive charge of 36. Let me write that, you have plus 36. Here you have plus 19. And you have 36 electrons, you have 36 electrons- I don't know, I've lost track of it, but in your outermost shell, in your fourth, you're going to have the 2S and then you're going to have the 6P. So you have 8 in your outermost shell. So that'd be 1, 2, 3, 4, 5, 6, 7, 8. So one way to think about it, if you have more positive charge in the center, and you have more negative charge on that outer shell, so that's going to bring that outer shell inward. It's going to have more I guess you could imagine one way, more Coulomb attraction right over there. And because of that, that outer most shell is going to drawn in. Krypton is going to be smaller, is going to have a smaller atomic radius than potassium. So the trend, as you go to the right is that you are getting, and the general trend I would say, is that you are getting smaller as you go to the right in a period. That's the reason why the smallest atom of all, the element with the smallest atom is not hydrogen, it's helium. Helium is actually smaller than hydrogen, depending on how you, depending on what technique you use to measure it. That's because, if we take the simplest case, hydrogen, you have 1 proton in the nucleus and then you have 1 electron in that 1S shell, and in helium you have 2, 2 protons in the nucleus and I'm not drawing the neutrons and obviously there's different isotopes, different numbers of neutrons, but you have 2 electrons now in your outer most shell. So you have more, I guess you could say, you could have more Coulomb attraction here. This is plus 2 and then these 2 combined are negative 2. They're going to be drawn inward. So, that's the trend as we go to the right, as we go from the left to the right of the periodic table, we're getting smaller. Now what do you think is going to happen as we go down the period table? As we go down the periodic table in a given group? Well, as we go down a group, each new element down the group, we're adding, we're in a new period. We're adding a new shell. So you're adding more and more and more shells. Here you have just the first shell, now the second shell and each shell is getting further and further and further away. So as you go down the periodic table, you are getting, you are getting larger. You're having a larger atomic radius depending on how you are measuring it. So what's the general trend? Well if you get larger as you go down, that means you're getting smaller as you go up. You get smaller, smaller as you go up. So, what are going to be, what's going to be the smallest ones? Well, we've already said helium is the smallest. So what are going to be some of the largest? What are going to be some of the largest atoms? Well that's going to be the atoms down here at the bottom left. So, these are going to be large, these are going to be small. So, large over here, small over here and the general trend, as you go from the bottom left to the top right you are getting, you are getting smaller." + }, + { + "Q": "Do you think 4th grade and 3rd grade are easy?I do", + "A": "im 7th grade and 5th and the rest are easy", + "video_name": "NWJinKmWzx8", + "transcript": "- [Voiceover] What I hope to do in this video is get a little more practice and intuition when we're multiplying multi-digit numbers. So let's say that we wanted to calculate what 7,000 times six is. 7,000 times six. Now, for some of you, it might just jump out at you. That, hey look, if I have seven of anything, and here I have seven thousands, and I multiply that by six, I'm now going to have seven times six of that thing, or 42 of that thing, and in this case we have 42 thousands. So you might just be able to cut to the chase and say, hey look, six times seven thousands is going to be 42 thousands. And that's great if you can just cut to the chase like that, and another way to think about it is like, look, six times seven is 42, and then since we're talking about thousands, we're not just talking about seven, we're talking about seven thousands. I have three zeros here, so I'm going to have 42 thousands, three zeros there. But I want to make sure that we really understand what is going on here. This will also help us with a little bit of practice of our multiplication properties. So 7,000 is the same thing as 1,000 times seven, or seven times 1,000. It's seven thousands. Or you could view it as a thousand sevens, either way. So this is the same thing as 1,000 times seven times six. And so you could view it as, you could do 1,000 times seven first, which would be 7,000, and then times six. Or you could do the seven times six first, and this is, this right over here is the associative property of multiplication. It sounds very fancy, but it just says that, hey look, we can multiply the seven times six first, before we multiply by the thousand. So we could rewrite this as 1,000 times, and if we're going to do the seven times six first we can put the parentheses around that, times seven, times six. Seven times six. Notice, it's 1,000 times seven times six. I could do 1,000 times seven first to get 7,000, or I could do the seven times six first to get, and you know where there is going, so if you multiply the seven times six first, you're going to get 42, and you're gong to have 1,000 times 42. 1,000 times 42. So you can view this as a thousand 42's or maybe a little bit more intuitively you could view this as 42 thousands. So, once again, we get to 42,000. And so the whole reason, some of y'all might have just been able to do this immediately in your head and that's all good, but it's good to understand what's actually going on here. And the reason why I also broke it up that way, this way, is that the exercises on Kahn Adacemy make you do this to make sure that you really are understanding how to break up these numbers and how you could re-associate when you multiply. Let's do another one. Let's say that we wanted to figure out, let's say that we wanted to figure out, let me give ourselves some space, let's say that we wanted to figure out what 56 times eight is. And there is a bunch of different ways that you could do it. You could say that, look, 56, this is the same thing as 50, five tens, that's 50 plus six ones, so 50 plus six, and all of that times eight. And then you could distribute the eight and you could say, look, this is going to be 50 times eight, So it's going to be 50 times eight, plus six times eight. Plus six times eight. And 50 times eight? Well, five times eight is 40, but we're not just saying five, we're saying five tens, so five tens times eight is going to be 40 tens, or it's going to be 400. Another way to think about it, five times eight is 40, but we're not talking about five, we're talking about five tens, so it's going to be 40 tens. So 50 times eight is 400, and then six times eight is, of course, equal to 48. So this is going to be equal to 448. This is actually how I do things in my head. When I do it in my head I obviously am not writing things down like this but I think, okay, 56 times eight, I could break that up into 50 and six, and eight times 50, well that's 400, or 40 tens you could say. Eight times five tens is going to be 40 tens, or 400. And then eight times six is going to be 48, so it's going to be 400 plus 48. Once you get some practice you're going to be able to do things like this in your head. And if it helps, we can also visualize this looking at an area of a rectangle. So imagine this rectangle right over here, and let's say that this dimension right over here is eight, it is eight units tall. So that's the eight. And this entire dimension, this entire length here is 56. So the area of this rectangle is going to be 56 times eight, which is what we set to figure out, and to do that, well we could break it up into 50 and six, so this first section right over here, this has length, we could say this has length 50, that has length 50, and then this second section, this has length six. This has length six. And the reason why we broke it up this way is cause we can, maybe in our heads, or without too much work, figure out what eight times 50 is and then separately figure out what eight times six is. So separately figure out the areas of these two pieces of the big rectangle and then add them together. So what's eight times five tens? Well it's going to be 40 tens, or 400. This is going to be 400 square units, is going to be the area of this yellow part. And then what's eight times six? Well we know that's going to be 48 square units. So the entire rectangle is going to be the eight times the 50, or the 50 times the eight, the 400, this area, the yellow area, plus the magenta area, plus the eight times the six, the 48, which is 448. 448 is going to be the area of the whole thing. Eight times 56." + }, + { + "Q": "At 0:21, what does Sal mean by \" arbrutary\"?", + "A": "Any old An arbitrary angle measure is a random angle or just any old angle", + "video_name": "0gzSreH8nUI", + "transcript": "Thought I would do some more example problems involving triangles. And so this first one, it says the measure of the largest angle in a triangle is 4 times the measure of the second largest angle. The smallest angle is 10 degrees. What are the measures of all the angles? Well, we know one of them. We know it's 10 degrees. Let's draw an arbitrary triangle right over here. So let's say that is our triangle. We know that the smallest angle is going to be 10 degrees. And I'll just say, let's just assume that this right over here is the measure of the smallest angle. It's 10 degrees. Now let's call the second largest angle-- let's call that x. So the second largest angle, let's call that x. So this is going to be x. And then the first sentence, they say the measure of the largest angle in a triangle is 4 times the measure of the second largest angle. So the second largest angle is x. 4 times that measure is going to be 4x. So the largest angle is going to be 4x. And so the one thing we know about the measures of the angles inside of a triangle is that they add up to 180 degrees. So we know that 4x plus x plus 10 degrees is going to be equal to 180 degrees. It's going to be equal to 180. And 4x plus x, that just gives us 5x. And then we have 5x plus 10 is equal to 180 degrees. Subtract 10 from both sides. You get 5x is equal to 170. And so x is equal to 170/5. And let's see, it'll go into it-- what is that, 34 times? Let me verify this. So 5 goes into-- yeah, it should be 34 times because it's going to go into it twice as many times as 10 would go into it. 10 would go into 170 17 times. 5 would go into 170 34 times. So we could verify it. Go into 170. 5 goes into 17 three times. 3 times 5 is 15. Subtract, you get 2. Bring down the 0. 5 goes into 20 four times, and then you're not going to have a remainder. 4 times 5 is 20. No remainder. So it's 34 times. So x is equal to 34. So the second largest angle has a measure of 34 degrees. This angle up here is going to be 4 times that. So 4 times 34-- let's see, that's going to be 120 degrees plus 16 degrees. This is going to be 136 degrees. Is that right? 4 times 4 is 16, 4 times 3 is 120, 16 plus 120 is 136 degrees. So we're done. The three measures, or the sizes of the three angles, are 10 degrees, 34 degrees, and 136 degrees. Let's do another one. So let's see. We have a little bit of a drawing here. And what I want to do is-- and we could think about different things. We could say, let's solve for x. I'm assuming that 4x is the measure of this angle. 2x is the measure of that angle right over there. We can solve for x. And then if we know x, we can figure out what the actual measures of these angles are, assuming that we can figure out x. And the other thing that they tell us is that this line over here is parallel to this line over here. And it was very craftily drawn. Because it's parallel, but one stops here, and then one starts up there. So the first thing I want to do-- if they're telling us that these two lines are parallel, there's probably going to be something involving transversals or something. It might be something involving-- the other option is something involving triangles. And at first, you might say, wait, is this angle and that angle vertical angles? But you have to be very careful. This is not the same line. This line is parallel to that line. This line, it's bending right over there, so we can't make any type of assumption like that. So the interesting thing-- and I'm not sure if this will lead in the right direction-- is to just make it clear that these two are part of parallel lines. So I could continue this line down like this. And then I can continue this line up like that. And then that starts to look a little bit more like we're used to when we're dealing with parallel lines. And then this line segment, BC-- or we could even say line BC, if we were to continue it on. If we were to continue it on and on, even pass D, then this is clearly a transversal of those two parallel lines. This is clearly a transversal. And so if this angle right over here is 4x, it has a corresponding angle. Half of the-- or maybe most of the work on all of these is to try to see the parallel lines and see the transversal and see the things that might be useful for you. So that right there is the transversal. These are the parallel lines. That's one parallel line. That is the other parallel line. You can almost try to zone out all of the other stuff in the diagram. And so if this angle right over here is 4x, it has a corresponding angle where the transversal intersects the other parallel line. This right here is its corresponding angle. So let me draw it in that same yellow. This right over here is a corresponding angle. So this will also be 4x. And we see that this angle-- this angle and this angle, this angle that has measure 4x and this angle that measures 2x-- we see that they're supplementary. They're adjacent to each other. Their outer sides form a straight angle. So they're supplementary, which means that their measures add up to 180 degrees. They kind of form-- they go all the way around like that if you add the two adjacent angles together. So we know that 4x plus 2x needs to be equal to 180 degrees, or we get 6x is equal to 180 degrees. Divide both sides by 6. You get x is equal to 30, or x is equal to-- well, I shouldn't say-- well, x could be 30. And then this angle right over here is 2 times x. So it's going to be 60 degrees. So this angle right over here is going to be 60 degrees. And this angle right over here is 4 times x. So it is 120 degrees, and we're done." + }, + { + "Q": "Is hydrogen an balanced element??", + "A": "What is the difference between the diatomic elemental molecule H2 and an atom of He?", + "video_name": "FmQoSenbtnU", + "transcript": "In the last few videos we learned that the configuration of electrons in an atom aren't in a simple, classical, Newtonian orbit configuration. And that's the Bohr model of the electron. And I'll keep reviewing it, just because I think it's an important point. If that's the nucleus, remember, it's just a tiny, tiny, tiny dot if you think about the entire volume of the actual atom. And instead of the electron being in orbits around it, which would be how a planet orbits the sun. Instead of being in orbits around it, it's described by orbitals, which are these probability density functions. So an orbital-- let's say that's the nucleus it would describe, if you took any point in space around the nucleus, the probability of finding the electron. So actually, in any volume of space around the nucleus, it would tell you the probability of finding the electron within that volume. And so if you were to just take a bunch of snapshots of electrons -- let's say in the 1s orbital. And that's what the 1s orbital looks like. You can barely see it there, but it's a sphere around the nucleus, and that's the lowest energy state that an electron can be in. If you were to just take a number of snapshots of electrons. Let's say you were to take a number of snapshots of helium, which has two electrons. Both of them are in the 1s orbital. It would look like this. If you took one snapshot, maybe it'll be there, the next snapshot, maybe the electron is there. Then the electron is there. Then the electron is there. Then it's there. And if you kept doing the snapshots, you would have a bunch of them really close. And then it gets a little bit sparser as you get out, as you get further and further out away from the electron. But as you see, you're much more likely to find the electron close to the center of the atom than further out. Although you might have had an observation with the electron sitting all the way out there, or sitting over here. So it really could have been anywhere, but if you take multiple observations, you'll see what that probability function is describing. It's saying look, there's a much lower probability of finding the electron out in this little cube of volume space than it is in this little cube of volume space. And when you see these diagrams that draw this orbital like this. Let's say they draw it like a shell, like a sphere. And I'll try to make it look three-dimensional. So let's say this is the outside of it, and the nucleus is sitting some place on the inside. They're just saying -- they just draw a cut-off -- where can I find the electron 90% of the time? So they're saying, OK, I can find the electron 90% of the time within this circle, if I were to do the cross-section. But every now and then the electron can show up outside of that, right? Because it's all probabilistic. So this can still happen. You can still find the electron if this is the orbital we're talking about out here. Right? And then we, in the last video, we said, OK, the electrons fill up the orbitals from lowest energy state to high energy state. You could imagine it. If I'm playing Tetris-- well I don't know if Tetris is the thing-- but if I'm stacking cubes, I lay out cubes from low energy, if this is the floor, I put the first cube at the lowest energy state. And let's say I could put the second cube But I only have this much space to work with. So I have to put the third cube at the next highest energy state. In this case our energy would be described as potential energy, right? This is just a classical, Newtonian physics example. But that's the same idea with electrons. Once I have two electrons in this 1s orbital -- so let's say the electron configuration of helium is 1s2 -- the third electron I can't put there anymore, because there's only room for two electrons. The way I think about it is these two electrons are now going to repel the third one I want to add. So then I have to go to the 2s orbital. And now if I were to plot the 2s orbital on top of this one, it would look something like this, where I have a high probability of finding the electrons in this shell that's essentially around the 1s orbital, right? So right now, if maybe I'm dealing with lithium right now. So I only have one extra electron. So this one extra electron, that might be where I observed that extra electron. But every now and then it could show up there, it could show up there, it could show up there, but the high probability is there. So when you say where is it going to be 90% of the time? It'll be like this shell that's around the center. Remember, when it's three-dimensional you would kind of cover it up. So that's what they drew here. They do the 1s. It's just a red shell. And then the 2s. The second energy shell is just this blue shell over it. And you can see it a little bit better in, actually, the higher energy orbits, the higher energy shells, where the seventh s energy shell is this red area. Then you have the blue area, then the red, and the blue. And so I think you get the idea that each of those are energy shells. So you kind of keep overlaying the s energy orbitals around each other. But you probably see this other stuff here. And the general principle, remember, is that the electrons fill up the orbital from lowest energy orbital to higher energy orbital. So the first one that's filled up is the 1s. This is the 1. This is the s. So this is the 1s. It can fit two electrons. Then the next one that's filled up is 2s. It can fill two more electrons. And then the next one, and this is where it gets interesting, you fill up the 2p orbital. 2p orbital. That's this, right here. 2p orbitals. And notice the p orbitals have something, p sub z, p sub x, p sub y. What does that mean? Well, if you look at the p-orbitals, they have these dumbbell shapes. They look a little unnatural, but I think in future videos we'll show you how they're analogous to standing waves. But if you look at these, there's three ways that you can configure these dumbbells. One in the z direction, up and down. One in the x direction, left or right. And then one in the y direction, this way, forward and backwards, right? And so if you were to draw-- let's say you wanted to draw the p-orbitals. So this is what you fill next. And actually, you fill one electron here, another electron here, then another electron there. Then you fill another electron, and we'll talk about spin and things like that in the future. But, there, there, and there. And that's actually called Hund's rule. Maybe I'll do a whole video on Hund's rule, but that's not relevant to a first-year chemistry lecture. But it fills in that order, and once again, I want you to have the intuition of what this would look like. Look. I should put look in quotation marks, because it's very abstract. But if you wanted to visualize the p orbitals-- let's say we're looking at the electron configuration for, let's say, carbon. So the electron configuration for carbon, the first two electrons go into, so, 1s1, 1s2. So then it fills-- sorry, you can't see everything. So it fills the 1s2, so carbon's configuration. It fills 1s1 then 1s2. And this is just the configuration for helium. And then it goes to the second shell, which is the second period, right? That's why it's called the periodic table. We'll talk about periods and groups in the future. And then you go here. So this is filling the 2s. We're in the second period right here. That's the second period. One, two. Have to go off, so you can see everything. So it fills these two. So 2s2. And then it starts filling up the p orbitals. So then it starts filling 1p and then 2p. And we're still on the second shell, so 2s2, 2p2. So the question is what would this look like if we just wanted to visualize this orbital right here, the p orbitals? So we have two electrons. So one electron is going to be in a-- Let's say if this is, I'll try to draw some axes. That's too thin. So if I draw a three-dimensional volume kind of axes. If I were to make a bunch of observations of, say, one of the electrons in the p orbitals, let's say in the pz dimension, sometimes it might be here, sometimes it might be there, sometimes it might be there. And then if you keep taking a bunch of observations, you're going to have something that looks like this bell shape, this barbell shape right there. And then for the other electron that's maybe in the x direction, you make a bunch of observations. Let me do it in a different, in a noticeably different, color. It will look like this. You take a bunch of observations, and you say, wow, it's a lot more likely to find that electron in kind of the dumbell, in that dumbbell shape. But you could find it out there. You could find it there. You could find it there. This is just a much higher probability of finding it in here than out here. And that's the best way I can think of to visualize it. Now what we were doing here, this is called an electron configuration. And the way to do it-- and there's multiple ways that are taught in chemistry class, but the way I like to do it -- is you take the periodic table and you say, these groups, and when I say groups I mean the columns, these are going to fill the s subshell or the s orbitals. You can just write s up here, just right there. These over here are going to fill the p orbitals. Actually, let me take helium out of the picture. The p orbitals. Let me just do that. Let me take helium out of the picture. These take the p orbitals. And actually, for the sake of figuring out these, you should take helium and throw it right over there. Right? The periodic table is just a way to organize things so it makes sense, but in terms of trying to figure out orbitals, you could take helium. Let me do that. The magic of computers. Cut it out, and then let me paste it right over there. Right? And now you see that helium, you get 1s and then you get 2s, so helium's configuration is -- Sorry, you get 1s1, then 1s2. We're in the first energy shell. Right? So the configuration of hydrogen is 1s1. You only have one electron in the s subshell of the first energy shell. The configuration of helium is 1s2. And then you start filling the second energy shell. The configuration of lithium is 1s2. That's where the first two electrons go. And then the third one goes into 2s1, right? And then I think you start to see the pattern. And then when you go to nitrogen you say, OK, it has three in the p sub-orbital. So you can almost start backwards, right? So we're in period two, right? So this is 2p3. Let me write that down. So I could write that down first. 2p3. So that's where the last three electrons go into the p orbital. Then it'll have these two that go into the 2s2 orbital. And then the first two, or the electrons in the lowest energy state, will be 1s2. So this is the electron configuration, right here, of nitrogen. And just to make sure you did your configuration right, what you do is you count the number of electrons. So 2 plus 2 is 4 plus 3 is 7. And we're talking about neutral atoms, so the electrons should equal the number of protons. The atomic number is the number of protons. So we're good. Seven protons. So this is, so far, when we're dealing just with the s's and the p's, this is pretty straightforward. And if I wanted to figure out the configuration of silicon, right there, what is it? Well, we're in the third period. One, two, three. That's just the third row. And this is the p-block right here. So this is the second row in the p-block, right? One, two, three, four, five, six. Right. We're in the second row of the p-block, so we start off with 3p2. And then we have 3s2. And then it filled up all of this p-block over here. So it's 2p6. And then here, 2s2. And then, of course, it filled up at the first shell before it could fill up these other shells. So, 1s2. So this is the electron configuration for silicon. And we can confirm that we should have 14 electrons. 2 plus 2 is 4, plus 6 is 10. 10 plus 2 is 12 plus 2 more is 14. So we're good with silicon. I think I'm running low on time right now, so in the next video we'll start addressing what happens when you go to these elements, or the d-block. And you can kind of already guess what happens. We're going to start filling up these d orbitals here that have even more bizarre shapes. And the way I think about this, not to waste too much time, is that as you go further and further out from the nucleus, there's more space in between the lower energy orbitals to fill in more of these bizarro-shaped orbitals. But these are kind of the balance -- I will talk about standing waves in the future -- but these are kind of a balance between trying to get close to the nucleus and the proton and those positive charges, because the electron charges are attracted to them, while at the same time avoiding the other electron charges, or at least their mass distribution functions. Anyway, see you in the next video." + }, + { + "Q": "Can 2 sperms fertilize one egg?", + "A": "The correct answer is if that happens hydatiform mole will occur. either partial or complete.", + "video_name": "VYSFNwTUkG0", + "transcript": "- We're gonna talk about the ovarian cycle. The ovaries are two structures in a female's reproductive system that produce her eggs. Each month her eggs go through a maturation process called the ovarian cycle, and that cycle creates a secondary oocyte than can be then fertilized by a sperm to result in a pregnancy. The ovarian cycle is also responsible for what we commonly know as the menstrual cycle. Basically, the primary oocytes that are destined to be ovulated will develop in the ovaries, complete meiosis one just before ovulation, and then they'll be ejected out of the ovary as a secondary oocyte to be picked up by the fimbriae and swept into the uterine tube to hope for fertilization. So let's start from the beginning. Inside the ovaries, eggs develop in structures called follicles, these purple circles here. And they start off as primordial follicles. And so what a follicle is- I'll just blow that up for you- It's one primary oocyte, so an egg cell, surrounded by a layer of cells called granulosa cells. And the granulosa cells develop and become more numerous as the follicle matures. Now the granulosa cells also secrete a few hormones. Estrogen, a little progesterone and some inhibin, and we'll talk about the functions of those a little bit later on. So let's put a timeline on this. Now the ovarian cycle lasts 28 days. This is day zero here at the primordial follicle, where we're going counter-clockwise. All the way over here, this is day 13. Here, where the secondary oocyte gets ejected, or ovulated, that's day 14. And then the rest of the time spent getting back to the primordial follicle stage are days 15 through 28. So now you have an idea of about how long this all takes. So you remember when we said that the granulosa cells produce hormones? Well, as the follicles develop over the first 13 days, and you can see the changes between the one here and the one here. It's got a lot more purple cells around here. Those are granulosa cells. So the number of granulosa cells goes up, and since they produce hormones, what do you think happens to the hormone levels in the blood? They go up. So that's sort of just a general point. So keep that in mind, but first we'll jump back to these. We know these are primordial follicles here. The next stage of development are these guys here, and these are called primary follicles. And in the primary follicles, the layers of granulosa cells and the oocyte, the egg, start to be separated by this other layer that starts to form between them. That's called the zona pellucida, and I'll draw it here in light blue. And even though the egg I've drawn in blue, there's still a layer of zona pellucida, even though the egg is originally drawn in blue because I wanted to draw the egg in blue. There's still a layer of zone pellucida around it. Now even though the zona pellucida is there separating the granulosa cells from the actual egg, the granulosa cells can still nourish the egg through gap junctions that go through the zona pellucida and into the egg. Gap junctions are just little passageways from one cell to another cell where they can exchange nutrients or other signals. And actually, through those gap junctions, the granulosa cells send through little chemicals that keep those primary oocytes stuck at that meiosis one stage, 'cause you remember at this point all of these primary oocytes are stuck in meiotic arrest. They're not dividing and reducing their chromosome copy number. So as we develop from our primordial to our primary to our next follicle here, called our pre-antral follicle, and you'll see why it's called that in a minute, the granulosa cells are actually starting to divide and become a lot greater in number. You can see that there's a pretty big difference in granulosa cell number from our primary follicles to our pre-antral follicle here. And remember the granulosa cells are shaded in in purple here. So while the granulosa cells are proliferating, this wall on the outside of the follicle called the theca starts to form. Theca cells have receptors for luteinizing hormone from the anterior pituitary, and when luteinizing hormone, or LH, binds these theca cells, they produce a hormone called androstenedione. And when the thecas get androstenedione, they give it to the granulosa cells, who then convert it to estrogen and release it into the blood. So the blood estrogen levels start to go up at this point. And so that's what these red and blue bits running down the middle of the ovary are, blood vessels, arteries and veins. And if they look a little bit weird to you, or unusual, that's just because they're cut in cross-section as well. Now you might be wondering what an antral refers to, like what you see in the pre-antral follicle and this early antral follicle here. It actually refers to the antrum, which will be formed in the next step. This space here is called an antrum. And the antrum is just basically fluid that's being produced by the granulosa cells. And it's that antrum and the fluid in the antrum that pushes against the edges of the follicle and causes it to expand. Now just so you're aware, during this ovarian cycle, multiple follicles are actually forming. It's not just this one pre-antral follicle, and then this one early antral follicle, and this one mature follicle. You're getting a lot of these happening at one time. But only one of the biggest ones is the one that eventually gets ovulated, because you only ovulate one egg every 28 days. And that one that gets ovulated is called the dominant follicle. So let's just say that what we're seeing here is an example of the dominant follicle's development. Because the rest of the ones that were developing along this pathway sort of degenerate and die off in a process called atresia. So I'll write that at the bottom here. And atresia just means to degenerate. So another note. In the ones that undergo atresia, both the follicle and the eggs they contain die off. And that means that a woman loses anywhere between 15 to 25 eggs per menstrual cycle to atresia, while only one gets ovulated. So you can kind of imagine how you go from two to four million eggs when you were born to having zero after about 35-ish years of ovulation. It's not just that one egg you lose by ovulation. You lose quite a few. So anyway, back to the development of the dominant follicle. It enlarges mostly due to the expanding antrum, as I mentioned earlier. And granulosa cells actually start to form this bit of a mound here that protrudes into the middle of the antrum. This mound of granulosa cells is called the cumulus oophorus. As part of the development of the dominant follicle, the cumulus oophorus and the egg sort of separate together from the wall of the follicle and float around in the middle of the antrum, like a little island. And the follicle increases in size. So the actual follicle is increasing in size as it gets filled with more and more fluid from the granulosa cells. And the granulosa cells are just producing fluid as a by-product of their metabolism and creation of hormones. Eventually this dominant follicle, which at this point is called the mature follicle, it starts to balloon out the side of the ovary, kind of like this. Just starts to push out against the edge of the ovary. And then because the edge of the ovary and the wall of the mature follicle are in such close proximity, enzymes within the follicle break down that common wall between them, and the egg pops out onto the surface of the ovary, because now this wall is broken down. And by the way, an enzyme is a protein that carries out a specific task. The task here is to break down the wall between the mature follicle and the ovary, and that happens on day 14. So it takes day zero to 13 of build up to get to this event. When this happens, some women feel a little bit of pelvic pain. And actually sometimes, by chance, two or more follicles reach maturity, and they all pop out. And that's how you get twins or triplets or quadruplets or octuplets, when they all pop out and get fertilized by different sperm each. Because they're all subsequently swept up into the uterine tubes where sperm can fertilize them. So now you have the egg out here, but what about the old follicle it was in? The follicle actually collapses a little and transforms into a structure called the corpus luteum. And in this transformation the granulosa cells get a lot bigger and start to produce more estrogen, progesterone and that other hormone, inhibin, that we mentioned before. Just briefly, inhibin lowers the amount of FSH, follicle stimulating hormone, that comes from the anterior pituitary. And it does that because follicle stimulating hormone actually propagates this whole process of follicle maturation, as you can imagine from the name. So if you didn't know this before, these are the exact follicles that follicle stimulating hormone refers to. At least in a female. Anyway, if the egg doesn't get fertilized, then the corpus luteum reaches a maximum size in about 10 days. So that's about day 25, which it's probably sitting at in this diagram. And then it degenerates by apoptosis. That's a process that cells use to sort of self-destruct and die off. And here I'm abbreviating corpus luteum as CL, just so you know what I mean. But if the egg is fertilized, i.e., it travels into the uterine tubes and gets fertilized by a sperm, then the corpus luteum persists, I mean it keeps living, because we want it to keep producing estrogen and progesterone. That's because estrogen and progesterone prepare the inner lining of the uterus, that's called the endometrium, for implantation, which would be really handy since we have a fertilized egg now that needs to develop. And that's where it does it, by implanting in the endometrium of the uterus. So just a final note. Ovulation doesn't happen forever. At about age 50 to 51, females undergo something called menopause. First menstrual cycles become less and less regular. In other words, they don't happen every 28 days like they do when you're under the age of 50. And then ultimately, they stop happening entirely. And that cessation of ovulation is called menopause. The main cause of menopause is sometimes referred to as ovarian failure. Basically the ovaries lose the ability to respond to signalling hormones from the brain called gonadotropins. And we know these as LH and FSH. And this happens because most, or all of the follicles and eggs have already gone through that process that we talked about called atresia. In other words, they've degenerated." + }, + { + "Q": "The solution to IXI=5 ARE THE same distance from 0. What can you say abou the solutions to Ix-2I=5?", + "A": "They are the same distance from whatever makes the expression inside the | | zero. |x| - the expression is 0 when x=0, so the solutions are the same distance from 0. |x - 2| - the expression is 0 when x - 2 = 0. That happens when x = 2, so the solutions are the same distance from 2.", + "video_name": "D1cKk48kz-E", + "transcript": "Solve for c and graph the solution. We have negative 5c is less than or equal to 15. So negative 5c is less than or equal to 15. I just rewrote it a little bit bigger. So if we want to solve for c, we just want to isolate the c right over here, maybe on the left-hand side. It's right now being multiplied by negative 5. So the best way to just have a c on the left-hand side is we can multiply both sides of this inequality by the inverse of negative 5, or by negative 1/5. So we want to multiply negative 1/5 times negative 5c. And we also want to multiply 15 times negative 1/5. I'm just multiplying both sides of the inequality by the inverse of negative 5, because this will cancel out with the negative 5 and leave me just with c. Now I didn't draw the inequality here, because we have to remember, if we multiply or divide both sides of an inequality by a negative number, you have to flip the inequality. And we are doing that. We are multiplying both sides by negative 1/5, which is the equivalent of dividing both sides by negative 5. So we need to turn this from a less than or equal to a greater than or equal. And now we can proceed solving for c. So negative 1/5 times negative 5 is 1. So the left-hand side is just going to be c is greater than or equal to 15 times negative 1/5. That's the same thing as 15 divided by negative 5. And so that is negative 3. So our solution is c is greater than or equal to negative 3. And let's graph it. So that is my number line. Let's say that is 0, negative 1, negative 2, negative 3. And then I could go above, 1, 2. And so c is greater than or equal to negative 3. So it can be equal to negative 3. So I'll fill that in right over there. Let me do it in a different color. So I'll fill it in right over there. And then it's greater than as well. So it's all of these values I am filling in in green. And you can verify that it works in the original inequality. Pick something that should work. Well, 0 should work. 0 is one of the numbers that we filled in. Negative 5 times 0 is 0, which is less than or equal to 15. It's less than 15. Now let's try a number that's outside of it. And I haven't drawn it here. I could continue with the number line in this direction. We would have a negative 4 here. Negative 4 should not be included. And let's verify that negative 4 doesn't work. Negative 4 times negative 5 is positive 20. And positive 20 is not less than 15, so it's good that we did not include negative 4. So this is our solution. And this is that solution graphed. And I wanted to do that in that other green color. That's what it looks like." + }, + { + "Q": "I was taught that a double bond is a pi bond (and only a pi bond) not a sigma bond + a pi bond, which is correct? Plus I'm not sure I understand even after reading some of the comments how to get the hybridisation from the normal electronic configuration", + "A": "Whoever taught you was not correct, a double bond does have both a sigma bond and a pi bond. You can t find the hybridisation from the electron configuration. You can tell from steric number, which is the number of bonded atoms plus the number of lone pairs. 4 = sp3 3 = sp2 2 = sp", + "video_name": "ROzkyTgscGg", + "transcript": "Voiceover: In an earlier video, we saw that when carbon is bonded to four atoms, we have an SP3 hybridization with a tetrahedral geometry and an ideal bonding over 109.5 degrees. If you look at one of the carbons in ethenes, let's say this carbon right here, we don't see the same geometry. The geometry of the atoms around this carbon happens to be planar. Actually, this entire molecule is planar. You could think about all this in a plane here. And the bond angles are close to 120 degrees. Approximately, 120 degree bond angles and this carbon that I've underlined here is bonded to only three atoms. A hydrogen, a hydrogen and a carbon and so we must need a different hybridization for each of the carbon's presence in the ethylene molecule. We're gonna start with our electron configurations over here, the excited stage. We have carbons four, valence electron represented. One, two, three and four. In the video on SP3 hybridization, we took all four of these orbitals and combined them to make four SP3 hybrid orbitals. In this case, we only have a carbon bonded to three atoms. We only need three of our orbitals. We're going to promote the S orbital. We're gonna promote the S orbital up and this time, we only need two of the P orbitals. We're gonna take one of the P's and then another one of the P's here. That is gonna leave one of the our P orbitals unhybridized. Each one of these orbitals has one electron and it's like that. This is no longer an S orbital. This is an SP2 hybrid orbital. This is no longer a P orbital. This is an SP2 hybrid orbital and same with this one, an SP2 hybrid orbital. We call this SP2 hybridization. Let me go and write this up here. and use a different color here. This is SP2 hybridization because we're using one S Orbital and two P orbitals to form our new hybrid orbitals. This carbon right here is SP2 hybridized and same with this carbon. Notice that we left a P orbital untouched. We have a P orbital unhybridized like that. In terms of the shape of our new hybrid orbital, let's go ahead and get some more space down here. We're taking one S orbital. We know S orbitals are shaped like spheres. We're taking two P orbitals. We know that a P orbital is shaped like a dumbbell. We're gonna take these orbitals and hybridized them to form three SP2 hybrid orbitals and they have a bigger front lobe and a smaller back lobe here like that. Once again, when we draw the pictures, we're going to ignore the smaller back lobe. This gives us our SP2 hybrid orbitals. In terms of what percentage character, we have three orbitals that we're taking here and one of them is an S orbital. One out of three, gives us 33% S character in our new hybrid SP2 orbital and then we have two P orbitals. Two out of three gives us 67% P character. 33% S character and 67% P character. There's more S character in an SP2 hybrid orbital than an SP3 hybrid orbital and since the electron density in an S orbital is closer to the nucleus. We think about the electron density here being closer to the nucleus that means that we could think about this lobe right here being a little bit shorter with the electron density being closer to the nucleus and that's gonna have an effect on the length of the bonds that we're gonna be forming. Let's go ahead and draw the picture of the ethylene molecule now. We know that each of the carbons in ethylenes. Just going back up here to emphasize the point. Each of these carbons here is SP2 hybridized. That means each of those carbons is going to have three SP2 hybrid orbitals around it and once unhybridized P orbital. Let's go ahead and draw that. We have a carbon right here and this is an SP2 hybridized orbitals. We're gonna draw in. There's one SP2 hybrid orbital. Here's another SP2 hybrid orbital and here's another one. Then we go back up to here and we can see that each one of those orbitals. Let me go ahead and mark this. Each one of those SP2 hybrid orbitals has one electron in it. Each one of these orbitals has one electron. I go back down here and I put in the one electron in each one of my orbitals like that. I know that each of those carbons is going to have an unhybridized P orbital here. An unhybridized P orbital with one electron too. Let me go ahead and draw that in. I'll go ahead and use a different color. We have our unhybridized P orbital like that and there's one electron in our unhybridized P orbital. Each of the carbons was SP2 hybridized. Let me go ahead and draw the dot structure right here again so we can take a look at it. The dot structure for ethylene. Let's do the other carbon now. The carbon on the right is also SP2 hybridized. We can go ahead and draw in an SP2 hybrid orbital and there's one electron in that orbital and then there's another one with one electron and then here's another one with one electron. This carbon being SP2 hybridized also has an unhybridized P orbital with one electron. Go ahead and draw in that P orbital with its one electron. We also have some hydrogens. We have some hydrogens to think about here. Each carbon is bonded to two hydrogens. Let me go ahead and put in the hydrogens. The hydrogen has a valance electron in an unhybridized S orbital. I'm going ahead and putting in the S orbital and the one valance electron from hydrogen like this. When we take a look at what we've drawn here, we can see some head on overlap of orbitals, which we know from our earlier video is called a sigma bond. Here's the head on overlap of orbitals. That's a sigma bond. here's another head on overlap of orbitals. The carbon carbon bond, here's also a head on overlap of orbitals and then we have these two over here. We have a total of five sigma bonds in our molecules. Let me go ahead and write that over here. There are five sigma bonds. If I would try to find those on my dot structure this would be a sigma bond. This would be a sigma bond. One of these two is a sigma bond and then these over here. A total of five sigma bonds and then we have a new type of bonding. These unhybridized P orbitals can overlap side by side. Up here and down here. We get side by side overlap of our P orbitals and this creates a pi bond. A pi bond, let me go ahead and write that here. A pi bond is side by side overlap. There is overlap above and below this sigma bond here and that's going to prevent free rotation. When we're looking at the example of ethane, we have free rotation about the sigma bond that connected the two carbons but because of this pi bond here, this pi bond is going to prevent rotations so we don't get different confirmations of the ethylene molecules. No free rotation due to the pi bonds. When you're looking at the dot structure, one of these bonds is the pi bonds, I'm just gonna say it's this one right here. If you have a double bond, one of those bonds, the sigma bond and one of those bonds is a pi bond. We have a total of one pi bond in the ethylene molecule. If you're thinking about the distance between the two carbons, let me go ahead and use a different color for that. The distance between this carbon and this carbon. It turns out to be approximately 1.34 angstroms, which is shorter than the distance between the two carbons in the ethane molecule. Remember for ethane, the distance was approximately 1.54 angstroms. A double bond is shorter than a single bond. One way to think about that is the increased S character. This increased S character means electron density is closer to the nucleus and that's going to make this lobe a little bit shorter than before and that's going to decrease the distance between these two carbon atoms here. 1.34 angstroms. Let's look at the dot structure again and see how we can analyze this using the concept of steric number. Let me go ahead and redraw the dot structure. We have our carbon carbon double bond here and our hydrogens like that. If you're approaching this situation using steric number remember to find the hybridization. We can use this concept. Steric number is equal to the number of sigma bonds plus number of lone pairs of electrons. If my goal was to find the steric number for this carbon. I count up my number of sigma bonds. That's one, two and then I know when I double bond one of those is sigma and one of those is pi. One of those is a sigma bond. A total of three sigma bonds. I have zero lone pairs of electrons around that carbon. Three plus zero, gives me a steric number of three. I need three hybrid orbitals and we've just seen in this video that three SP2 hybrid orbitals form if we're dealing with SP2 hybridization. If we get a steric number of three, you're gonna think about SP2 hybridization. One S orbital and two P orbitals hybridizing. That carbon is SP2 hybridized and of course, this one is too. Both of them are SP2 hybridized. Let's do another example. Let's do boron trifluoride. BF3. If you wanna draw the dot structure of BF3, you would have boron and then you would surround it with your flourines here and you would have an octet of electrons around each flourine. I go ahead and put those in on my dot structure. If your goal is to figure out the hybridization of this boron here. What is the hybridization stage of this boron? Let's use the concept of steric number. Once again, let's use steric number. Find the hybridization of this boron. Steric number is equal to number of sigma bonds. That's one, two, three. Three sigma bonds plus lone pairs of electrons. That's zero. Steric number of three tells us this boron is SP2 hybridized. This boron is gonna have three SP2 hybrid orbitals and one P orbital. One unhybridized P orbital. Let's go ahead and draw that. We have a boron here bonded to three flourines and also it's going to have an unhybrized P orbital. Now, remember when you are dealing with Boron, it has one last valance electron and carbon. Carbon have four valance electrons. Boron has only three. When you're thinking about the SP2 hybrid orbitals that you create. SP2 hybrid orbital, SP2, SP2 and then one unhybridized P orbital right here. Boron only has three valance electrons. Let's go ahead and put in those valance electrons. One, two and three. It doesn't have any electrons in its unhybridized P orbital. Over here when we look at the picture, this has an empty orbital and so boron can accept a pair of electrons. We're thinking about its chemical behavior, one of the things that BF3 can do, the Boron can accept an electron pair and function as a lewis acid. That's one way in thinking about how hybridizational allows you to think about the structure and how something might react. This boron turns out to be SP2 hybridized. This boron here is SP2 hybridized and so we can also talk about the geometry of the molecule. It's planar. Around this boron, it's planar and so therefore, your bond angles are 120 degrees. If you have boron right here and you're thinking about a circle. A circle is 360 degrees. If you divide a 360 by 3, you get 120 degrees for all of these bond angles. In the next video, we'll look at SP hybridization." + }, + { + "Q": "I believe the title of this section is a little misleading. I sort of was led to think I was just going to rotate the shape on a plane and find the same shape somewhere else on the plane (rotations do not affect area, etc.) but this video is talking about what shape you would create if you have a constant cross-section of the original shape.", + "A": "Reported it as a mistake in the video, Title should be somewhere along the lines of revolutions not rotations", + "video_name": "vdpyWeiHXmU", + "transcript": "- What I want to do in this video is get some practice visualizing what happens if we were to try to rotate two dimensional shapes in three dimensions. Well what do I mean by that? Let's say I started with a right triangle. So let's say my right triangle looks like this. So let's say it looks like that. Right over there. And so this is a right angle. And let's say that this width right over here is three units and let's say that this length is five units and now I'm gonna do something interesting. I'm gonna take this two dimensional right triangle and I'm gonna try to rotate it in three dimensions around this line, around the line that I'm doing as a dotted magenta line. So I'm gonna rotate it around this line right over there. So if I were to rotate it around this line, what type of a shape am I going to get? And I encourage you -- It's going to be a three dimensional shape. I encourage you to think about it, maybe take out a piece of paper, draw it, or just try to imagine it in your head. Well to think about it in three dimensions, what I'm going to do is try to look at this thing in three dimensions. So let me draw this same line but I'm gonna draw it at an angle so we can visualize the whole thing in three dimensions. So imagine if this was sitting on the ground. So that's our magenta line, and then I can draw my triangle. So my triangle would look something like this. So it would look like this. So once again this is five units, this is three units, this a right triangle. I'm gonna rotate it around the line, so what's it gonna look like? Well this and this right over here is gonna rotate around and it's gonna form a circle with a radius of three, right? So it's gonna form, so it intersects, if that was on the ground it's gonna be three again. And let me draw it down so it's gonna keep going down. Whoops. We don't want to press the wrong button. So it's gonna look something like this. That's what the base is gonna look like. But then this end right over here is just gonna stay at a point because this is right on that magenta line. So it's gonna stay at a point. And so if you were to look at the intersect so it would look something like this. So it would look like this and then you'd have another thing that goes like this and so if you were to take a section like this it would have a little smaller circle here based on what this distance is. So what is the shape, what is the shape that I am drawing? Well what you see, what it is, it's a cone. It's a cone and if I shade it in you might see the cone a little bit better. So let me shade it in so you see the cone. So what you end up getting is a cone where it's base, so I'm shading it in so that hopefully helps a little bit, so what you end up getting is a cone where the base has a radius of three units. So let me draw this. This right over here is the radius of the base and it is three units. I could also draw it like this. So the cone is gonna look like this. And this is the tip of the cone and it's gonna look just like this. And once again let me shade it a little bit so that you can appreciate that this is a three dimensional shape. So draw the cone so you can shade it and we can even construct the original so that, well or we can construct the original shape so you see how it constructs so it makes this, the line, that magenta line, is gonna do this type of thing. It's gonna go through the center of the base, it's gonna go through the center of the base just like that. And our original shape, our original right triangle, if you just took a cross section of it that included that line you would have your original shape. Let me do this in orange. So the original shape is right over there. So what do you get? You get a cone where the radius of the base is three units. Interesting." + }, + { + "Q": "is Maurice Ravel French?", + "A": "Yes. Joseph Maurice Ravel (7 March 1875 \u00e2\u0080\u0093 28 December 1937) was a French composer, pianist and conductor. Hope this helps! Take care", + "video_name": "qfqbomqKZqY", + "transcript": "(Maurice Ravel's \"Daphnis et Chloe\") Gerard: Maurice Ravel, one of the great composers of the 20th century. He was commissioned by Diaghilev to do a ballet, \"Daphnis et Chloe.\" Premiered two years after The Firebird. Very much the same cast of characters, the same choreographers, the same dancers, the same huge orchestra except Ravel added a chorus besides the huge orchestra. A kind of similar type story, in this case it was the story about the love between Daphnis and Chloe. Chloe was abducted and then through the intervention of Pan, who was the God of playing the flute, and his love for the Syrinx. Chloe was found, and Daphnis and Chloe lived happily ever after. More or less something like that. From this great ballet that Ravel wrote, he created two suites. Ravel used basically the first part of the ballet for Suite One and the second part of the ballet for Suite Two. Suite No. Two is the one that's done the most often. The piece begins in the most remarkable way. The woodwinds, two flutes and then two clarinets, play these incredibly fast notes. One of the most important orchestral excerpts for these instruments, and hard and fast and soft. The harp playing glissandos underneath that and the double basses just underpinning the whole thing with a melodic gesture, not a melody certainly. It is an incredible moment. It creates such an atmosphere. (Suite Two, Maurice Ravel's \"Daphnis et Chloe\") It is just about to be daybreak. You can just feel how the sun is about to come up. Ravel uses his melody. (piano playing) This melody can go on forever. (piano playing) It's a kind of melody that you can repeat over and over again, it just keeps going. If you think of that little gesture, this little melody, and if you orchestrate it and put it in different instruments and do it in interesting ways, it can be a glorious moment. (orchestral music) The little bird comes in the piccolo, and the solo violins. (Maurice Ravel's \"Daphnis et Chloe\") As the daybreak and the sun is coming up and birds are there, shepherds start to be seen coming through. Obviously, they're getting up and getting ready to take care of their sheep and doing their job in a sense. You can hear the little solo from the piccolo and the little solo from the E-flat clarinet. We have a group of herdsmen come in. For the herdsmen, Ravel uses a slightly different melody. You can hear the similarities. (piano playing) (orchestra playing) We have all of these filigree, the glissandos of the harp, these melodies that are being developed but nothing, no great gestures yet. Some new material comes in, played by the viola and the clarinet. You can feel that what's happening now, of course, is that Daphnis is trying to find Chloe. If you didn't know that, it wouldn't matter. It's what was needed at the moment. You can tell musically, after all these beautiful gestures, you can't just do beautiful gestures forever, and he gets a little agitated section. In the story, Daphnis is looking for Chloe and she appears surrounded by the shepherds. (Maurice Ravel's \"Daphnis et Chloe\") In the next juncture of this ballet, we get to the part where, of course, Chloe has been abducted. Now, Pan is helping Daphnis to find her. In fact, they reverse roles. Daphnis becomes Pan, and Chloe becomes the Syrinx that Pan loved. It leads to the central section of the piece which is this incredible flute solo. The greatest flute solo probably ever written. Very simple accompaniment, pizzicato strings, second and fourth horns, and harp, and this beautiful flute solo. (flute playing) The flute solo becomes more and more agitated, and eventually Chloe falls into Daphnis' arms. It's this mini-concerto for flute section. You see the first flute playing a cascading scale, then the second flute picks it up. The alto flute is the final one. Starts with the piccolo and works its way down in the section. In fact, what Ravel does is he continues that and it continues to be a little flute section concerto right in the middle of this piece. A phenomenal use of the instruments. (Maurice Ravel's \"Daphnis et Chloe\") Finally it comes to an end and at this moment, the nymphs are falling in love and they pledge their love for each other and they dedicate some sheep to their joy together. It is represented by this somewhat gorgeous chorale. (orchestral chorale) This leads us to the first inkling of the fast section. Everything to now has been relatively slow. A build-up for the flutes and then it comes back again and then we have this little chorale for the sheep and the shepherds. The women of the company, in this case the dance company, enter to do a special general dance. It starts out where the music that is going to permeate the rest of the piece is sounded but only a few bars because immediately it comes back down and we hear that same beautiful chorale, the solo for the alto flute, and then the general dance begins. This is the dance that's basically in five. There are a few moments that are in three but basically in five. It's five beats per bar, accent on two, and that in itself is unusual. (Maurice Ravel's \"Daphnis et Chloe\") At one point, after this tremendous build-up, like any great composer, he could have ended it just there. Instead, he brings everything back down. Everything back down to the essence, which is the rhythm and in a very soft way, the snare drum and the double basses play this bom-ba-dom-ba-dom-ba-dom-ba-dom, ba-dom-ba-dom-ba-dom-ba-dom. It starts over again. It's so interesting that composers do this because you know it could end. He brings it back and then when it does end, it's even more exciting. The end of \"Daphnis et Chloe,\" Second Suite, is among the most exciting pieces of music one could ever hear. Using seven percussion, four trumpets, four flutes, oboes, English horn, clarinet, it's a huge orchestra. Full of what we know of Ravel, one of the greatest orchestrators of all time. (Second Suite of Maurice Ravel's \"Daphnis et Chloe\")" + }, + { + "Q": "find the derivatives of f(x)=arc sin 1/x", + "A": "f(x) = arcsin(1/x) f (x) = d/dx (1/x)/\u00e2\u0088\u009a[1 - (1/x)\u00c2\u00b2] f (x) = -1/x\u00c2\u00b2/\u00e2\u0088\u009a[1 - (1/x)\u00c2\u00b2] f (x) = -1/x\u00c2\u00b2\u00e2\u0088\u009a[1 - (1/x)\u00c2\u00b2]`", + "video_name": "FJ7AMaR9miI", + "transcript": "In terms of k, where k does not equal 0, what is the y-intercept of the line tangent to the curve f of x is equal to 1/x at the point of the curve where x is equal to k? So let's just think about what they're asking. So if I were to draw myself-- let me draw some quick axes right over here. So that's my y-axis. This is my x-axis right over here. And the graph f of x is equal to 1/x would look something like this. So it looks something like this. So it kind of spikes up there, and then it comes down, and then it goes like this. And I'm just doing a rough approximation of it. So it would look something like that. And then on the negative side, it looks something like this. So this is my hand-drawn version of roughly what this graph looks like. So this right over here is f of x is equal to 1/x. Now, we are concerning ourselves with the point x equals k. So let's say-- I mean, it could be anything that's non-zero, but let's just say that this is k right over here. So that is the point k 1/k. We can visualize the line tangent to the curve there. So it might look something like this. And we need to figure out its y-intercept. Where does it intercept the y-axis? So we need to figure out this point right over here. Well, the best way to do it, if we can figure out the slope of the tangent line here, the slope of the tangent line is just the derivative of the line at that point. If we could figure out the slope of the tangent line, we already know that line contains the point. Let me do this in a different color. We know it contains the point k comma 1/k. So if we know its slope, we know what point it contains, we can figure out what its y-intercept is. So the first step is just, well, what's the slope of the tangent line? Well, to figure out the slope of the tangent line, let's take the derivative. So if we write f of x, instead writing it as 1/x, I'll write it as x to the negative 1 power. That makes it a little bit more obvious that we're about to use the power rule here. So the derivative of f at any point x is going to be equal to-- well, it's going to be the exponent here is negative 1. So negative 1 times x to the-- now we decrement the exponent to the negative 2 power. Or I could say it's negative x to the negative 2. Now, what we care about is the slope when x equals k. So f prime of k is going to be equal to negative k to the negative 2 power. Or another way of thinking about it, this is equal to negative 1 over k squared. So this right over here is the slope of the tangent line at that point. Now, let's just think about what the equation of the tangent line is. And we could think about it in slope-intercept form. So we know the equation of a line in slope-intercept form is y is equal to mx plus b, where m is the slope and b is the y-intercept. So if we can get it in this form, then we know our answer. We know what the y-intercept is going to be. It's going to be b. So let's think about it a little bit. This equation, so we could say y is equal to our m, our slope of the tangent line, when x is equal to k, we just figure out to be this business. It equals this thing right over here. So let me write that in blue. Negative 1 over k squared times x plus b. So how do we solve for b? Well, we know what y is when x is equal to k. And so we can use that to solve for b. We know that y is equal to 1/k when x is equal to k. So this is going to be equal to negative 1-- that's not the same color. Negative 1 over k squared times k plus b. Now, what does this simplify to? See, k over k squared is the same thing as 1/k, so this is going to be negative 1/k. So this part, all of this simplifies to negative 1/k. So how do we solve for b? Well, we could just add 1/k to both sides and we are left with-- if you add 1/k here-- actually, let me just do that. Plus 1/k, left-hand side, you're left with 2/k, and on the right-hand side, you're just left with b, is equal to b. So we're done. The y-intercept of the line tangent to the curve when x equals k is going to be 2/k. If we wanted the equation of the line, well, we've done all the work. Let's write it out. It'll be satisfying. It's going to be y is equal to negative 1/k squared x plus our y-intercept, plus 2/k. And we're done." + }, + { + "Q": "This came up on the practice section can anyone tell me what I am doing wrong?\n'Which of the following ordered pairs represents a solution to the equation below?'\n(\u00e2\u0088\u00922,\u00e2\u0088\u00922)(\u00e2\u0088\u00921,\u00e2\u0088\u00922)(0,\u00e2\u0088\u00921)(1,3)(2,7)\ny=2x+1\n\nEvery time I put the first number in the y place and the second number in the x place they come up different.\nI understood the question as meaning something like is y the same as two ? plus 1\nIs that wrong?", + "A": "What the question is asking you is, when you put the first number of the pair in for x, and the second number in for y, does the equation make a true statement? I ll do one for you: -2 = 2(-2) + 1? -2 =-4 +1? -2 = -3? NO. So the pair (-2,-2) does not represent a solution to the equation.", + "video_name": "HUn4XwV7o9I", + "transcript": "Is 3 comma negative 4 a solution to the equation 5x plus 2y is equal to 7? So they're saying, does x equal 3, y equal negative 4, satisfy this equation, or this relationship right here? So one way to do it is just to substitute x is equal to 3 and y equals negative 4 into this and see if 5 times x plus 2 times y does indeed equal 7. So we have 5 times 3 plus 2 times negative 4. This is equal to 15. 15 plus negative 8, which does indeed equal 7. So it does satisfy the equation. So it is on the line. It is a solution. x equals 3, y equals negative 4 is a solution to this equation. So we've essentially answered our question. It is. Now, another way to do it, and I'm not going to go into the details here, is you could actually graph the line. So maybe the line might look something like this, I'm not going to do it in detail. And you see, if you have a very good drawing of it, you see whether the point lies on the line. If the point, when you graph the point, does lie on the line, it would be a solution. If the point somehow ends up not being on the line then you'd know it isn't a solution. But to do this, you would have to have a very good drawing and so you could very precisely determine whether it's on the line. If you do the substitution method, if you just substitute the values into the equation and see if it comes out mathematically, this will always be exact. So this is all we really had to do in this example. So it definitely is a solution to the equation." + }, + { + "Q": "I have two Questions:\n\nDoes the production of lactic acid in skeletal muscle contribute to lowering the pH and thus increase O2 delivery?\n\nAlso, because there is low CO2 in the lungs and therefore less HCO3- and H2CO3 because of Le'Chatelier's principle, does that mean that the lungs have a more basic environment than the rest of the body?\n\nThanks for any info on this!", + "A": "I think that you raise a good point for #2. In the lungs HCO3- and H2CO3 are converted back into CO2 and H2O, which removes protons from the blood, which should raise the pH of the blood. So I guess yes, the blood in the lungs will be more alkali than the blood in the muscles. Great question!", + "video_name": "dHi9ctwDUnc", + "transcript": "So we've talked a little bit about the lungs and the tissue, and how there's an interesting relationship between the two where they're trying to send little molecules back and forth. The lungs are trying to send, of course, oxygen out to the tissues. And the tissues are trying to figure out a way to efficiently send back carbon dioxide. So these are the core things that are going on between the two. And remember, in terms of getting oxygen across, there are two major ways, we said. The first one, the easy one is just dissolved oxygen, dissolved oxygen in the blood itself. But that's not the major way. The major way is when oxygen actually binds hemoglobin. In fact, we call that HbO2. And the name of that molecule is oxyhemoglobin. So this is how the majority of the oxygen is going to get delivered to the tissues. And on the other side, coming back from the tissue to the lungs, you've got dissolved carbon dioxide. A little bit of carbon dioxide actually, literally comes just right in the plasma. But that's not the majority of how carbon dioxide gets back. The more effective ways of getting carbon dioxide back, remember, we have this protonated hemoglobin. And actually remember, when I say there's a proton on the hemoglobin, there's got to be some bicarb floating around in the plasma. And the reason that works is because when they get back to the lungs, the proton, that bicarb, actually meet up again. And they form CO2 and water. And this happens because there's an enzyme called carbonic anhydrase inside of the red blood cells. So this is where the carbon dioxide actually gets back. And of course, there's a third way. Remember, there's also some hemoglobin that actually binds directly to carbon dioxide. And in the process, it forms a little proton as well. And that proton can go do this business. It can bind to a hemoglobin as well. So there's a little interplay there. But the important ones I want you to really kind of focus in on are the fact that hemoglobin can bind to oxygen. And also on this side, that hemoglobin actually can bind to protons. Now, the fun part about all this is that there's a little competition, a little game going on here. Because you've got, on the one side, you've got hemoglobin binding oxygen. And let me draw it twice. And let's say this top one interacts with a proton. Well, that protons going to want to snatch away the hemoglobin. And so there's a little competition for hemoglobin. And here, the oxygen gets left out in the cold. And the carbon dioxide does the same thing, we said. Now, we have little hemoglobin bound to carbon dioxide. And it makes a proton in the process. But again, it leave oxygen out in the cold. So depending on whether you have a lot of oxygen around, if that's the kind of key thing going on, or whether you have a lot of these kinds of products the proton or the carbon dioxide. Depending on which one you have more of floating around in the tissue in the cell, will determine which way that reaction goes. So keeping this concept in mind, then I could actually step back and say, well, I think that oxygen is affected by carbon dioxide and protons. I could say, well, these two, carbon dioxide and protons, are actually affecting, let's say, are affecting the, let's say, the affinity or the willingness of hemoglobin to bind, of hemoglobin for oxygen. That's one kind of statement you could make by looking at that kind of competition. And another person come along and they say, well, I think oxygen actually is affecting, depending on which one, which perspective you take. You could say, oxygen is affecting maybe the affinity of hemoglobin for the carbon dioxide and proton of hemoglobin for CO2 and protons. So you could say it from either perspective. And what I want to point out is that actually, in a sense, both of these are true. And a lot of times we think, well, maybe it's just saying the same thing twice. But actually, these are two separate effects. And they have two separate names. So the first one, talking about carbon dioxide and protons, their effect is called the Bohr effect. So you might see that word or this description. This is the Bohr effect. And the other one, looking at it from the other prospective, looking at it from oxygen's perspective, this would be the Haldane effect. That's just the name of it, Haldane effect. So what is the Bohr effect and the Haldane effect? Other than simply saying that the things compete for hemoglobin. Well, let me actually bring up a little bit of the canvas. And let's see if I can't diagram this out. Because sometimes I think a little diagram would really go a long way in explaining these things. So let's see if I can do that. Let's use a little graph and see if we can illustrate the Bohr effect on this graph. So this is the partial pressure of oxygen, how much is dissolved in the plasma. And this is oxygen content, which is to say, how much total oxygen is there in the blood. And this, of course, takes into account mostly the amount of oxygen that's bound to hemoglobin. So as I slowly increase the partial pressure of oxygen, see how initially, not too much is going to be binding to the hemoglobin. But eventually as a few of the molecules bind, you get cooperativity. And so then, slowly the slope starts to rise. And it becomes more steep. And this is all because of cooperativity. Oxygen likes to bind where other oxygens have already bound. , And then it's going to level off. And the leveling off is because hemoglobin is starting to get saturated. So there aren't too many extra spots available. So you need lots and lots of oxygen dissolved in the plasma to be able to seek out and find those extra remaining spots on hemoglobin. So let's say we choose two spots. One spot, let's say, is a high amount of oxygen dissolved in the blood. And this, let's say, is a low amount of oxygen dissolved in the blood. I'm just kind of choosing them arbitrarily. And don't worry about the units. And if you were to think of where in the body would be a high location, that could be something like the lungs where you have a lot of oxygen dissolved in blood. And low would be, let's say, the thigh muscle where there's a lot of CO2 but not so much oxygen dissolved in the blood. So this could be two parts of our body. And you can see that. Now, if I want to figure out, looking at this curve how much oxygen is being delivered to the thigh, then that's actually pretty easy. I could just say, well, how much oxygen was there in the lungs, or in the blood vessels that are leaving the lungs. And there's this much oxygen in the blood vessels leaving the lungs. And there's this much oxygen in the blood vessels leaving the thigh. So the difference, whenever oxygen is between these two points, that's the amount of oxygen that got delivered. So if you want to figure out how much oxygen got delivered to any tissue you can simply subtract these two values. So that's the oxygen delivery. But looking at this, you can see an interesting point which is that if you wanted to increase the oxygen delivery. Let's say, you wanted for some reason to increase it, become more efficient, then really, the only way to do that is to have the thigh become more hypoxic. As you move to the left on here, that's really becoming hypoxic, or having less oxygen. So if you become more hypoxic, then, yes, you'll have maybe a lower point here, maybe a point like this. And that would mean a larger oxygen delivery. But that's not ideal. You don't want your thighs to become hypoxic. That could start aching and hurting. So is there another way to have a large oxygen delivery without having any hypoxic tissue, or tissue that has a low amount of oxygen in it. And this is where the Bohr effect comes into play. So remember, the Bohr effect said that, CO2 and protons affect the hemoglobin's affinity for oxygen. So let's think of a situation. I'll do it in green. And in this situation, where you have a lot of carbon dioxide and protons, the Bohr effect tells us that it's going to be harder for oxygen to bind hemoglobin. So if I was to sketch out another curve, initially, it's going to be even less impressive, with less oxygen bound to hemoglobin. And eventually, once the concentration of oxygen rises enough, it will start going up, up, up. And it does bind hemoglobin eventually. So it's not like it'll never bind hemoglobin in the presence of carbon dioxide and protons. But it takes longer. And so the entire curve looks shifted over. These conditions of high CO2 and high protons, that's not really relevant to the lungs. The lungs are thinking, well, for us, who cares. We don't really have these conditions. But for the thigh, it is relevant because the thigh has a lot of CO2. And the thigh has a lot of protons. Again, remember, high protons means low pH. So you can think of it either way. So in the thigh, you're going to get, then, a different point. It's going to be on the green curve not the blue curve. So we can draw it at the same O2 level, actually being down here. So what is the O2 content in the blood that's leaving the thigh? Well, then to do it properly, I would say, well, it would actually be over here. This is the actual amount. And so O2 deliver is actually much more impressive. Look at that. So O2 delivery is increased because of the Bohr effect. And if you want to know exactly how much it's increased, I could even show you. I could say, well, this amount from here down to here. Literally the vertical distance between the green and the blue lines. So this is the extra oxygen delivered because of the Bohr effect. So this is how the Bohr effect is so important at actually helping us deliver oxygen to our tissues. So let's do the same thing, now, but for the Haldane effect. And to do this, we actually have to switch things around. So our units and our axes are going to be different. So we're going to have the amount of carbon dioxide there. And here, we'll do carbon dioxide content in the blood. So let's think through this carefully. Let's first start out with increasing the amount of carbon dioxide slowly but surely. And see how the content goes up. And here, as you increase the amount of carbon dioxide, the content is kind of goes up as a straight line. And the reason it doesn't take that S shape that we had with the oxygen is that there's no cooperativity in binding the hemoglobin. It just goes up straight. So that's easy enough. Now, let's take two points like we did before. Let's take a point, let's say up here. This will be a high amount of CO2 in the blood. And this will be a low amount of CO2 in the blood. So you'd have a low amount, let's say right here, in what part of the tissue? Well, low CO2, that sounds like the lungs because there's not too much CO2 there. But high CO2, it probably is the thighs because the thighs like little CO2 factories. So the thigh has a high amount and the lungs have a low amount. So if I want to look at the amount of CO2 delivered, we'd do it the same way. We say, OK, well, the thighs had a high amount. And this is the amount of CO2 in the blood, remember. And this is the amount of CO2 in the blood when it gets to the lungs. So the amount of CO2 that was delivered from the thigh to the lungs is the difference. And so this is how much CO2 delivery we're actually getting. So just like we had O2 delivery, we have this much CO2 delivery. Now, read over the Haldane effect. And let's see if we can actually sketch out another line. In the presence of high oxygen, what's going to happen? Well, if there's a lot of oxygen around, then it's going to change the affinity of hemoglobin for carbon dioxide and protons. So it's going to allow less binding of protons and carbon dioxide directly to the hemoglobin. And that means that you're going to have less CO2 content for any given amount of dissolved CO2 in the blood. So the line still is a straight line, but it's actually, you notice, it's kind of slope downwards. So where is this relevant? Where do you have a lot of oxygen? Well, it's not really relevant for the thighs because the thighs don't have a lot of oxygen. But it is relevant for the lungs. It is very relevant there. So now you can actually say, well, let's see what happens. Now that you have high O2, how much CO2 delivery are you getting? And you can already see it. It's going to be more because now you've got this much. You've got going all the way over here. So this is the new amount of CO2 delivery. And it's gone up. And in fact, you can even show exactly how much it's gone up by, by simply taking this difference. So this difference right here between the two, this is the Haldane effect. This is the visual way that you can actually see that Haldane effect. So the Bohr effect and the Haldane effect, these are two important strategies our body has for increasing the amount of O2 delivery and CO2 delivery going back and forth between the lungs and the tissues." + }, + { + "Q": "16.88 rounded to the nearest hundredth is", + "A": "The answer is 16.90", + "video_name": "19yOv4P2ccw", + "transcript": "We're asked to round 152, 137, 245, and 354 to the nearest 100, which is another way of saying round each of these numbers to the nearest multiple of 100. So let's think about them one by one. So let's draw a number line here. And here I'm counting up by hundreds. What I've marked here, 100, 200, 300, 400, these are all multiples of 100. I could keep going up. I could go to 500, 600, so on and so forth. Now, let's start with 152. Well, where does 152 sit? So halfway in between is 150. 152 is going to be right to the right of that. So that's 152 right over here. So what are our two options? We might round up. The multiple of 100 above 152 is 200. The multiple of 100 below 152 is 100. So which direction do we go in? Do we round up to 200, or do we round down to 100? Well, if we're rounding to the nearest 100, we want to look at one place to the right of that. We want to look at the tens place to decide which multiple of 100 it is closer to. And the rules are very similar to when we're rounding to the nearest tens place or really any place. We look one place to the right of it. So in this case, we look at the tens place. And we say, if this is 5 or larger, we round up. And this is 5 or larger, so we're going to round up to 200. So this we're going to round. 152, we're going to round to 200, which also makes sense. 152 is a little bit closer to 200. It's 48 away from to 200 than it to 100. It's 52 away from 100. So it makes sense to go up to the nearest multiple of 100. Now, let's think about 137. And I encourage you to now pause the video and try to round each of these other three numbers to the nearest 100. Well, 137 is going to sit someplace right over here. 137 is going to be right over there. So two options-- we can round down to 100. That's the multiple of 100 below 137. Or we could round up to 200. Well, 137, just looking at it, is clearly closer to 100. Or we could apply our rule. If we're rounding to the nearest 100, we want to look at one place to the right of that. We want to look at the tens place. If this is 5 or larger, we round up. If it's less than 5, we round down. So in this case, we would round down to 100. Let's do the same thing with 245. If you haven't paused it and tried it yourself, once again, I want to emphasize. That'll make it really valuable for you to try it on your own. So let's plot where 245 is. So 245 is right around-- this is 250, so 245 might be right around here. Now, let's apply that rule. If we're trying to round to the nearest 100, we would look to one place to the right. We'll look at the tens place. We could ignore the ones place. We look at the tens place here. If it is greater than or equal to 5, we round up. If it is less than 5, we round down. So here, we're clearly going to round down. And when we round down, we're going to round down to the multiple of 100 that's directly below 245. Well, we're going to round down to 200. We had two options. If we rounded up, we would have gone to 300. If we rounded down, we would go to 200. We're clearly closer to 200. And we can verify that with the rule. The tens place, we're in the 40's here. The tens place is a 4. We're going to round down. Now let's think about 354. If we were to plot that, this is 350. 354 might be right over here. So if we rounded down, we would go to 300. If we rounded up, we would go to 400. Now let's apply our rule, and let's also think about it on the number line. This is all about finding the multiple of 100 that it's closest to. If you're trying to round to the nearest 100, you want to look at the tens place, a place to the right of the place you're rounding to. If the tens place is a 5 or larger, you're going to round up. This is a 5 or larger so we're going to round up to 400. And that also makes sense. The rule is really valuable. If you're right at 350, right in between, then you would need the rule to say, hey, let's look at this 5 But 354 is also closer to 400 than it is to 300. It's 54 away from 300. It's 46 away from 400. So it makes sense that we would round up to 400." + }, + { + "Q": "So the numbers were 13 and 15... but they could just as well have been 29 and 31, and the shared secret would be the same... so how could you ever find the original numbers?", + "A": "They don t need to find the original numbers. The whole point is to generate the shared secret (which they can to generate a key for a symmetric key cipher).", + "video_name": "M-0qt6tdHzk", + "transcript": "Now this is our solution. First Alice and Bob agree publicly on a prime modulus and a generator, in this case 17 and 3. Then Alice selects a private random number, say 15, and calculates three to the power 15 mod 17 and sends this result publicly to Bob. Then Bob selects his private random number, say 13, and calculates 3 to the power 13 mod 17 and sends this result publicly to Alice. And now the heart of the trick; Alice takes Bob's public result and raises it to the power of her private number to obtain the shared secret, which in this case is 10. Bob takes Alice's public result and raises it to the power of his private number resulting in the same shared secret. Notice they did the same calculation, though it may not look like it at first. Consider Alice, the 12 she received from Bob was calculated as 3 to the power 13 mod 17. So her calculation was the same as 3 to the power 13 to the power 15 mod 17. Now consider Bob, the 6 he received from Alice was calculated as 3 to the power 15 mod 17. So his calculation was the same as 3 to the power 15 to the power 13. Notice they did the same calculation with the exponents in a different order. When you flip the exponent the result doesn't change. So they both calculated 3 raised to the power of their private numbers. Without one of these private numbers, 15 or 13, Eve will not be able to find the solution. And this is how it's done; While Eve is stuck grinding away at the discrete logarithm problem, and with large enough numbers, we can say it's practically impossible for her to break the encryption in a reasonable amount of time. This solves the Key Exchange problem. It can be used in conjunction with a pseudorandom generator to encrypt messages between people who have never met." + }, + { + "Q": "What is the oxidation state of hydrogen in acids? Is it +1 or -1 ?", + "A": "The oxidation state of hydrogen in acids is +1. In fact, in the majority of its compounds hydrogen is in the +1 state (other than, of course, hydrogen gas). The -1 state of hydrogen in pretty much limited to a class of chemicals called hydrides. For example, NaH, or NaBH\u00e2\u0082\u0084. Since hydrides react with water, I don t know of any that you would find in aqueous solution.", + "video_name": "CCsNJFsYSGs", + "transcript": "Let's see if we can come up with some general rules of thumb or some general trends for oxidation states by looking at the periodic table. So first, let's just focus on the alkali metals. And I'll box them off. We'll think about hydrogen in a second. Well, I'm going to box-- I'm going to separate hydrogen because it's kind of a special case. But if we look at the alkali metals, the Group 1 elements right over here, we've already talked about the fact they're not too electronegative. They have that one valence electron. They wouldn't mind giving away that electron. And so for them, that oxidation state might not even be a hypothetical charge. These are very good candidates for actually forming ionic bonds. And so it's very typical that when these are in a molecule, when these form bonds, that these are the things that are being oxidized. They give away an electron. So they get to-- a typical oxidation state for them would be positive 1. If we go one group over right over here to the alkaline earth metals, two valence electrons, still not too electronegative. So they're likely to fully give or partially give away two electrons. So if you're forced to assign an ionic-- if you were to say, well, none of this partial business, just give it all away or take it, you would say, well, these would typically have an oxidation state of positive 2. In a hypothetical ionic bonding situation, they would be more likely to give the two electrons because they are not too electronegative, and it would take them a lot to complete their valence shell to get all the way to 8. Now, let's go to the other side of the periodic table to Group 7, the halogens. The halogens right over here, they're quite electronegative, sitting on the right-hand side of the periodic table. They're one electron away from being satisfied from a valence electron point of view. So these are typically reduced. They typically have an oxidation state of negative 1. And I keep saying typically, because these are not going to always be the case. There are other things that could happen. But this is a typical rule of thumb that they're likely to want to gain an electron. If we move over one group to the left, Group 6-- and that's where the famous oxygen sits-- we already said that oxidizing something is doing to something what oxygen would have done, that oxidation is taking electrons away from it. So these groups are typically oxidized. And oxygen is a very good oxidizing agent. Or another way of thinking about it is oxygen normally takes away electrons. These like to take away electrons, typically two electrons. And so their oxidation state is typically negative 2-- once again, just a rule of thumb-- or that their charge is reduced by two electrons. So these are typically reduced. These are typically oxidized. Now, we could keep going. If we were to go right over here to the Group 5 elements, typical oxidation state is negative 3. And so you see a general trend here. And that general trend-- and once again, it's not even a hard and fast rule of thumb, even for the extremes, but as you get closer and closer to the middle of the periodic table, you have more variation in what these typical oxidation states could be. Now, I mentioned that I put hydrogen aside. Because if you really think about it, hydrogen, yes, hydrogen only has one electron. And so you could say, well, maybe it wants to give away that electron to get to zero electrons. That could be a reasonable configuration for hydrogen. But you can also view hydrogen kind of like a halogen. So you could kind of view it kind of like an alkali metal. But in theory, it could have been put here on the periodic table as well. You could have put hydrogen here, because hydrogen, in order to complete its first shell, it just needs one electron. So in theory, hydrogen could have been put there. So hydrogen actually could typically could have a positive or a negative 1 oxidation state. And just to see an example of that, let's think about a situation where hydrogen is the oxidizing agent. And an example of that would be lithium hydride right Now, in lithium hydride, you have a situation where hydrogen is more electronegative. A lithium is not too electronegative. It would happily give away an electron. And so in this situation, hydrogen is the one that's oxidizing the lithium. Lithium is reducing the hydrogen. Hydrogen is the one that is hogging the electron. So the oxidation state on the lithium here is a positive 1. And the oxidation state on the hydrogen here is a negative. So just, once again, I really want to make sure we get the notation. Lithium has been oxidized by the hydrogen. Hydrogen has been reduced by the lithium. Now, let's give an example where hydrogen plays the other role. Let's imagine hydroxide. So the hydroxide anion-- so you have a hydrogen and an oxygen. And so essentially, you could think of a water molecule that loses a hydrogen proton but keeps that hydrogen's electron. And this has a negative charge. This has a negative 1 charge. But what's going on right over here? And actually, let me just draw that, because it's fun to think about it. So this is a situation where oxygen typically has-- 1, 2, 3, 4, 5, 6 electrons. And when it's water, you have 2 hydrogens like that. And then you share. And then you have covalent bond right over there sharing that pair, covalent bond sharing that right over there. To get to hydroxide, the oxygen essentially nabs both of these electrons to become-- so you get-- that pair, that pair. Now you have-- let me do this in a new color. Now, you have this pair as well. And then you have that other covalent bond to the other hydrogen. And now this hydrogen is now just a hydrogen proton. This one now has a negative charge. So this is hydroxide. And so the whole thing has a negative charge. And oxygen, as we have already talked about, is more electronegative than the hydrogen. So it's hogging the electrons. So when you look at it right over here, you would say, well, look, hydrogen, if we had to, if we were forced to-- remember, oxidation states is just an intellectual tool which we'll find useful. If you had to pretend this wasn't a covalent bond, but an ionic bond, you'd say, OK, then maybe this hydrogen would fully lose an electron, so it would get an oxidation state of plus 1. It would be oxidized by the oxygen. And that the oxygen actually has fully gained one electron. And you could say, well, if we're forced to, we could say-- if we're forced to think about this is an ionic bond, we'll say it fully gains two electrons. So we'll have an oxidation state of negative 2. And once again, the notation, when you do the superscript notation for oxidation states and ionic charge, you write the sign after the number. And this is just the convention. And now, with these two examples, the whole point of it is to show that hydrogen could have a negative 1 or a positive 1 oxidation state. But there's also something interesting going on here. Notice, the oxidation states of the molecules here, they add up to the whole-- or the oxidation state of each of the atoms in a molecule, they add up to the entire charge of the molecule. So if you add a positive 1 plus negative 1, you get 0. And that makes sense because the entire molecule lithium hydride is neutral. It has no charge. Similarly, hydrogen, plus 1 oxidation state; oxygen, negative 2 oxidation number or oxidation state-- you add those two together, you have a negative 1 total charge for the hydroxide anion, which is exactly the charge that we have right over there." + }, + { + "Q": "hey i am studying for a algebra test why am i having to study in trig and geo?", + "A": "Usually you learn algebra to apply it in different math topics such as trigonometry and geometry. Connecting topics can help you understand them more.", + "video_name": "Ei54NnQ0FKs", + "transcript": "Voiceover:Let's say you're studying some type of a little hill or rock formation right over here. And you're able to figure out the dimensions. You know that from this point to this point along the base, straight along level ground, is 60 meters. You know the steeper side, steeper I guess surface or edge of this cliff or whatever you wanna call it, is 20 meters. And then the longer side here, I guess the less steep side, is 50 meters long. So you're able to measure that. But now what you wanna do is use your knowledge of trigonometry, given this information, to figure out how steep is this side. What is the actual inclination relative to level ground? Or another way of thinking about it, what is this angle theta right over there? And I encourage you to pause the video and think about it on your own. Well it might be ringing a bell. Well you know three sides of a triangle and then we want to figure out an angle. And so the thing that jumps out in my head, well maybe the law of cosines could be useful. Let me just write out the law of cosines, before we try to apply it to this triangle right over here. So the law of cosines tells us that C-squared is equal to A-squared, plus B-squared, minus two A B, times the cosine of theta. And just to remind ourselves what the A, B's, and C's are, C is the side that's opposite the angle theta. So if I were to draw an arbitrary triangle right over here. And if this is our angle theta, then this determines that C is that side, and then A and B could be either of these two sides. So A could be that one and B could be that one. Or the other way around. As you can see, A and B essentially have the same role in this formula right over here. This could be B or this could be A. So what we wanna do is somehow relate this angle... We wanna figure out what theta is in our little hill example right over here. So if this is going to be theta, what is C going to be? Well C is going to be this 20 meter side. And then we could set either one of these to be A or B. We could say that this A is 50 meters and B is 60 meters. And now we could just apply the law of cosines. So the law of cosines tells us that 20-squared is equal to A-squared, so that's 50 squared, plus B-squared, plus 60 squared, minus two times A B. So minus two times 50, times 60, times 60, times the cosine of theta. This works out well for us because they've given us everything. There's really only one unknown. There's theta here. So let's see if we can solve for theta. So 20 squared, that is 400. 50 squared is 2,500. 60 squared is 3,600. And then 50 times... Let's see, two times 50 is 100, times 60, this is all equal to 6,000. So let's see, if we simplify this a little bit we're going to get 400 is equal to 2,500 plus 3,600. Let's see, that'd be 6,100. That's equal to 6,000... Let me do this in a new color. So when I add these two, I get 6,100. Did I do that right? So it's 2,000 plus 3,000, plus 5,000. 500 plus 600 is 1,100. So I get 6,100 minus 6,000, times the cosine of theta. And let's see, now we can subtract 6,100 from both sides. So I'm just gonna subtract 6,100 from both sides so that I get closer to isolating the theta. So let's do that. So this is going to be negative 5,700. Is that right? 5,700 plus... Yes, that is right. Right, because if this was the other way around, if this was 6,100 minus 400 it would be positive 5,700. Alright. And then these two of course cancel out. And this is going to be equal to negative 6,000 times the cosine of theta. Now we can divide both sides by negative 6,000. And we get... I'm just gonna swap the sides. We get cosine of theta is equal to... Let's see we could divide the numerator and the denominator by essentially negative 100. So these are both going to become positive. So cosine of theta is equal to 57 over 60. And actually that can be simplified even more. Three goes into 57, is that 19 times? Yep, so this is actually... This could be simplified. This is equal to 19 over 20. We actually didn't have to do that simplification step because we're about to use our calculators, but that makes the math a little more tractable. Right, 3 goes into 57, yeah, 19 times. And so now we can take the inverse cosine of both sides. So we could get theta is equal to the inverse cosine, or the arc cosine, of 19 over 20. So let's get our calculator out and see if we get something that makes sense. So we wanna do the inverse cosine of 19 over 20. And we deserve a drum roll. We get 18.19 degrees. And I already verified that my calculator is in degree mode. So it gets 18.19 degrees. So if we wanted to round, this is approximately equal to 18.2 degrees, if we wanna round to the nearest tenth. So that essentially gives us a sense of how steep this slope actually is." + }, + { + "Q": "Does the \"Angular Momentum\" in the term \"Angular Moment Quantum Number\" share any similar concept at all to the \"Angular Momentum\" in Physics?", + "A": "Yes angular momentum in quantum theories is called angular momentum because it acts just like angular momentum in the macroscopic world.", + "video_name": "KrXE_SzRoqw", + "transcript": "- [Voiceover] In the Bohr model of the hydrogen atom, the one electron of hydrogen is in orbit around the nucleus at a certain distance, r. So in the Bohr model, the electron is in orbit. In the quantum mechanics version of the hydrogen atom, we don't know exactly where the electron is, but we can say with high probability that the electron is in an orbital. An orbital is the region of space where the electron is most likely to be found. For hydrogen, imagine a sphere, a three-dimensional volume, a sphere, around the nucleus. Somewhere in that region of space, somewhere in that sphere, we're most likely to find the one electron of hydrogen. So we have these two competing visions. The Bohr model is classical mechanics. The electron orbits the nucleus like the planets around the sun, but quantum mechanics says we don't know exactly where that electron is. The Bohr model turns out to be incorrect, and quantum mechanics has proven to be the best way to explain electrons in orbitals. We can describe those electrons in orbitals using the four quantum numbers. Let's look at the first quantum number here. This is called the principal quantum number. The principal quantum number is symbolized by n. n is a positive integer, so n could be equal to one, two, three, and so on. It indicates the main energy level occupied by the electron. This tells us the main energy level. You might hear this referred to as a shell sometimes, so we could say what kind of shell the electron is in. As n increases, the average distance of the electron from the nucleus increases, and therefore so does the energy. For example, if this was our nucleus right here, and let's talk about n is equal to one. For n is equal to one, let's say the average distance from the nucleus is right about here. Let's compare that with n is equal to two. n is equal to two means a higher energy level, so on average, the electron is further away from the nucleus, and has a higher energy associated with it. That's the idea of the principal quantum number. You're thinking about energy levels or shells, and you're also thinking about average distance from the nucleus. All right, our second quantum number is called the angular momentum quantum number. The angular momentum quantum number is symbolized by l. l indicates the shape of the orbital. This will tell us the shape of the orbital. Values for l are dependent on n, so the values for l go from zero all the way up to n minus one, so it could be zero, one, two, or however values there are up to n minus one. For example, let's talk about the first main energy level, or the first shell. n is equal to one. There's only one possible value you could get for the angular momentum quantum number, l. n minus one is equal to zero, so that's the only possible value, the only allowed value of l. When l is equal to zero, we call this an s orbital. This is referring to an s orbital. The shape of an s orbital is a sphere. We've already talked about that with the hydrogen atom. Just imagine this as being a sphere, so a three-dimensional volume here. The angular momentum quantum number, l, since l is equal to zero, that corresponds to an s orbital, so we know that we're talking about an s orbital here which is shaped like a sphere. So the electron is most likely to be found somewhere in that sphere. Let's do the next shell. n is equal to two. If n is equal to two, what are the allowed values for l? l goes zero, one, and so on all the way up to n minus one. l is equal to zero. Then n minus one would be equal to one. So we have two possible values for l. l could be equal to zero, and l could be equal to one. Notice that the number of allowed values for l is equal to n. So for example, if n is equal to one, we have one allowed value. If n is equal to two, we have two allowed values. We've already talked about what l is equal to zero, what that means. l is equal to zero means an s orbital, shaped like a sphere. Now, in the second main energy level, or the second shell, we have another value for l. l is equal to one. When l is equal to one, we're talking about a p orbital. l is equal to one means a p orbital. The shape of a p orbital is a little bit strange, so I'll attempt to sketch it in here. You might hear several different terms for this. Imagine this is a volume. This is a three-dimensional region in here. You could call these dumbbell shaped or bow-tie, whatever makes the most sense to you. This is the orbital, this is the region of space where the electron is most likely to be found if it's found in a p orbital here. Sometimes you'll hear these called sub-shells. If n is equal to two, if we call this a shell, then we would call these sub-shells. These are sub-shells here. Again, we're talking about orbitals. l is equal to zero is an s orbital. l is equal to one is a p orbital. Let's look at the next quantum number. Let's get some more space down here. This is the magnetic quantum number, symbolized my m sub l here. m sub l indicates the orientation of an orbital around the nucleus. This tells us the orientation of that orbital. The values for ml depend on l. ml is equal to any integral value that goes from negative l to positive l. That sounds a little bit confusing. Let's go ahead and do the example of l is equal to zero. l is equal to zero up here. Let's go ahead and write that down here. If l is equal to zero, what are the allowed values for ml? There's only one, right? The only possible value we could have here is zero. When l is equal to zero ... Let me use a different color here. If l is equal to zero, we know we're talking about an s orbital. When l is equal to zero, we're talking about an s orbital, which is shaped like a sphere. If you think about that, we have only one allowed value for the magnetic quantum number. That tells us the orientation, so there's only one orientation for that orbital around the nucleus. And that makes sense, because a sphere has only one possible orientation. If you think about this as being an xyz axis, (clears throat) excuse me, and if this is a sphere, there's only one way to orient that sphere in space. So that's the idea of the magnetic quantum number. Let's do the same thing for l is equal to one. Let's look at that now. If we're considering l is equal to one ... Let me use a different color here. l is equal to one. Let's write that down here. If l is equal to one, what are the allowed values for the magnetic quantum number? ml is equal to -- This goes from negative l to positive l, so any integral value from negative l to positive l. Negative l would be negative one, so let's go ahead and write this in here. We have negative one, zero, and positive one. So we have three possible values. When l is equal to one, we have three possible values for the magnetic quantum number, one, two, and three. The magnetic quantum number tells us the orientations, the possible orientations of the orbital or orbitals around the nucleus here. So we have three values for the magnetic quantum number. That means we get three different orientations. We already said that when l is equal to one, we're talking about a p orbital. A p orbital is shaped like a dumbbell here, so we have three possible orientations for a dumbbell shape. If we went ahead and mark these axes here, let's just say this is x axis, y axis, and the z axis here. We could put a dumbbell on the x axis like that. Again, imagine this as being a volume. This would be a p orbital. We call this a px orbital. It's a p orbital and it's on the x axis here. We have two more orientations. We could put, again, if this is x, this is y, and this is z, we could put a dumbbell here on the y axis. There's our second possible orientation. Finally, if this is x, this is y, and this is z, of course we could put a dumbbell on the z axis, like that. This would be a pz orbital. We could write a pz orbital here, and then this one right here would be a py orbital. We have three orbitals, we have three p orbitals here, one for each axis. Let's go to the last quantum number. The last quantum number is the spin quantum number. The spin quantum number is m sub s here. When it says spin, I'm going to put this in quotations. This seems to imply that an electron is spinning on an axis. That's not really what's happening, but let me just go ahead and draw that in here. I could have an electron ... Let me draw two different versions here. I could have an electron spin like a top, if you will, this way, or I could have an electron spin around that axis going this way. Again, this is not actually what's happening in reality. The electrons don't really spin on an axis like a top, but it does help me to think about the fact that we have two possible values for this spin quantum number. You could spin one way, so we could say the spin quantum number is equal to a positive one-half. Usually you hear that called spin up, so spin up, and we'll symbolize this with an arrow going up in later videos here. Then the other possible value for the spin quantum number, so the spin quantum number is equal to a negative one-half. You usually hear that referred to as spin down, and you could put an arrow going down. Again, electrons aren't really spinning in a physical sense like this, but, again, if you think about two possible ways for an electron to spin, then you get these two different, these two possible spin quantum numbers, so positive one-half or negative one-half. Those are the four quantum numbers, and we're going to use those to, again, think about electrons in orbitals." + }, + { + "Q": "How do I work this problem x^2-7x+10", + "A": "x\u00c2\u00b2-7x+10 = x\u00c2\u00b2-5x-2x+10 = x\u00c2\u00b2+(-5-2)x + (-5*-2) = (x-5)(x-2) (Since, x+(a+b)x+ab = (x+a)(x+b))", + "video_name": "2ZzuZvz33X0", + "transcript": "We're asked to solve for s. And we have s squared minus 2s minus 35 is equal to 0. Now if this is the first time that you've seen this type of what's essentially a quadratic equation, you might be tempted to try to solve for s using traditional algebraic means, but the best way to solve this, especially when it's explicitly equal to 0, is to factor the left-hand side, and then think about the fact that those binomials that you factor into, that they have to be equal to 0. So let's just do that. So how can we factor this? We've seen it in several ways. I'll show you the standard we've been doing it, by grouping, and then there's a little bit of a shortcut when you have a 1 as a coefficient over here. So when you do something by grouping, when you factor by grouping, you think about two numbers whose sum is going to be equal to negative 2. So you think about two numbers whose sum, a plus b, is equal to negative 2 and whose product is going to be equal to negative 35. a times b is equal to negative 35. So if the product is a negative number, one has to be positive, one has to be negative. And so if you think about it, ones that are about two apart, you have 5 and negative 7, I think that'll work. 5 plus negative 7 is equal to negative 2. So to factor by grouping, you split this middle term. We can split this into a-- let me write it this way. We have s squared, and then this middle term right here, I'll do it in pink. This middle term right there I can write it as plus 5s minus 7s and then we have the minus 35. And then, of course, all of that is equal to 0. Now, we call it factoring by grouping because we group it. So we can group these first two terms. And these first two terms, they have a common factor of s. So let's factor that out. You have s times s plus 5. That's the same thing as s squared plus 5s. Now, in these second two terms right here, you have a common factor of negative 7, so let's factor that out. So you have negative 7 times s plus 5. And, of course, all of that is equal to 0. Now, we have two terms here, where both of them have s plus 5 as a factor. So we can factor that out. So let's do that. So you have s plus 5 times this s right here, right? S plus 5 times s will give you this term. And then you have minus that 7 right there. I undistributed the s plus 5. And then this is going to be equal to 0. Now that we've factored it, we just have to think a little bit about what happens when you take the product of two numbers? I mean, s plus 5 is a number. s minus 7 is another number. And we're saying that the product of those two numbers is equal to zero. If ever told you that I had two numbers, if I told you that I had the numbers a times b and that they equal to 0, what do we know about either a or b or both of them? Well, at least one of them has to be equal to 0, or both of them have to be equal to 0. So, the fact that this number times that number is equal to zero tells us that either s plus 5 is equal to 0 or-- and maybe both of them-- s minus 7 is equal to 0. I'll do that in just green. And so you have these two equations, and actually, we could say and/or. It could be or/and, either way, and both of them could be equal to 0. So let's see how we can solve for this. Well, we can just subtract 5 from both sides of this equation right there. And so you get, on the left-hand side, you have s is equal to negative 5. That is one solution to the equation, or you can add 7 to both sides of that equation, and you get s is equal to 7. So if s is equal to negative 5, or s is equal to 7, then we have satisfied this equation. We can even verify it. If you make s equal to negative 5, you have positive 25 plus 10, which is minus 35. That does equal zero. If you have 7, 49 minus 14 minus 35 does equal zero. So we've solved for s. Now, I mentioned there's an easier way to do it. And when you have something like this, where you have 1 as the leading coefficient, you don't have to do this two-step factoring. Let me just show you an example. If I just have x plus a times x plus b, what is that equal to? x times x is x squared, x times b is bx. a times x is plus ax. a times b is ab. So you get x squared plus-- these two can be added-- plus a plus bx plus ab. And that's the pattern that we have right here. We have 1 as a leading coefficient here, we have 1 as a leading coefficient here. So once we have our two numbers that add up to negative 2, that's our a plus b, and we have our product that gets to negative 35, then we can straight just factor it into the product of those two things. So it will be-- or the product of the binomials, where those will be the a's and the b's. So we figured it out. It's 5 and negative 7. 5 plus negative 7 is negative 2. 5 times negative 7 is negative 35. So we could have just straight factored at this point. 2, well, actually this was the case of s. So we could have factored it straight to the case of s plus 5 times s minus 7. We could have done that straight away and would've gotten to that right there. And, of course, that whole thing was equal to zero. So that would've been a little bit of a shortcut, but factoring by grouping is a completely appropriate way to do it as well." + }, + { + "Q": "What is in-between the pericardial walls", + "A": "Not very much, just the regular fluid that is all around your body. The interesting thing is that there are no cells between the walls, it is just that small amount of fluid", + "video_name": "bm65xCS5ivo", + "transcript": "So you're probably feeling pretty comfortable with the diagram of the heart, but let me just go ahead and label a few things just to make sure we're all on the same page. So blood flows from the right atrium to the right ventricle and then goes to the lungs and then the left atrium to the left ventricle. So that's usually the flow of blood. And one of the things that keeps the blood flowing in the right direction, we know, is the valves. And two of the valves I'm actually going to give you new names, something slightly different from what we have been referring to them by. These are the atrioventricular valves, and you can take a guess as to which ones I'm referring to. Atrioventricular valves are the two valves between the atria and the ventricles. So one will be the tricuspid valve, and the other is the mitral valve. And just to orient us, this is the tricuspid, the T. And this is our mitral, or M. And the atrioventricular valves, these two valves, if you look at them, they're both facing downwards. And one of the things that you might be wondering is, well, how is it that they aren't just flopping back and forth? And these valves, in particular, have a very interesting strategy. And that is that they actually are tethered to the walls. So they're held down here like that, and they have on the other end of those tethers a little muscle there. Now, this makes perfect sense if you think about it, because the ventricles are very strong. We know the ventricles are really, really strong. And so if the ventricles are squeezing, there's a good chance that that blood is going to shoot up in any direction it can go. It's going to go back perhaps through the mitral valve if it can go there, or it'll go through that tricuspid valve if it can go there. But the reason that it won't is that these papillary muscles are basically kind of sending out little lifelines, these chordae tendineae lifelines, to keep the valve from flipping backwards. So these chordae tendineae, these cords, are important for that reason. They're keeping the valve from flipping backwards. So these are all chordae tendineae, and these are all the papillary muscles. And these are particularly important, then, we can tell, for when you're trying to make sure that the ventricles don't screw up the valves. And now let's say that by accident our ventricle is just too strong, too powerful. Let's say that it broke one of these cords. Let's say it broke this one right here. And that's because our ventricle was just forcing too much blood back, and it just snapped the cord. What would happen? Well, this would basically kind of start flipping back and forth. It would flip this way and this way. And then on the next heartbeat, blood would start going the wrong direction, because this valve is not able to keep that nice tight seal. And so blood would basically kind of go this way when it wasn't supposed to. And all of a sudden, our flow of blood is now going in the wrong direction. So the chordae tendineae and the papillary muscles do a really, really important job in preventing that from happening. So let's move our attention to another area. Let's focus on this right here, which is the interventricular septum. And you can think of septum as basically a wall, interventricular septum. In this interventricular septum, the one thing I want to point out, which is maybe fairly obvious when you look at it-- you might think, well, why did you even have to say it? That's pretty obvious. This area is really thin, and this area is really thick by comparison. So the two areas are not equal in size. This is much thicker. And the reason I wanted to bring that up is because this first area in blue is called the membranous part, literally like a membrane. And the bottom, the red part, is the muscular part. This is the strong muscular part. So you have two different areas in that interventricular septum, the wall between the ventricles. And one of the interesting things about the membranous part, in particular, is that a lot of babies are born with holes in that membranous part. So when I say a lot, I don't mean the majority of babies, by any means. But one of the most common defects, if there is going to be a defect, would be that you would actually have a communication between these two so that blood could actually, again, flow from a place that it's not supposed to go, the left ventricle, into a place it shouldn't be going, the right ventricle. So blood can actually flow through those holes, and that is a problem. That is called a VSD. And actually, you might hear that term at some point. So I just wanted to point out where that happens. And while I'm writing VSD, you can take a stab at guessing what it might stand for. Ventricular, and S is septal. Again, septal just means wall. And D is defect. So a VSD is most common in that membranous part, more so than that muscular part. Now, let's move on again to one final thing I want to point out, which is I want to zoom in on the walls. So here in a gray box I'm going to kind of highlight what's going on this wall and how many layers there are in this wall. Let me draw out a little rectangle to correspond to that little rectangle I drew on the heart itself. So there are three layers to the heart muscle. And actually, I'm going to go through all three layers. And we'll start from the inside and work our way out. So on the inside, you have what's called the endocardium. And I'm actually going to draw the endocardium all the way around here. It goes all the way around the valves, so now you already learned that the valves now have endocardium. It goes around the ventricle and, as I showed you in the beginning, also around the atrium. And it goes all the way up and covers both the right and left side. The endocardium is very, very similar in many ways to the inner lining of the blood vessels, actually. So it's a really thin layer. It's not a very thick layer. It's the layer that all the red blood cells are kind of bumping up against. So when the red blood cells are entering the chambers of the heart, the part that they're going to see is going to be the endocardium. So this is what it looks like, and this is that green layer all the way around that I've drawn now. So if I was to draw it kind of in a blown-up version, it might look like this. And it's a few cell layers thick. And like I said, on the inside you have some red blood cells bumping along. So maybe this is one red blood cell, and this is maybe another one. And they would bump into that endocardium. Now, if you go a little bit deeper to the endocardium, what do you get to next? Well, next you get to myocardium. And that would be, let's say, the biggest chunk of our wall. And that would look something like this. And that myocardium you can kind of appreciate without even having me point it out, because it's the most common part of this entire thing. So this is our myocardium, and let me go back and actually label the endocardium as well. And on the other side-- and actually, just notice that the words are all pretty similar. Myo means muscle. And actually, while I'm on myocardium, let me just point out one more thing. The myocardium is where all of the contractile muscle is going to be, so that's where a lot of the work is being done. And that's also where a lot of the energy is being used up. So when the heart needs oxygen, it's usually the myocardium, because that's the part that's doing all of the work. OK. Now, on the other side of myocardium, what do we have on the outside? Well, we have a layer called the pericardium, and let me try to draw that in for you. Pericardium is something like this, kind of a thin layer. And the interesting thing about pericardium is that there's actually two layers to it. So there's actually something like this where you have two layers, an inner layer and an outer layer. And between the two layers you have literally a gap. There's a gap right there. And in that gap, you might have a little bit of fluid. But it's not actually cells. I guess that's the biggest point. It's not actually cells. It's more just a little bit of fluid that hangs out there. So this whole thing is called the pericardium. Now, you may be wondering how in the world do you get a layer that has a gap within it. So let me actually try to show you what happens in a fetus. So let's say you have a little fetus heart, a tiny little heart like this, and it gets a little bit bigger like this. And then it finally gets into an adult heart, something like that. So this would be the adult heart, right? Well, at the same time that the heart is growing, you actually also have a sac, almost like a little balloon. And this balloon actually begins to envelope the heart, so this growing heart kind of grows right into the balloon. And so this balloon kind of starts going around it like that, and you get something like this. And then eventually, as the heart gets really big, you get something like this. You basically have this kind of inner layer of the balloon that's pancaked out that doesn't even look like a balloon anymore. It's very flat, and then it kind of folds back on itself like that. And it comes all the way around. And now you can see why even though it's continuous-- it's not like it breaks. It is continuous here-- you can see how if you actually just were to look at one chunk of it, like we're looking at right here, you can see how it would actually look like a pancake. And so on our heart actually, it literally would be something like this, like a very thin kind of pancake. And I'm not doing a very, very good job making it look thin, but you can imagine what it is that it could look like if I was to zoom in on it. Basically, something like that, where you have two layers that are basically just kind of turned in on themselves. And both layers put together are called your pericardium. Now, there are actually separate names for the two layers. So for example, the layer that's kind of hugging up against the heart, this layer that I'm drawing right now, this layer is called the visceral pericardium. So you call that the visceral pericardium. And the name visceral, this right here, would be visceral. And the reason it's called visceral is because viscera refers to organs, so that's called the visceral pericardium. And then this outer layer, the one I'm drawing now, is called the parietal pericardium. And that's the layer that actually is on the outside, so let me label that as well. So that's this guy. That would be the parietal pericardium. So now you can actually see the layers of the heart-- the endocardium, myocardium, and pericardium. And actually, just to throw you a curve ball, because I'm pretty sure you can handle it, this visceral pericardium, another name for it, just because you might see it sometime, is the epicardium. Sometimes you might see the name epicardium. And don't get thrown off. It's really just the visceral pericardium. It's just the outermost layer of that heart before you get to the parietal layer." + }, + { + "Q": "100% of 100 squares is 1 whole right ?", + "A": "I m pretty sure this is what you re asking: 100% of 100 squares is 100 squares, just as 20% of 100 squares is 20 squares. 100% is the total amount so out of 100 squares it s 100. So yes, I think you re correct.", + "video_name": "Lvr2YsxG10o", + "transcript": "We're asked to shade 20% of the square below. Before doing that, let's just even think about what percent means. Let me just rewrite it. 20% is equal to-- I'm just writing it out as a word-- 20 percent, which literally means 20 per cent. And if you're familiar with the word century, you might already know that cent comes from the Latin for the word hundred. This literally means you can take cent, and that literally means 100. So this is the same thing as 20 per 100. If you want to shade 20%, that means, if you break up the square into 100 pieces, we want to shade 20 of them. 20 per 100. So how many squares have they drawn here? So if we go horizontally right here, we have one, two, three, four, five, six, seven, eight, nine, ten squares. If we go vertically, we have one, two, three, four, five, six, seven, eight, nine, ten. So this is a 10 by 10 square. So it has 100 squares here. Another way to say it is that this larger square-- I guess that's the square that they're talking about. This larger square is a broken up into 100 smaller squares, so it's already broken up into the 100. So if we want to shade 20% of that, we need to shade 20 of every 100 squares that it is broken into. So with this, we'll just literally shade in 20 squares. So let me just do one. So if I just do one square, just like that, I have just shaded 1 per 100 of the squares. 100 out of 100 would be the whole. I've shaded one of them. That one square by itself would be 1% of the entire square. If I were to shade another one, if I were to shade that and that, then those two combined, that's 2% of the It's literally 2 per 100, where 100 would be the entire square. If we wanted to do 20, we do one, two, three, four-- if we shade this entire row, that will be 10%, right? One, two, three, four, five, six, seven, eight, nine, ten. And we want to do 20, so that'll be one more row. So I can shade in this whole other row right here. And then I would have shaded in 20 of the 100 squares. Or another way of thinking about it, if you take this larger square, divide it into 100 equal pieces, I've shaded in 20 per 100, or 20%, of the entire larger square. Hopefully, that makes sense." + }, + { + "Q": "would you use the same formula for an isoscles triangle", + "A": "No. Area is still 1/2 s * h but it has a base length a, and two sides length b. So a + 2b = x/3.", + "video_name": "IFU7Go6Qg6E", + "transcript": "Let's say that I have a 100 meter long wire. So that is my wire right over there. And it is 100 meters. And I'm going to make a cut someplace on this wire. And so let's say I make the cut right over there. With the left section of wire-- I'm going to obviously cut it in two-- with the left section, I'm going to construct an equilateral triangle. And with the right section, I'm going to construct a square. And my question for you and for me is, where do we make this cut in order to minimize the combined areas of this triangle and this square? Well, let's figure out. Let's define a variable that we're trying to minimize, or that we're trying to optimize with respect to. So let's say that the variable x is the number of meters that we decide to cut from the left. So if we did that, then this length for the triangle would be x meters, and the length for the square would be, well, if we use x up for the left hand side, we're going to have 100 minus x for the right hand side. And so what would the dimensions of the triangle and the square Well, the triangle sides are going to be x over 3, x over 3, and x over 3 as an equilateral triangle. And the square is going to be 100 minus x over 4 by 100 minus x over 4. Now it's easy to figure out an expression for the area of the square in terms of x. But let's think about what the area of an equilateral triangle might be as a function of the length of its sides. So let me do a little bit of an aside right over here. So let's say we have an equilateral triangle. Just like that. And its sides are length s, s, and s. Now we know that the area of a triangle is 1/2 times the base times the height. So in this case, the height we could consider to be altitude, if we were to drop an altitude just like this. This length right over here, this is the height. And this would be perpendicular, just like that. So our area is going to be equal to one half times our base is s. 1/2 times s times whatever our height is, times our height. Now how can we express h as a function of s? Well, to do that we just have to remind ourselves that what we've drawn over here is a right triangle. It's the left half of this equilateral triangle. And we know what this bottom side of this right triangle is. This altitude splits this side exactly into two. So this right over here has length s over 2. So to figure out what h is, we could just use the Pythagorean theorem. We would have h squared plus s over 2 squared plus s over 2 squared is going to be equal to the hypotenuse squared, is going to be equal to s squared. So you would get h squared plus s squared over 4 is equal to s squared. Subtract s squared over 4 from both sides, and you get h squared is equal s squared minus s squared over 4. Now, to do this I could call it s squared. I could call this 4s squared over 4, just to be able to have a common denominator. And 4s squared minus s squared over 4 is going to be equal to 3s squared over 4. So we get h squared is equal to 3s squared over 4. Now we can take the principal root of both sides, and we get h is equal to the square root of 3 times s over 2. So now we can just substitute back right over here and we get our area. Our area is equal to 1/2 s times h. Well h is this business. So it's s times this. So it's 1/2 times s times the square root of 3s over 2, which is going to be equal to s times s is s squared. So this is going to be square root of 3 s squared over 2 times 2 over 4. So this is the area of an equilateral triangle as the function of the length of its sides. So what's the area of this business going to be? So the area of our little equilateral triangle-- let me write a combined area. And let me do that in a neutral color. So let me do that in white. So the combined area, I'll write it A sub c is going to be equal to the area of my triangle A sub t plus the area of my square. Well, the area of my triangle, we know what it's going to be. It's going to be square root of 3 times the length of a side squared divided by 4. So it's going to be square root of 3. Let me do that in that same yellow color. It's going to be-- make sure I switch colors. It's going to be square root of 3 over 4 times a side squared, times x over 3 squared. All I did is the length of a side is x over 3. We already know what the area is. It's the square root of 3 over 4 times the length of a side squared. And then the area of this square right over here, the area of the square is just going to be 100 minus x over 4 squared. So our area, our combined area-- and maybe I can write it like this-- our combined area as a function of where we make the cut is all of this business right over here. And this is what we need to minimize. So we need to minimize that right over there. And I will do that in the next video." + }, + { + "Q": "If a white dwarf isn't undergoing fusion, why does it radiate anything at all? Shouldn't it just skip straight to a black dwarf?", + "A": "Leftover heat.", + "video_name": "EdYyuUUY-nc", + "transcript": "In the last video, we started with a star in its main sequence, like the sun. And inside the core of that star, you have hydrogen fusion going on. So that is hydrogen fusion, and then outside of the core, you just had hydrogen. You just hydrogen plasma. And when we say plasma, it's the electrons and protons of the individual atoms have been disassociated because the temperatures and pressures are so high. So they're really just kind of like this soup of electrons and protons, as opposed to proper atoms that we associate with at lower temperatures. So this is a main sequence star right over here. And we saw in the last video that this hydrogen is fusing into helium. So we start having more and more helium here. And as we have more and more helium, the core becomes more and more dense, because helium is a more massive atom. It is able to pack more mass in a smaller volume. So this gets more and more dense. So core becomes more dense. And so while the core is becoming more and more dense, that actually makes the fusion happen faster and faster. Because it's more dense, more gravitational pressure, more mass wanting to get to it, more pressure on the hydrogen that's fusing, so it starts to fuse hotter. So let me write this, so the fusion, so hydrogen fuses faster. And actually, we even see this in our sun. Our sun today is brighter and hotter. It's fusing faster than it was when it was born 4.5 or 4.6 billion years ago. But eventually you're going to get to the point so that the core, you only have helium. So there's going to be some point where the entire core is all helium. And it's going to be way denser than this core over here. All of that mass over there has now been turned into helium. A lot of it has been turned into energy. But most of it is now in helium, and it's going to be at a much, much smaller volume. And the whole time, the temperature is increasing, the fusion is getting faster and faster. And now there's this dense volume of helium that's not fusing. You do have, and we saw this in this video, a shell around it of hydrogen that is fusing. So this right here is hydrogen fusion going on. And then this over here is just hydrogen plasma. Now the unintuitive thing, or at least this was unintuitive to me at first, is what's going on the core is that the core is getting more and more dense. It's fusing at a faster rate. And so it's getting hotter and hotter. So the core is hotter, fusing faster, getting more and more dense. I kind of imagine it's starting to collapse. Every time it collapses, it's getting hotter and more dense. But at the same time that's happening, the star itself is getting bigger. And this is actually not drawn to scale. Red giants are much, much larger than main sequence stars. But the whole time that this is getting more dense, the rest of the star is, you could kind of view it as getting less dense. And that's because this is generating so much energy that it's able to more than offset, or better offset the gravitational pull into it. So even though this is hotter, it's able to disperse the rest of the material in the sun over a larger volume. And so that volume is so big that the surface, and we saw this in the last video, the surface of the red giant is actually cooler-- let me write that a little neater-- is actually cooler than the surface of a main sequence star. This right here is hotter. And just to put things in perspective, when the sun becomes a red giant, and it will become a red giant, its diameter will be 100 times the diameter that it is today. Or another way to be put it, it will have the same diameter as the Earth's orbit around the current sun. Or another way to view it is, where we are right now will be on the surface or near the surface or maybe even inside of that future sun. Or another way to put it, when the sun becomes a red giant, the Earth's going to be not even a speck out here. And it will be liquefied and vaporized at that point in time. So this is super, super huge. And we've even thought about it. Just for light to reach the current sun to our point in orbit, it takes eight minutes. So that's how big one of these stars are. To get from one side of the star to another side of the star, it'll take 16 minutes for light to travel, if it was traveling that diameter, and even slightly longer if it was to travel it in a circumference. So these are huge, huge, huge stars. And we'll talk about other stars in the future. They're even bigger than this when they become supergiants. But anyway, we have the hydrogen in the center-- sorry. We have the helium in the center. Let me write this down. We have a helium core in the center. We're fusing faster and faster and faster. We're now a red giant. The core is getting hotter and hotter and hotter until it gets to the temperature for ignition of helium. So until it gets to 100 million Kelvin-- remember the ignition temperature for hydrogen was 10 million Kelvin. So now we're at 100 million Kelvin, factor of 10. And now, all of a sudden in the core, you actually start to have helium fusion. And we touched on this in the last video, but the helium is fusing into heavier elements. And some of those heavier elements, and predominately, it will be carbon and oxygen. And you may suspect this is how heavier and heavier elements form in the universe. They form, literally, due to fusion in the core of stars. Especially when we're talking about elements up to iron. But anyway, the core is now experiencing helium fusion. It has a shell around it of helium that is not quite there, does not quite have the pressures and temperatures to fuse yet. So just regular helium. But then outside of that, we do have the pressures and temperatures for hydrogen to continue to fuse. So out here, you do have hydrogen fusion. And then outside over here, you just have the regular hydrogen plasma. So what just happened here? When you have helium fusion all of a sudden-- now this is, once again, providing some type of energetic outward support for the core. So it's going to counteract the ever-increasing contraction of the core as it gets more and more dense, because now we have energy going outward, energy pushing things outward. But at the same time that that is happening, more and more hydrogen in this layer is turning into helium, is fusing into helium. So it's making this inert part of the helium core even larger and larger and denser, even larger and larger, and putting even more pressure on this inside part. And so what's actually going to happen within a few moments, I guess, especially from a cosmological point of view, this helium fusion is going to be burning super-- I shouldn't use-- igniting or fusing at a super-hot level. But it's contained due to all of this pressure. But at some point, the pressure won't be able to contain it, and the core is going to explode. But it's not going to be one of these catastrophic explosions where the star is going to be destroyed. It's just going to release a lot of energy all of a sudden into the star. And that's called a helium flash. But once that happens, all of a sudden, then now the star is going to be more stable. And I'll use that in quotes without writing it down because red giants, in general, are already getting to be less stable than a main sequence star. But once that happens, you now will have a slightly larger volume. So it's not being contained in as small of a tight volume. That helium flash kind of took care of that. So now you have helium fusing into carbon and oxygen. And there's all sorts of other combinations of things. Obviously, there's many elements in between helium and carbon and oxygen. But these are the ones that dominate. And then outside of that, you have helium forming. You have helium that is not fusing. And then outside of that, you have your fusing hydrogen. Over here, you have hydrogen fusing into helium. And then out here in the rest of the radius of our super-huge red giant, you just have your hydrogen plasma out here. Now what's going to happen as this star ages? Well, if we fast forward this a bunch-- and remember, as a star gets denser and denser in the core, and the reactions happen faster and faster, and this core is expelling more and more energy outward, the star keeps growing. And the surface gets cooler and cooler. So if we fast forward a bunch, and this is what's going to happen to something the mass of our sun, if it's more massive, then at some point, the core of carbon and oxygen that's forming can start to fuse into even heavier elements. But in the case of the sun, it will never get to that 600 million Kelvin to actually fuse the carbon and the oxygen. And so eventually you will have a core of carbon and oxygen, or mainly carbon and oxygen surrounded by fusing helium surrounded by non-fusing helium surrounded by fusing hydrogen, which is surrounded by non-fusing hydrogen, or just the hydrogen plasma of the sun. But eventually all of this fuel will run out. All of the hydrogen will run out in the stars. All of this hydrogen, all of this fusing hydrogen will run out. All of this fusion helium will run out. This is the fusing hydrogen. This is the inert helium, which will run out. It'll be used in kind of this core, being fused into the carbon and oxygen, until you get to a point where you literally just have a really hot core of carbon and oxygen. And it's super-dense. This whole time, it will be getting more and more dense as heavier and heavier elements show up in the course. So it gets denser and denser and denser. But the super dense thing will not, in the case of the sun-- and if it was a more massive star, it would get there-- but in the case of the sun, it will not get hot enough for the carbon and the oxygen to form. So it really will just be this super-dense ball of carbon and oxygen and all of the other material in the sun. Remember, it was superenergetic. It was releasing tons and tons of energy. The more that we progressed down this, the more energy was releasing outward, and the larger the radius of the star became, and the cooler the outside of the star became, until the outside just becomes this kind of cloud, this huge cloud of gas around what once was the star. And in the center-- so I could just draw it as this huge-- this is now way far away from the star, much even bigger than the radius or the diameter of a red giant. And all we'll have left is a mass, a superdense mass of, I would call it, inert carbon or oxygen. This is in the case of the sun. And at first, when it's hot, and it will be releasing radiation because it's so hot. We'll call this a white dwarf. This right here is called a white dwarf. And it'll cool down over many, many, many, many, many, many, many, years, until it becomes, when it's completely cooled down, lost all of its energy-- it'll just be this superdense ball of carbon and oxygen, at which point, we would call it a black dwarf. And these are obviously very hard to observe because they're not emitting light. And they don't have quite the mass of something like a black hole that isn't even emitting light, but you can see how it's affecting things around it. So that's what's going to happen to the sun. In the next few videos, we're going to talk about what would happen to things less massive than the sun and what would happen to things more massive can imagine the more massive. There would be so much pressure on these things, because you have so much mass around it, that these would begin to fuse into heavier and heavier elements until we get to iron." + }, + { + "Q": "what dose the little 2 mean", + "A": "The little 2 is an exponent. The exponent is another way to write a number, like 5^2(5 with a little two on the top), it means 5 * 5, when you see a tiny number next to a big number(has to be on the top), it means the big numbers times itself that many times(the small number). So 250^600 would mean 250 multiplied by 250 600 times.", + "video_name": "aoXUWSwiDzE", + "transcript": "Let's do some practice problems dealing with variable expressions. So these first problems say write the following in a more condensed form by leaving out the multiplication symbol or leaving out a multiplication symbol. So here we have 2 times 11x, so if we have 11 x's and then we're going to have 2 times those 11 x's, we're going to have 22 x's. So another way you could view this, 2 times 11x, you could view this as being equal to 2 times 11, and all of that times x, and that's going to be equal to 22 x's. You had 11 x's, you're going to have 2 times as many x's, so you're going to have 22 x's. Let's see, you have 1.35 times y. Now here we're just going to do a straight simplifying how we write it. So 1.35 times y-- I'll do it in a different color-- 1.35 times y-- that's a little dot there. In algebra we can just get rid of that dot symbol. If we have a variable following a number, we know that means 1.35 times that variable. So that, we could rewrite as just being equal to 1.35y. We've condensed it by getting rid of the multiplication sign. Let's see, here we have 3 times 1/4. Well, this is just straight up multiplying a fraction. So in problem 3-- this was problem 1, this is problem 2, problem 3-- 3 times 1/4, that's the same thing as 3 over 1 times 1/4. Multiply the numerators, you get 3. Multiply the denominators, 1 times 4, you get 4. So number 3, I got 3/4. And then finally, you have 1/4 times z. We could do the exact same thing we did up here in problem number 2. This was the same thing as 1.35y. That's the same thing as 1.35 times y. So down here we could rewrite this as either being equal to 1/4z, or we could view this as being equal to 1 over 4 times z over 1, which is the same thing as z times 1, over 4 times 1, or the same thing as z over 4. So all of these are equivalent. Now, what do they want us to do down here? Evaluate the following expressions for a is equal to 3, b is equal to 2, c is equal to 5, and d is equal to minus 4-- or, actually, I should say negative 4 is the correct terminology. Negative 4. So we just substitute. Every time we see an a, we're going to put a minus 3 there, or a negative 3 there. Every time we see a b, we'll put a positive 2 there. Every time we see a c, we'll put a 5 there. And every time we see a d, we'll put a minus 4 there. And I'll do a couple of these. I won't do all of them, just for the sake of time. So let's say problem number 5. They gave us 2 times a plus 3 times b. Well, this is the same thing as 2 times-- instead of an a, we know that a is going to be equal to negative 3. So 2 times minus 3, plus 3 times b-- what's b? They're telling us that b is equal to 2-- so 3 times 2. And what is this equal to? 2 times minus 3-- let me do it in a different color-- 2 times negative 3 is negative 6, plus 3 times 2. 3 times 2 is 6. That's positive 6. So that is equal to 0. And notice the order of operations. We did the multiplications, we did the two multiplications before we added the two numbers. Multiplication and division takes precedence over addition and subtraction. Let's do problem 6. I'll do that right here. So you have 4 times c. 4 times-- now what's c equal to? They tell us c is equal to 5. So 4 times 5, that's our c, plus d. d is minus or negative 4. So we have 4 times 5 is 20, plus negative 4-- that's the same thing as minus 4, so that is equal to 16. Problem 6. Now, let's do one of the harder ones down here. This problem 10 looks a little bit more daunting. Problem 10 right there. So we have a minus 4b in the numerator, if you can read it, it's kind of small. a is minus 3. So we have minus 3-- or negative 3-- minus 4 times b. b is 2. So 4 times 2. Remember, this right here is a, that right there is b. They're telling me up here. And then all of that over-- all of that is over 3c plus 2d. So 3 times-- what was c? c is 5 plus 2 times d. What is d? d is negative 4. So let's figure this out. So we have to do order of operations. Multiplication comes first before addition and subtraction. So this is going to be equal to minus 3 minus 4 times 2, minus 8, all of that over-- 3 times 5 is 15, plus 2 times negative 4 is negative 8, or 15 plus negative 8 is 15 minus 8. And now, the numerator becomes negative 3 minus 8, which is negative 11. And the denominator is 15 minus 8, which is 7. So problem 10, we simplified it to negative 11 over 7. Right there. Let's do a couple of these over here. OK, we see some exponents. I'll pick one of the harder ones. Let's do this one over here, problem 18. So 2x squared minus 3x squared, plus 5x minus 4. OK, well, this wasn't that hard. All of them are dealing with x. But what we could do here-- let me write this down. 2x squared minus 3x squared, plus 5x minus 4. And they tell us that x is equal to negative 1. One thing we could do is simplify this before we even substitute for negative 1. So what's 2 of something minus 3 of something? This is 2x squareds minus 3x squareds. So 2 of something minus 3 of something, that's going to be minus 1 of that something. So that right there-- or negative 1 of that something-- that would be negative 1x squared plus 5x minus 4. And they tell us x is equal to negative 1. So this is negative 1 times x squared, negative 1 squared, plus 5 times x, which is negative 1, minus 4. So what is this? Negative 1 squared is just 1. That's just 1. So this whole expression simplifies to negative 1 plus 5 times negative 1-- we do the multiplication first, of course. So that's minus 5, or negative 5 minus 4. So negative 1 minus 5 is negative 6, minus 4 is equal to negative 10. And I'll do these last two just to get a sample of all of the types of problems in this variable expression section. The weekly cost, c, of manufacturing x remote controls-- so the cost is c, x is the remote controls-- is given by this formula. The cost is equal to 2,000 plus 3 times the number of remote controls, where cost is given in dollars. Question a, what is the cost of producing 1,000 remote controls? Well, the number of remote controls is x. So for part a-- I could write it over here-- the cost is going to be equal to-- just use this formula-- 2,000 plus 3 times the number of remote controls. x is the number of remote controls. So 3 times 1,000. So it's going to be equal to 2,000 plus 3 times 1,000 is 3,000, which is equal to $5,000. So that's part a. $5,000. Now part b. What is the cost of producing 2,000 remote controls? Well, the cost-- just use the same formula-- is equal to 2,000 plus 3 times the number of remote controls. So 3 times 2,000. So that's equal to 2,000 plus 3 times 2,000 is 6,000. So that's equal to $8,000. Now we're at problem 22. The volume of a box without a lid is given by the formula, volume is equal to 4x times 10 minus x squared, where x is a length in inches, and v is the volume in cubic inches. What is the volume when x is equal to 2? So part a, x is equal to 2. We just substitute 2 wherever we see an x here. So the volume is going to be equal to 4 times x, which is 2, times 10 minus 2 squared. And so this is going to be equal to 4 times 2 is 8, times 10 minus 2 is 8 squared. So that's 8 times-- so this is equal to 8 times-- 8 squared is 64. You could say this is going to be 8 to the third power. And 64 times 8-- 4 times 8 is 32. 6 times 8 is 48, plus 3 is 51. So it's 512. Now, what is the volume when x is equal to 3? I'll do it here in pink. b, when is equal to 3, then the volume is equal to 4 times 3-- x is equal to 3 now-- times 10 minus 3 squared. 4 times 3 is 12, times 10 minus 3 is 7 squared. So it's equal to 12 times 49. And just to get the exact answer, let's multiply that. So 49 times 12-- 2 times 9 is 18, 2 times 4 is 8, plus 1 is 9. Put a 0. 1 times 9 is 9. 1 times 4 is 4. And then we add the two. We get 8, 9 plus 9 is 18, and then 1 plus 4 is 588." + }, + { + "Q": "But what if you choose something like S and choose your number shift as 7.Then what do you do?", + "A": "It wraps around the alphabet. So if you shift A by 1 to get B, you would shift Z by 1 to get A.", + "video_name": "sMOZf4GN3oc", + "transcript": "SPEAKER 1: The first well known cipher, a substitution cipher, was used by Julius Caesar around 58 BC. It is now referred to as the Caesar Cipher. Caesar shifted each letter in his military commands in order to make them appear meaningless should the enemy intercept it. Imagine Alice and Bob decided to communicate using the Caesar Cipher First, they would need to agree in advance on a shift to use-- say, three. So to encrypt her message, Alice would need to apply a shift of three to each letter in her original message. So A becomes D, B becomes E, C becomes F, and so on. This unreadable, or encrypted message, is then sent to Bob openly. Then Bob simply subtracts the shift of three from each letter in order to read the original message. Incredibly, this basic cipher was used by military leaders for hundreds of years after Caesar. JULIUS CAESAR: I have fought and won. But I haven't conquered over man's spirit, which is indomitable. SPEAKER 1: However, a lock is only as strong as its weakest point. A lock breaker may look for mechanical flaws. Or failing that, extract information in order to narrow down the correct combination. The process of lock breaking and code breaking are very similar. The weakness of the Caesar Cipher was published 800 years later by an Arab mathematician named Al-Kindi. He broke the Caesar Cipher by using a clue based on an important property of the language a message is written in. If you scan text from any book and count the frequency of each letter, you will find a fairly consistent pattern. For example, these are the letter frequencies of English. This can be thought of as a fingerprint of English. We leave this fingerprint when we communicate without realizing it. This clue is one of the most valuable tools for a codebreaker. To break this cipher, they count up the frequencies of each letter in the encrypted text and check how far the fingerprint has shifted. For example, if H is the most popular letter in the encrypted message instead of E, then the shift was likely three. So they reverse the shift in order to reveal the original message. This is called frequency analysis, and it was a blow to the security of the Caesar cipher." + }, + { + "Q": "I think this is a really dumb question, but I don't get how the sun goes through the atmosphere.", + "A": "(facepalm) The Sun is 93 million miles away. It s not in our atmosphere at all.", + "video_name": "05qDIjKevJo", + "transcript": "In the last video, we talk about how seasons on Earth are not caused by how close Earth is to the sun in its orbit. And we also hint at the fact that it's actually caused by the tilt of the earth. And so in this video, I want to show you how the tilt of the earth causes the seasons to happen. So let's draw-- so I'm going to try to draw as many diagrams as possible here. Because at least for my brain they help me visualize what's actually going on. So we could imagine a top view first. So let's have a top view. That is the sun right over there. And let me draw the earth's orbit. So Earth's orbit maybe looks something like that. Let me draw it almost, it is almost circular. So I'll draw it as something that's pretty close to a circle right over here. And I'm going to draw Earth at different points in its orbit. And I'm going to try to depict the tilt of its rotational axis. And obviously, this is not drawn anywhere near close to scale. Earth is much further away from the sun, and much, much smaller than the sun as well. So I'll draw the earth at that point. And at this point, the earth will be tilted away from the sun. So Earth's tilt does not change if you think about the direction, or at least over the course of a year, if we think about relatively small periods of time. It does not change relative to the direction that it's pointing at in the universe. And we'll talk about that in a second. So let's say right over here we are pointed away from the sun. So we're up and out of this page. So if I wanted to put some perspective on an arrow it would be up and-- actually it would be more like up and out of this page. So that's the direction. If you were to come straight out of the North Pole. And if you were to go straight out of the South Pole you'd go below that circle right over there. And if I wanted to draw the same position, but if we're looking sideways along the plane, the orbital plane, or the plane of Earth's orbit. So if we're looking at it from that direction. So let me do it this way. If we're looking at directly sideways, this is the sun right over here, and this is Earth at that position. This is Earth right over there. If I were to draw an arrow pointing straight out of the North Pole it would look something like this. So this arrow and this arrow they are both popping straight out of the North Pole. And so when we talk about the tilt of the earth, we're talking about the tilt of its orbital axis, kind of this pole that could go straight between the South Pole and the North Pole. The angle between that and a pole that would actually be at a 90-degree angle, or perpendicular, to the plane of its orbit. And so compared to if it was just straight up and down, relative to the plane of the orbit. So this right here is the angle of Earth's tilt. Let me draw that a little bit bigger just so it becomes a little bit clearer. So if this is the plane of the orbit, we're looking sideways along the plane of the orbit. And this is Earth right over here. My best attempt to draw a circle. That is Earth. Earth does not rotate. Its axis of rotation is not perpendicular to the plane of the orbit. So this is how Earth would orbit. This is how Earth would rotate if it was. Earth rotates, Earth's rotational axis is at an angle to that vertical relative to the plane of its orbit, I guess you could say it. It rotates at an angle like this. So this would be the North Pole. That is the South Pole. And so it rotates like this. And that angle relative to being vertical with respect to the orbital plane, this angle right here for Earth right now is 23.4 degrees. And if we're talking about relatively short periods of time, like our lifespans, that is constant. But it is actually changing over long periods of time. That is changing between-- and these are rough numbers-- it is changing between 22.1 degrees and 24.5 degrees, if my sources are correct. But that gives a rough estimate of what it's changing between. But I want to make it clear, this is not happening overnight. The period to go from roughly a 22-degree angle to a 24 and 1/2-degree angle and back to a 22-degree angle is 41,000 years. And this long-term change in the tilt, this might play into some of the long-term climactic change. Maybe it might contribute, on some level, to some of the ice ages that have formed over Earth's past. But for the sake of thinking about our annual seasons you don't have to worry too much, or you don't have to worry at all really about this variation. You really just have to know that it is tilted. And right now it is tilted at an angle of 23.4 degrees. Now you might say OK, I understand what the tilt is. But how does that change the seasons in either the Northern or that Southern Hemisphere? And to do that, I'm going to imagine the earth when the Northern Hemisphere is most tilted away from the sun, and when it is most tilted towards the sun. So remember this tilt, the direction this arrow points into relative to the rest of the universe, if we assume that this tilt is at 23.4%, it's not changing throughout the year. But depending on where it is in the orbit it's either going to be tilting away from the sun, as it is in this example right over here. Or it will be tilting towards the sun. I'll do the towards the sun in this magenta color, or it would be tilting towards the sun. So six months later when the earth is over here, it's going to, relative to the rest of the universe, it will be tilted in that same direction, up out of this page and to the right. Just like it was over here. But now that it's on the other side of the sun that makes it tilt a little bit more towards the sun. If I were to draw it right over here, it is now tilted towards the sun. And what I want to think about is how much sunlight will different parts of the planet receive. And I'll focus on the Northern Hemisphere. But you can make a similar argument for the Southern Hemisphere. I want to think about how much sunlight they receive when it's tilted away or tilted towards the sun. And so let's think about those two situations. So first of all, let's think about this situation here where we are tilted away from the sun. So let me zoom in a little bit. So this is the situation, where we're tilted away from the sun. So if this is the vertical, so let me draw it. I could actually just use this diagram. But let me make it. So we're tilted away from the sun like this. I'm going to do this in a different color. So if we have an arrow coming straight out of the North Pole it would look like this. And we are rotating around like that. So we're out of the page on the left-hand side, and then into the page on the right-hand side. And so we're rotating towards the east, constantly. So this arrow is in the direction of the east. So when we're at this point in Earth's orbit, and actually let me copy and paste this. And I'm going to use the same exact diagram for the different seasons. So let me copy. And then let me paste this exact diagram. I'll do it over here for two different points. So when we are here in Earth's orbit where is the sunlight coming from? Well, it's going to be coming from the left, at least the way I've drawn the diagram right over here. So the sunlight is coming from the left in this situation. And so if you think about it, what part of the earth is being lit by sunlight? Or what part of the earth is in daylight, the way I've drawn it right over here? Well, the part that is facing the sun. So all of this right over here is going to be in daylight. As we rotate whatever part of the surface of the earth enters into this yellow part right over here will be in daylight. But let's think about what's happening So let me draw the equator, which separates our Northern and Southern Hemispheres. So this is the equator. And then let me go into the Northern Hemisphere. And I want to show you why when the North Pole is pointed away from the sun why this is our winter. So when we're pointed away from the sun-- Well, if we go to the Arctic Circle-- so let me go right over here. Let me go to some point in the Arctic Circle. As it goes, as the earth rotates every 24 hours, this point on the globe will just rotate around just like that. It will just keep rotating around just like that. And so my question is, that point in the Arctic Circle, as it rotates will it ever see sunlight? Well, no, it will never see sunlight. Because the North Pole is tilted away from the sun. So what I'm drawing, what I'm shading here in purple, that part of the earth, when it's completely tilted away, will never see sunlight. Or at least it won't see sunlight while it's tilted away, while it's in this position, or in this position in the orbit. I won't say never, because once it becomes summer they will be able to see it. So no sunlight, no day, I guess you could say, no daylight. If you go to slightly more southern latitudes, so let's say you go over here. So maybe that's the latitude of something like, I don't know, New York or San Francisco or something like that. Let's think about what it would see as the earth rotates every 24 hours. So this would be daylight, daylight, daylight, daylight, then nighttime, nighttime, nighttime, nighttime, nighttime. This is now going behind the globe nighttime, nighttime, nighttime, nighttime, nighttime, daylight, daylight, daylight, daylight. So if you just compare this. So let me do the daylight in orange. And then nighttime I will do in this bluish purplish color. So night time over here. So if you go to really northern latitudes, like the Arctic Circle, they don't get any daylight when we are tilted away from the earth. And if we go to slightly still northern latitudes, but not as north as the Arctic Circle, it does get daylight. But it gets a lot less daylight. It spends a lot less time in the daylight than in the night time. So notice if you say that this circumference represents the positions over 24 hours, it spends much less time in the daylight than it does in the nighttime. So because, while the Northern Hemisphere is tilted away from the earth, the latitudes in the northern hemisphere are getting less daylight. They are also getting less energy from the sun. And so that's what leads to winter, or just being generally colder. And to see what happens in the summer let's just go the other side. So now we're going to the other side of our orbit This is going to be six months later. And notice the actual direction, relative to the rest of the universe, has not changed. We're still pointed in that same direction. We still have a 23.4 degree tilt relative to, I guess, being straight up and down. But now once we're over here the light from the sun is going to be coming from the right. Just like that. And now, if on this diagram at least, this is the side of the earth that is going to be getting the sunlight. And let me draw the equator again, or my best attempt to draw the equator. I'll draw the equator in that same color actually, in that green color. So this separates the Northern and the Southern Hemisphere. And now let's think about the Arctic Circle. So let's say I'm sitting here in the Arctic Circle. As the day goes on, as 24 hours go around, I'll keep rotating around here. But notice the whole time I am inside of the sun. I'm getting no nighttime. There is no night in the Arctic Circle while we are tilted towards the sun. And if we still do that fairly northern latitude, but not as far as the Arctic Circle, maybe in San Francisco or New York, or something like that . If we go to that latitude, notice how much time we spend in the sun. So maybe we just enter. So this is right at sunrise. And then as the day goes on we're in sunlight, sunlight, sunlight, sunlight, sunlight, sunlight, sunlight, sunlight. Then we hit sunset. Then we hit nighttime, nighttime, then we hit nighttime, and then we get sunrise again. And so when you look at the amount of time that something in the Northern Hemisphere spends in the daylight versus sunlight, you'll see it spends a lot more time in the daylight when the Northern Hemisphere is tilted towards the sun. So this is more day, less night. So it is getting more energy from the sun. So when it is tilted towards the sun it is getting more energy from the sun. So things will generally be warmer. And so you are now talking about summer in the Northern Hemisphere. And the arguments for the Southern Hemisphere are identical. You could even play it right over here. When the Northern Hemisphere is tilted away from the sun, then the Southern Hemisphere is tilted towards the sun. And so for example, the South Pole will have all daylight and no nighttime. And southern latitudes will have more daylight than nighttime. And so the south will have summer. So this is summer in the south, in the Southern Hemisphere. And it's winter in the north. And then down here the Southern Hemisphere is pointed away from the sun. So this is winter in the Southern Hemisphere. And you might be saying, hey Sal, what about you haven't talked a lot about spring and fall. Well let's think about it. Well, if we're talking about the Northern Hemisphere, this over here, we decided, was winter in the Northern Hemisphere. And we're going to rotate around the sun. And at some point, we're going to get over here. And then because of this tilt we aren't pointed away or towards the sun. We're kind of pointed I guess sideways relative to the direction of the sun. But this doesn't favor one hemisphere over the other. So when we're over here in-- and this will actually be the spring now- when we're in the spring, both hemispheres are getting the equal amount of daylight and sunlight, or for a given latitude above or below the equator, they're getting the same amount. And the same thing is true over here when we get to-- so this is spring. This is the summer in the Northern Hemisphere. Now this will be the fall in the Northern Hemisphere. And once again, we're tilted in this direction. And so the Northern Hemisphere isn't tilted away or towards the sun. And so both hemispheres are going to get the same amount of radiation from the sun. So you really see the extremes in the winters and the summers. Now one thing I do want to make clear, and I started off with just the length of day and nighttime. Because frankly, that's maybe a little bit, or at least in my brain, a little bit easier to visualize. But that by itself does not account for all of the difference between summer and winter. Another cause, and actually this is probably the biggest cause, is if you think about the total amount of sun. So let's talk about the Northern Hemisphere winter. And let's say there's a certain amount of sunlight that is reaching the earth. So this is the total amount of sunlight that's reaching the earth at any point in time. You see that much more of that is hitting the Southern Hemisphere than the Northern Hemisphere here. All of these, if you imagine it, all of these rays right over here are hitting the Southern Hemisphere. So a majority of the rays are hitting the Southern Hemisphere. And much fewer are hitting the Northern Hemisphere. So actually a smaller amount of the radiation period, at even a given period in time, not even talking about the amount of time you are facing the sun. But at any given moment in time more energy is hitting the Southern Hemisphere than the Northern. And the opposite is true when the tilt is then towards the sun. And now a disproportionate amount of the sun's energy is hitting the Northern Hemisphere. So if you draw a bunch of, if you just think that this is all of the energy from the sun, most of it, all of these rays up here, are hitting the Northern Hemisphere. And only these down here are hitting the Southern And on top of that, what makes it even more extreme is that the actual angle, and of course, this is to some degree is due to the fact that where the angle of the sun relative to the horizon, or where you are on Earth. But even more than that if you are on, let's say that this is the land, and we're talking about the winter in the Northern Hemisphere. So let's say you're talking about, let's say we're up over here at this northern latitude. And we're just looking at the sun here. And over here, you could see even when we are closest to the sun the sun is not directly overhead. When we're closest to the sun the sun still is pretty low on the horizon. So it may be right over here when we're closest to the sun in the winter the sun might be right over here. But if you look at that same latitude in the summer when it is closest to the sun, the sun is more close to being directly overhead. It still won't be directly overhead. Because we are still at a relatively northern latitude. But the sun is going to be much higher in the sky. And these are all related to each other. It's kind of connected with this idea that more energy is hitting one hemisphere or the other. But also, when you have a, I guess you could say, a steeper angle from the rays of the sun with the earth, it's actually going to be dissipated less by the atmosphere. And let me just make it clear how this is. So in the summer-- so let's say that that's the land. And let's say that-- let me draw the atmosphere in white-- so all of this area right over here, this is the atmosphere. And obviously there's not a hard boundary for the atmosphere. But let's just say this is the densest part of the atmosphere. In the summer, when the sun is higher in the sky, the rays from the sun are dissipated by less atmosphere. So they have to get through this much atmosphere. And they're bounced off. And they heat some of that atmosphere. And they're absorbed before they get to the ground. In the winter when the sun is lower in the sky, so maybe the sun is out here. Let me draw it a little bit. So when the sun is lower in the sky relative to this point, you see that the rays of sunlight have to travel through a lot more atmosphere. So they get dissipated much more before they get to this point on the planet. So all in all it is the tilt that is causing the changes in the season. But it's causing it for multiple reasons. One is when you're tilted, we'll say when you're tilted towards the sun, you're getting more absolute hours of daylight. Not only are you getting more absolute hours of daylight, but at any given moment, most or more of the sun's total rays that are hitting the earth are hitting the Northern Hemisphere as opposed to the Southern Hemisphere. And the stuff that's hitting the places that have summer, it has to go through less atmosphere. So it gets dissipated less." + }, + { + "Q": "Give the geometric interpretation of equation 3y+15=0 as in\n1) One Variable &\n2) Two Varibles", + "A": "The equation 3y + 15 = 0 simplifies to y = -5. This is linear equation in one variable, so the geometric interpretation of this is a straight line. In this case it is parallel to the x-axis and crosses the y-axis at the point (0, -5).", + "video_name": "2VeqrZ_PMiY", + "transcript": "Use substitution to solve for x and y. And they give us a system of equations here. y is equal to negative 5x plus 8 and 10x plus 2y is equal to negative 2. So they've set it up for us pretty well. They already have y explicitly solved for up here. So they tell us, this first constraint tells us that y must be equal to negative 5x plus 8. So when we go to the second constraint here, every time we see a y, we say, well, the first constraint tells us that y must be equal to negative 5x plus 8. So everywhere we see a y, we can substitute it with negative 5x plus 8. Because that's what the first constraint tells us. y is equal to that. I don't want to be repetitive, but I really want you to internalize that's all it's saying. y is that. So every time we see a y in the second constraint, we can substitute it with that. So let's do it. So the second equation over here is 10x plus 2. And instead of writing a y there, and I've said it multiple times already, we can write a negative 5x plus 8. The first constraint tells us that's what y is. So negative 5x plus 8 is equal to negative 2. Now, we have one equation with one unknown. We can just solve for x. We have 10x plus. So we can multiply it. We can distribute this 2 onto both of these terms. So we have 2 times negative 5x is negative 10x. And then 2 times 8 is 16. So plus 16 is equal to negative 2. Now we have 10x minus 10x. Those guys cancel out. 10x minus 10x is equal to 0. So these guys cancel out. And we're just left with 16 equals negative 2, which is crazy. We know that 16 does not equal negative 2. This is an inconsistent result. And that's because these two lines actually don't intersect. And we could see that by actually graphing these lines. Whenever you get something like some number equalling some other number that they're clearly not equal to, that means it's an inconsistent result, It's an inconsistent system, and that these lines actually don't intersect. So let me just graph these just to make it clear. This first equation is already in slope y-intercept form. So it looks something like this. That's our x-axis. This is our y-axis. And it's negative 5x plus 8, so 1, 2, 3, 4, 5, 6, 7, 8. And then it has a very steep downward slope. Every time you move forward 1, you have to go down 5. So it looks something like that. That's this first equation right over there. The second equation, let me rewrite it in slope y-intercept form. So it's 10x plus 2y is equal to negative 2. Let's subtract 10x from both sides. You get 2y is equal to negative 10x minus 2. Let's divide both sides by 2. You get y is equal to negative 5x, negative 5x minus 1. So it's y-intercept is negative 1. It's right over there. And it has the same slope as this first line. So it looks like this. It's parallel. It's just shifted down a bit. So it just looks like that. So they're parallel lines. They have the same slope, different y-intercepts. We get an inconsistent result. They don't intersect. And the telltale sign of that, when you're doing it algebraically, is you get something wacky like this. This is why it's called inconsistent. It's not consistent for 16 to be equal to negative 2. These don't intersect. There's no solution to both of these constraints, no x and y that satisfies both of them." + }, + { + "Q": "Isn't FOILing basically just doing the distributive property?", + "A": "Yes... It is an expansion of the distributive property. If you want to multiply: (x+3)(x+5), the distributive property allows us to distribute the whole binomial (x+3) across the 2nd binomial. We get: x(x+3) +5(x+3). We then apply the distributive property to distribute both the x and the 5, and we get: x^2 + 3x + 5x + 15. This is the same sequence of terms as if you apply FOIL. FOIL let s us skip the 1st step of creating x(x+3) +5(x+3). Hope this helps.", + "video_name": "ZMLFfTX615w", + "transcript": "Multiply (3x+2) by (5x-7). So we are multiplying two binomials. I am actually going to show you two really equivalent ways of doing this. One that you might hear in a classroom and it is kind of a more mechanical memorizing way of doing it which might be faster but you really don't know what you are doing and then there is the one where you are essentially just applying something what you already know and kind of a logical way. So I will first do the memorizing way that you might be exposed to and they'll use something called FOIL. So let me write this down here. So you can immediately see that whenever someone gives you a new mnemonic to memorize, that you are doing something pretty mechanical. So FOIL literally stands for First Outside, let me write it this way.....F O I L where the F in FOIL stands for First, the O in FOIL stand for Outside, the I stands for Inside and then the L stands for Last. The reason why I don't like these things is that when you are 35 years old, you are not going to remember what FOIL stood for and then you are not going to remember how to multiply this binomial. But lets just apply FOIL. So First says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x. So (3x. 5x). The Outside part tells us to multiply the outside terms. So in this case, you have 3x on the outside and you have -7 on the outside. So that is +3x(-7). The inside, well the inside terms here are 2 and 5x. So, (+2.5x) and then finally you have the last terms. You have the 2 and the -7. So the last terms are 2 times -7. 2(-7). So what you are essentially doing is just making sure that you are multipying each term by every other term here. What we are essentially doing is multiplying, doing the distributive property twice. We are multiplying the 3x times (5x-7). So 3x times (5x-7) is (3x . 5x) plus (3x - 7). And we are multiplying the 2 times (5x-7) to give us these terms. But anyway, lets just multiply these out just to get our answer. 3x times 5x is same thing as (3 times 5) ( x times x) which is the same thing as 15x square. You can just do this x to the first time to x to the first. You multiply the x to get x squared. 3 times 5 is 15. This term right here 3 times -7 is -21 and then you have your x right over here. And then you have this term which is 2 times 5 which is 10 times x. So +10x. And then finally you have this term here in blue. 2 times -7 is -14. And we aren't done yet, we can simplify this a little bit. We have two like terms here. We have this...let me find a new color. We have 2 terms with a x to the first power or just an x term right over here. So we have -21 of something and you add 10 or in another way, you have 10 of something and you subtract 21 of them, you are going to have -11 of that something. We put the other terms here, you have 15... 15x squared and then you have your -14 and we are done. Now I said I would show you another way to do it. I want to show you why the distributive property can get us here without having to memorize FOIL. So the distributive property tells us that if we 're... look if we are multipying something times an expression, you just have to multiply times every term in the expression. So we can distribute, we can distribute the 5x onto the 3..., or actually we could...well, let me view it this way... we could distribute the 5x-7, this whole thing onto the 3x+2. Let me just change the order since we are used to distributing something from the left. So this is the same thing as (5x-7)(3x+2). I just swapped the two expressions. And we can distribute this whole thing times each of these terms. Now what happens if I take (5x-7) times 3x? Well, thats just going to be 3x times (5x-7). So I have just distributed the 5x-7 times 3x and to that I am going to add 2 times 5x-7. I have just distributed the 5x-7 onto the 2. Now, you can do the distributive property again. We can distribute the 3x onto the 5x. We can distribute the 3x onto the 5x. And we can distribute the 3x onto the -7. We can distribute the 2 onto the 5x, over here and we can distribute the 2 on that -7. Now if we do it like this what do we get ? 3x times 5x, that's this right over here. If we do 3x times -7, that's this term right over here. If you do 2 times 5x, that's this term right over here. If you do 2 times -7, that is this term right over here. So we got the exact same result that we got with FOIL. Now, FOIL can be faster if you just wanted to do it and kind of skip to this step. I think its important that you know that this is how it actually works. Just in case you do forget this when you are 35 or 45 years old and you are faced with multiplying binomial, you just have to remember the distributive property." + }, + { + "Q": "what was the Bernoulli Principle", + "A": "ok that was the Bernoulli Principle! very helpful now i know what this guy is talking about!", + "video_name": "aLJzEl5st8s", + "transcript": "So this is an airplane here. OK, so you probably already knew that. If you've flown in one, or maybe just seen them fly. But even if you've seen them or been in one, do you know how they work? Is it magic? [CHANTS GIBBERISH] Are there invisible fairies that hold the plane aloft? All right, men. We've got a busy morning and lots of flights to carry. Or is it science? Well, you can guess that the answer is indeed science. That's ridiculous. What? So to discuss how an airplane flies, we first have to talk about the forces on an airplane which push it around in all sorts of different directions. Now we're going to focus on airplanes today because they're awesome, but most of these forces apply to any other vehicle. The first force acts on all these vehicles-- really, it acts on everything. It's the weight force, which points down towards the center of Earth. Weights is equal to the mass of the airplane-- m right here-- times the acceleration due to gravity. Here on Earth, g is equal to 9.81 meters per second squared. Now that's only for Earth. The acceleration due to gravity really depends on the mass of the planet that your are on. The larger the planet, the higher the gravity. So 9.81 meters per second squared here on Earth. The moon, however-- it's smaller than Earth. So the acceleration due to gravity is only 1/6 that on Earth-- 1.6 meters per second squared. This is why astronauts can bounce high on the moon, but not on Earth. This isn't nearly as much fun. Obviously, there has to be another force opposing the weight and pushing the airplane up. This force is called lift. Lift operates perpendicular to the airplane's wings, which are right here in this side view. Now, if these are our only two forces, our aircraft will be able to go up and down, but it won't go anywhere. So we have to have a force that pushes the airplane forward, and this is called thrust. All vehicles have thrust, otherwise they wouldn't go anywhere like our airplane. Why didn't you buy a car with thrust? I'm sorry. We can at least roll down the hill. On an aircraft, this thrust is produced by engines. There are two main types of engines. We have propellers, like this little guy right here. And jet engines, like our first model. Whatever the type of engines, they all work by the same principle. Let's draw a little side view of an engine here. The engines accelerate air out the back this direction. And by Newton's third law, there's an equal and opposite reaction, and that's the thrust force pushing the aircraft forward. This is really the same thing that happens when you blow up a balloon and you let it go. The air come out the back and the balloon moves forward. We have a force that opposes the thrust. It's called drag. It points opposite the direction of flight. The major type of drag is pressure drag, which is the force caused by the air smacking into the airplane. So we try to minimize this type of drag by making the airplane as aerodynamic as possible. That means that it has smooth lines in the air flows nice and cleanly over the front here. You can feel the pressure drag when you stick your hand out the window of a moving car. Uh, honey, honey. Your hand-- your hand, please. When your hand this horizontal, it's aerodynamic and you really don't feel a lot of drag. But if you slowly turn your hand vertical, you really feel the drag increasing. So these are our four forces on the airplane, but perhaps you're thinking-- So this really cool and everything, but how do we increase and decrease the airplanes lift to move up and down? That's a great question. Let's look at the equation for the magnitude of lift per unit wing area. We'll call that L. L equals 1/2 times rho times cl times v squared. That simple. OK, OK, I'll tell you what each of these things mean. So rho-- it's not a P. It's the Greek letter, rho. Rho is the density of the air, which is a measure of the number of air molecules in a certain volume. Density of the air varies with altitude and temperature, so you go higher up. There, the air is thinner, and so the density is lower. If we want to simplify things, we generally use the standard density, which is 1.2754 kilograms per meters cubed. v here is the speed of the aircraft, or how fast it's traveling. And cl is something called the coefficient of lift. It's a number that gives us some information about the shape of the aircraft's wings-- these things right here. The coefficient of lift changes with the angle of attack. Angle of what? Aircraft can pitch up and down, and even if they're pitched up, they're still traveling in a horizontal direction like that. Now the angle formed here by the horizontal direction of travel and the direction of the aircraft's nose is called the angle of attack, and we denote that with the Greek letter alpha. So we can make a little plot here of that. We're going to put coefficient of lift up on the y-axis, and the angle of attack down on the x-axis. So as the airplane starts to pitch up-- if I can get a little hand here-- thank you. As the aircraft starts to pitch up, the coefficient of lift increases. This is a good thing because we have more lift. As we continue to increase, we eventually reach a point where we keep pitching up but the lift starts decreasing. This is something called stall, and it's not a good thing. So we generally avoid try to pitching up this much. There's a similar equation for the drag per unit wing area, D. D equals 1/2 rho. Not cl-- that wouldn't make any sense. cd, as you can guess, is the coefficient of drag times the velocity squared. The coefficient of drag is-- it's another number that tells us something about the wings, and it also varies with the angle of attack. So as the angle of attack increases-- oh, thank you-- the coefficient of drag increases as well. Thank you very much. This is because as the aircraft is pitching up, there is more wing area perpendicular to the flow. Now, this reminds me of something that we talked about earlier. Exactly. This is very similar to whenever you hold your hand out the window of a car. And so, that's pretty much everything you need know about how an aircraft flies. So the next time you're on an airplane or you just see one, you can really know exactly what it is that's keeping it up in the air. Nope. No, it's not them either. Ah, there you are. Now you got it." + }, + { + "Q": "What about something like p over 5 is equal to 230", + "A": "p/5=230 p/5*5=230*5 p=1,150", + "video_name": "a3acutLstF8", + "transcript": "Let's get some practice solving some equations, and we're gonna set up some equations that are a little bit hairier than normal, they're gonna have some decimals and fractions in them. So let's say I had the equation 1.2 times c is equal to 0.6. So what do I have to multiply times 1.2 to get 0.6? And it might not jump out immediately in your brain but lucky for us we can think about this a little bit methodically. So one thing I like to do is say okay, I have the c on the left hand side, and I'm just multiplying it by 1.2, it would be great if this just said c. If this just said c instead of 1.2c. So what can I do there? Well I could just divide by 1.2 but as we've seen multiple times, you can't just do that to the left hand side, that would change, you no longer could say that this is equal to that if you only operate on one side. So you have to divide by 1.2 on both sides. So on your left hand side, 1.2c divided by 1.2, well that's just going to be c. You're just going to be left with c, and you're going to have c is equal to 0.6 over 1.2 Now what is that equal to? There's a bunch of ways you could approach it. The way I like to do it is, well let's just, let's just get rid of the decimals. Let's just multiply the numerator and denominator by a large enough number so that the decimals go away. So what happens if we multiply the numerator and the denominator by... Let's see if we multiply them by 10, you're gonna have a 6 in the numerator and 12 in the denominator, actually let's do that. Let's multiply the numerator and denominator by 10. So once again, this is the same thing as multiplying by 10 over 10, it's not changing the value of the fraction. So 0.6 times 10 is 6, and 1.2 times 10 is 12. So it's equal to six twelfths, and if we want we can write that in a little bit of a simpler way. We could rewrite that as, divide the numerator and denominator by 6, you get 1 over 2, so this is equal to one half. And if you look back at the original equation, 1.2 times one half, you could view this as twelve tenths. Twelve tenths times one half is going to be equal to six tenths, so we can feel pretty good that c is equal to one half. Let's do another one. Let's say that we have 1 over 4 is equal to y over 12. So how do we solve for y here? So we have a y on the right hand side, and it's being divided by 12. Well the best way I can think of of getting rid of this 12 and just having a y on the right hand side is multiplying both sides by 12. We do that in yellow. So if I multiply the right hand side by 12, I have to multiply the left hand side by 12. And once again, why did I pick 12? Well I wanted to multiply by some number, that when I multiply it by y over 12 I'm just left with y. And so y times 12 divided by 12, well that's just going to be 1. And then on the left hand side you're going to have 12 times one fourth, which is twelve fourths. So you get 12 over 4, is equal to y. Or you could say y is equal to 12 over 4, y is equal to, let me do that just so you can see what I'm doing, just flopping the sides, doesn't change what's being said, y is equal to 12 over 4. Now what is twelve fourths? Well, you can view this as 12 divided by 4, which is 3, or you could view this as twelve fourths which would be literally, 3 wholes. So you could say this would be equal to 3. Y is equal to 3, and you can check that. One fourth is equal to 3 over 12, so it all works out. That's the neat thing about equations, you can always check to see if you got the right answer. Let's do another one, can't stop. 4.5 is equal to 0.5n So like always, I have my n already on the right hand side. But it's being multiplied by 0.5, it would be great if it just said n. So what can I do? Well I can divide both sides, I can divide both sides by 0.5, once again, if I do it to the right hand side I have to do it to the left hand side. And why am I dividing by 0.5? So I'm just left with an n on the right hand side. So this is going to be, so on the left hand side, I have 4.5 over 0.5, let me just, I don't want to skip too many steps. 4.5 over 0.5, is equal to n, because you have 0.5 divided by 0.5, you're just left with an n over here. So what does that equal to? Well 4.5 divided by 0.5, there's a couple ways to view this. You could view this as forty-five tenths divided by five tenths, which would tell you okay, this is going to be 9. Or if that seems a little bit confusing or a little bit daunting, you can do what we did over here. You could multiply the numerator and the denominator by the same number, so that we get rid of the decimals. And in this case, if you multiply by 10 you can move the decimal one to the right. So once again, it has to be multiplying the numerator and the denominator by the same thing. We're multiplying by 10 over 10, which is equivalent to 1, which tells us that we're not changing the value of this fraction. So let's see, this is going to be 45 over 5, is equal to n. And some of you might say wait wait wait, hold on a second, you just told us whatever we do to one side of the equation, we have to do to the other side of the equation and here you are, you're just multiplying the left hand side of this equation by 10 over 10. Now remember, what is 10 over 10? 10 over 10 is just 1. Yes, if I wanted to, I could multiply the left hand side by 10 over 10, and I could multiply the right hand side by 10 over 10, but that's not going to change the value of the right hand side. I'm not actually changing the values of the two sides. I'm just trying to rewrite the left hand side by multiplying it by 1 in kind of a creative way. But notice, n times 10 over 10, well that's still going to just be n. So I'm not violating this principle of whatever I do to the left hand side I do to the right hand side. You can always multiply one side by 1 and you can do that as many times as you want. Like the same way you can add 0 or subtract 0 from one side, without necessarily having to show you're doing it to the other side, because it doesn't change the value. But anyways, you have n is equal to 45 over 5, well what's 45 over 5? Well that's going to be 9. So we have 9 is equal to, why did I switch to green? We have 9 is equal to n, or we could say n is equal to 9. And you could check that: 4.5 is equal to 0.5 times 9, yup half of 9 is 4.5 Let's do one more, because once again I can't stop. Alright, let me get some space here, so we can keep the different problems apart that we had. So let's do, let's have a different variable now. Let's say we have g over 4 is equal to 3.2. Well I wanna get rid of this dividing by 4, so the easiest way I can think of doing that is multiplying both sides by 4. So I'm multiplying both sides by 4, and the whole reason is 4 divided by 4 gives me 1, so I'm gonna have g is equal to, what's 3.2 times 4? Let's see 3 times 4 is 12, and two tenths times 4 is eight tenths, so it's gonna be 12 and eight tenths. G is going to be 12.8, and you can verify this is right. 12.8 divided by 4 is 3.2." + }, + { + "Q": "Does the heart pump the same blood back to the body? If it does, why does it have to go get oxygen from the lungs when it already went through the process? Also, if it is the same blood, why does it turn blue when blood already went through the pulmonary circulation?", + "A": "As the blood travels through the body, oxygen is transferred to the tissues. Blood has to pass through the lungs to get more oxygen. You can think of the hemoglobin in the blood like buckets that hold oxygen - the buckets keep going around and around. When the buckets travel through the body they drop off oxygen and when they go to the lungs they pick more up.", + "video_name": "7b6LRebCgb4", + "transcript": "- [Instructor] Let's talk a little bit about arteries and veins and the roles they play in the circulatory system. So I want you to pause this video and first think to yourself, Do you have a sense of what arteries and veins are? Well one idea behind arteries and veins are that well, in most of these drawings, arteries are drawn in red, and I even made the artery word here in red. And veins are drawn in blue. And so maybe that represents how much oxygen they have. And so one possible explanation is that arteries carry oxygenated blood, oxygenated, oxygenated blood, while veins carry deoxygenated blood. So blood that has less oxygen now. Now this is actually incorrect. It is, many times, the case that arteries are the ones carrying oxygenated blood and veins are carrying the deoxygenated blood. But as we will see, this is not always the case. And since we're already talking about oxygenated blood and deoxygenated blood and the colors red and blue, it's worth addressing another misconception. Many times it is said that deoxygenated blood looks blue, and the reason why people believe that is if you look at your wrist and you're able to see some of the vessels in there, you will see some blue vessels. And those, or at least they look blue when you're looking from the outside of your skin. And those, indeed, are veins. And so that's where the misconception has come from, that veins, which, in your arm, are carrying deoxygenated blood. That that deoxygenated blood is blue. It turns out that it is not blue. It is just a deeper red. And the reason why the veins look blue is because of the optics of light going through your skin and then seeing the outside of the veins and then reflecting back. That is not the color of the actual blood. So so far I have not given you a clear definition of what arteries versus veins are. A better definition, so let me cross these two out, are that arteries carry blood away from the heart. Away from the heart. And veins carry blood towards the heart. Towards the heart. And I can get a zoomed in image of the heart right here and that will make it a little bit clearer. And you can also see, or we're about to see, why this first definition, or this first distinction between arteries and veins does not always hold. So let's just imagine some blood that is being pumped away from the heart. So right when it gets pumped away from the heart, it'll be right over here. It gets pumped through the aorta, and you can see the aorta branches, so some blood can go up towards your head, and if it didn't, you would pass out and die. And then a lot of the blood goes down towards the rest of your body. And that, indeed, is the most oxygenated blood. And so it'll flow through your body. And these arteries will keep branching and branching into smaller vessels, all the way until they form these very small branches. And it's that place, especially, where they will lose a lot of their oxygen to the fluid and the cells around them. And then the blood is less oxygenated. And then even though deoxygenated blood is not blue, it often gets depicted as blue in a lot of diagrams. So I will do the same. And these vessels start building into your veins. And these really small vessels that really bridge between arteries and veins, where a lot of the gas and nutrient exchange occurs, these are called capillaries. And so after going through the capillaries, the blood will then come back to the heart and now it's coming towards the heart through the veins. It comes into the right atrium, then the right ventricle. Then that gets pumped towards the lungs. And this is the exception to the first incorrect definition of arteries and veins that we looked at. This right over here, is an artery. Even though it's carrying less oxygenated or deoxygenated blood, it's an artery because it's carrying blood away from the heart. But in this case, it's not carrying it to the rest of the body, it is carrying it to the lungs. That is why it is called the pulmonary artery, even though it's carrying less oxygenated blood. So that it goes to the lungs and then, in the lungs, there's more gas exchange that occurs. The blood gets oxygenated and then it comes back to the heart. And so it comes back to the heart in these vessels right over here, and that even though these are carrying highly oxygenated blood, these are considered veins because they're carrying blood towards the heart. So these are pulmonary veins. And then the cycle starts again. The pulmonary veins bring the oxygenated blood into the left atrium and the left ventricle, and then that pumps it to the rest of the body to the aorta, for your systemic circulation. You have your pulmonary circulation, which circulates the blood to, through and from the lungs. And you have your systemic circulation, which takes the blood to and from the rest of the body. So now that we have this main distinction between arteries and veins, what are some other interesting things that we know about it? Well one thing to keep in mind is that since arteries are being pumped directly by the heart towards the rest of the body, they have high pressure. I'll write that in caps. High pressure. And so if you were to have an accident of some type, which you do not want to have, and you were to accidentally cut an artery, because of that high pressure, it would actually spurt blood, a lot more than if you were to cut a vein. And most of the times where you get a cut, you're really just cutting capillaries. Like if you were to prick your finger, it's usually a series of capillaries that get cut, and that's why the blood would come out very very slowly. Now if arteries are high pressure, veins are low pressure. Low, low pressure. And one way to think about it is the arteries, the blood is being pumped directly by the heart. But then once it goes through the capillaries and comes back through the veins, it's kind of sluggishly making its way back to the heart. It's not being directly pumped. And that's why in veins, because you don't have that high pressure to bring everything back to the heart, you have these valves that make sure that for the most part, the blood is going in one direction. I'm going to draw the blood in red in the veins, just so we don't keep going with that misconception, that blood in the veins is blue somehow. Now related to the fact that the blood in the arteries is under higher pressure, in order to transport a fixed volume of blood in a certain amount of time, you need less volume. And so that's why arteries are low volume. And on the other hand, veins are high volume. And to appreciate the difference, the blood volume in arteries are only approximately 15% of the entire blood volume in your body, while the blood volume in veins are closer to 65%, approximately 65%. And if you're wondering where the rest of the blood is, about five percent is in capillaries, five percent is in your heart, and about 10% is in your lungs. So I will leave you there. The big take away: arteries are the vessels that take blood away from the heart. Veins are the vessels that take blood towards the heart." + }, + { + "Q": "How will this help me later in life?", + "A": "Whichever job you are in, you are sure to find your self using math to solve stuff. So almost everything in math will help you later in life.", + "video_name": "Tm98lnrlbMA", + "transcript": "I'm here with Jesse Ro, whose a math teacher at Summit San Jose and a Khan Academy teaching fellow and you had some interesting ideas or questions. Yeah, one question that students ask a lot when they start Algebra is why do we need letters, why can't we just use numbers for everything? Why letters? So why do we have all these Xs and Ys and Zs and ABCs when we start dealing with Algebra? Yeah, exactly. That's interesting, well why don't we let people think about that for a second. So Sal, how would you answer this question? Why do we need letters in Algebra? So why letters. So there are a couple of ways I'd think about it. One is if you have an unknown. So if I were to write X plus three is equal to ten the reason why we're doing this is that we don't know what X is It's literally an unknown. And so we're going to solve for it in some way. But it did not have to be the letter X. We could have literally written blank plus three is equal to ten. Or we could have written Question Mark plus three is equal to ten. So it didn't have to be letters, but we needed some type of symbol. It literally could've been Smiley Face plus three is equal to ten. But until you know it, you need some type of a symbol to represent whatever that number is. Now we can go and solve this equation and then know what that symbol represents. But if we knew it ahead of time, it wouldn't be an unknown. It wouldn't be something that we didn't know. So that's one reason why I would use letters and where just numbers by itself wouldn't be helpful. The other is when you're describing relationships between numbers. So I could do something like - I could say - that whenever you give me a three, I'm going to give you a four. And I could say, if you give me a five, I'm going to give you a six. And i could keep going on and on forever. If you give me a 7.1, I'm going to give you an 8.1. And I could keep listing this on and on forever. Maybe you could give me any number, and I could tell you what I'm going to give you. But I would obviously run out of space and time if I were to list all of them. And we could do that much more elegantly if we used letters to describe the relationship. Maybe what you give me we call X, and what I give you we call Y. And so I say, look, whatever you give me, I'm going to add one to it. And that's what I'm going to give back to you. And so now, this very simple equation here can describe an infinite number of relationships between X or an infinite number of corresponding Ys and Xs. So now someone knows whatever X you give me you give me three, I add one to it, and I'm going to give you four. You give me 7.1, I'm going to add one to it and give you 8.1. So there is no more elegant way that you could've done it than by using symbols. With that said, I didn't have to use Xs and Ys. This is just a convention that kind of comes to use from history. I could've defined what you give me as Star and what I give you as Smiley Face and this also would've been a valid way to express this. So the letters are really just symbols. Nothing more." + }, + { + "Q": "And wouldn't it be safer for you to both deny? then you would both keep the same sentences. You are taking the risk of taking on another year if you both confess.", + "A": "But they cannot talk together... remember that.", + "video_name": "UkXI-zPcDIM", + "transcript": "On the same day, police have made two at first unrelated arrests. They arrest a gentleman named Al. And they caught him red handed selling drugs. So it's an open and shut case. And the same day, they catch a gentleman named Bill. And he is also caught red handed, stealing drugs. And they bring them separately to the police station. And they tell them, look, this is an open and shut case. You're going to get convicted for drug dealing and you're going to get two years. And they tell this to each of them individually. They were selling the same type of drugs, just happened to be that. But they were doing it completely independently. Two years for drugs is what's going to happen assuming nothing else. But then the district attorney has a chance to chat with each of these gentleman separately and while he's chatting with them he reinforces the idea this is an open and shut case for the drug dealing. They're each going to get two years, if nothing else happens. But then he starts to realize that these two characters look like-- he starts to have a suspicion, for whatever reason, that these were the two characters that actually committed a much more serious offense. That they had committed a major armed robbery a few weeks ago. And all the district attorney has to go on is his hunch, his suspicion. He has no hard evidence. So what he wants to do is try to get a deal with each of these guys so that they have an incentive to essentially snitch on each other. So what he tells each of them is, look, you're going to get two years for drug dealing. That's kind of guaranteed. But he says, look, if you confess and the other doesn't then you will get 1 year. And the other guy will get 10 years. So he's telling Al, look, we caught Bill, too, just If you confess that it was you and Bill who performed that armed robbery your term is actually going to go down from two years to one year. But Bill is obviously going to have to spend a lot more time in jail. Especially because he is not cooperating with us. He is not confessing. But then, the other statement is also true. If you deny and the other confesses now it switches around. You will get 10 years, because you're not cooperating. And the other, your co-conspirator, will get a reduced sentence-- will get the one year. So this is like telling Al, look, if you deny that you were the armed robber and Bill snitches you out, then you're going to get 10 years in prison. And Bill's only going to get one year in prison. And if both of you essentially confess, you will both get three years. So this scenario is called the prisoner's dilemma. Because we'll see in a second there is a globally optimal scenario for them where they both deny and they both get two years. But we'll see, based on their incentives, assuming they don't have any unusual loyalty to each other-- and these are hardened criminals here. They're not brothers or related to each other in any way. They don't have any kind of loyalty pact. We'll see that they will rationally pick, or they might rationally pick, a non-optimal scenario. And to understand that I'm going to draw something called a payoff matrix. So let me do it right here for Bill. So Bill has two options. He can confess to the armed robbery or he can deny that he had anything-- that he knows anything about the armed robbery. And Al has the same two options. Al can confess and Al can deny. And since it's called a payoff matrix, let me draw some grids here. Let me draw some grids and let's think about all of the different scenarios and what the payoffs would be. If Al confesses and Bill confesses then we're in scenario four. They both get three years in jail. So they both will get three for Al and three for Bill. Now, if Al confesses and Bill denies, then we are in scenario two from Al's point of view. Al is only going to get one year. But Bill is going to get 10 years. Now, if the opposite thing happens, if Bill confesses and Al denies, then it goes the other way around. Al's going to get 10 years for not cooperating. And Bill's going to have a reduced sentence of one year for cooperating. And then if they both deny, they're in scenario one, where they're both just going to get their time for the drug dealing. So Al will get two years, and Bill will get two years. Now, I alluded to this earlier in the video. What is the globally optimal scenario for them? Well, it's this scenario, where they both deny having anything to do with the armed robbery. Then they both get two years. But what we'll see is actually somewhat rational, assuming that they don't have any strong loyalties to each other, or strong level of trust with the other party, to not go there. And it's actually rational for both of them to confess. And the confession is actually a Nash equilibrium. And we'll talk more about this, but a Nash equilibrium is where each party has picked a choice given the choices of the other party. So when we think of, or each party has to pick the optimal choice, given whatever choice the other party picks. And so from Al's point of view, he says, well, look, I don't know whether Bill is confessing or denying. So let's say he confesses. What's better for me to do? If he confesses, and I confess, then I get three years. If he confesses and I deny I get 10 years. So if he confesses, it's better for me to confess as well. So this is a preferable scenario to this one down here. Now, I don't know that Bill confessed. He might deny. If I assume Bill denied, is it better for me to confess and get one year or deny and get two years? Well, once again, it's better for me to confess. And so regardless of whether Bill confesses or denies, so this once again, the optimal choice for Al to pick, taking into account Bill's choices, is to confess. If Bill confesses, Al is better off confessing. And if Bill denies, Al is better off confessing. Now, we look at it from Bill's point of view. And it's completely symmetric. If Bill says, well, I don't know if Al is confessing or denying. If Al confesses, I can confess and get three years or I can deny and get 10 years. Well, three years in prison is better than 10. So I will go-- I would go for the three years if I know Al is confessing. But I don't know that Al is definitely confessing. He might deny. If Al is denying, I could confess and get one year or I could deny and get two years. Well, once again, I would want to confess and get the one year. So Bill, taking into account each of the scenarios that Al might take, it's always better for him to confess. And so this is interesting. They are rationally deducing that they should get to this scenario, this Nash equilibrium state, as opposed to this globally optimal state. They're both getting three years by both confessing as opposed to both of them getting two years by both denying. The problem with this one is this is an unstable state. If one of them assumes that the other one has-- if one of them assumes that they're somehow in that state temporarily, they say, well, I can always improve my scenario by changing what I want to do. If Al thought that Bill was definitely denying, Al could improve his circumstance by moving out of that state and confessing and only getting one here. Likewise, if Bill thought that maybe Al is likely to deny, he realizes that he can optimize by moving in this direction. Instead of denying, getting, two and two, he could move in that direction right over there. So this is an unstable optimal scenario. But this Nash equilibrium, this state right over here, is actually very, very, very stable. If they assume, it's better for each of them to confess regardless of what the other ones does. And assuming all of the other actors have chosen their strategy, there's no incentive for Bill. So if assuming everyone else has changed their strategy, you can only move in that direction. If you're Bill, you can go from the Nash equilibrium of confessing to denying, but you're worse off. So you won't want to do that. Or you could move in this direction, which would be Al changing his decision. But once again, that gives a worse outcome for Al. You're going from three years to 10 years. So this is the equilibrium state, the stable state, that both people will pick something that is not optimal globally." + }, + { + "Q": "I am confused about the way you write electron configuration. ex. 1s^2 2s^2 and so on. I'm pretty sure the \"1s\" means first shell and \"2s\" means second shell. (correct me if I'm wrong) but what is the exponent? (^2) what does that represent?", + "A": "The number in front tells you the shell The letter tells you what type of orbital it is The superscript number tells you how many electrons are in that orbital 1s^2 means 2 electrons are in the 1s orbital 2s^2 means 2 electrons are in the 2s orbital Note the s doesn t stand for shell", + "video_name": "FmQoSenbtnU", + "transcript": "In the last few videos we learned that the configuration of electrons in an atom aren't in a simple, classical, Newtonian orbit configuration. And that's the Bohr model of the electron. And I'll keep reviewing it, just because I think it's an important point. If that's the nucleus, remember, it's just a tiny, tiny, tiny dot if you think about the entire volume of the actual atom. And instead of the electron being in orbits around it, which would be how a planet orbits the sun. Instead of being in orbits around it, it's described by orbitals, which are these probability density functions. So an orbital-- let's say that's the nucleus it would describe, if you took any point in space around the nucleus, the probability of finding the electron. So actually, in any volume of space around the nucleus, it would tell you the probability of finding the electron within that volume. And so if you were to just take a bunch of snapshots of electrons -- let's say in the 1s orbital. And that's what the 1s orbital looks like. You can barely see it there, but it's a sphere around the nucleus, and that's the lowest energy state that an electron can be in. If you were to just take a number of snapshots of electrons. Let's say you were to take a number of snapshots of helium, which has two electrons. Both of them are in the 1s orbital. It would look like this. If you took one snapshot, maybe it'll be there, the next snapshot, maybe the electron is there. Then the electron is there. Then the electron is there. Then it's there. And if you kept doing the snapshots, you would have a bunch of them really close. And then it gets a little bit sparser as you get out, as you get further and further out away from the electron. But as you see, you're much more likely to find the electron close to the center of the atom than further out. Although you might have had an observation with the electron sitting all the way out there, or sitting over here. So it really could have been anywhere, but if you take multiple observations, you'll see what that probability function is describing. It's saying look, there's a much lower probability of finding the electron out in this little cube of volume space than it is in this little cube of volume space. And when you see these diagrams that draw this orbital like this. Let's say they draw it like a shell, like a sphere. And I'll try to make it look three-dimensional. So let's say this is the outside of it, and the nucleus is sitting some place on the inside. They're just saying -- they just draw a cut-off -- where can I find the electron 90% of the time? So they're saying, OK, I can find the electron 90% of the time within this circle, if I were to do the cross-section. But every now and then the electron can show up outside of that, right? Because it's all probabilistic. So this can still happen. You can still find the electron if this is the orbital we're talking about out here. Right? And then we, in the last video, we said, OK, the electrons fill up the orbitals from lowest energy state to high energy state. You could imagine it. If I'm playing Tetris-- well I don't know if Tetris is the thing-- but if I'm stacking cubes, I lay out cubes from low energy, if this is the floor, I put the first cube at the lowest energy state. And let's say I could put the second cube But I only have this much space to work with. So I have to put the third cube at the next highest energy state. In this case our energy would be described as potential energy, right? This is just a classical, Newtonian physics example. But that's the same idea with electrons. Once I have two electrons in this 1s orbital -- so let's say the electron configuration of helium is 1s2 -- the third electron I can't put there anymore, because there's only room for two electrons. The way I think about it is these two electrons are now going to repel the third one I want to add. So then I have to go to the 2s orbital. And now if I were to plot the 2s orbital on top of this one, it would look something like this, where I have a high probability of finding the electrons in this shell that's essentially around the 1s orbital, right? So right now, if maybe I'm dealing with lithium right now. So I only have one extra electron. So this one extra electron, that might be where I observed that extra electron. But every now and then it could show up there, it could show up there, it could show up there, but the high probability is there. So when you say where is it going to be 90% of the time? It'll be like this shell that's around the center. Remember, when it's three-dimensional you would kind of cover it up. So that's what they drew here. They do the 1s. It's just a red shell. And then the 2s. The second energy shell is just this blue shell over it. And you can see it a little bit better in, actually, the higher energy orbits, the higher energy shells, where the seventh s energy shell is this red area. Then you have the blue area, then the red, and the blue. And so I think you get the idea that each of those are energy shells. So you kind of keep overlaying the s energy orbitals around each other. But you probably see this other stuff here. And the general principle, remember, is that the electrons fill up the orbital from lowest energy orbital to higher energy orbital. So the first one that's filled up is the 1s. This is the 1. This is the s. So this is the 1s. It can fit two electrons. Then the next one that's filled up is 2s. It can fill two more electrons. And then the next one, and this is where it gets interesting, you fill up the 2p orbital. 2p orbital. That's this, right here. 2p orbitals. And notice the p orbitals have something, p sub z, p sub x, p sub y. What does that mean? Well, if you look at the p-orbitals, they have these dumbbell shapes. They look a little unnatural, but I think in future videos we'll show you how they're analogous to standing waves. But if you look at these, there's three ways that you can configure these dumbbells. One in the z direction, up and down. One in the x direction, left or right. And then one in the y direction, this way, forward and backwards, right? And so if you were to draw-- let's say you wanted to draw the p-orbitals. So this is what you fill next. And actually, you fill one electron here, another electron here, then another electron there. Then you fill another electron, and we'll talk about spin and things like that in the future. But, there, there, and there. And that's actually called Hund's rule. Maybe I'll do a whole video on Hund's rule, but that's not relevant to a first-year chemistry lecture. But it fills in that order, and once again, I want you to have the intuition of what this would look like. Look. I should put look in quotation marks, because it's very abstract. But if you wanted to visualize the p orbitals-- let's say we're looking at the electron configuration for, let's say, carbon. So the electron configuration for carbon, the first two electrons go into, so, 1s1, 1s2. So then it fills-- sorry, you can't see everything. So it fills the 1s2, so carbon's configuration. It fills 1s1 then 1s2. And this is just the configuration for helium. And then it goes to the second shell, which is the second period, right? That's why it's called the periodic table. We'll talk about periods and groups in the future. And then you go here. So this is filling the 2s. We're in the second period right here. That's the second period. One, two. Have to go off, so you can see everything. So it fills these two. So 2s2. And then it starts filling up the p orbitals. So then it starts filling 1p and then 2p. And we're still on the second shell, so 2s2, 2p2. So the question is what would this look like if we just wanted to visualize this orbital right here, the p orbitals? So we have two electrons. So one electron is going to be in a-- Let's say if this is, I'll try to draw some axes. That's too thin. So if I draw a three-dimensional volume kind of axes. If I were to make a bunch of observations of, say, one of the electrons in the p orbitals, let's say in the pz dimension, sometimes it might be here, sometimes it might be there, sometimes it might be there. And then if you keep taking a bunch of observations, you're going to have something that looks like this bell shape, this barbell shape right there. And then for the other electron that's maybe in the x direction, you make a bunch of observations. Let me do it in a different, in a noticeably different, color. It will look like this. You take a bunch of observations, and you say, wow, it's a lot more likely to find that electron in kind of the dumbell, in that dumbbell shape. But you could find it out there. You could find it there. You could find it there. This is just a much higher probability of finding it in here than out here. And that's the best way I can think of to visualize it. Now what we were doing here, this is called an electron configuration. And the way to do it-- and there's multiple ways that are taught in chemistry class, but the way I like to do it -- is you take the periodic table and you say, these groups, and when I say groups I mean the columns, these are going to fill the s subshell or the s orbitals. You can just write s up here, just right there. These over here are going to fill the p orbitals. Actually, let me take helium out of the picture. The p orbitals. Let me just do that. Let me take helium out of the picture. These take the p orbitals. And actually, for the sake of figuring out these, you should take helium and throw it right over there. Right? The periodic table is just a way to organize things so it makes sense, but in terms of trying to figure out orbitals, you could take helium. Let me do that. The magic of computers. Cut it out, and then let me paste it right over there. Right? And now you see that helium, you get 1s and then you get 2s, so helium's configuration is -- Sorry, you get 1s1, then 1s2. We're in the first energy shell. Right? So the configuration of hydrogen is 1s1. You only have one electron in the s subshell of the first energy shell. The configuration of helium is 1s2. And then you start filling the second energy shell. The configuration of lithium is 1s2. That's where the first two electrons go. And then the third one goes into 2s1, right? And then I think you start to see the pattern. And then when you go to nitrogen you say, OK, it has three in the p sub-orbital. So you can almost start backwards, right? So we're in period two, right? So this is 2p3. Let me write that down. So I could write that down first. 2p3. So that's where the last three electrons go into the p orbital. Then it'll have these two that go into the 2s2 orbital. And then the first two, or the electrons in the lowest energy state, will be 1s2. So this is the electron configuration, right here, of nitrogen. And just to make sure you did your configuration right, what you do is you count the number of electrons. So 2 plus 2 is 4 plus 3 is 7. And we're talking about neutral atoms, so the electrons should equal the number of protons. The atomic number is the number of protons. So we're good. Seven protons. So this is, so far, when we're dealing just with the s's and the p's, this is pretty straightforward. And if I wanted to figure out the configuration of silicon, right there, what is it? Well, we're in the third period. One, two, three. That's just the third row. And this is the p-block right here. So this is the second row in the p-block, right? One, two, three, four, five, six. Right. We're in the second row of the p-block, so we start off with 3p2. And then we have 3s2. And then it filled up all of this p-block over here. So it's 2p6. And then here, 2s2. And then, of course, it filled up at the first shell before it could fill up these other shells. So, 1s2. So this is the electron configuration for silicon. And we can confirm that we should have 14 electrons. 2 plus 2 is 4, plus 6 is 10. 10 plus 2 is 12 plus 2 more is 14. So we're good with silicon. I think I'm running low on time right now, so in the next video we'll start addressing what happens when you go to these elements, or the d-block. And you can kind of already guess what happens. We're going to start filling up these d orbitals here that have even more bizarre shapes. And the way I think about this, not to waste too much time, is that as you go further and further out from the nucleus, there's more space in between the lower energy orbitals to fill in more of these bizarro-shaped orbitals. But these are kind of the balance -- I will talk about standing waves in the future -- but these are kind of a balance between trying to get close to the nucleus and the proton and those positive charges, because the electron charges are attracted to them, while at the same time avoiding the other electron charges, or at least their mass distribution functions. Anyway, see you in the next video." + }, + { + "Q": "I am having trouble converting 2x=12+2y into slope intercept form", + "A": "All we have to do is isolate the variable y : 2x = 12 + 2y Remember that slope-intercept form is: y = mx + b Let s flip our equation to make it easier to work with: 12 + 2y = 2x Subtract 12 from both sides: 2y = 2x - 12 Now divide both sides by 2: y = (2x - 12)/2 y = x - 6 Comment if you have questions.", + "video_name": "V6Xynlqc_tc", + "transcript": "We're asked to convert these linear equations into slope-intercept form and then graph them on a single coordinate plane. We have our coordinate plane over here. And just as a bit of a review, slope-intercept form is a form y is equal to mx plus b, where m is the slope and b is the intercept. That's why it's called slope-intercept form. So we just have to algebraically manipulate these equations into this form. So let's start with line A, so start with a line A. So line A, it's in standard form right now, it's 4x plus 2y is equal to negative 8. The first thing I'd like to do is get rid of this 4x from the left-hand side, and the best way to do that is to subtract 4x from both sides of this equation. So let me subtract 4x from both sides. The left hand side of the equation, these two 4x's cancel out, and I'm just left with 2y is equal to. And on the right-hand side I have negative 4x minus is 8, or negative 8 minus 4, however you want to do it. Now we're almost at slope-intercept form. We just have to get rid of this 2, and the best way to do that that I can think of is divide both sides of this equation by 2. So let's divide both sides by 2. So we divide the left-hand side by 2 and then divide the right-hand side by 2. You have to divide every term by 2. And then we are left with y is equal to negative 4 divided by 2 is negative 2x. Negative 8 divided by 2 is negative 4, negative 2x minus 4. So this is line A, let me graph it right now. So line A, its y-intercept is negative 4. So the point 0, negative 4 on this graph. If x is equal to 0, y is going to be equal to negative 4, you can just substitute that in the graph. So 0, 1, 2, 3, 4. That's the point 0, negative 4. That's the y-intercept for line A. And then the slope is negative 2x. So that means that if I change x by positive 1 that y goes down by negative 2. So let's do that. So if I go over one in the positive direction, I have to go down 2, that's what a negative slope's going to do, negative 2 slope. If I go over 2, I'm going to have to go down 4. If I go back negative 1, so if I go in the x direction negative 1, that means in the y direction I go positive two, because two divided by negative one is still negative two, so I go over here. If I go back 2, I'm going to go up 4. Let me just do that. Back 2 and then up 4. So this line is going to look like this. Do my best to draw it, that's a decent job. That is line A right there. All right, let's do line B. So line B, they say 4x is equal to negative 8, and you might be saying hey, how do I get that into slope-intercept form, I don't see a y. And the answer is you won't be able to because you this can't be put into slope-intercept form, but we can simplify it. So let's divide both sides of this equation by 4. So you divide both sides of this equation by 4. And you get x is equal to negative 2. So this just means, I don't care what your y is, x is just always going to be equal to negative 2. So x is equal to negative 2 is right there, negative 1, negative 2, and x is just always going to be equal to negative 2 in both directions. And this is the x-axis, that's the y-axis, I forgot to label them. Now let's do this last character, 2y is equal to negative eight. So line C, we have 2y is equal to negative 8. We can divide both sides of this equation by 2, and we get y is equal to negative 4. So you might say hey, Sal, that doesn't look like this form, slope-intercept form, but it is. It's just that the slope is 0. We can rewrite this as y is equal to 0x minus 4, where the y-intercept is negative 4 and the slope is 0. So if you move an arbitrary amount in the x direction, the y is not going to change, it's just going to stay at negative 4. Let me do a little bit neater. y is just going to stay at negative 4. Or you can just interpret it as y is equal to negative 4 no matter what x is. So then we are done." + }, + { + "Q": "The H2CO3 is carbonic acid, right?", + "A": "Correct, H2CO3 is carbonic acid.", + "video_name": "OCD4Dr3kmmA", + "transcript": "Let me do a little experiment. Let's say I have oxygen here, and we know that oxygen is about 21% of the atmosphere. And I decide to take a cup, let's say a cup like this-- simple cup of water. And I leave it out on the counter. And it's about room temperature, about 25 degrees Celsius here. And I want to know, how much oxygen is really going to enter that cup at that surface layer? So let's say I want to measure the concentration of oxygen in that surface layer of water. Well, you know, I say 21% so, of course, there's some molecules of oxygen here. And it's only 21%, it's not like it's the majority. So I've got to draw some other molecules. This could be nitrogen or some other molecule, let's say. But I'm focused on the blue dots, because the blue dots are the oxygen dots. And so over time I let this kind of sit out. And maybe I come back and check, and a little bit of oxygen has entered my surface layer of water. In fact, if I measured it, I could say, well, the concentration, C, at that level is 0.27 millimoles per liter. And this number is literally just something that I would have to measure, right? I would actually measure the concentration there, and that's the measure of oxygen. So I've learned about Henry's law, and I can think well, you know, I know the partial pressure now. And I can rearrange the formula so that it looks something like this. I can say, well Henry's law basically is like that. So if I know the pressure and I know the concentration, I should be able to figure out the constant for myself. I can figure it out and kind of give it in units that I like. So I'm going to write the units of the KH down here. I can say, well, 769 liters times atmospheres over moles. And that's something that I've just calculated. I've just taken two numbers and I've divided them by each other. So this is my calculation for oxygen. And so far, so good, right? But now I decide to challenge myself and say, let's do this again. But instead of with oxygen, I'm going to create an environment that's 21% carbon dioxide, which is way more carbon dioxide than we actually have. But imagine I could actually do that. I actually find a way to crank up the carbon dioxide, and I do the exact same thing. I take a cup of water and I keep it out at room temperature, 25 degrees Celsius. And I say, OK, let's see how much carbon dioxide goes into my cup. I've got my carbon dioxide out here. And over time more and more molecules kind of settle in here. And, of, course the atmosphere is not going to run out of carbon dioxide molecules. They're just going to keep replacing them. But they keep settling into this top layer, this surface layer, of my water. So it's actually looking already really different than what was happening on the other side. We only had a little bit of oxygen but now I've got tons of carbon dioxide. And I don't want to make it uneven. I mentioned before, we have nitrogen-- so let me still draw a bunch of nitrogen-- that will outnumber the carbon dioxide dramatically. Because we have here about, let's say, 79% nitrogen and we only had 21% carbon dioxide. So it'll look something like that. But there's lots and lots of carbon dioxide there. In fact, if I was to calculate the concentration on this side, the concentration would be pretty high. It would be 7.24 millimoles per liter. Again, these numbers, I'm assuming that I'm doing the experiment. This is the number I would find if I actually did the experiment. So it's a much bigger number than I had over here. On the oxygen side, the number was actually pretty small, not very impressive. And yet on the carbon oxide side, much, much higher. Now that's kind of funny. It might strike you as kind of a funny thing. Because look, these partial pressures are basically the same. I mean, not even basically, they're exactly the same. There's no difference in the partial pressure. And yet the concentrations are different. So if you keep the P the same, the only way to make for different concentrations is if you have a different constant. So let me actually move on and figure out what the constant is. So what do you think the constant on this side would be, higher or lower? Let's see if we can figure it out together. The K sub H on this side is going to be lower. It's going to be lower. It's 29 liters times atmosphere divided by moles. So it's a much lower number. And I don't want you to get so distracted by this bit. This is kind of irrelevant to what we're talking about. It's just the units, and we can change the units to whatever we want. But it's this part-- it's the fact that the number itself on the carbon dioxide side is lower. Now let's think back to this idea of Henry's law. Henry's law told us that the partial pressure, this number, tells you about what's going to be going into the water, and that the K sub H tells you about what's going out of the water. And so if what's going in on both sides is equivalent, then really the difference is going to be what's the leaving. And on this side, on the first side of our experiment, we had lots of oxygens leaving this water. They didn't like being in water. They were leaving readily. And so you didn't see that, but they were actually constantly leaving. And on the carbon dioxide side, you had maybe a little bit of leaving, but not very much. The carbon dioxide was actually very comfortable with the water. In fact, to see that as a chemical formula, you might recall this. Remember, there's this formula where CO2 binds with water and it forms H2CO3. Well, think about that. If it's a binding to the water then it's not going to want to leave. It's pretty comfortable being in the water. And so the moment that carbon dioxide goes into water, it does something like this. It binds to the water. It turns into bicarbonate and protons. And so it's a very comfortable being in water, and that's why it's not leaving. In fact, I can take this one step further and even compare I could say, well, 769 divided by 29 equals about 26. So that's another way of saying that carbon dioxide is 26 times more soluble than oxygen. I'll put that in parentheses-- than oxygen. And I should make sure I make it very clear. This is at 25 degrees Celsius, and this is in water. Now, you might say, well, that's fine for 25 degrees Celsius. But what about body temperature? What's happening in our actual body? What's happening in our lungs? So in our lungs, we have 37 degrees Celsius. And instead of water-- actually, I shouldn't be writing water-- instead of water, it's blood, which is slightly different than water. The consistency is different. And so these K sub H values are actually temperature dependent. And they're going to change as you increase the temperature. So at this new temperature, it turns out that carbon dioxide is about 22 times more soluble than oxygen. So it's still pretty impressive. Sometimes you might even see 24 times, depending on what numbers you read. But this is an impressive difference. And actually, what I wanted to get to is the fact that it goes back to the idea of what's going in and what's coming out. And the net difference is why you end up with a huge difference in concentrations between carbon dioxide and oxygen." + }, + { + "Q": "What is Hemophelia?", + "A": "Hemophilia is an inherited blood disorder in which the blood s clotting ability is impaired. Therefore there is no mechanism in the body to stop the bleeding if a hemophiliac get a cut or even bruise (bleeding under the skin). The primary gene that codes for hemophilia is x-linked, or carried on the x-chromosome so it affects males more often than females as they don t have another x-chromosome to mask the expression of the gene.", + "video_name": "-ROhfKyxgCo", + "transcript": "By this point in the biology playlist, you're probably wondering a very natural question, how is gender determined in an organism? And it's not an obvious answer,because throughout the animal kingdom, it's actually determined in different ways. In some creatures,especially some types of reptiles,it's environmental Not all reptiles,but certain cases of it. It could be maybe the temperature in which the embryo develops will dictate whether it turns into a male or female or other environmental factors. And in other types of animals,especially mammals, of which we are one example,it's a genetic basis. And so your next question is,hey,Sal,so-- let me write this down,in mammals it's genetic-- so,OK,maybe they're different alleles,a male or a female allele. But then you're like,hey,but there's so many different characteristics that differentiate a man from a woman. Maybe it would have to be a whole set of genes that have to work together. And to some degree,your second answer would be more correct. It's even more than just a set of genes. It's actually whole chromosomes determine it. So let me draw a nucleus.That's going to be my nucleus. And this is going to be the nucleus for a man. So 22 of the pairs of chromosomes are just regular non-sex-determining chromosomes. So I could just do,that's one of the homologous,2,4,6,8,10,12,14. I can just keep going.And eventually you have 22 pairs. So these 22 pairs right there,they're called autosomal. And those are just our standard pairs of chromosomes that code for different things. Each of these right here is a homologous pair,homologous, which we learned before you get one from each of your parents. They don't necessarily code for the same thing, for the same versions of the genes,but they code for the same genes. If eye color is on this gene,it's also on that gene, on the other gene of the homologous pair. Although you might have different versions of eye color on either one and that determines what you display. But these are just kind of the standard genes that have nothing to do with our gender. And then you have these two other special chromosomes. I'll do this one. It'll be a long brown one,and then I'll do a short blue one. And the first thing you'll notice is that they don't look homologous. How could they code for the same thing when the blue one is short and the brown one's long? And that's true.They aren't homologous. And these we'll call our sex-determining chromosomes. And the long one right here, it's been the convention to call that the x chromosome. Let me scroll down a little bit. And the blue one right there,we refer to that as the y chromosome. And to figure out whether something is a male or a female, it's a pretty simple system. If you've got a y chromosome,you are a male. So let me write that down. So this nucleus that I drew just here-- obviously you could have the whole broader cell all around here-- this is the nucleus for a man. So if you have an x chromosome-- and we'll talk about in a second why you can only get that from your mom-- an x chromosome from your mom and a y chromosome from your dad, you will be a male. If you get an x chromosome from your mom and an x chromosome from your dad,you're going to be a female. And so we could actually even draw a Punnett square. This is almost a trivially easy Punnett square, but it kind of shows what all of the different possibilities are. So let's say this is your mom's genotype for her sex-determining chromosome. She's got two x's. That's what makes her your mom and not your dad. And then your dad has an x and a y-- I should do it in capital--and has a Y chromosome. And we can do a Punnett square. What are all the different combinations of offspring? Well,your mom could give this X chromosome, in conjunction with this X chromosome from your dad. This would produce a female. Your mom could give this other X chromosome with that X chromosome. That would be a female as well. Well,your mom's always going to be donating an X chromosome. And then your dad is going to donate either the X or the Y. So in this case,it'll be the Y chromosome. So these would be female,and those would be male. And it works out nicely that half are female and half are male. But a very interesting and somewhat ironic fact might pop out at you when you see this. what determines whether someone is or Who determines whether their offspring are male or female? Is it the mom or the dad? Well,the mom always donates an X chromosome, so in no way does- what the haploid genetic makeup of the mom's eggs of the gamete from the female, in no way does that determine the gender of the offspring. It's all determined by whether--let me just draw a bunch of-- dad's got a lot of sperm,and they're all racing towards the egg. And some of them have an X chromosome in them and some of them have a Y chromosome in them. And obviously they have others. And obviously if this guy up here wins the race. Or maybe I should say this girl.If she wins the race, then the fertilized egg will develop into a female. If this sperm wins the race, then the fertilized egg will develop into a male. And the reason why I said it's ironic is throughout history, and probably the most famous example of this is Henry the VIII. I mean it's not just the case with kings. It's probably true,because most of our civilization is male dominated, that you've had these men who are obsessed with producing a male heir to kind of take over the family name. And,in the case of Henry the VIII,take over a country. And they become very disappointed and they tend to blame their wives when the wives keep producing females,but it's all their fault. Henry the VIII,I mean the most famous case was with Ann Boleyn. I'm not an expert here,but the general notion is that he became upset with her that she wasn't producing a male heir. And then he found a reason to get her essentially decapitated, even though it was all his fault. He was maybe producing a lot more sperm that looked like that than was looking like this. He eventually does produce a male heir so he was-- and if we assume that it was his child-- then obviously he was producing some of these,but for the most part, it was all Henry the VIII's fault. So that's why I say there's a little bit of irony here. Is that the people doing the blame are the people to blame for the lack of a male heir? Now one question that might immediately pop up in your head is, Sal,is everything on these chromosomes related to just our sex-determining traits or are there other stuff on them? So let me draw some chromosomes. So let's say that's an X chromosome and this is a Y chromosome. Now the X chromosome,it does code for a lot more things, although it is kind of famously gene poor. It codes for on the order of 1,500 genes. And the Y chromosome,it's the most gene poor of all the chromosomes. It only codes for on the order of 78 genes. I just looked this up,but who knows if it's exactly 78. But what it tells you is it does very little other than determining what the gender is. And the way it determines that, it does have one gene on it called the SRY gene. You don't have to know that. SRY,that plays a role in the development of testes or the male sexual organ.So if you have this around, this gene right here can start coding for things that will eventually lead to the development of the testicles. And if you don't have that around,that won't happen, so you'll end up with a female. And I'm making gross oversimplifications here. But everything I've dealt with so far,OK, this clearly plays a role in determining sex. But you do have other traits on these genes. And the famous cases all deal with specific disorders. So,for example,color blindness. The genes,or the mutations I should say. So the mutations that cause color blindness. Red-green color blindness,which I did in green, which is maybe a little bit inappropriate. Color blindness and also hemophilia. This is an inability of your blood to clot. Actually, there's several types of hemophilia. But hemophilia is an inability for your blood to clot properly. And both of these are mutations on the X chromosome. And they're recessive mutations.So what does that mean? It means both of your X chromosomes have to have-- let's take the case for hemophilia-- both of your X chromosomes have to have the hemophilia mutation in order for you to show the phenotype of having hemophilia. So,for example,if there's a woman, and let's say this is her genotype. She has one regular X chromosome and then she has one X chromosome that has the-- I'll put a little superscript there for hemophilia-- she has the hemophilia mutation.She's just going to be a carrier. Her phenotype right here is going to be no hemophilia. She'll have no problem clotting her blood. The only way that a woman could be a hemophiliac is if she gets two versions of this, because this is a recessive mutation. Now this individual will have hemophilia. Now men,they only have one X chromosome. So for a man to exhibit hemophilia to have this phenotype, he just needs it only on the one X chromosome he has. And then the other one's a Y chromosome. So this man will have hemophilia. So a natural question should be arising is, hey, you know this guy-- let's just say that this is a relatively infrequent mutation that arises on an X chromosome-- the question is who's more likely to have hemophilia? A male or a female? All else equal,who's more likely to have it? Well if this is a relatively infrequent allele,a female, in order to display it,has to get two versions of it. So let's say that the frequency of it-- and I looked it up before this video-- roughly they say between 1 in 5,000 to 10,000 men exhibit hemophilia. So let's say that the allele frequency of this is 1 in 7,000, the frequency of Xh,the hemophilia version of the X chromosome. And that's why 1 in 7,000 men display it, because it's completely determined whether-- there's a 1 in 7,000 chance that this X chromosome they get is the hemophilia version. Who cares what the Y chromosome they get is, cause that essentially doesn't code at all for the blood clotting factors and all of the things that drive hemophilia. Now,for a woman to get hemophilia,what has to happen? She has to have two X chromosomes with the mutation. Well the probability of each of them having the mutation is 1 in 7,000 So the probability of her having hemophilia is 1 in 7,000 times 1 in 7,000,or that's 1 in what,49 million. So as you can imagine,the incidence of hemophilia in women is much lower than the incidence of hemophilia in men. And in general for any sex-linked trait, if it's recessive, if it's a recessive sex-linked trait,which means men, if they have it,they're going to show it, because they don't have another X chromosome to dominate it. Or for women to show it, she has to have both versions of it. The incidence in men is going to be, so let's say that m is the incidence in men.I'm spelling badly. Then the incidence in women will be what? You could view this as the allele frequency of that mutation on the X chromosome. So women have to get two versions of it. So the woman's frequency is m squared. And you might say,hey,that looks like a bigger number. I'm squaring it. But you have to remember that these numbers, the frequency is less than 1, so in the case of hemophilia,that was 1 in 7,000. So if you square 1 in 7,000,you get 1 in 49 million. Anyway,hopefully you found that interesting and now you know how we all become men and women. And even better you know whom to blame when some of these, I guess, male-focused parents are having trouble getting their son." + }, + { + "Q": "I don't mean to be inappropriate, but was Ernst Rohm a homosexual?", + "A": "Yes, he was gay", + "video_name": "ZrbbKMnPDUk", + "transcript": "We have now seen the Nazis come to power in 1933 - and by mid 1933 they are the only allowed party in Germany. And Hitler is fundamentally the dictator of Germany. But they aren't happy with just that consolidated power, they want to ensure that they stay in power. And so as we get into 1934 they continue to consolidate their power and now more directly start eliminating their opponents. And when I say eliminating, they are actually directly killing these individuals. And there is an entire Nazi power apparatus that is at play, but the three figures here are the ones that are most notable other than Hitler. This is Joseph Goebbels who is the head of Nazi Propaganda, Hermann G\u00f6ring, who is the head of the Nazi's secret police, the Gestapo, which later goes under the control of the SS, under Heinrich Himmler. He is a major player in the rearmament of Germany for war footing. He eventually helds the Luftwaffe which is the German airforce during the World War II. We have Heinrich Himmler who is the head of Schutzstaffel, more famously known as the SS, which is the paramilitary group of the Nazi party, used to intimidate and eliminate (as we will see) their opponents and they are also responsible for the execution of the actual the planning and the execution - of the actual holocaust. And so as we get into 1934 Hitler is eager to eliminate more of his opponents - and we are not just talking about opponents outside of the Nazi party, we are also talking about rivals within the Nazi party. And the most notable of these was Ernst R\u00f6hm who as you see at this picture was clearly a Nazi. He was head of the Storm Battalion or the SA, which was the paramilitary group that the SS splintered out of and it was far more independent than Hitler would have liked and it was not popular amongst the German people because of its violence - and Hitler and his lieutenants viewed R\u00f6hm as a potential rival to Hitler's authority within the Nazi party. And so in 1934, June 30th in particular - or we could say June 30th to July 2nd - you have what's called the Night of the Long Knives and it really should be called the Nights of the Long Knives but the Night of the Long Knives, where under the pretext of a supposed a coup d'\u00e9tat or a putsch on the part of Ernst R\u00f6hm The SS and the Gestapo start rounding up R\u00f6hm, his allies and any perceived enemies of Hitler. Start rounding them up and eliminating them and shooting them... And Ernst R\u00f6hm is arrested and when he refuses to kill himself in his jail cell he is shot at point-blank range. Gregor Strasser, who is a former rival of Hitler within the Nazi party, he is eliminated. Kurt von Schleicher who is a previous chancellor of Germany, perceived as a rival to Hitler - he and his wife are gunned down. Gustav Ritter von Kahr who is in no way anymore a rival to Hitler but he was one of the major actors for putting down the Beer Hall Putsch - the fail the Beer Hall Putch of 1923. He is hacked - he is hacked to get dead. And these are just four of the more notable folks. On those few nights over 85 major officials in Germany were eliminated. And as we will see this is just the beginning of the consolidation of Nazi power and now they are doing it through extra legal means." + }, + { + "Q": "09:00 You're placing the oxygen on the same side as Br but you're saying that it's inversion. How come?\nAlso, I really don't understand that triangle type bond, what's that?", + "A": "See videos on stereochemistry. This video is way too advanced if you don t understand chirality or wedge and dash representations of bonds.", + "video_name": "3LiyCxCTrqo", + "transcript": "- [Instructor] Let's look at the mechanism for an SN2 reaction. On the left we have an alkyl halide and we know that this bromine is a little bit more electronegative than this carbon so the bromine withdraws some electron density away from that carbon which makes this carbon a little bit positive, so we say partially positive. That's the electrophilic center so this on the left is our electrophile. On the right, we know that this hydroxide ion which we could get from something like sodium hydroxide, is a negative one formal charge on the oxygen which makes it a good nucleophile. Let me write down here. This is our nucleophile on the right and on the left is our electrophile which I'm also gonna refer to as a substrate in this video. This alkyl halide is our substrate. We know from an earlier video that the nucleophile will attack the electrophile because opposite charges attract. This negative charge is attracted to this partially positive charge. Lone pair of electrons on the oxygen will attack this partially positive carbon. At the same time, the two electrons in this bond come off onto the bromine. Let me draw the bromine over here. The bromine had three lone pairs of electrons on it and it's gonna pick up another lone pair of electrons. Let me show those electrons in magenta. This bond breaks and these two electrons come off onto the bromine which gives the bromine a negative one formal charge. This is the bromide anion. And we're also forming a bond between the oxygen and this carbon and this bond comes from this lone pair of electrons which I've just marked in blue here. Those two electrons in blue form this bond and we get our product which is an alcohol. The SN2 mechanism is a concerted mechanism because the nucleophile attacks the electrophile, at the same time we get loss of a leaving groups. There's only one step in this mechanism. Let's say we did a series of experiments to determine the rate law for this reaction. Remember from general chemistry, rate laws are determined experimentally. Capital R is the rate of the reaction and that's equal to the rate constant k times the concentration of our alkyl halide. And it's determined experimentally this is to the first power times the concentration of the hydroxide ion also to the first power. So what does this mean? This means if we increased the concentration of our alkyl halide. If we increase the concentration of our alkyl halide by a factor of two, what happens to the rates of the reaction? Well, the rate of the reaction is proportional to the concentration of the alkyl halide to the first power. Two to the first is equal to two which means the overall rate of the reaction would increase by a factor of two. Doubling the concentration of your alkyl halide while keeping this concentration, the hydroxide ion concentration the same should double the rate of the reaction. And also, if we kept the concentration of alkyl halide the same and we double the concentration of hydroxide, that would also increase the rate by a factor of two. And this experimentally determined rate law makes sense with our mechanism. If we increase the concentration of the nucleophile or we increase the concentration of the electrophile, we increase the frequency of collisions between the two which increases the overall rate of the reaction. The fact that our rate law is proportional to the concentration of both the substrate and the nucleophile fits with our idea of a one step mechanism. Finally, let's take a look at where this SN2 comes from. We keep on saying an SN2 mechanism, an SN2 reaction. The S stands for substitution. Let me write in here substitution because our nucleophile is substituting for our leaving group. We can see in our final products here, the nucleophile has substituted for the leaving group. The N stands for nucleophilic because of course it is our nucleophile that is doing the substituting. And finally the two here refers to the fact that this is bimolecular which means that the rate depends on the concentration of two things. The substrate and the nucleophile. That's different from an SN1 mechanism where the rate is dependent only on the concentration of one thing. The rate of the reaction also depends on the structure of the alkyl halide, on the structure of the substrate. On the left we have a methyl halide followed by a primary alkyl halide. The carbon bonded to our bromine is directly attached to one alkyl group followed by a secondary alkyl halide, the carbon bonded to the bromine is bonded to two alkyl groups, followed by a tertiary alkyl halide. This carbon is bonded to three alkyl groups. Turns out that the methyl halides and the primary alkyl halide react the fastest in an SN2 mechanism. Secondary alkyl halides react very slowly and tertiary alkyl halides react so, so slowly that we say they are unreactive toward an SN2 mechanism. And this makes sense when we think about the mechanism because remember, the nucleophile has to attack the electrophile. The nucleophile needs to get close enough to the electrophilic carbon to actually form a bond and steric hindrance would prevent that from happening. Something like a tertiary alkyl halide has this big bulky methyl groups which prevent the nucleophile for attacking. Let's look at a video so we can see this a little bit more clearly. Here's our methyl halide with our carbon directly bonded to a halogen which I'm seeing as yellow. And here's our nucleophile which could be the hydroxide ion. The nucleophile approaches the electrophile for the side opposite of the leaving group and you can see with the methyl halide there's no steric hindrance. When we move to a primary alkyl halide, the carbon bonded to the halogen has only one alkyl group bonded to it, it's still easy for the nucleophile to approach. When we move to a secondary alkyl halide, so for a secondary you can see that the carbon bonded to the halogen has two methyl groups attached to it now. It gets a little harder for the nucleophile to approach in the proper orientation. These bulky methyl groups make it more difficult for the nucleophile to get close enough to that electrophilic carbon. When we go to a tertiary alkyl halide, so three alkyl groups. There's one, there's two and there's three. There's a lot more steric hindrance and it's even more difficult for our nucleophile to approach. As we saw on the video, for an SN2 reaction we need decreased steric hindrance. So, if we look at this alkyl halide, the carbon that is directly bonded to our halogen is attached to only one alkyl group. This is a primary alkyl halide and that makes this a good SN2 reaction. The decreased steric hindrance allows the nucleophile to attack the electrophile." + }, + { + "Q": "Help with this is greatly appreciated. I got lost when he divided by -3 by 3 and 5/6 by 3. Also, how come its just 6/3 giving us 5/2 and not 5/6 divided by 3?", + "A": "Can you give me a time-stamp or the exact equation? I can t seem to understand your writing.", + "video_name": "5fkh01mClLU", + "transcript": "In this video I'm going to do a bunch of examples of finding the equations of lines in slope-intercept form. Just as a bit of a review, that means equations of lines in the form of y is equal to mx plus b where m is the slope and b is the y-intercept. So let's just do a bunch of these problems. So here they tell us that a line has a slope of negative 5, so m is equal to negative 5. And it has a y-intercept of 6. So b is equal to 6. So this is pretty straightforward. The equation of this line is y is equal to negative 5x plus 6. That wasn't too bad. Let's do this next one over here. The line has a slope of negative 1 and contains the point 4/5 comma 0. So they're telling us the slope, slope of negative 1. So we know that m is equal to negative 1, but we're not 100% sure about where the y-intercept is just yet. So we know that this equation is going to be of the form y is equal to the slope negative 1x plus b, where b is the y-intercept. Now, we can use this coordinate information, the fact that it contains this point, we can use that information to solve for b. The fact that the line contains this point means that the value x is equal to 4/5, y is equal to 0 must satisfy this equation. So let's substitute those in. y is equal to 0 when x is equal to 4/5. So 0 is equal to negative 1 times 4/5 plus b. I'll scroll down a little bit. So let's see, we get a 0 is equal to negative 4/5 plus b. We can add 4/5 to both sides of this equation. So we get add a 4/5 there. We could add a 4/5 to that side as well. The whole reason I did that is so that cancels out with that. You get b is equal to 4/5. So we now have the equation of the line. y is equal to negative 1 times x, which we write as negative x, plus b, which is 4/5, just like that. Now we have this one. The line contains the point 2 comma 6 and 5 comma 0. So they haven't given us the slope or the y-intercept explicitly. But we could figure out both of them from these So the first thing we can do is figure out the slope. So we know that the slope m is equal to change in y over change in x, which is equal to-- What is the change in y? Let's start with this one right here. So we do 6 minus 0. Let me do it this way. So that's a 6-- I want to make it color-coded-- minus 0. So 6 minus 0, that's our change in y. Our change in x is 2 minus 2 minus 5. The reason why I color-coded it is I wanted to show you when I used this y term first, I used the 6 up here, that I have to use this x term first as well. So I wanted to show you, this is the coordinate 2 comma 6. This is the coordinate 5 comma 0. I couldn't have swapped the 2 and the 5 then. Then I would have gotten the negative of the answer. But what do we get here? This is equal to 6 minus 0 is 6. 2 minus 5 is negative 3. So this becomes negative 6 over 3, which is the same thing as negative 2. So that's our slope. So, so far we know that the line must be, y is equal to the slope-- I'll do that in orange-- negative 2 times x plus our y-intercept. Now we can do exactly what we did in the last problem. We can use one of these points to solve for b. We can use either one. Both of these are on the line, so both of these must satisfy this equation. I'll use the 5 comma 0 because it's always nice when you have a 0 there. The math is a little bit easier. So let's put the 5 comma 0 there. So y is equal to 0 when x is equal to 5. So y is equal to 0 when you have negative 2 times 5, when x is equal to 5 plus b. So you get 0 is equal to -10 plus b. If you add 10 to both sides of this equation, let's add 10 to both sides, these two cancel out. You get b is equal to 10 plus 0 or 10. So you get b is equal to 10. Now we know the equation for the line. The equation is y-- let me do it in a new color-- y is equal to negative 2x plus b plus 10. We are done. Let's do another one of these. All right, the line contains the points 3 comma 5 and negative 3 comma 0. Just like the last problem, we start by figuring out the slope, which we will call m. It's the same thing as the rise over the run, which is the same thing as the change in y over the change in x. If you were doing this for your homework, you wouldn't I just want to make sure that you understand that these are all the same things. Then what is our change in y over our change in x? This is equal to, let's start with the side first. It's just to show you I could pick either of these points. So let's say it's 0 minus 5 just like that. So I'm using this coordinate first. I'm kind of viewing it as the endpoint. Remember when I first learned this, I would always be tempted to do the x in the numerator. No, you use the y's in the numerator. So that's the second of the coordinates. That is going to be over negative 3 minus 3. This is the coordinate negative 3, 0. This is the coordinate 3, 5. We're subtracting that. So what are we going to get? This is going to be equal to-- I'll do it in a neutral color-- this is going to be equal to the numerator is negative 5 over negative 3 minus 3 is negative 6. So the negatives cancel out. You get 5/6. So we know that the equation is going to be of the form y is equal to 5/6 x plus b. Now we can substitute one of these coordinates in for b. So let's do. I always like to use the one that has the 0 in it. So y is a zero when x is negative 3 plus b. So all I did is I substituted negative 3 for x, 0 for y. I know I can do that because this is on the line. This must satisfy the equation of the line. Let's solve for b. So we get zero is equal to, well if we divide negative 3 by 3, that becomes a 1. If you divide 6 by 3, that becomes a 2. So it becomes negative 5/2 plus b. We could add 5/2 to both sides of the equation, plus 5/2, plus 5/2. I like to change my notation just so you get familiar with both. So the equation becomes 5/2 is equal to-- that's a 0-- is equal to b. b is 5/2. So the equation of our line is y is equal to 5/6 x plus b, which we just figured out is 5/2, plus 5/2. We are done. Let's do another one. We have a graph here. Let's figure out the equation of this graph. This is actually, on some level, a little bit easier. What's the slope? Slope is change in y over change it x. So let's see what happens. When we move in x, when our change in x is 1, so that is our change in x. So change in x is 1. I'm just deciding to change my x by 1, increment by 1. What is the change in y? It looks like y changes exactly by 4. It looks like my delta y, my change in y, is equal to 4 when my delta x is equal to 1. So change in y over change in x, change in y is 4 when change in x is 1. So the slope is equal to 4. Now what's its y-intercept? Well here we can just look at the graph. It looks like it intersects y-axis at y is equal to negative 6, or at the point 0, negative 6. So we know that b is equal to negative 6. So we know the equation of the line. The equation of the line is y is equal to the slope times x plus the y-intercept. I should write that. So minus 6, that is plus negative 6 So that is the equation of our line. Let's do one more of these. So they tell us that f of 1.5 is negative 3, f of negative 1 is 2. What is that? Well, all this is just a fancy way of telling you that the point when x is 1.5, when you put 1.5 into the function, the function evaluates as negative 3. So this tells us that the coordinate 1.5, negative 3 is on the line. Then this tells us that the point when x is negative 1, f of x is equal to 2. This is just a fancy way of saying that both of these two points are on the line, nothing unusual. I think the point of this problem is to get you familiar with function notation, for you to not get intimidated if you see something like this. If you evaluate the function at 1.5, you get negative 3. So that's the coordinate if you imagine that y is equal to f of x. It would be equal to negative 3 when x is 1.5. Anyway, I've said it multiple times. Let's figure out the slope of this line. The slope which is change in y over change in x is equal to, let's start with 2 minus this guy, negative 3-- these are the y-values-- over, all of that over, negative 1 minus this guy. Let me write it this way, negative 1 minus that guy, minus 1.5. I do the colors because I want to show you that the negative 1 and the 2 are both coming from this, that's why I use both of them first. If I used these guys first, I would have to use both the x and the y first. If I use the 2 first, I have to use the negative 1 first. That's why I'm color-coding it. So this is going to be equal to 2 minus negative 3. That's the same thing as 2 plus 3. So that is 5. Negative 1 minus 1.5 is negative 2.5. 5 divided by 2.5 is equal to 2. So the slope of this line is negative 2. Actually I'll take a little aside to show you it doesn't matter what order I do this in. If I use this coordinate first, then I have to use that coordinate first. Let's do it the other way. If I did it as negative 3 minus 2 over 1.5 minus negative 1, this should be minus the 2 over 1.5 minus the negative 1. This should give me the same answer. This is equal to what? Negative 3 minus 2 is negative 5 over 1.5 minus negative 1. That's 1.5 plus 1. That's over 2.5. So once again, this is equal the negative 2. So I just wanted to show you, it doesn't matter which one you pick as the starting or the endpoint, as long as If this is the starting y, this is the starting x. If this is the finishing y, this has to be the finishing x. But anyway, we know that the slope is negative 2. So we know the equation is y is equal to negative 2x plus some y-intercept. Let's use one of these coordinates. I'll use this one since it doesn't have a decimal in it. So we know that y is equal to 2. So y is equal to 2 when x is equal to negative 1. Of course you have your plus b. So 2 is equal to negative 2 times negative 1 is 2 plus b. If you subtract 2 from both sides of this equation, minus 2, minus 2, you're subtracting it from both sides of this equation, you're going to get 0 on the left-hand side is equal to b. So b is 0. So the equation of our line is just y is equal to negative 2x. Actually if you wanted to write it in function notation, it would be that f of x is equal to negative 2x. I kind of just assumed that y is equal to f of x. But this is really the equation. They never mentioned y's here. So you could just write f of x is equal to 2x right here. Each of these coordinates are the coordinates of x and f of x. So you could even view the definition of slope as change in f of x over change in x. These are all equivalent ways of viewing the same thing." + }, + { + "Q": "what are the communist countries that still exist today", + "A": "The main that comes to mind is of course China. But China actually implement some forms of liberalization and since the 70s permits private property, capital and business to function, so it is goign in the direction of being some kind of hybrid version. Then we have Cuba, which is communist since 1959. North Korea is also a communist country with maybe the tightest regime in the world. And also we have Laos and Vietnam, which are communist since the mid 70s after there bloody conflicts there.", + "video_name": "MmRgMAZyYN0", + "transcript": "Thought I would do a video on communism just because I've been talking about it a bunch in the history videos, and I haven't given you a good definition of what it means, or a good understanding of what it means. And to understand communism-- let me just draw a spectrum here. So I'm going to start with capitalism. And this is really just going to be an overview. People can do a whole PhD thesis on this type of thing. Capitalism, and then I'll get a little bit more-- and then we could progress to socialism. And then we can go to communism. And the modern versions of communism are really kind of the brainchild of Karl Marx and Vladimir Lenin. Karl Marx was a German philosopher in the 1800s, who, in his Communist Manifesto and other writings, kind of created the philosophical underpinnings for communism. And Vladimir Lenin, who led the Bolshevik Revolution in the-- and created, essentially, the Soviet Union-- he's the first person to make some of Karl Marx's ideas more concrete. And really every nation or every country which we view as communist has really followed the pattern of Vladimir Lenin. And we'll talk about that in a second. But first, let's talk about the philosophical differences between these things, and how you would move. And Karl Marx himself viewed communism as kind of a progression from capitalism through socialism to communism. So what he saw in capitalism-- and at least this part of what he saw was right-- is that you have private property, private ownership of land. That's the main aspect of capitalism. And this is the world that most of us live in today. The problem that he saw with capitalism is he thought, well, look, when you have private property, the people who start accumulating some capital-- and when we talk about capital, we could be talking about land, we could be talking about factories, we could be talking about any type of natural resources-- so the people who start getting a little bit of them-- so let me draw a little diagram here. So let's say someone has a little bit of capital. And that capital could be a factory, or it could be land. So let me write it. Capital. And let's just say it's land. So let's say someone starts to own a little bit of land. And he owns more than everyone else. So then you just have a bunch of other people who don't own land. But they need, essentially-- and since this guy owns all the land, they've got to work on this guy's land. They have to work on this guy's land. And from Karl Marx's point of view, he said, look, you have all of these laborers who don't have as much capital. This guy has this capital. And so he can make these laborers work for a very small wage. And so any excess profits that come out from this arrangement, the owner of the capital will be able to get it. Because these laborers won't be able to get their wages to go up. Because there's so much competition for them to work on this guy's farm or to work on this guy's land. He really didn't think too much about, well, maybe the competition could go the other way. Maybe you could have a reality eventually where you have a bunch of people with reasonable amounts of capital, and you have a bunch of laborers. And the bunch of people would compete for the laborers, and maybe the laborers could make their wages go up, and they could eventually accumulate their own capital. They could eventually start their own small businesses. So he really didn't think about this reality too much over here. He just saw this reality. And to his defense-- and I don't want to get in the habit of defending Karl Marx too much-- to his defense, this is what was happening in the late 1800s, especially-- we have the Industrial Revolution. Even in the United States, you did have kind of-- Mark Twain called it the Gilded Age. You have these industrialists who did accumulate huge amounts of capital. They really did have a lot of the leverage relative to the laborers. And so what Karl Marx says, well, look, if the guy with all the capital has all the leverage, and this whole arrangement makes some profits, he's going to be able to keep the profits. Because he can keep all of these dudes' wages low. And so what's going to happen is that the guy with the capital is just going to end up with more capital. And he's going to have even more leverage. And he'll be able to keep these people on kind of a basic wage, so that they can never acquire capital for themselves. So in Karl Marx's point of view, the natural progression would be for these people to start organizing. So these people maybe start organizing into unions. So they could collectively tell the person who owns the land or the factory, no, we're not going to work, or we're going to go on strike unless you increase our wages, or unless you give us better working conditions. So when you start talking about this unionization stuff, you're starting to move in the direction of socialism. The other element of moving in the direction of socialism is that Karl Marx didn't like this kind of high concentration-- or this is socialists in general, I should say-- didn't like this high concentration of wealth. That you have this reality of not only do you have these people who could accumulate all of this wealth-- and maybe, to some degree, they were able to accumulate it because they were innovative, or they were good managers of land, or whatever, although the Marxists don't give a lot of credit to the owners of capital. They don't really give a lot of credit to saying maybe they did have some skill in managing some type of an operation. But the other problem is is that it gets handed over. It gets handed over to their offspring. So private property, you have this situation where it just goes from maybe father to son, or from parent to a child. And so it's not even based on any type of meritocracy. It's really just based on this inherited wealth. And this is a problem that definitely happened in Europe. When you go back to the French Revolution, you have generation after generation of nobility, regardless of how incompetent each generation would be, they just had so much wealth that they were essentially in control of everything. And you had a bunch of people with no wealth having to work for them. And when you have that type of wealth disparity, it does lead to revolutions. So another principle of moving in the socialist direction is kind of a redistribution of wealth. So let me write it over here. So redistribution. So in socialism, you can still have private property. But the government takes a bigger role. So you have-- let me write this. Larger government. And one of the roles of the government is to redistribute wealth. And the government also starts having control of the major factors of production. So maybe the utilities, maybe some of the large factories that do major things, all of a sudden starts to become in the hands of the government, or in the words of communists, in the hands of the people. And the redistribution is going on, so in theory, you don't have huge amounts of wealth in the hands of a few people. And then you keep-- if you take these ideas to their natural conclusion, you get to the theoretical communist state. And the theoretical communist state is a classless, and maybe even a little bit-- a classless society, and in Karl Marx's point of view-- and this is a little harder to imagine-- a stateless society. So in capitalism, you definitely had classes. You had the class that owns the capital, and then you had the labor class, and you have all of these divisions, and they're different from each other. He didn't really imagine a world that maybe a laborer could get out of this, they could get their own capital, then maybe they could start their own business. So he just saw this tension would eventually to socialism, and eventually a classless society where you have a central-- Well, he didn't even go too much into the details but you have kind of equal, everyone in society has ownership over everything, and society somehow figures out where things should be allocated, and all of the rest. And it's all stateless. And that's even harder to think about in a concrete fashion. So that's Karl Marx's view of things. But it never really became concrete until Vladimir Lenin shows up. And so the current version of communism that we-- The current thing that most of us view as communism is sometimes viewed as a Marxist-Leninist state. These are sometimes used interchangeably. Marxism is kind of the pure, utopian, we're eventually going to get to a world where everyone is equal, everyone is doing exactly what they want, there's an abundance of everything. I guess to some degree, it's kind of describing what happens in Star Trek, where everyone can go to a replicator and get what they want. And if you want to paint part of the day, you can paint part of the day, and you're not just a painter, you can also do whatever you want. So it's this very utopian thing. Let me write that down. So pure Marxism is kind of a utopian society. And just in case you don't know what utopian means, it's kind of a perfect society, where you don't have classes, everyone is equal, everyone is leading these kind of rich, diverse, fulfilling lives. And it's also, utopian is also kind of viewed as unrealistic. It's kind of, if you view it in the more negative light, is like, hey, I don't know how we'll ever be able to get there. Who knows? I don't want to be negative about it. Maybe we will one day get to a utopian society. But Leninist is kind of the more practical element of communism. Because obviously, after the Bolshevik Revolution, 1917, in the Russian Empire, the Soviet Union gets created, they have to actually run a government. They have to actually run a state based on these ideas of communism. And in a Leninist philosophy-- and this is where it starts to become in tension with the ideas of democracy-- in a Leninist philosophy, you need this kind of a party system. So you need this-- and he calls this the Vanguard Party. So the vanguard is kind of the thing that's leading, the one that's leading the march. So this Vanguard Party that kind of creates this constant state of revolution, and its whole job is to guide society, is to kind of almost be the parent of society, and take it from capitalism through socialism to this ideal state of communism. And it's one of those things where the ideal state of communism was never-- it's kind of hard to know when you get there. And so what happens in a Leninist state is it's this Vanguard Party, which is usually called the Communist Party, is in a constant state of revolution, kind of saying, hey, we're shepherding the people to some future state without a real clear definition of what that future state is. And so when you talk about Marxist-Leninist, besides talking about what's happening in the economic sphere, it's also kind of talking about this party system, this party system where you really just have one dominant party that it will hopefully act in the interest of the people. So one dominant communist party that acts in the interest of the people. And obviously, the negative here is that how do you know that they actually are acting in the interest of people? How do you know that they actually are competent? What means are there to do anything if they are misallocating things, if it is corrupt, if you only have a one-party system? And just to make it clear, the largest existing communist state is the People's Republic of China. And although it is controlled by the Communist Party, in economic terms it's really not that communist anymore. And so it can be confusing. And so what I want to do is draw a little bit of a spectrum. On the vertical axis, over here, I want to put democratic. And up here, I'll put authoritarian or totalitarian. Let me put-- well, I'll put authoritarian. I'll do another video on the difference. And they're similar. And totalitarian is more an extreme form of authoritarian, where the government controls everything. And you have a few people controlling everything and it's very non-democratic. But authoritarian is kind of along those directions. And then on this spectrum, we have the capitalism, socialism, and communism. So the United States, I would put-- I would put the United States someplace over here. I would put the United States over here. It has some small elements of socialism. You do have labor unions. They don't control everything. You also have people working outside of labor unions. It does have some elements of redistribution. There are inheritance taxes. There are-- I mean it's not an extreme form of redistribution. You can still inherit private property. You still have safety nets for people, you have Medicare, Medicaid, you have welfare. So there's some elements of socialism. But it also has a very strong capitalist history, private property, deep market, so I'd stick the United States over there. I would put the USSR-- not current Russia, but the Soviet Union when it existed-- I would put the Soviet Union right about there. So this was the-- I would put the USSR right over there. I would put the current state of Russia, actually someplace over here. Because they actually have fewer safety nets, and they kind of have a more-- their economy can kind of go crazier, and they actually have a bigger disparity in wealth than a place like the United States. So this is current Russia. And probably the most interesting one here is the People's Republic of China, the current People's Republic of China, which is at least on the surface, a communist state. But in some ways, it's more capitalist than the United States, in that they don't have strong wealth redistribution. They don't have kind of strong safety nets for people. So you could put some elements of China-- and over here, closer to the left. And they are more-- less democratic than either the US or even current Russia, although some people would call current Russia-- well, I won't go too much into it. But current China, you could throw it here a little bit. So it could be even a little bit more capitalist than the United States. Definitely they don't even have good labor laws, all the rest. But in other ways, you do have state ownership of a lot, and you do have state control of a lot. So in some ways, they're kind of spanning this whole range. So this right over here is China. And even though it is called a communist state, in some ways, it's more capitalist than countries that are very proud of their capitalism. But in a lot of other ways, especially with the government ownership and the government control of things, and this one dominant party, so it's kind of Leninist with less of the Marxist going on. So in that way, it is more in the communist direction. So hopefully that clarifies what can sometimes be a confusing topic." + }, + { + "Q": "Do we borrow money from the Federal Reserve?", + "A": "Well, where do you think government treasury bonds come from?", + "video_name": "-05OfTp6ZEE", + "transcript": "Before we talk about the debt ceiling, it's important to realize the difference between the deficit and the debt. Because these words are thrown around and it's clear that they're related, but sometimes people might confuse one for the other. The deficit is how much you overspend in a given year, while the debt is the total amount, the cumulative amount, of debt you you've gotten over many, many years. So let's take a look, I guess a very simplified example, let's say you have some type of a country. And that country spends, in a given year, $10. But it's only bringing in $6 in tax revenue. So it's bringing in taxes. It's only bringing in $6. So this country in this year, where it spends $10, even though it only has $6 to spend, it has a $4 deficit. Def is the short for deficit. And well, let me just write it out. You might think it's defense or something. It has a $4 deficit. And you might say, well, how does it spend more money that it brings in? How can it actually continue to spend this much? Where will it get the $4 from? And the answer is, it will borrow that $4. Our little country will borrow it. And so the debt, maybe going into this year, the country already had some debt. Maybe it already had $100 of debt. And so in this situation, it would have to borrow another $4 of debt. And so exiting this year, it would have $104 of debt. If the country runs the same $4 deficit the year after this, then the debt will increase to $108. If it runs another $4 deficit, than the debt will increase to $112. Now that we have that out of the way, let's think about what the debt ceiling is. So you could imagine, the United States actually does. It's continuing to run a deficit. It's continuing to spend more than it brings in. And actually, for the United States, these ratios are appropriate. For every dollar that the United States spends right now, 40% is borrowed. Or another way to think about it, it only has 60% of every dollar that it needs to spend right now. So it has to go out into the debt markets and borrow 40% to keep spending at its current rate. And so if it's continuing to borrow, you could imagine that the debt keeps on increasing. So let me draw a little graph here. So that axis is time. This axis right over here is the total cumulative amount of debt that we have. We continue to have to borrow 40% of every dollar And so our debt is continuing to increase. And Congress has the power, or Congress has the authority, to essentially limit how much debt we have. So right now we have a current debt limit of $14.3 trillion. And even though Congress has this authority, the way that it's worked in the past, is this kind of just a rubber stamp. Congress has just always allowed the debt ceiling to go up and up and up to fund our borrowing costs. And if you think about it, that kind of makes sense because right now Congress is the one that decides where to spend the money. What are the obligations. And so the debt ceiling is like, OK, we've already agreed what you have to spend your money on. Congress is the one that figures out what we spend our money on, and what our taxes are. And so they say, look, we've already determined how much you have to borrow. It would seem kind of ridiculous for us after we've determined how much you borrow to say that you cannot borrow it. You cannot you cannot actually do what we've told you to do. And so historically, Congress has just kind of gone with the flow. They said, OK, yeah we've told you we need to borrow more money to execute-- the executive branch has to run the government-- for you to actually run the government based on the budget we told you. So they just keep upping it. And the last time the debt ceiling was raised was actually very recently, February 12, 2010. It was raised from $12.3 trillion, point actually $12.4 trillion to the $14.3 trillion. And this happens pretty regularly. It's happened 10 times since 2001, 74 times since 1962. So it's just a regular operating thing. And right now the Obama administration says, look, we've actually come up against our debt ceiling. We want to raise it, and ideally for the Obama administration, they want to raise it by about $2.4 trillion. So they want to raise it to $16.7 trillion, which will kind of put it off the table for a little bit. Put it past the elections so that we don't have to debate this anymore. The Republicans on the other the side, want to essentially use this, and this is a little bit unusual, to use this as leverage to essentially reduce the deficit. And not only to reduce the deficit, but it's in particular to reduce the deficit through spending cuts. And so that's why it's become this big game of chicken and why we're going up against this limit. Now, one thing that you may or may not realize is that we've actually already hit the debt limit, the current debt limit. And we hit that debt limit on May 16, 2011. I'm making this video at the end of July in 2011. And the only reason why the country's continuing to operate, and the only reason why the country has been able to continue to pay interest on its obligations, and pay issue social security checks, and support Medicare, and buy fuel for aircraft carriers, and all the rest, is that Geithner, who's the Treasury Secretary, has been able to find cash in other places, cash normally set aside for employee pensions and all the rest. And has essentially done a little bit of a bookkeeping, taking money from one place to feed another. But what he said, what he's publicly said, is that he won't be able to do that anymore as of August 2, So this right here is the date that everyone is paying attention to, August 2, 2011. According to Geithner, at that point, he won't be able to find random pockets of cash here and there and shuffle it around. And what he calls extraordinary measures. And at that point, the United States will not be able to fulfill all of its obligations. And so if you think about all of the obligations of the United States, this is a huge oversimplification here. So this bar represent all of the obligations. Some of those obligations are things like interest on the debt that it already owes. It already owes a huge amount of debt, $14.3 trillion. And things like social security, Medicare, defense, and then all of the other stuff that the country has to support, all of their other obligations. So if as of August 2, 2011, we cannot issue any more debt, and Geithner doesn't have any extra cash laying around with these extraordinary measures, then, if those are the only options on the table, The only option is to somehow reduce some of these things by 40%. Because 40% of every dollar we used to spend on all of these obligations, 40% are borrowed. And so something over here is going to give. We're not going to fulfill our obligations to one or more of these things, all of these things that we are legally obligated to fulfill. That Congress has said, these are the things that the United States should be spending its money on. And so at that point, it is perceived that we would have to default. And a default actually would be on any of its obligations. But in particular, we could be, especially if we have to cut everything by 40%. And we don't want to see retirees not be able to get evicted from their houses, or aircraft carriers not have fuel, or whatever else. We might defer, or try to restructure, or do something weird with our debt. In which case, we would be defaulting. And I want to be clear, a default, it's usually referred to not fully paying the interest on debt that you owe. But a default would be any of its obligations. The United States has this AAA rating. If the United States says it's going to give you a Social Security check, you trust that. If the United States says that it's going to pay for that Medicare payment, you trust that. If it says it's going to give you an interest payment, you trust that. All of a sudden, if United States does not fulfill any of those obligations, then all of the obligations becomes suspect. And the reason why this is a big deal, as you can imagine, if you borrow money, you've always been good at paying back that money, you're going to pay lower interest than other people would have to pay. But all of a sudden, for whatever reason, one day you default. You either delay your payment, or you say you don't have the cash to pay your payments, then people's like, wow you're a much riskier person to lend money to. So now I'm going to increase the interest rates on you. And so the perception is if the United States were to default on its debt, or any of its obligations, that interest rates would go up. And the reason why this would really not great is because it would make the debt and the deficit even worse. Then this chunk is going to have to grow. Our obligations are on debt. As new debt gets issued, we're gonna have to pay more and more interest. So it's going to just make matters worse. It's going to make the deficit worse. And on top of that, it's not just that the government's debt, the interest on the government's debt will go up, but interest on all debt in the United States will probably go up. Because government debt is perceived to be the safest, it's the benchmark. A lot of other debt contractors are actually tied to government debt. So you'll have interest rates throughout the economy go up, which is exactly what you do not want to happen when you are either in a recession, or when you are recovering from a recession." + }, + { + "Q": "I was confused by the answer because I did the equation as (fx - gx) and got a different answer. The equation must be (gx -fx) to answer the problem.\n(fx - gx) does not equal (g - f)(x).", + "A": "I think this is mainly because of the commutative property. In addition you can move things around, but you can t do this when it comes to subtraction. That s why you got completely different answers in both of your expressions. It s interesting that you took the time to figure that out :) Good job.", + "video_name": "KvMyZY9upuA", + "transcript": "- [Instructor] We're told that f of x is equal to two x times the square root of five minus four. And we're also told that g of x is equal to x squared plus two x times the square root of five minus one. And they want us to find g minus f of x. So pause this video, and see if you can work through that on your own. So the key here is to just realize what this notation means. G minus f of x is the same thing as g of x minus f of x. And so, again, if this was helpful to you, once again I encourage you to pause the video. All right, now let's work through this again. So this is going, or, I guess the first time, but now that we know that this is equal to g of x minus f of x. So what is g of x? Well, that's the same thing as x squared plus two x times the square root of five minus one. And what is f of x? Well, it's going to be two x times the square root of five minus four. And we are subtracting f of x from g of x. So let's subtract, this is f of x, from g of x. And so now it's just going to be a little bit of algebraic simplification. So this is going to be equal to, this is equal to x squared plus two x times the square root of five minus one. And now we just have to distribute this negative sign. So negative one times two x times the square root of five is, we're gonna have minus two x times the square root of five. And then the negative of negative four is positive four. Now let's see if we can simplify this some. So this is going to be equal to, we only have one x squared term, so that's that one there. So we have x squared. Now let's see, we have two x times the square root of five. And then we have another, oh, and then we subtract two x times the square root of five. So these two cancel out with each other. So those cancel out. And then we have minus one plus four. So if we have negative one and then we add four to it, we're going to have positive three. So if we just fact, if we take this and this into consideration, four minus one is going to be equal to three, and we're done. That's what g minus f of x is equal to, x squared plus three." + }, + { + "Q": "Where can I get a printed pattern for my hexaflexagon?", + "A": "There are lots of templates online, just print one out.", + "video_name": "AmN0YyaTD60", + "transcript": "Hexaflexagons-- they're cool, hip, and hexa-fun to play with, right? Wrong. Hexaflexagons are not toys. With the increasing number of hexaflexagons finding their way into homes and schools, it's important to be aware of proper flexagation regulations when engaging in flexagon construction and use. Taking proper precautions can help avoid a flexa-catastrophe. Do not wear loose clothing when engaging in flexagation. If you have long hair, tie it back, so it doesn't get caught in a flexagation device. Ties are also a common source of incidents. Stay alert. Never flexagate while under the influence. When using a hexaflexagon, sudden unexpected sides may appear, and drugs like alcohol can slow reaction time. If you aren't sure what kind of flexagon you're dealing with, it's safer to temporarily disable the flexagon. Flexagons can be disarmed by using scissors to cut them apart. You can cut across the original seam where the paper strip was taped together, which may appear on the edge or through the face of the flexagon. In an emergency, however, flexagons can be cut apart right through a triangle, or on three edges if you want to retain symmetry, or into nine separate triangles if you really want to be safe. You can even cut them in half down the length of the paper strip like this, into two separate-- Once you cut your flexagon apart, you can figure out what kind it is. If it has nine triangles, that's 18 triangle sides. So at six triangles per hexagon side, that's three sides of trihexaflexagon. Note that some flexagons might be made from a double strip of triangles that have been folded in half, so that marker doesn't bleed through. Don't let yourself be fooled by the extra triangles. Avoid danger during hexaflexagon construction. If you're not working from a printed pattern, you might start your flexagon by picking a point on the edge of a strip of paper, folding that 180 degree angle into thirds to create 360 degree angles, and then using the equilateral triangle that results as a guide to fold the rest of the strip of paper, zigzagging back and forth. Without proper attention and focus, this could easily lead to becoming unreasonably amused with the springy spring of happy triangles that results. Always keep your hexaflexagon in good working order. Pre-creasing all the triangles both ways before configuring them into hexaflexagonal formation will help your flexagon operate properly and avoid accidents. Keep a close watch on the chirality of your hexaflexagon. That is, whether it is right or left handed. Notice how in this hexaflexagon, water flows clockwise down under the flaps, even if you flip it over or flex it. Well, in this hexaflexagon, it flows counter-clockwise. They're mirror images. The chirality is decided when you fold and tape your triangles into a twisty loop, and once taped, it is impossible to change from one to the other without cutting it apart, at least in three-dimensional euclidean space. A change in chirality could be a sign that your flexagon has been flipped through four-dimensional space and is possibly a highly dangerous multi-dimensional portal. With experience, a hexaflexagon master can construct a hexaflexagon in mere seconds. Some forgo tape and scissors entirely by folding a double strip that's too long and tucking the extra in. This is an advanced technique that should not be attempted without prior training. Beware topological changes. This family seems safe from this philosoraptor, because they live on separate planets with a cold, empty vacuum of space between them. But after a single flex, the unfortunate victims are now doomed, protected only by the inconsequential barrier of their domicile. Your stars might explode, your frowns may become smiles, your most pointy of triangles might become the roundest of circles. Perfectly healthy snakes may turn into snake loops, or worse, become decapitated. Either state is fatal for the snake, as having no head can lead to starvation. This can be avoided by simply marking where connections will be across neighboring triangles first. Afterwards, the lines can be filled in however you like. Be aware that with the trihexaflexagon, there are two variations to each face. So you can simply draw one side where triangles connect, and flip and draw the other. But in the hexa-hexaflexagon, the main three faces each appear four different ways. If you use hexaflexagons, keep an eye out for signs of dependency. Overuse can lead to addiction and possibly an overdose. Some users of hexaflexagons report confusion, mind-blown syndrome, hexaflexaperplexia, hexaflexadyslexia, hexaflexaperfectionism, and hexaflexa-Mexican-food-cravings. If you find yourself experiencing any of these symptoms, stop flexagon use immediately, and see the head of your math department. With proper precautions, flexagating can be a great part of your life. Follow these simple safety guidelines, and you should be ready for a fun and safe hexaflexagon experience." + }, + { + "Q": "Wow I thought violas were smaller and higher pitched than the violins! Are their more violins or violas in an orchestra? And how many if you know?", + "A": "I think you just got the two mixed up! Violas are bigger and lower, as you noticed, and there are more violins in an orchestra. There are two violin sections - the violin 1 and violin 2. There would probably be about 20 violins and then 8 or so violas. Hope this helps!", + "video_name": "ivbT-wvzK58", + "transcript": "(\"Symphony No. 4 in F minor\" by Pyotr Ilyich Tchaikovsky) - This large instrument is a viola. It's actually larger than most violas. Most violas are about this, the body part comes to about here. This one is here and it doesn't seem like a lot but it's huge. When you need to reach something, not many people can play this instrument. Violins are pretty much all the same size. Violas do come in many different shapes and sizes. If you look around in an orchestra, some of the mark cut down like this and they look awfully pretty. And some of the mark kinda big down here and small up here which is kinda nice because it's easier to get up here. You don't have to worry about this shoulder here. It's easier. (\"Symphony No. 3 in E-flat major\" by Robert Schumann) Violins are tuned like this. (tune playing) Okay, violas sound like this. (tune playing) They're much lower. It's like if you have a singing group and you have sopranos and you have altos, the violins and then the violas are the alto voice, and the cello's the tenor and base is the bass. (\"Symphony No. 9 in E minor\" by Antonin Dvorak) A bunch of years ago, I was playing a double viola concerto that was written for me and my stand partner of (mumbles). She had just acquired a million-dollar instrument, Gaspar Cassado, from Italy. I was playing something that got me my job. I mean, I played and I got it myself but the instrument that I was playing when I got in Philharmonic, I also got a major job in another orchestra at some point. On that instrument, the sound just didn't compare to this Gaspar. So the orchestra, administrators I guess, and the powers that be said, \"Go find yourself an instrument.\" So the orchestra owns this instrument. I went out and I asked around and the people put the word out and I tried many instruments. There was one that I had found and loved it but there was... Once we went to find out the background about it, there was little suspect. Anyway, a lot of politics and instruments that I didn't know about, but found this one. It's so big that it is... I said, \"Don't show me anything 17 inches or over.\" And this is actually 17 1/2 plus but the guy who was joining to make (mumbles) said, \"You have to hear this instrument.\" So I picked it up and went, \"Oh no (laughs), I really love it.\" (\"Symphony No. 9 in E minor\" by Antonin Dvorak) I always knew I was gonna be a musician. I started because I used to go to the young people's concert when I was two 1/2. I saw people playing and the story goes. I don't remember but my mom said I used to roll up the programs and put one here and put one here and pretend and then when I came time to play, my mom said, \"Would you like to play?\" I asked her, I remember like it was yesterday. We were standing in the garage and she was putting garbage in the garbage can and she had one, the can in her hand, the top in her hand like this and I said, \"I want to play violin.\" And she said, \"Well, don't you want to start with piano?\" And I said, \"No, I want to do this.\" So she found me a teacher shortly thereafter and got a feel and I started playing and then, I guess I got kinda good. I auditioned for Juilliard Pre-College at eight but I didn't get in but they sent me to a teacher, Fannie Chase, who is no longer with us, but she taught me every technical thing that I know on the violin or viola. Then I reauditioned five years later and I got in when I was 13 to Juilliard. But by the time I was done I guess I was a little bit bored. I used to like to play the second violin parts which are always the harmony parts, not the melody parts. I even, with my own children, I would always make them sing the melody of something so that I could sing the harmony. Like a commercial on TV, I would always make them sing the high things, so I could sing the middle thing. But it was the same thing for me in string quartets, I always wanted to play the middle voice. So when I was ready, I was not happy with it. I would guess I was a little bored. Somebody suggested to my mom, \"\"In Pre-College Juilliards, we need some more violas. \"And she's tall and she got long arms \"and maybe she could play the viola.\" So I said, \"All right, I'll try it.\" And I tried it and I loved it. My teacher was very inspiring. It was only nine months with that one particular teacher, Eugene Becker. (\"Symphony No. 4 in F minor\" by Pyotr Ilyich Tchaikovsky) That's what I always did. It's what I always knew I would do. Not only did I always know I was gonna be a musician but I always knew I was gonna be in the Philharmonic. Don't ask me. It's just the way it always was. It wasn't, \"Oh, I hope I get this audition\", or, \"I hope I can get in to Juilliards.\" \"I hope\", it's just, \"Okay, now I'm gonna take \"this audition and now I'm gonna do this.\" It just kinda... I mean, I'm lucky I guess. (\"Of Paradise and Light\" by Augusta Read Thomas)" + }, + { + "Q": "I don't understand how Bob decrypts the message.", + "A": "He is given a key beforehand.", + "video_name": "FlIG3TvQCBQ", + "transcript": "For over 400 years, the problem remained. How could Alice design a cipher that hides her fingerprint, thus stopping the leak of information? The answer is randomness. Imagine Alice rolled a 26 sided die to generate a long list of random shifts, and shared this with Bob instead of a code word. Now, to encrypt her message, Alice uses the list of random shifts instead. It is important that this list of shifts be as long as the message, as to avoid any repetition. Then she sends it to Bob, who decrypts the message using the same list of random shifts she had given him. Now Eve will have a problem, because the resulting encrypted message will have two powerful properties. One, the shifts never fall into a repetitive pattern. And two, the encrypted message will have a uniform frequency distribution. Because there is no frequency differential and therefore no leak, it is now impossible for Eve to break the encryption. This is the strongest possible method of encryption, and it emerged towards the end of the 19th century. It is now known as the one-time pad. In order to visualize the strength of the one-time pad, we must understand the combinatorial explosion which takes place. For example, the Caesar Cipher shifted every letter by the same shift, which was some number between 1 and 26. So if Alice was to encrypt her name, it would result in one of 26 possible encryptions. A small number of possibilities, easy to check them all, known as brute force search. Compare this to the one-time pad, where each letter would be shifted by a different number between 1 and 26. Now think about the number of possible encryptions. It's going to be 26 multiplied by itself five times, which is almost 12 million. Sometimes it's hard to visualize, so imagine she wrote her name on a single page, and on top of it stacked every possible encryption. How high do you think this would be? With almost 12 million possible five-letter sequences, this stack of paper would be enormous, over one kilometer high. When Alice encrypts her name using the one-time pad, it is the same as picking one of these pages at random. From the perspective of Eve, the code breaker, every five letter encrypted word she has is equally likely to be any word in this stack. So this is perfect secrecy in action." + }, + { + "Q": "I don't really get that. Dose that mean when your simplifeying a number like 3/8 whatever number comes like suppose you have to multiply the 3 by a 2 will you have to multiply 8 by a 2 as well?", + "A": "To create an equivalent fraction, you must multiply both top and bottom by the same value. So, yes... if you multiply the 3 in 3/8 by 2, you would also multiply the 8 by 2. 3/8 * 2/2 = 6/16", + "video_name": "dCQbfaQZtaY", + "transcript": "You are filling up trays to make ice cubes. You notice that each tray holds the exact same amount of water but has a different number of ice cubes that it makes. The blue tray makes 8 equally sized ice cubes. The pink tray makes 16 equally sized ice cubes. So let me draw the blue tray here. The blue tray, I'll draw it like this. That is not blue. Let me draw it in blue. So the blue tray makes 8. So this is the blue tray right over here. And let me draw it and divide it into 8 sections to represent the 8 equally sized ice cubes. And I'll try to do it. That's about in half, and then I'm going to put each of those in half, and then each of those in half. So there we go. This is pretty close to 8 equally sized cubes, except for this one there. Let me actually make it a little bit cleaner. So there's the half, half, half, half, half. All right, so this is looking a little bit better, so 8 equally sized ice cubes. This is the blue tray. Now, the pink tray takes the same amount of water, so I'm going to make it the exact same length. So the pink tray is the exact same length, but it has 16 equally sized ice cubes. So what I'm going to do is I'm going to make these same sections here, but then I'm going to cut these sections in two. So that's 8, and to 16, I got to divide these into two. So almost done. This is the hard part. That last one wasn't that neat. Almost done. All right. There we go. Set up the problem. Now, you put 3 ice cubes from the blue tray into your drink. So, let's do that. So, 1, 2, 3 ice cubes from the blue tray into your drink. How many ice cubes from the pink tray would you need to equal the same exact amount of ice? So there's a bunch of ways to do it. We can think about it with numbers, or we can think about it visually. Let's first think about it with numbers. So how much of this tray have I pulled out? Well, I have 8 equally sized cubes and I took 3 of them out, so this literally represents 3/8. So the question is 3 over 8, if I take 3 out of 8 equally sized cubes, that's the same quantity as taking what? I want to do that in white. That's the same quantity as taking how many cubes out of 16, out of 16 equally sized cubes? Well, let's look at it over here visually. So if we want the exact same amount of ice, so we're gonna have 1, 2, 3, 4, 5, 6. We have 6/16, so this is equal to 6/16. Now, does that make actual sense? Well, sure. To go from 3/8 to 6/16, you multiply the numerator by 2, and you multiply the denominator by 2. Now, does that actually make sense? Well, sure it does. Because for the pink ice tray, you have 2 ice cubes for every 1 that you have in the blue ice So the blue ice tray, you have 8 equally sized cubes. Well, for each of those, you're going to have 2 in the pink ice tray, so you multiply by 2 to have 16 equally sized cubes. And out of the blue tray, if you take 3, well, that equivalent amount for each of those cubes, you would get 2 from the pink ice tray. So you're multiplying by 2 right over there. So the answer to the question, how many ice cubes from the pink tray would you need to equal the same amount of ice? Well, that is you would need 6 cubes." + }, + { + "Q": "why didn't the Germans go through Switzerland or Italy then??\nthey would have a better chance of winning the war .", + "A": "Tanks can t go through the Alps. Not to mention it is SUPER COLD there. If your mindset is not Hannibal or Napoleon, it would be harder than death doing it", + "video_name": "huOnuYAyv6w", + "transcript": "In the last video, we left off with the assassination of Archduke Franz Ferdinand of Austria, the heir to the Austro-Hungarian empire. And so you could imagine, the Austro-Hungarian empire did not take that well. They already did not enjoy the kingdom of Serbia trying to, essentially, provoke this nationalistic movement. And they viewed them as this small, little, weak country right below them. And so they use this to, essentially, issue an ultimatum. Essentially say, look, immediately bring all the people who did this to justice, all the people who might have conspired with the Gavrilo Princips to allow this assassination attempt to occur, and do it or else, and accept full responsibility. And actually, the kingdom of Serbia was in no mood to get into a war with Austria-Hungary, and so they tried their best to comply, but their best wasn't enough. So then July 28, a month later, you have the Austrians declaring war on the Serbians. Now up until this point, the Austro-Hungarian empire is thinking, OK, Serbia is a small, little kingdom here. It has some ties and obviously, it's a Slavic nation so it has some linguistic ties with the Russian empire. The Russian empire also had some political ties to it, but Russia is not going to get into a war with us over this. We're justified in attacking them. They've just killed the heir to our throne. So we're going to go in there and Russia is probably not interested in actually creating a larger skirmish here. That was a severe miscalculation on the part of the Austrians. The Russians were not happy about this. They felt close ties to the Serbians and they felt a need to protect the Serbians, or you could argue, that maybe they wanted to mobilize just to scare the Austrians. Whatever it might be, whether it was Russia wanted to get into a war, whether they were really looking to protect the Serbians, or whether they were looking to mobilize to scare the Austrians from actually attacking, the Russians began to mobilize. So the Russians began to gather their troops. So the Russians mobilize. And now, this is where all of the alliances start to come into effect. If you remember about the alliances we talked about several videos ago, if you go to 1879, you have the Dual Alliance Treaty between Germany and Austria-Hungary to protect each other if Russia attacks and actually, if Russia attacks or mobilizes. So now, Germany is like, hey, maybe I am obligated to protect Austria-Hungary from Russia. You also have to remember in 1892, the Franco-Russian Military Convention. Military assistance both ways in the event of attack. So Germany is thinking, look, we signed this treaty and we, to some degree, are maybe eager for war because we've been militarizing so much. And I can't just fight Russia. I also know that France and Russia have this alliance right over here. So Germany in quite surprising quickness decides to declare war on both. So on August 1, Germany declares war on Russia. And then on August 3, Germany declares war on France because they know or at least they feel that they can't declare war on only one of them. And they wanted to do it very quickly because they didn't want Russia a chance to mobilize too much. And the fact that they were able to do this so quickly-- we're talking three days after Austria declares war on Serbia and then another two days declare war on France-- kind of shows that Germany was already in a war footing. It's not a joke to all of a sudden invade or declare war on countries. So Germany was, essentially, preparing for this. And this right over here gave them the excuse to, essentially, declare war. So they declare war on both of these characters, Germany against Russia. Germany is declaring war on France. Now, the easiest way for them to move into France-- so they're literally going on the offense here-- is for them to go through Belgium. But the Germans were aware. They're aware that there's this 75-year-old treaty, the Treaty of London in 1839, Article 7 said that Britain was to protect the neutrality of Belgium. And Germany was in no interest to get into a war with the British. The British had a powerful military, especially a very powerful navy. The Germans said, hey, let's just take on the Russians and the French for now. And so they actually reached out to the British and said, hey, this little treaty that you got here from 1839, this 75-year-old treaty, you don't really take this seriously, right? I mean if we have to go through Belgium, you're not really going to hold true to this treaty? And the British said, no, we actually take that very seriously. Obviously, the British didn't want the Germans to be aggressive here. The British didn't want the Germans to be able to invade France. And so on August 4 when Germany, essentially, rolls through Belgium, Germany invades Belgium to get to France, this gave the legal justification for the British to declare war on Germany. And so in a matter of only-- I mean not even, we're talking a few months here from the assassination of Franz Ferdinand of Austria, you essentially have all of the major powers of Europe. And then as we'll see because they had these empires, in not too long most of the world is at war with each other." + }, + { + "Q": "What is relative permeability? Does any one knows have brief expalanation", + "A": "It s just a constant needed to make the equations work, like G or epsilon naught . It relates the proportionality of the current to the magnetic field and distance.", + "video_name": "Ri557hvwhcM", + "transcript": "So not only can a magnetic field exert some force on a moving charge, we're now going to learn that a moving charge or a current can actually create a magnetic field. So there is some type of symmetry here. And as we'll learn later when we learn our calculus and we do this in a slightly more rigorous way, we'll see that magnetic fields and electric fields are actually two sides of the same coin, of electromagnetic fields. But anyway, we won't worry about that now. And I think it's enough to ponder right now that a current can actually induce a magnetic field. And actually, just a moving electron creates a magnetic field. And it does it in a surface of a sphere-- I won't go into all Because the math gets a little bit fancy there. But what you might encounter in your standard high school physics class-- that's not getting deeply into vector calculus-- is that if you just have a wire-- let me draw a wire. That's my wire. And it's carrying some current I, it turns out that this wire will generate a magnetic field. And the shape of that magnetic field is going to be co-centric circles around this wire. Let me see if I can draw that. So here I'll draw it just like how I do when I try to do rotations of solids in the calculus video. So the magnetic field would go behind and in front and it goes like that. Or another way you can think about it is if-- let's go down here-- is on the left side of this wire. If you say that the wire's in the plane of this video, the magnetic field is popping out of your screen. And on this side, on the right side, the magnetic field is popping into the screen. It's going into the screen. And you could imagine that, right? You could imagine if, on this drawing up here, you could say this is where it intersects the screen. All of this is kind of behind the screen. And all of this is in front of the screen. And this is where it's popping out. And this is where it's popping into the screen. Hopefully that makes a little bit of sense. And how did I know that it's rotating this way? Well, it actually does come out of the cross product when you do it with a regular charge and all of that. But we're not going to go into that right now. And so there's a different right hand rule that you can use. And it's literally you hold this wire, or you imagine holding this wire, with your right hand with your thumb going in the direction of the current. And if you hold this wire with your thumb going in the direction of the current, your fingers are going to go in the direction of the magnetic field. So let me see if I can draw that. I will draw it in blue. So if this is my thumb, my thumb is going along the top of the wire. And then my hand is curling around the wire. Those are the veins on my hand. This is my nail. So as you can see, if I was holding that same wire-- let me draw the wire. So if I was holding that same wire, we see that my thumb is going in the direction of the current. So this is a slightly new thing to memorize. And what is the magnetic field doing? Well, it's going in the direction of my fingers. My fingers are popping out on this side of the wire. They're coming out on this side of the wire. And they're going in, or at least my hand is going in, on that side. It's going into the screen. Hopefully that makes sense. Now, how can we quantify? Well, before we quantify, let's get a little bit more of the intuition of what's happening. It turns out that the closer you get to the wire, the stronger the magnetic field, and the further you get out, the weaker the magnetic field. And that kind of makes sense if you imagine the magnetic field spreading out. I don't want to go into too sophisticated analogies. But if you imagine the magnetic field spreading out, and as it goes further and further out it kind of gets distributed over a wider and wider circumference. And actually the formula I'm going to give you kind of has a taste for that. So the formula for the magnetic field-- and it really is defined with a cross product and things like that, but for our purposes we won't worry about that. You'll just have to know that this is the shape if the current is going in that direction. And, of course, if the current was going downwards then the magnetic field would just reverse directions. But it would still be in co-centric circles around the wire. But anyway, what is the magnitude of that field? The magnitude of that magnetic field is equal to mu-- which is a Greek letter, which I will explain in a second-- times the current divided by 2 pi r. So this has a little bit of a feel for what I was saying before. That the further you go out-- where r is how far you are from the wire-- the further you go out, if r gets bigger, the magnitude of the magnetic field is going to get weaker. And this 2 pi r, that looks a lot like the circumference of a circle. So that gives you a taste for it. I know I haven't proved anything rigorously. But it at least gives you a sense of, look there's a little formula for circumference of a circle here. And that kind of makes sense, right? Because the magnetic field at that point is kind of a circle. The magnitude is equal at an equal radius around that wire. Now what is this mu, this thing that looks like a u? Well, that's the permeability of the material that the wire's in. So the magnetic field is actually going to have a different strength depending on whether this wire is going through rubber, whether it's going through a vacuum, or air, or metal, or water. And for the purposes of your high school physics class, we assume that it's going through air normally. And the value for air is pretty close to the value for a vacuum. And it's called the permeability of a vacuum. And I forget what that value is. I could look it up. But it actually turns out that your calculator has that value. So let's do a problem, just to put some numbers to the formula. So let's say I had this current and it is-- I don't know, the current is equal to-- I'm going 2 amperes. And let's say that I just pick a point right here that is-- let's say that that's 3 meters away from the wire in question. So my question to you is what is the magnitude in the direction of the magnetic field right there? Well, the magnitude is easy. We just substitute in this equation. So the magnitude of the magnetic field at this point is equal to-- and we assume that the wire's going through air or a vacuum-- the permeability of free space-- that's just a constant, though it looks fancy-- times the current times 2 amperes divided by 2 pi r. What's r? It's 3 meters. So 2 pi times 3. So it equals the permeability of free space. The 2 and the 2 cancel out over 3 pi. So how do we calculate that? Well, we get out our trusty TI-85 calculator. And I think you'll be maybe pleasantly surprised or shocked to realize that-- I deleted everything just so you can see how I get there-- that it actually has the permeability of free space stored in it. So what you do is you go to second and you press constant, which is the 4 button. It's in the built-in constants. Let's see, it's not one of those. You press more. It's not one of those, press more. Oh look at that. Mu not. The permeability of free space. That's what I need. And I have to divide it by 3 pi. Divide it by 3-- and then where is pi? There it is. It's over the power sign. Divided by 3 pi. It equals 1.3 times 10 to the negative seventh. It's going to be teslas. The magnetic field is going to be equal to 1.3 times 10 to the minus seventh teslas. So it's a fairly weak magnetic field. And that's why you don't have metal objects being thrown around by the wires behind your television set. But anyway, hopefully that gives you a little bit-- and just so you know how it all fits together. We're saying that these moving charges, not only can they be affected by a magnetic field, not only can a current be affected by a magnetic field or just a moving charge, it actually creates them. And that kind of creates a little bit of symmetry in your head, hopefully. Because that was also true of electric field. A charge, a stationary charge, is obviously pulled or pushed by a static electric field. And it also creates its own static electric field. So it's always in the back of your mind. Because if you keep studying physics, you're going to actually prove to yourself that electric and magnetic fields are two sides of the same coin. And it just looks like a magnetic field when you're in a different frame of reference, When something is whizzing past you. While if you were whizzing along with it, then that thing would look static. And then it might look a little bit more like an electric field. But anyway, I'll leave you there now. And in the next video I will show you what happens when we have two wires carrying current parallel to each other. And you might guess that they might actually attract or repel each other. Anyway, I'll see you in the next video." + }, + { + "Q": "When in history were we first able to calculate the size of the sun?", + "A": "The distance to the sun was accurately determined in the late 1700 s. Once the distance is known, the diameter is easy to calculate using basic geometry.", + "video_name": "GZx3U0dbASg", + "transcript": "My goal in this video and the next video is to start giving a sense of the scale of the earth and the solar system. And as we see, as we start getting into to the galaxy and the universe, it just becomes almost impossible to imagine. But we'll at least give our best shot. So I think most of us watching this video know that this right here is earth. And just to get a sense of scale here, I think probably the largest distance that we can somehow relate to is about 100 miles. You can get into a car for an hour, hour and a half, and go about 100 miles. And on the earth that would be about this far. It would be a speck that would look something like that. That is 100 miles. And also to get us a bit of scale, let's think about a speed that at least we can kind of comprehend. And that would be, maybe, the speed of a bullet. Maybe we can't comprehend it, but I'll say this is the fastest thing that we could maybe comprehend. It goes about-- and there are different types of bullets depending on the type of gun and all of that-- about 280 meters per second, which is about 1,000 kilometers per hour. And this is also roughly the speed of a jet. So just to give a sense of scale here, the earth's circumference-- so if you were to go around the planet-- is about 40,000 kilometers. So if you were to travel at the speed of a bullet or the speed of a jetliner, at 1,000 kilometers an hour, it would take you 40 hours to circumnavigate the earth. And I think none of this information is too surprising. You might have taken a 12- or 15-hour flight that gets you-- not all the way around the earth-- but gets you pretty far. San Francisco to Australia, or something like that. So right now these aren't scales that are too crazy. Although, even for me, the earth itself is a pretty mind-blowingly large object. Now, with that out of the way let's think about the sun. Because the sun starts to approach something far huger. So this obviously here is the sun. And I think most people appreciate that the sun is much larger than the earth, and that it's pretty far away from the earth. But I don't think most people, including myself, fully appreciate how large the sun is or how far it is away from the earth. So just to give you a sense, the sun is 109 times the circumference of the earth. So if we do that same thought exercise there-- if we said, OK, if I'm traveling at the speed of a bullet or the speed of a jetliner, it would take me 40 hours to go around the earth. Well, how long would it take to go around the sun? So if you were to get on a jet plane and try to go around the sun, or if you were to somehow ride a bullet and try to go around the sun-- do a complete circumnavigation of the sun-- it's going to take you 109 times as long as it would have taken you to do the earth. So it would be 100 times-- I could do 109, but just for approximate-- it's roughly 100 times the circumference of the earth. So 109 times 40 is equal to 4,000 hours. And just to get a sense of what 4,000 is-- actually, since I have the calculator out, let's do the exact calculation. It's 109 times the circumference of the earth times 40 hours. That's what it would take to do the circumference of the Earth. So it's 4,360 hours to circumnavigate the sun, going at the speed of a bullet or a jetliner. And so that is-- 24 hours in the day-- that is 181 days. It would take you roughly half a year to go around the sun at the speed of a jetliner. Let me write this down. Half a year. The sun is huge. Now, that by itself may or may not be surprising--and actually let me give you a sense of scale here, because I have this other diagram of a sun. And we'll talk more about the rest of the solar system in the next video. But over here, at this scale, the sun, at least on my screen-- if I were to complete it, it would probably be about 20 inches in diameter. The earth is just this little thing over here, smaller than a raindrop. If I were to draw it on this scale, where the sun is even smaller, the earth would be about that big. Now, what isn't obvious, because we've all done our science projects in third and fourth grade--or we always see these diagrams of the solar system that look something like this-- is that these planets are way further away. Even though these are depicted to scale, they're way further away from the sun than this makes it look. So the earth is 150 million kilometers from the sun. So if this is the sun right here, at this scale you wouldn't even be able to see the earth. It wouldn't even be a pixel. But it would be 150 million kilometers from the earth. And this distance right here is called an astronomical unit-- and we'll be using that term in the next few videos just because it's an easier way to think about distance-- sometimes abbreviated AU, astronomical unit. And just to give a sense of how far this is, light, which is something that we think is almost infinitely fast and that is something that looks instantaneous, that takes eight minutes to travel from the sun to the earth. If the sun were to disappear, it would take eight minutes for us to know that it disappeared on earth. Or another way, just to put it in the sense of this jet airplane-- let's get the calculator back out. So we're talking about 150 million kilometers. So if we're going at 1,000 kilometers an hour, it would take us 150,000 hours at the speed of a bullet or at the speed of a jet plane to get to the sun. And just to put that in perspective, if we want it in days, there's 24 hours per day. So this would be 6,250 days. Or, if we divided by 365, roughly 17 years. If you were to shoot a bullet straight at the sun it would take 17 years to get there, if it could maintain its velocity somehow. So this would take a bullet or a jet plane 17 years to get to the sun. Or another way to visualize it-- this sun right over here, on my screen it has about a five- or six-inch diameter. If I were to actually do it at scale, this little dot right here, which is the earth, this speck-- I would have to put this back about 50 feet away from the sun. 50 or 60 feet away from the sun. If you were to look at the solar system-- and obviously there's other things in the solar system, and we'll talk more about them in the next video-- you wouldn't even notice this speck. This is a little dust thing flying around this sun. And as we go further and further out of this solar system, you're going to see even this distance starts to become ridiculously small. Or another way to think about it-- if the sun was about this size, then the earth at this scale would be about 200 feet away from it. So you could imagine, if you had a football field-- let me draw a football field. These are the end zones-- one end zone, another end zone. And if you were to stick something-- maybe the size of a medicine ball, a little bit bigger than a basketball, at one end zone-- this little speck would be about 60 yards away, roughly 60 meters away. So it's this little speck. You wouldn't even notice it on the scale of a football field, something this size. Anyway, I'm going to leave you there. Hopefully that just starts to blow your mind when you think about the scale of the sun, the earth, and how far the earth is away from the sun. And then we're going to see even those distances, even those scales, are super small when you start thinking about the rest of the solar system. And especially when we start going beyond the solar system." + }, + { + "Q": "Hi! I was wondering: what happens if there is a negative number inside the square root when you -4ac from b^2? I have had problems like that before on math homework. I've tried many ways to get answers but non of them work\u00e2\u0080\u00a6 here's and example!\n3x^2 \u00e2\u0080\u0093 5x + 4 = 0\nThanks!", + "A": "If there is a negative number inside the radical, then there is NO real solutions. This would only be solvable through complex numbers (imaginary numbers).", + "video_name": "iulx0z1lz8M", + "transcript": "Use the quadratic formula to solve the equation, 0 is equal to negative 7q squared plus 2q plus 9. Now, the quadratic formula, it applies to any quadratic equation of the form-- we could put the 0 on the left hand side. 0 is equal to ax squared plus bx plus c. And we generally deal with x's, in this problem we're dealing with q's. But the quadratic formula says, look, if you have a quadratic equation of this form, that the solutions of this equation are going to be x is going to be equal to negative b plus or minus the square root of b squared minus 4ac-- all of that over 2a. And this is actually two solutions here, because there's one solution where you take the positive square root and there's another solution where you take the negative root. So it gives you both roots of this. So if we look at the quadratic equation that we need to solve here, we can just pattern match. We're dealing with q's, not x's, but this is the same general idea. It could be x's if you like. And if we look at it, negative 7 corresponds to a. That is our a. It's the coefficient on the second degree term. 2 corresponds to b. It is the coefficient on the first degree term. And then 9 corresponds to c. It's the constant. So, let's just apply the quadratic formula. The quadratic formula will tell us that the solutions-- the q's that satisfy this equation-- q will be equal to negative b. b is 2. Plus or minus the square root of b squared, of 2 squared, minus 4 times a times negative 7 times c, which is 9. And all of that over 2a. All of that over 2 times a, which is once again negative 7. And then we just have to evaluate this. So this is going to be equal to negative 2 plus or minus the square root of-- let's see, 2 squared is 4-- and then if we just take this part right here, if we just take the negative 4 times negative 7 times 9, this negative and that negative is going to cancel out. So it's just going to become a positive number. And 4 times 7 times 9. 4 times 9 is 36. 36 times 7. Let's do it up here. 36 times 7. 7 times 6 is 42. 7 times 3, or 3 times 7 is 21. Plus 4 is 25. 252. So this becomes 4 plus 252. Remember, you have a negative 7 and you have a minus out front. Those cancel out, that's why we have a positive 252 for that part right there. And then our denominator, 2 times negative 7 is negative 14. Now what does this equal? Well, we have this is equal to negative 2 plus or minus the square root of-- what's 4 plus 252? It's just 256. All of that over negative 14. And what's 256? What's the square root of 256? It's 16. You can try it out for yourself. This is 16 times 16. So the square root of 256 is 16. So we can rewrite this whole thing as being equal to negative 2 plus 16 over negative 14. Or negative 2 minus-- right? This is plus 16 over negative 14. Or minus 16 over negative 14. If you think of it as plus or minus, that plus is that plus right there. And if you have that minus, that minus is that minus right there. Now we just have to evaluate these two numbers. Negative 2 plus 16 is 14 divided by negative 14 is negative 1. So q could be equal to negative 1. Or negative 2 minus 16 is negative 18 divided by negative 14 is equal to 18 over 14. The negatives cancel out, which is equal to 9 over 7. So q could be equal to negative 1, or it could be equal to 9 over 7. And you could try these out, substitute these q's back into this original equation, and verify for yourself that they satisfy it. We could even do it with the first one. So if you take q is equal to negative 1. Negative 7 times negative 1 squared-- negative 1 squared is just 1-- so this would be negative 7 times 1, right? That's negative 1 squared. Negative 1 times 2 is minus 2 plus 9. So it's negative 7 minus 2, which is negative 9, plus 9, does indeed equal 0. So this checks out. And I'll leave it up to you to verify that 9 over 7 also works out." + }, + { + "Q": "how is (2,-1) a solution for -3x-4y=-2 and y=2x-5", + "A": "If a point is a solution to the system of equations, then it will make both equations be true. Use substitution to test the ordered pair in each equation. The ordered pair tells you X = 2 and Y = -1 1st equation: -3(2) - 4(-1)=-2 -6 + 4=-2 -2=-2 (the ordered pair makes this equation true) 2nd equation: -1 = 2(2) - 5 -1 = 4 - 5 -1 = -1 (the ordered pair makes this equation true as well) Since the ordered pair satisfies both equations, it is a solution to the system.", + "video_name": "GWZKz4F9hWM", + "transcript": "So that it's less likely that we get shown up by talking birds in the future, we've set a little bit of exercise for solving systems of equations with substitution. And so this is the first exercise or the first problem that they give us. -3x-4y=-2 and y=2x-5 So let me get out my little scratch pad and let me rewrite the problem. So this is -3x-4y=-2 and then they tell us y=2x-5. So, what's neat about this is that they've already solved the second equation. They've already made it explicitly solved for y which makes it very easy to substitute for. We can take this constraint, the constraint on y in terms of x and substitute it for y in this first blue equation and then solve for x. So let's try it out. So this first blue equation would then become -3x-4 but instead of putting a y there the second constraint tells us that y needs to be equal to 2x-5. So it's 4(2x-5) and all of that is going to be equal to -2. So now we get just one equation with one unknown. and now we just have to solve for x. So, let's see if we can do that. So, it's -3x and then this part right over here we have a -4, be careful, we have a -4 we want to distribute. We are going to multiply -4*2x which is -8x and -4*-5 is positive 20 and thats going to equal -2. And now we can combine all the x terms so -3x-8x, that's going to be -11x and then we have -11x+20=-2. Now to solve for x, we'll subtract 20 from both sides to get rid of the 20 on the left hand side. On the left hand side, we're just left with the -11x and then on the right hand side we are left with -22. Now we can divide both sides by -11. And we are left with x is equal to 22 divided by 11 is 2, and the negatives cancel out. x = 2. So we are not quite done yet. We've done, I guess you can say the hard part, we have solved for x but now we have to solve for y. We could take this x value to either one of these equations and solve for y. But this second one has already explicitly solved for y so let's use that one, so it says y = 2 times and instead of x, we now know that the x value where these two intersect, you could view it that way is going to be equal to 2, so 2 * 2 - 5 let's figure out the corresponding y value. So you get y=2(2)-5 and y = 4 - 5 so y = -1. And you can verify that it'll work in this top equation If y = -1 and x=2, this top equation becomes -3(2) which is -6-4(-1) which would be plus 4. And -6+4 is indeed -2. So it satisfies both of these equations and now we can type it in to verify that we got it right, although, we know that we did, so x=2 and y=-1. So, let's type it in... x=2 and y=-1. Excellent, now we're much less likely to be embarassed by talking birds." + }, + { + "Q": "I still don't understand why at the end, the answer is negative.Why did the triangle double towards the negative axis? Also, how is the x co-ordinate the cosine of the angle? Thanks.", + "A": "As Sal mentions at 3:16, he s using the unit circle definition of the trigonometric functions. I could try to explain that in this answer, but I think you re better off watching the series of videos on the unit circle definition under the Basic Trigonometry section of Trigonometry and Precalculus. Hopefully those ll help you make more sense of this.", + "video_name": "D_smr0GBPvA", + "transcript": "We have triangle ABC here, which looks like a right triangle. And we know it's a right triangle because 3 squared plus 4 squared is equal to 5 squared. And they want us to figure out what cosine of 2 times angle So that's this angle-- ABC. Well, we can't immediately evaluate that, but we do know what the cosine of angle ABC is. We know that the cosine of angle ABC-- well, cosine is just adjacent over hypotenuse. It's going to be equal to 3/5. And similarly, we know what the sine of angle ABC is. That's opposite over hypotenuse. That is 4/5. So if we could break this down into just cosines of ABC and sines of ABC, then we'll be able to evaluate it. And lucky for us, we have a trig identity at our disposal that does exactly that. We know that the cosine of 2 times an angle is equal to cosine of that angle squared minus sine of that angle squared. And we've proved this in other videos, but this becomes very helpful for us here. Because now we know that the cosine-- Let me do this in a different color. Now, we know that the cosine of angle ABC is going to be equal to-- oh, sorry. It's the cosine of 2 times the angle ABC. That's what we care about. 2 times the angle ABC is going to be equal to the cosine of angle ABC squared minus sine of the angle ABC squared. And we know what these things are. This thing right over here is just going to be equal to 3/5 squared. Cosine of angle a ABC is 3/5. So we're going to square it. And this right over here is just 4/5 squared. So it's minus 4/5 squared. And so this simplifies to 9/25 minus 16/25, which is equal to 7/25. Sorry. It's negative. Got to be careful there. 16 is larger than 9. Negative 7/25. Now, one thing that might jump at you is, why did I get a negative value here when I doubled the angle here? Because the cosine was clearly a positive number. And there you just have to think of the unit circle-- which we already know the unit circle definition of trig functions is an extension of the Sohcahtoa definition. Y-axis. Let me draw a unit circle here. My best attempt. So that's our unit circle. So this angle right over here looks like something like this. And so you see its x-coordinate-- which is the cosine of that angle-- looks positive. But then, if you were to double this angle, it would take you out someplace like this. And then, you see-- by the unit circle definition-- the x-coordinate, we are now sitting in the second quadrant. And the x-coordinate can be negative. And that's essentially what happened in this problem." + }, + { + "Q": "What is displacement?", + "A": "It s the distance from and object, just with a direction as well.", + "video_name": "oRKxmXwLvUU", + "transcript": "Now that we know a little bit about vectors and scalars, let's try to apply what we know about them for some pretty common problems you'd, one, see in a physics class, but they're also common problems you'd see in everyday life, because you're trying to figure out how far you've gone, or how fast you're going, or how long it might take you to get some place. So first I have, if Shantanu was able to travel 5 kilometers north in 1 hour in his car, what was his average velocity? So one, let's just review a little bit about what we know about vectors and scalars. So they're giving us that he was able to travel 5 kilometers to the north. So they gave us a magnitude, that's the 5 kilometers. That's the size of how far he moved. And they also give a direction. So he moved a distance of 5 kilometers. Distance is the scalar. But if you give the direction too, you get the displacement. So this right here is a vector quantity. He was displaced 5 kilometers to the north. And he did it in 1 hour in his car. What was his average velocity? So velocity, and there's many ways that you might see it defined, but velocity, once again, is a vector quantity. And the way that we differentiate between vector and scalar quantities is we put little arrows on top of vector quantities. Normally they are bolded, if you can have a typeface, and they have an arrow on top of them. But this tells you that not only do I care about the value of this thing, or I care about the size of this thing, I also care about its direction. The arrow isn't necessarily its direction, it just tells you that it is a vector quantity. So the velocity of something is its change in position, including the direction of its change in position. So you could say its displacement, and the letter for displacement is S. And that is a vector quantity, so that is displacement. And you might be wondering, why don't they use D for displacement? That seems like a much more natural first letter. And my best sense of that is, once you start doing calculus, you start using D for something very different. You use it for the derivative operator, and that's so that the D's don't get confused. And that's why we use S for displacement. If someone has a better explanation of that, feel free to comment on this video, and then I'll add another video explaining that better explanation. So velocity is your displacement over time. If I wanted to write an analogous thing for the scalar quantities, I could write that speed, and I'll write out the word so we don't get confused with displacement. Or maybe I'll write \"rate.\" Rate is another way that sometimes people write speed. So this is the vector version, if you care about direction. If you don't care about direction, you would have your rate. So this is rate, or speed, is equal to the distance that you travel over some time. So these two, you could call them formulas, or you could call them definitions, although I would think that they're pretty intuitive for you. How fast something is going, you say, how far did it go over some period of time. These are essentially saying the same thing. This is when you care about direction, so you're dealing with vector quantities. This is where you're not so conscientious about direction. And so you use distance, which is scalar, and you use rate or speed, which is scalar. Here you use displacement, and you use velocity. Now with that out of the way, let's figure out what his average velocity was. And this key word, average, is interesting. Because it's possible that his velocity was changing over that whole time period. But for the sake of simplicity, we're going to assume that it was kind of a constant velocity. What we are calculating is going to be his average velocity. But don't worry about it, you can just assume that it wasn't changing over that time period. So his velocity is, his displacement was 5 kilometers to the north-- I'll write just a big capital. Well, let me just write it out, 5 kilometers north-- over the amount of time it took him. And let me make it clear. This is change in time. This is also a change in time. Sometimes you'll just see a t written there. Sometimes you'll see someone actually put this little triangle, the character delta, in front of it, which explicitly means \"change in.\" It looks like a very fancy mathematics when you see that, but a triangle in front of something literally means \"change in.\" So this is change in time. So he goes 5 kilometers north, and it took him 1 hour. So the change in time was 1 hour. So let me write that over here. So over 1 hour. So this is equal to, if you just look at the numerical part of it, it is 5/1-- let me just write it out, 5/1-- kilometers, and you can treat the units the same way you would treat the quantities in a fraction. 5/1 kilometers per hour, and then to the north. Or you could say this is the same thing as 5 kilometers per hour north. So this is 5 kilometers per hour to the north. So that's his average velocity, 5 kilometers per hour. And you have to be careful, you have to say \"to the north\" if you want velocity. If someone just said \"5 kilometers per hour,\" they're giving you a speed, or rate, or a scalar quantity. You have to give the direction for it to be a vector quantity. You could do the same thing if someone just said, what was his average speed over that time? You could have said, well, his average speed, or his rate, would be the distance he travels. The distance, we don't care about the direction now, is 5 kilometers, and he does it in 1 hour. His change in time is 1 hour. So this is the same thing as 5 kilometers per hour. So once again, we're only giving the magnitude here. This is a scalar quantity. If you want the vector, you have to do the north as well. Now, you might be saying, hey, in the previous video, we talked about things in terms of meters per second. Here, I give you kilometers, or \"kil-om-eters,\" depending on how you want to pronounce it, kilometers per hour. What if someone wanted it in meters per second, or what if I just wanted to understand how many meters he travels in a second? And there, it just becomes a unit conversion problem. And I figure it doesn't hurt to work on that right now. So if we wanted to do this to meters per second, how would we do it? Well, the first step is to think about how many meters we are traveling in an hour. So let's take that 5 kilometers per hour, and we want to convert it to meters. So I put meters in the numerator, and I put kilometers in the denominator. And the reason why I do that is because the kilometers are going to cancel out with the kilometers. And how many meters are there per kilometer? Well, there's 1,000 meters for every 1 kilometer. And I set this up right here so that the kilometers cancel out. So these two characters cancel out. And if you multiply, you get 5,000. So you have 5 times 1,000. So let me write this-- I'll do it in the same color-- 5 times 1,000. So I just multiplied the numbers. When you multiply something, you can switch around the order. Multiplication is commutative-- I always have trouble pronouncing that-- and associative. And then in the units, in the numerator, you have meters, and in the denominator, you have hours. Meters per hour. And so this is equal to 5,000 meters per hour. And you might say, hey, Sal, I know that 5 kilometers is the same thing as 5,000 meters. I could do that in my head. And you probably could. But this canceling out dimensions, or what's often called dimensional analysis, can get useful once you start doing really, really complicated things with less intuitive units than something But you should always do an intuitive gut check right here. You know that if you do 5 kilometers in an hour, that's a ton of meters. So you should get a larger number if you're talking about meters per hour. And now when we want to go to seconds, let's do an intuitive gut check. If something is traveling a certain amount in an hour, it should travel a much smaller amount in a second, or 1/3,600 of an hour, because that's how many seconds there are in an hour. So that's your gut check. We should get a smaller number than this when we want to say meters per second. But let's actually do it with the dimensional analysis. So we want to cancel out the hours, and we want to be left with seconds in the denominator. So the best way to cancel this hours in the denominator is by having hours in the numerator. So you have hours per second. So how many hours are there per second? Or another way to think about it, 1 hour, think about the larger unit, 1 hour is how many seconds? Well, you have 60 seconds per minute times 60 minutes per hour. The minutes cancel out. 60 times 60 is 3,600 seconds per hour. So you could say this is 3,600 seconds for every 1 hour, or if you flip them, you would get 1/3,600 hour per second, or hours per second, depending on how you want to do it. So 1 hour is the same thing as 3,600 seconds. And so now this hour cancels out with that hour, and then you multiply, or appropriately divide, the numbers right here. And you get this is equal to 5,000 over 3,600 meters per-- all you have left in the denominator here is second. Meters per second. And if we divide both the numerator and the denominator-- I could do this by hand, but just because this video's already getting a little bit long, let me get my trusty calculator out. I get my trusty calculator out just for the sake of time. 5,000 divided by 3,600, which would be really the same thing as 50 divided by 36, that is 1.3-- I'll just round it over here-- 1.39. So this is equal to 1.39 meters per second. So Shantanu was traveling quite slow in his car. Well, we knew that just by looking at this. 5 kilometers per hour, that's pretty much just letting the car roll pretty slowly." + }, + { + "Q": "A small question, I think I get how an increase in volume increases entropy, but now I understand a little more this concept... can somebody explain to me how come an increase in temperature increases this combination of states?", + "A": "When the temperature of a gas is increased, the molecules have higher kinetic energies. When the molecules move around faster they can configure themselves in more ways than they could have if they had less kinetic energy. More specifically, molecules in an ideal gas follow Maxwell-Boltzmann statistics which imply that the molecules velocity distribution broadens as the temperature of the system is increased.", + "video_name": "xJf6pHqLzs0", + "transcript": "I've now supplied you with two definitions of the state variable entropy. And it's S for entropy. The thermodynamic definition said that the change in entropy is equal to the heat added to the system divided by the temperature at which the heat is added. So obviously, if the temperature is changing while we add the heat, which is normally the case, we're going to have to do a little bit of calculus. And then you can view this as the mathematical, or the statistical, or the combinatorical definition of entropy. And this essentially says that entropy is equal to some constant times the natural log of the number of states the system can take on. And this is the case when all the states are equally probable, which is a pretty good assumption. If you have just a gazillion molecules that could have a gazillion gazillion states, you can assume they're all roughly equally likely. There's a slightly more involved definition if they had different probabilities, but we won't worry about that now. So given that we've seen these two definitions, it's a good time to introduce you to the second law of thermodynamics. And that's this. And it's a pretty simple law, but it explains a whole range of phenomena. It tells us that the change in entropy for the universe when any process is undergone is always greater than or equal to 0. So that tells us that when anything ever happens in the universe, the net effect is that there's more entropy in the universe itself. And this seems very deep, and it actually is. So let's see if we can apply it to see why it explains, or why it makes sense, relative to some examples. So let's say I have two reservoirs that are in contact with each other. So I have T1. And let's call this our hot reservoir. And then I have T2. I'll call this our cold reservoir. Well, we know from experience. What happens if I put a hot cup of water, and it's sharing a wall with a cold glass of water, or cold cube of water, what happens? Well, their temperatures equalize. If these are the same substance, we'll end up roughly in between, if they're in the same phase. So essentially, we have a transfer of heat from the hotter substance to the colder substance. So we have some heat, Q, that goes from the hotter substance to the colder substance. You don't see, in everyday reality, heat going from a colder substance to a hotter substance. If I put an ice cube in, let's say, some hot tea, you don't see the ice cube getting colder and the hot tea getting hotter. You see them both getting to some equal temperature, which essentially the T is giving heat to the ice cube. Now in this situation there are reservoirs, so I'm assuming that their temperatures stay constant. Which would only be the case if they were both infinite, which we know doesn't exist in the real world. In the real world, T1's temperature as it gave heat would go down, and T2's temperature would go up. But let's just see whether the second law of thermodynamics says that this should happen. So what's happening here? What's the net change in entropy for T1? So the second law of thermodynamics says that the change in entropy for the universe is greater than 0. But in this case, that's equal to the change in entropy for T1 plus the change in entropy for-- oh, I shouldn't-- instead of T1, let me call it just 1. For system 1, that's this hot system up here, plus the change in entropy for system 2. So what's the change in entropy for system 1? It loses Q1 at a high temperature. So this equals minus the heat given to the system is Q over some hot temperature T1. And then we have the heat being added to the system T2. So plus Q over T2. This is the change in entropy for the system 2, right? This guy loses the heat, and is at temperature 1, which is a higher temperature. This guy gains the heat, and he is at a temperature 2, which is a colder temperature. Now, is this going to be greater than 0? Let's think about it a little bit. If I divide-- let me rewrite this. So I can rearrange them, so that we can write this as Q over T2 minus this one. I'm just rearranging it. Minus Q over T1. Now, which number is bigger? T2 to T1? Well, T1 is bigger, right? This is bigger. Now, if I have a bigger number, bigger than this-- when we use the word bigger, you have to compare it to something. Now, T1 is bigger than this. We have the same number in the numerator in both cases, right? So if I take, let's say, 1 over some, let's say, 1/2 minus 1/3, we're going to be bigger than 0. This is a larger number than this number, because this has a bigger denominator. You're dividing by a larger number. That's a good way to think about it. You're dividing this Q by some number here to get something, and then you're subtracting this Q divided by a larger number. So this fraction is going to be a smaller absolute number. So this is going to be greater than 0. So that tells us the second law of thermodynamics, it verifies this observation we see in the real world, that heat will flow from the hot body to the cold body. Now, you might say, hey, Sal. I have a case that will show you that you are wrong. You could say, look. If I put an air conditioner in a room-- Let's say this is the room, and this is outside. You'll say, look what the air conditioner does. The room is already cold, and outside is already hot. But what the air conditioner does, is it makes the cold even colder, and it makes the hot even hotter. It takes some Q and it goes in that direction. Right? It takes heat from the cold room, and puts it out into the hot air. And you're saying, this defies the second law of thermodynamics. You have just disproved it. You deserve a Nobel Prize. And I would say to you, you're forgetting one small fact. This air conditioner inside here, it has some type of a compressor, some type of an engine, that's actively doing this. It's putting in work to make this happen. And this engine right here-- I'll do it in magenta-- it's also expelling some more heat. So let's call that Q of the engine. So if you wanted to figure out the total entropy created for the universe, it would be the entropy of the cold room plus the change in entropy for outside-- I'll call it outside, maybe I'll call this, for the room. So you might say, OK. This change in entropy for the room, it's giving away heat-- let's see the room is roughly at a constant temperature for that one millisecond we're looking at it. It's giving away some Q at some temperature T1. And then-- so that's a minus. And then this the outside is gaining some heat at some temperature T2. And so you'll immediately say, hey. This number right here is a smaller number than this one. Because the denominator is higher. So if you just look at this, this would be negative entropy, and you'd say hey, this defies the second law of thermodynamics. But what you have to throw in here is another notion. You have to throw in here the notion that the outside is also getting this heat from the engine over the outside temperature. And this term, I can guarantee you-- I'm not giving you numbers right now-- will make this whole expression positive. This term will turn the total net entropy to the universe to be positive. Now let's think a little bit how about what entropy is and what entropy isn't in terms of words. So when you take an intro chemistry class, the teacher often says, entropy equals disorder. Which is not incorrect. It is disorder, but you have to be very careful what we mean by disorder. Because the very next example that's often given is that they'll say, look. A clean room-- let's say your bedroom is clean, and then it And they'll say, look. The universe became more disordered. The dirty room has more disorder than the clean room. And this is not a case of entropy increase. So this is not a good example. Why is that? Because clean and dirty are just states of the room. Remember, entropy is a macro state variable. It's something you use to describe a system where you're not in the mood to sit there and tell me what exactly every particle is doing. And this is a macro variable that actually tells me how much time would it take for me to tell you what every particle is doing. It actually tells you how many states there are, or how much information I would have to give you to tell you the exact state. Now, when you have a clean room and a dirty room, these are two different states of the same room. If the room has the same temperature, and it has the same number of molecules in it and everything, then they have the same entropy. So clean to dirty, it's not more entropy. Now, for example, I could have a dirty, cold room. And let's say I were to go into that room and, you know, I work really hard to clean it up. And by doing so, I add a lot of heat to the system, and my sweat molecules drop all over the place, and so there's just more stuff in that room, and it's all warmed up to me-- so to a hot, clean room with sweat in it-- so it's got more stuff in here that can be configured in more ways, and because it's hot, every molecule in the room can take on more states, right? Because the average kinetic energy is up, so they can kind of explore the spaces of how many kinetic energies it can have. There's more potential energies that each molecule can take on. This is actually an increase in entropy. From a dirty, cold room to a hot, clean room. And this actually goes well with what we know. I mean, when I go into room and I start cleaning it, I am in putting heat into the room. And the universe is becoming more-- I guess we could say it's the entropy is increasing. So where does the term disorder apply? Well, let's take a situation where I take a ball. I take a ball, and it falls to the ground. And then it hits the ground. And there should have been a question that you've been asking all the time, since the first law of thermodynamics. So the ball hits the ground, right? It got thrown up, it had some potential energy at the top, then that all gets turned into kinetic energy and it hits the ground, and then it stops. And so your obvious question is, what happened to all that energy, right? Law of conservation of energy. Where did all of it go? It had all that kinetic energy right before it hit the ground, then it stopped. It seems like it disappeared. But it didn't disappear. So when the ball was falling, it had a bunch of-- you know, everything had a little bit of heat. But let's say the ground was reasonably ordered. The ground molecules were vibrating with some kinetic energy and potential energies. And then our ball molecules were also vibrating a little bit. But most of their motion was downwards, right? Most of the ball molecules' motion was downwards. Now, when it hits the ground, what happens-- let me show you the interface of the ball. So the ball molecules at the front of the ball are going to look like that. And there's a bunch of them. It's a solid. It will maybe be some type of lattice. And then it hits the ground. And when it hits the ground-- so the ground is another solid like that-- All right, we're looking at the microstate. What's going to happen? These guys are going to rub up against these guys, and they're going to transfer their-- what was downward kinetic energy, and a very ordered downward kinetic energy-- they're going to transfer it to these ground particles. And they're going to bump into the ground particles. And so when this guy bumps into that guy, he might start moving in that direction. This guy will start oscillating in that direction, and go back and forth like that. That guy might bounce off of this guy, and go in that direction, and bump into that guy, and go into that direction. And then, because that guy bumped here, this guy bumps here, and because this guy bumps here, this guy bumps over there. And so what you have is, what was relatively ordered motion, especially from the ball's point of view, when it starts rubbing up against these molecules of the ground, it starts making the kinetic energy, or their movement, go in all sorts of random directions. Right? This guy's going to make this guy go like that, and that guy go like that. And so when the movement is no longer ordered, if I have a lot of molecules-- let me do it in a different color-- if I have a lot of molecules, and they're all moving in the exact same direction, then my micro state looks like my macro state. The whole thing moves in that direction. Now, if I have a bunch of molecules, and they're all moving in random directions, my ball as a whole will be stationary. I could have the exact same amount of kinetic energy at the molecular level, but they're all going to be bouncing into each other. And in this case, we described the kinetic energy as internal energy, or we describe it as temperature, where temperature is the average kinetic energy. So in this case, when we talk about, the world is becoming more disordered, you think about the order of maybe the velocities or the energies of the molecules. Before they were reasonably ordered, the molecules-- they might have been vibrating a little bit, but they were mainly going down in the ball. But when they bump into the ground, all of a sudden they start vibrating in random directions a little bit more. And they make the ground vibrate in more random directions. So it makes-- at the microstate-- everything became just that much more disordered. Now there's an interesting question here. There is some probability you might think-- Look, this ball came down and hit the ground. Why doesn't the ball just-- isn't there some probability that if I have a ground, that these molecules just rearrange themselves in just the right way to just hit these ball molecules in just the right way? There's some probability, just from the random movement, that at get some second, all the ground molecules just hit the ball molecules just right to send the ball back up. And the answer is yes. There's actually some infinitesimally small chance that that happens. That you could have a ball that's sitting on the ground-- and this is interesting-- could have a ball that's sitting on the ground, and while you're looking, you'll probably have to wait a few gazillion years for it to happen, if it happens at all-- it could just randomly pop up. And there's some random, very small chance that these molecules just randomly vibrate in just the right way to be ordered for a second, and then the ball will pop up. But the probability of this happening, relative to everything else, is essentially 0. So when people talk about order and disorder, the disorder is increasing, because now these molecules are going in more random directions, and they can take on more potential states. And we saw that here. And you know, on some level, entropy seems something kind of magical, but on some level, it seems relatively common sense. In that video-- I think was the last video-- I had a case where I had a bunch of molecules, and then I had this extra space here, and then I removed the wall. And we saw that these molecules will-- we know, there's always some modules that are bouncing off this wall before, because we probably had some pressure associated with it. And then as soon as we remove that wall, the molecule that would have bounced there just keeps going. There's nothing to stop it from there. In that direction, there's a lot of stuff. It could bump into other molecules, and it could bumping into these walls. But in this direction, the odds of it bumping into everything is, especially for these leading molecules, is So it's going to expand to fill the container. So that's kind of common sense. But the neat thing is that the second law of thermodynamics, as we saw in that video, also says that this will happen. That the molecules will all expand to fill the container. And that the odds of this happening are very low. That they all come back and go into a ordered state. Now there is some chance, just from the random movements once they fill, that they all just happen to come back here. But it's a very, very small probability. And even more-- and I want to make this very clear-- S is a macro state. We never talk about the entropy for an individual molecule. If we know what an individual molecule is doing, we shouldn't be worrying about entropy. We should be worrying about the system as a whole. So even if we're looking at the system, if we're not looking directly at the molecules, we won't even know that this actually happened. All we can do is look at the statistical properties of the molecules. How many molecules they are, what their temperature is, all their macro dynamics, the pressure, and say, you know what? A box that has these molecules has more state than a smaller box, than the box when we had the wall there. Even if, by chance, all of the molecules happened to be collecting over there, we wouldn't know that that happened, because we're not looking at the micro states. And that's a really important thing to consider. When someone says that a dirty room has a higher entropy than a clean room, they're looking at the micro states. And entropy essentially is a macro state variable. You could just say that a room has a certain amount of entropy. So entropy is associated with the room, and it's only useful when you really don't know exactly what's going on in the room. You just have a general sense of how much stuff there is in the room, what's the temperature of the room, what's the pressure in the room. Just the general macro properties. And then entropy will essentially tell us how many possible micro states that macro system can actually have. Or how much information-- and there's a notion of information entropy-- how much information would I have to give you to tell you what the exact micro state is of a system at that point in time. Hopefully you found this discussion a little bit useful, and it clears up some misconceptions about entropy, and gives you a little bit more intuition about what it actually is. See you in the next video." + }, + { + "Q": "what is 700.75 compared to 700.756", + "A": "700.75 is less because 700.756 has 6 thousandths more", + "video_name": "JJawhaMqaXg", + "transcript": "We're asked to order the following numbers from least to greatest, and I encourage you to pause this video and try to think of it on your own. Order these numbers from least to greatest. Well, let's work through it together now. So, none of these numbers have any places, any value to the left of the decimal point. They have no ones here. 0 ones, 0 ones, 0 ones, 0 ones, 0 ones. So let me then go to the next decimal place to the right. So I'm starting with the largest decimal places, and then I'm going to successively smaller decimal places. So I'll go to the tenths place. So this one right over here has 0 tenths. This has 0 tenths. This has 0 tenths. This one has 0 tenths. This one has 7 tenths, so this one actually has tenths. This has seven of them, so I'm going to leave this one here as the greatest. We're ordering from least to greatest. Now let's move to the hundredths place. So this one, this one, this one, this one all have 0 tenths. Let's look at the hundredths place. This has 7 hundredths. This has 7 hundredths. This one has 7 hundredths. This has no hundredths as well. This has 0 hundredths, so this one has neither tenths nor hundredths. So this one is going to be the smallest. This one has no tenths, no hundredths. This one actually has tenths. All of these-- these three in the middle-- have no tenths, but they have some hundredths, and they all have the same number of hundredths. 7 hundredths, 7 hundredths, 7 hundredths. So now let's look at the thousandths place. This one has 9 thousandths. This one has 0 thousandths. And this one has 0 thousandths as well. So out of these three, this one is the largest, because this one actually had some thousandths out of these three. Now let's go look at-- we have to pick between these two. Both of these have no tenths. Both of them have exactly 7 hundredths. Both of them have no thousandths, but this one has 9 ten thousandths while this one has no ten thousandths. So this one is less than that one. And now I'm done. I think I have ordered the numbers from least to greatest. And the key here is go to the place value that's most significant-- I guess you could say-- that has the most value. So that was the ones place. Compare them on that. Then go to successively smaller place values. Keep going to the right, and keep comparing them, and then you'll be able to order them." + }, + { + "Q": "so if you are at x and you take away 1/3x you have 2/3x?", + "A": "yes 1x - 1/3 x = 3/3 x - 1/3 x = 2/3 x", + "video_name": "wo7DSaPP8hQ", + "transcript": "We're told that for the past few months, Old Maple Farms has grown about 1,000 more apples than their chief rival in the region, River Orchards. Due to cold weather this year, the harvests at both farms were down by about a third. However, both farms made up for some of the shortfall by purchasing equal quantities of apples from farms in neighboring states. What can you say about the number of apples available at each farm? Does one farm have more than the other, or do they have the same amount? How do I know? So let's define some variables here. Let's let M be equal to number of apples at Maple Farms. And then who's the other guy? River Orchards. So let's let R be equal to the number of apples at River Orchards. So this first sentence, they say-- let me do this in a different color-- they say for the past few years, Old Maple Farms has grown about 1,000 more apples than their chief rival in the region, River Orchards. So we could say, hey, Maple is approximately Old River, or M is approximately River plus 1,000. Or since we don't know the exact amount-- it says it's about 1,000 more, so we don't know it's exactly 1,000 more-- we can just say that in a normal year, Old Maple Farms, which we denote by M, has a larger amount of apples than River Orchard. So in a normal year, M is greater than R, right? It has about 1,000 more apples than Old Maple Farms. Now, they say due to cold weather this year-- so let's talk about this year now-- the harvests at both farms were down about a third. So this isn't a normal year. Let's talk about what's going to happen this year. In this year, each of these characters are going to be down by 1/3. Now if I go down by 1/3, that's the same thing as being 2/3 of what I was before. Let me do an example. If I'm at x, and I take away 1/3x, I'm left with 2/3x. So going down by 1/3 is the same thing as multiplying the quantity by 2/3. So if we multiply each of these quantities by 2/3, we can still hold this inequality, because we're doing the same thing to both sides of this inequality, and we're multiplying by a positive number. If we were multiplying by a negative number, we would have to swap the inequality. So we can multiply both sides of this by 2/3. So 2/3 of M is still going to be greater than 2/3 of R. And you could even draw that in a number line if you like. Let's do this in a number line. This all might be a little intuitive for you, and if it is, I apologize, but if it's not, it never hurts. So that's 0 on our number line. So in a normal year, M is has 1,000 more than R. So in a normal year, M might be over here and maybe R is over here. I don't know, let's say R is over there. Now, if we take 2/3 of M, that's going to stick us some place around, oh, I don't know, 2/3 is right about there. So this is M-- let me write this-- this is 2/3 M. And what's 2/3 of R going to be? Well, if you take 2/3 of this, you get to right about there, that is 2/3R. So you can see, 2/3R is still less than 2/3M, or 2/3M is greater than 2/3R. Now, they say both farms made up for some of the shortfall by purchasing equal quantities of apples from farms in neighboring states. So let's let a be equal to the quantity of apples both purchased. So they're telling us that they both purchased the same amount. So we could add a to both sides of this equation and it will not change the inequality. As long as you add or subtract the same value to both sides, it will not change the inequality. So if you add a to both sides, you have a plus 2/3M is a greater than 2/3R plus a. This is the amount that Old Maple Farms has after purchasing the apples, and this is the amount that River Orchards has. So after everything is said and done, Old Maple Farms still has more apples, and you can see that here. Maple Farms, a normal year, this year they only had 2/3 of the production, but then they purchased a apples. So let's say a is about, let's say that a is that many apples, so they got back to their normal amount. So let's say they got back to their normal amount. So that's how many apples they purchased, so he got back to M. Now, if R, if River Orchards also purchased a apples, that same distance, a, if you go along here gets you to right about over there. So once again, this is-- let me do it a little bit different, because I don't like it overlapping, so let me So let's say this guy, M-- I keep forgetting their names-- Old Maple Farms purchases a apples, gets them that far. So that's a apples. But River Orchards also purchases a apples, so let's add that same amount. I'm just going to copy and paste it so it's the exact same amount. So River Orchards also purchases a, so it also purchases that same amount. So when all is said and done, River Orchards is going to have this many apples in the year that they had less production but they went and purchased it. So this, right here, is-- this value right here is 2/3R plus a. That's what River Orchards has. And then Old Maple Farms has this value right here, which is 2/3M plus a. Everything said and done, Old Maple Farms still has more apples." + }, + { + "Q": "how do you find the sin cos or tan of an angle without a triangle or calculator?", + "A": "(1) If the angle is 30, 45, or 60 degrees, then the ratios can be figured out exactly. (The special right triangles .) (2) You can use a Taylor series or the CORDIC algorithm to compute a specific value. Neither is something you want to attempt manually. (3) Back in the days... we used printed tables to look them up. Presumably those tables were computed using one of the techniques in (2).", + "video_name": "l5VbdqRjTXc", + "transcript": "We're asked to solve the right triangle shown below. Give the links to the nearest tenth. So when they say solve the right triangle, we can assume that they're saying, hey figure out the lengths of all the sides. So whatever a is equal to, whatever b is equal to. And also what are all the angles of the right triangle? They've given two of them. We might have to figure out this third right over here. So there's multiple ways to tackle this, but we'll just try to tackle side XW first, try to figure out what a is. And I'll give you a hint. You can use a calculator, and using a calculator, you can use your trigonometric functions that we've looked at a good bit now. So I'll give you a few seconds to think about how to figure out what a is. Well, what do we know? We know this angle y right over here. We know the side adjacent to angle y. And length a, this is the side that's the length of the side that is opposite to angle Y. So what trigonometric ratio deals with the opposite and the adjacent? So if we're looking at angle Y, relative to angle Y, this is the opposite. And this right over here is the adjacent. Well if we don't remember, we can go back to SohCahToa. Sine deals with opposite and hypotenuse. Cosine deals with adjacent and hypotenuse. Tangent deals with opposite over adjacent. So we can say that the tangent of 65 degrees, of that angle of 65 degrees, is equal to the opposite, the length of the opposite side, which we know has length a over the length of the adjacent side, which they gave us in the diagram, which has length five. And you might say, how do I figure out a? Well we can use our calculator to evaluate what the tangent of 65 degrees are. And then we can solve for a. And actually if we just want to get the expression explicitly solving for a, we could just multiply both sides of this equation times 5. So let's do that. 5 times, times 5. These cancel out, and we are left with, if we flip the equal around, we're left with a is equal to 5 times the tangent of 65 degrees. So now we can get our calculator out and figure out what this is to the nearest tenth. That's my handy TI-85 out and I have 5 times the tangent-- I didn't need to press that second right over there, just a regular tangent-- of 65 degrees. And I will get, if I round to the nearest tenth like they ask me to, I get 10.7. So a is approximately equal to 10.7. I say approximately because I rounded it down. This is not the exact number. But a is equal to 10.7. So we now know that this has length 10.7, approximately. There are several ways that we can try to tackle b. And I'll let you pick the way you want to. But then I'll just do it the way I would like to. So my next question to you is, what is the length of the side YW? Or what is the value of b? Well there are several ways to do it. This is the hypotenuse. So we could use trigonometric functions that deal with adjacent over hypotenuse or opposite over hypotenuse. Or we could just use the Pythagorean theorem. We know two sides of a right triangle. We can come up with the third side. I will go with using trigonometric ratios since that's what we've been working on a good bit. So this length of b, that's the length of the hypotenuse. So this side WY is the hypotenuse. And so what trigonometric ratios-- or we can decide what we want to use. We could use opposite and hypotenuse. We could use adjacent and hypotenuse. Since we know that XY is exactly 5 and we don't have to deal with this approximation, let's use that side. So what trigonometric ratios deal with adjacent and hypotenuse? Well we see from SohCahToa cosine deals with adjacent over hypotenuse. So we could say that the cosine of 65 degrees is equal to the length of the adjacent side, which is 5 over the length of the hypotenuse, which has a length of b. And then we can try to solve for b. You multiply both sides times b, you're left with b times cosine of 65 degrees is equal to 5. And then to solve for b, you could divide both sides by cosine of 65 degrees. This is just a number here. So we're just dividing-- we have to figure it out what our calculator, but this is just going to evaluate to some number. So we can divide both sides by that, by cosine of 65 degrees. And we're left with b is equal to 5 over the cosine of 65 degrees. So let us now use our calculator to figure out the length of b. Length of b is 5 divided by cosine of 65 degrees. And I get, if I round to the nearest tenth, 11.8. So b is approximately equal to, rounded to the nearest tenth, 11.8. So b is equal to 11.8. And then we're almost done solving this right triangle. And you could have figured this out using the Pythagorean theorem as well, saying that 5 squared plus 10.7 squared should be equal to b squared. And hopefully you would get the exact same answer. And the last thing we have to figure out is the measure of angle W right over here. So I'll give you a few seconds to think about what the measure of angle W is. Well here we just have to remember that the sum of the angles of a triangle add up to 180 degrees. So angle w plus 65 degrees, that's this angle right up here, plus the right angle, this is a right triangle, they're going to add up to 180 degrees. So all we need to do is-- well we can simplify the left-hand side right over here. 65 plus 90 is 155. So angle W plus 155 degrees is equal to 180 degrees. And then we get angle W-- if we subtract 155 from both sides-- angle W is equal to 25 degrees. And we are done solving the right triangle shown below." + }, + { + "Q": "When a boy jumps from bus there is a danger for him to fall\nA)in the direction of motion\nB)towards the moving bus\nC)opposite in the direction of motion\nD)away from the bus", + "A": "At the moment that he jumps from the bus, he would have velocity pushing him forward in the direction of the motion of the bus at the same magnitude of the velocity of the bus, and (assuming he jumps straight forward), velocity straight outwards perpendicular to the bus. This means that he would fall in both the direction of the motion and away from the bus. Hope this helps, and please correct if im wrong, as i often am :)", + "video_name": "jmSWImPs6fQ", + "transcript": "- [Instructor] Let's talk about how to handle a horizontally launched projectile problem. These, technically speaking, if you already know how to do projectile problems, there is nothing new, except that there's one aspect of these problems that people get stumped by all of the time. So I'm gonna show you what that is in a minute so that you don't fall into the same trap. What we mean by a horizontally launched projectile is any object that gets launched in a completely horizontal velocity to start with. So if something is launched off of a cliff, let's say, in this straight horizontal direction with no vertical component to start with, then it's a horizontally launched projectile. What could that be? I mean a boring example, it's just a ball rolling off of a table. If you just roll the ball off of the table, then the velocity the ball has to start off with, if the table's flat and horizontal, the velocity of the ball initially would just be horizontal. So if the initial velocity of the object for a projectile is completely horizontal, then that object is a horizontally launched projectile. A more exciting example. People do crazy stuff. Let's say this person is gonna cliff dive or base jump, and they're gonna be like \"whoa, let's do this.\" We're gonna do this, they're pumped up. They're gonna run but they don't jump off the cliff, they just run straight off of the cliff 'cause they're kind of nervous. Let's say they run off of this cliff with five meters per second of initial velocity, straight off the cliff. And let's say they're completely crazy, let's say this cliff is 30 meters tall. So that's like over 90 feet. That is kind of crazy. So 30 meters tall, they launch, they fly through the air, there's water down here, so they initially went this way, and they start to fall down, and they do something like pschhh, and then they splash in the water, hopefully they don't hit any boats or fish down here. That fish already looks like he got hit. He or she. Alright, fish over here, person splashed into the water. We want to know, here's the question you might get asked: how far did this person go horizontally before striking the water? This is a classic problem, gets asked all the time. And if you were a cliff diver, I mean don't try this at home, but if you were a professional cliff diver you might want to know for this cliff high and this speed how fast do I have to run in order to avoid maybe the rocky shore right here that you might want to avoid. Maybe there's this nasty craggy cliff bottom here that you can't fall on. So how fast would I have to run in order to make it past that? Alright, so conceptually what's happening here, the same thing that happens for any projectile problem, the horizontal direction is happening independently of the vertical direction. And what I mean by that is that the horizontal velocity evolves independent to the vertical velocity. Let me get the velocity this color. So say the vertical velocity, or the vertical direction is pink, horizontal direction is green. This vertical velocity is gonna be changing but this horizontal velocity is just gonna remain the same. These do not influence each other. In other words, this horizontal velocity started at five, the person's always gonna have five meters per second of horizontal velocity. So this horizontal velocity is always gonna be five meters per second. The whole trip, assuming this person really is a freely flying projectile, assuming that there is no jet pack to propel them forward and no air resistance. This person's always gonna have five meters per second of horizontal velocity up onto the point right when they splash in the water, and then at that point there's forces from the water that influence this acceleration in various ways that we're not gonna consider. How about vertically? Vertically this person starts with no initial velocity. So this person just ran horizontally straight off the cliff and then they start to gain velocity. So they're gonna gain vertical velocity downward and maybe more vertical velocity because gravity keeps pulling, and then even more, this might go off the screen but it's gonna be really big. So a lot of vertical velocity, this should keep getting bigger and bigger and bigger because gravity's influencing this vertical direction but not the horizontal direction. So how do we solve this with math? Let's write down what we know. What we know is that horizontally this person started off with an initial velocity. V initial in the x, I could have written i for initial, but I wrote zero for v naught in the x, it still means initial velocity is five meters per second. And we don't know anything else in the x direction. You might think 30 meters is the displacement in the x direction, but that's a vertical distance. This is not telling us anything about this horizontal distance. This horizontal distance or displacement is what we want to know. This horizontal displacement in the x direction, that's what we want to solve for, so we're gonna declare our ignorance, write that here. We don't know how to find it but we want to know that we do want to find so I'm gonna write it there. How about in the y direction, what do we know? We know that the, alright, now we're gonna use this 30. You might want to say that delta y is positive 30 but you would be wrong, and the reason is, this person fell downward 30 meters. Think about it. They started at the top of the cliff, ended at the bottom of the cliff. It means this person is going to end up below where they started, 30 meters below where they started. So this has to be negative 30 meters for the displacement, assuming you're treating downward as negative which is typically the convention shows that downward is negative and leftward is negative. So if you choose downward as negative, this has to be a negative displacement. What else do we know vertically? Well, for a freely flying object we know that the acceleration vertically is always gonna be negative 9.8 meters per second squared, assuming downward is negative. Now, here's the point where people get stumped, and here's the part where people make a mistake. They want to say that the initial velocity in the y direction is five meters per second. I mean people are just dying to stick these five meters per second into here because that's the velocity that you were given. But this was a horizontal velocity. That's why this is called horizontally launched projectile motion, not vertically launched projectile motion. So think about it. The initial velocity in the vertical direction here was zero, there was no initial vertical velocity. This person was not launched vertically up or vertically down, this person was just launched straight horizontally, and so the initial velocity in the vertical direction is just zero. People don't like that. They're like \"hold on a minute.\" They're like, this person is gonna start gaining, alright, this person is gonna start gaining velocity right when they leave the cliff, this starts getting bigger and bigger and bigger in the downward direction. But that's after you leave the cliff. We're talking about right as you leave the cliff. That moment you left the cliff there was only horizontal velocity, which means you started with no initial vertical velocity. So this is the part people get confused by because this is not given to you explicitly in the problem. The problem won't say, \"Find the distance for a cliff diver \"assuming the initial velocity in the y direction was zero.\" Now, they're just gonna say, \"A cliff diver ran horizontally off of a cliff. \"Find this stuff.\" And you're just gonna have to know that okay, if I run off of a cliff horizontally or something gets shot horizontally, that means there is no vertical velocity to start with, I'm gonna have to plug this initial velocity in the y direction as zero. So that's the trick. Don't fall for it now you know how to deal with it. So we want to solve for displacement in the x direction, but how many variables we know in the y direction? I mean we know all of this. But we can't use this to solve directly for the displacement in the x direction. We need to use this to solve for the time because the time is gonna be the same for the x direction and the y direction. So I find the time I can plug back in over to there, because think about it, the time it takes for this trip is gonna be the time it takes for this trip. It doesn't matter whether I call it the x direction or y direction, time is the same for both directions. In other words, the time it takes for this displacement of negative 30 is gonna be the time it takes for this displacement of whatever this is that we're gonna find. So let's solve for the time. Now, how will we do that? Think about it. We know the displacement, we know the acceleration, we know the initial velocity, and we know the time. But we don't know the final velocity and we're not asked to find the final velocity, we don't want to know it. So let's use a formula that doesn't involve the final velocity and that would look like this. So if we use delta y equals v initial in the y direction times time plus one half acceleration in the y direction times time squared. Alright, now we can plug in values. My displacement in the y direction is negative 30. My initial velocity in the y direction is zero. This is where it would happen, this is where the mistake would happen, people just really want to plug that five in over here. But don't do it, it's a trap. So, zero times t is just zero so that whole term is zero. Plus one half, the acceleration is negative 9.8 meters per second squared. And then times t squared, alright, now I can solve for t. I'm gonna solve for t, and then I'd have to take the square root of both sides because it's t squared, and what would I get? I'd have to multiply both sides by two. So I get negative 30 meters times two, and then I have to divide both sides by negative 9.8 meters per second squared, equals, notice if you would have forgotten this negative up here for negative 30, you come down here, this would be a positive up top. You'd have a negative on the bottom. You'd have to plug this in, you'd have to try to take the square root of a negative number. Your calculator would have been all like, \"I don't know what that means,\" and you're gonna be like, \"Er, am I stuck?\" So you'd start coming back here probably and be like, \"Let's just make stuff positive and see if that works.\" It would work because look at these negatives canceled but it's best to just know what you're talking about in the first place. So be careful: plug in your negatives and things will work out alright. So if you solve this you get that the time it took is 2.47 seconds. It's actually a long time. It might seem like you're falling for a long time sometimes when you're like jumping off of a table, jumping off of a trampoline, but it's usually like a fraction of a second. This is actually a long time, two and a half seconds of free fall's a long time. So we could take this, that's how long it took to displace by 30 meters vertically, but that's gonna be how long it took to displace this horizontal direction. We can use the same formula. We can say that well, if delta x equals v initial in the x direction, I'm just using the same formula but in the x direction, plus one half ax t squared. So the same formula as this just in the x direction. Delta x is just dx, we already gave that a name, so let's just call this dx. So I'm gonna scooch this equation over here. Dx is delta x, that equals the initial velocity in the x direction, that's five. Alright, this is really five. In the x direction the initial velocity really was five meters per second. How about the initial time? Oh sorry, the time, there is no initial time. The time here was 2.47 seconds. This was the time interval. The time between when the person jumped, or ran off the cliff, and when the person splashed in the water was 2.4, let me erase this, 2.47 seconds. So 2.47 seconds, and this comes over here. How about this ax? This ax is zero. Remember there's nothing compelling this person to start accelerating in x direction. If they've got no jet pack, there is no air resistance, there is no reason this person is gonna accelerate horizontally, they maintain the same velocity the whole way. So what do we get? If we solve this for dx, we'd get that dx is about 12.4, I believe. Let's see, I calculated this. 12.4-ish meters. Okay, so if these rocks down here extend more than 12 meters, you definitely don't want to do this. I mean if it's even close you probably wouldn't want do this. In fact, just for safety don't try this at home, leave this to professional cliff divers. I'm just saying if you were one and you wanted to calculate how far you'd make it, this is how you would do it. So, long story short, the way you do this problem and the mistakes you would want to avoid are: make sure you're plugging your negative displacement because you fell downward, but the big one is make sure you know that the initial vertical velocity is zero because there is only horizontal velocity to start with. That's not gonna be given explicitly, you're just gonna have to provide that on your own and your own knowledge of physics." + }, + { + "Q": "can someone please help me solve y-3x=2\ny=-2x+7", + "A": "So I you put y=-2x+7 in the first eq. you get -2x+7-3x=2 put the x s together and the # s together you get: -2x-3x = 2-7 -5x=-5 then divide by -5 you get x=1 Check your answer by plugging in the 1 for x and y= 5", + "video_name": "2VeqrZ_PMiY", + "transcript": "Use substitution to solve for x and y. And they give us a system of equations here. y is equal to negative 5x plus 8 and 10x plus 2y is equal to negative 2. So they've set it up for us pretty well. They already have y explicitly solved for up here. So they tell us, this first constraint tells us that y must be equal to negative 5x plus 8. So when we go to the second constraint here, every time we see a y, we say, well, the first constraint tells us that y must be equal to negative 5x plus 8. So everywhere we see a y, we can substitute it with negative 5x plus 8. Because that's what the first constraint tells us. y is equal to that. I don't want to be repetitive, but I really want you to internalize that's all it's saying. y is that. So every time we see a y in the second constraint, we can substitute it with that. So let's do it. So the second equation over here is 10x plus 2. And instead of writing a y there, and I've said it multiple times already, we can write a negative 5x plus 8. The first constraint tells us that's what y is. So negative 5x plus 8 is equal to negative 2. Now, we have one equation with one unknown. We can just solve for x. We have 10x plus. So we can multiply it. We can distribute this 2 onto both of these terms. So we have 2 times negative 5x is negative 10x. And then 2 times 8 is 16. So plus 16 is equal to negative 2. Now we have 10x minus 10x. Those guys cancel out. 10x minus 10x is equal to 0. So these guys cancel out. And we're just left with 16 equals negative 2, which is crazy. We know that 16 does not equal negative 2. This is an inconsistent result. And that's because these two lines actually don't intersect. And we could see that by actually graphing these lines. Whenever you get something like some number equalling some other number that they're clearly not equal to, that means it's an inconsistent result, It's an inconsistent system, and that these lines actually don't intersect. So let me just graph these just to make it clear. This first equation is already in slope y-intercept form. So it looks something like this. That's our x-axis. This is our y-axis. And it's negative 5x plus 8, so 1, 2, 3, 4, 5, 6, 7, 8. And then it has a very steep downward slope. Every time you move forward 1, you have to go down 5. So it looks something like that. That's this first equation right over there. The second equation, let me rewrite it in slope y-intercept form. So it's 10x plus 2y is equal to negative 2. Let's subtract 10x from both sides. You get 2y is equal to negative 10x minus 2. Let's divide both sides by 2. You get y is equal to negative 5x, negative 5x minus 1. So it's y-intercept is negative 1. It's right over there. And it has the same slope as this first line. So it looks like this. It's parallel. It's just shifted down a bit. So it just looks like that. So they're parallel lines. They have the same slope, different y-intercepts. We get an inconsistent result. They don't intersect. And the telltale sign of that, when you're doing it algebraically, is you get something wacky like this. This is why it's called inconsistent. It's not consistent for 16 to be equal to negative 2. These don't intersect. There's no solution to both of these constraints, no x and y that satisfies both of them." + }, + { + "Q": "I have literally been looking for over a week on how to do this through the quadratic formula and completing the square but I just don't get it.. I can't logicize this no matter how hard i try...", + "A": "If you just substitute the values into the quadratic formula then you will get the solutions for the equation(what x will equal to make the equation equal zero) is you do it on a calculator, then the answer may be an irrational number because the calculator tries to put the roots into decimal notation so it may be easier to do it by hand", + "video_name": "TV5kDqiJ1Os", + "transcript": "We're asked to complete the square to solve 4x squared plus 40x minus 300 is equal to 0. So let me just rewrite it. So 4x squared plus 40x minus 300 is equal to 0. So just as a first step here, I don't like having this 4 out front as a coefficient on the x squared term. I'd prefer if that was a 1. So let's just divide both sides of this equation by 4. So let's just divide everything by 4. So this divided by 4, this divided by 4, that divided by 4, and the 0 divided by 4. Just dividing both sides by 4. So this will simplify to x squared plus 10x. And I can obviously do that, because as long as whatever I do to the left hand side, I also do the right hand side, that will make the equality continue to be valid. So that's why I can do that. So 40 divided by 4 is 10x. And then 300 divided by 4 is what? That is 75. Let me verify that. 4 goes into 30 seven times. 7 times 4 is 28. You subtract, you get a remainder of 2. Bring down the 0. 4 goes into 20 five times. 5 times 4 is 20. Subtract zero. So it goes 75 times. This is minus 75 is equal to 0. And right when you look at this, just the way it's written, you might try to factor this in some way. But it's pretty clear this is not a complete square, or this is not a perfect square trinomial. Because if you look at this term right here, this 10, half of this 10 is 5. And 5 squared is not 75. So this is not a perfect square. So what we want to do is somehow turn whatever we have on the left hand side into a perfect square. And I'm going to start out by kind of getting this 75 out You'll sometimes see it where people leave the 75 on the left hand side. I'm going to put on the right hand side just so it kind of clears things up a little bit. So let's add 75 to both sides to get rid of the 75 from the left hand side of the equation. And so we get x squared plus 10x, and then negative 75 plus 75. Those guys cancel out. And I'm going to leave some space here, because we're going to add something here to complete the square that is equal to 75. So all I did is add 75 to both sides of this equation. Now, in this step, this is really the meat of completing the square. I want to add something to both sides of this equation. I can't add to only one side of the equation. So I want to add something to both sides of this equation so that this left hand side becomes a perfect square. And the way we can do that, and saw this in the last video where we constructed a perfect square trinomial, is that this last term-- or I should say, what we see on the left hand side, not the last term, this expression on the left hand side, it will be a perfect square if we have a constant term that is the square of half of the coefficient on the first degree So the coefficient here is 10. Half of 10 is 5. 5 squared is 25. So I'm going to add 25 to the left hand side. And of course, in order to maintain the equality, anything I do the left hand side, I also have to do to the right hand side. And now we see that this is a perfect square. We say, hey, what two numbers if I add them I get 10 and when I multiply them I get 25? Well, that's 5 and 5. So when we factor this, what we see on the left hand side simplifies to, this is x plus 5 squared. x plus 5 times x plus 5. And you can look at the videos on factoring if you find that confusing. Or you could look at the last video on constructing perfect square trinomials. I encourage you to square this and see that you get exactly this. And this will be equal to 75 plus 25, which is equal to 100. And so now we're saying that something squared is equal to 100. So really, this is something right over here-- if I say something squared is equal to 100, that means that that something is one of the square roots of 100. And we know that 100 has two square roots. It has positive 10 and it has negative 10. So we could say that x plus 5, the something that we were squaring, that must be one of the square roots of 100. So that must be equal to the plus or minus square root of 100, or plus or minus 10. Or we could separate it out. We could say that x plus 5 is equal to 10, or x plus 5 is equal to negative 10. On this side right here, I can just subtract 5 from both sides of this equation and I would get-- I'll just write it out. Subtracting 5 from both sides, I get x is equal to 5. And over here, I could subtract 5 from both sides again-- I subtracted 5 in both cases-- subtract 5 again and I can get x is equal to negative 15. So those are my two solutions that I got to solve this equation. We can verify that they actually work, and I'll do that in blue. So let's try with 5. I'll just do one of them. I'll leave the other one for you. I'll leave the other one for you to verify that it works. So 4 times x squared. So 4 times 25 plus 40 times 5 minus 300 needs to be equal to 0. 4 times 25 is 100. 40 times 5 is 200. We're going to subtract that 300. 100 plus 200 minus 300, that definitely equals 0. So x equals 5 worked. And I think you'll find that x equals negative 15 will also work when you substitute it into this right over here." + }, + { + "Q": "Who is actually learning things from this video? I know I am.", + "A": "It depends on your skill level, but yes i am sure lots of people are learning from this. Khan Academy has very good videos :)", + "video_name": "Ye13MIPv6n0", + "transcript": "So we have our scale again. And we've got some masses on the left hand side and some masses on the right hand side. And we see that our scale is balanced. We have the same total mass on the left hand side that we have on the right hand side. Instead of labeling the mystery masses as question mark, I've labeled them all x. And since they all have an x on it, we know that each of these have the same mass. But what I'm curious about is, what is that mass? What is the mass of each of these mystery masses, I guess we could say? And so I'll let think about that for a second. How would you figure out what this x value actually is? How many kilograms is the mass of each of these things? What could you do to either one or both sides of this scale? I'll give you a few seconds to think about that. So you might be tempted to say, well if I could end up with just one mystery mass on the left hand side, and if I keep my scale balanced, then that thing's going to be equal to whatever I have on the right hand side. And that part would actually be a true statement. But then to get only one of these mystery masses on the left hand side, you might say, well why don't I just remove two of them? You might just say, well why don't I just remove-- let me do it a good color for removing-- why don't I just remove that one and that one? And then I'll just be left with that right over there. But if you just removed these two, then the left hand side is going to become lighter or it's going to have a lower mass than the right hand side. So it's going to move up and the right hand side is going to move down. And then you might say, OK, I understand. Whatever I have to do to the left hand side, I have to do to the right hand side in order to keep my scale balanced. So you might say, well why don't I remove two of these mystery masses from the right hand side? But that's a problem too because you don't know what this mystery mass is. You could try to remove two from this, but how many of these blocks represent a mystery mass? We actually don't know. But you might then say, well let's see, I've got three of these things here. If I essentially multiply what I have here by 1/3 or if I only leave a 1/3 of the stuff here, and if I only leave a 1/3 of the stuff here, then the scale should be balanced. If this has the total mass as this, then 1/3 of this total mass is going to be the same thing as 1/3 of that total mass. So let's just keep only 1/3 of this here. So that's the equivalent to multiplying by 1/3. So if we're only going to keep 1/3 there, we're going to be left with only one of the masses. And if we only keep 1/3 here, let's see, we have one, two, three, four, five, six, seven, eight, nine masses. If we multiply this by 1/3, or if we only keep 1/3 of it there, 1/3 times 9 is 3. So we're going to remove these . And so we have 1/3 of what we originally had on the right hand side and 1/3 of what we originally had on the left hand side. And they will be balanced because we took 1/3 of the same total masses. And so what you're left with is just one of these mystery masses, this x thing right over here, whatever x might be. And you have three kilograms on the right hand side. And so you can make the conclusion, and the whole time you kept this thing balanced, that x is equal to 3." + }, + { + "Q": "where are some easy practice questions", + "A": "Go to the left tab and you ll find a practice skill.", + "video_name": "ory05j2jgBM", + "transcript": "- [Voiceover] What I wanna do in this video is compare the fractions 3/4 and 4/5, and I wanna do this visually. So what I'm gonna do is I'm gonna have two copies of the same whole, so let me just draw that, but I'm gonna divide the first one, so this is one whole right over here, this rectangle, when we draw the whole thing. So this is a whole, and right below that, we have the same whole. We have a rectangle of exactly the same size. Now you might notice that I've divided them into a different number of equal sections. In the top one, I've divided it into four equal sections because I am concerned with fourths so I've divided this top whole into fourths and I've divided this bottom whole, or this bottom bar or this bottom rectangle, into fifths, or five equal sections. So let's think about what 3/4 represent. So that's gonna be one of the fourths, right over here, two of the fourths, and then three of the fourths. And what is 4/5 going to be? Well, 4/5 is going to be one fifth, two fifths, three fifths, and four fifths. So when you look at them visually, remember, we're taking fractions of the same whole. This is 3/4 of that rectangle, this is 4/5 of a same-sized rectangle. It wouldn't make any sense if you're doing it for different shapes or different sized rectangles. We just divided them into different sections and you see that if you have four of the fifths, that that is going to be more than three of the fourths, and so 4/5 is greater than 3/4 or you could say 3/4 is less than 4/5, or any way you wanna think about it. The symbol you wanna use always opens to the larger number. 4/5 is larger than 3/4, so the large end of our symbol is facing the 4/5, so we would say 3/4 is less than 4/5." + }, + { + "Q": "what omega \"w\" '\"signifies\" in simple harmonic motion", + "A": "it represents angular velocity", + "video_name": "oqBHBO8cqLI", + "transcript": "And if you were covering your eyes because you didn't want to see calculus, I think you can open your eyes again. There shouldn't be any significant displays of calculus in this video. But just to review what we went over, we just said, OK if we have a spring-- and I drew it vertically this time-- but pretend like there's no gravity, or maybe pretend like we're viewing-- we're looking at the top of a table, because we don't want to look at the effect of We just want to look at a spring by itself. So this could be in deep space, or something else. But we're not thinking about gravity. But I drew it vertically just so that we can get more intuition for this curve. Well, we started off saying is if I have a spring and 0-- x equals 0 is kind of the natural resting point of the spring, if I just let this mass-- if I didn't pull on the spring at all. But I have a mass attached to the spring, and if I were to stretch the spring to point A, we said, well what happens? Well, it starts with very little velocity, but there's a restorative force, that's going to be pulling it back towards this position. So that force will accelerate the mass, accelerate the mass, accelerate the mass, until it gets right here. And then it'll have a lot of velocity here, but then it'll start decelerating. And then it'll decelerate, decelerate, decelerate. Its velocity will stop, and it'll come back up. And if we drew this as a function of time, this is what happens. It starts moving very slowly, accelerates. At this point, at x equals 0, it has its maximum speed. So the rate of change of velocity-- or the rate of change of position is fastest. And we can see the slope is very fast right here. And then, we start slowing down again, slowing down, until we get back to the spot of A. And then we keep going up and down, up and down, like that. And we showed that actually, the equation for the mass's position as a function of time is x of t-- and we used a little bit of differential equations to prove it. But this equation-- not that I recommend that you memorize anything-- but this is a pretty useful equation to memorize. Because you can use it to pretty much figure out anything-- about the position, or of the mass at any given time, or the frequency of this oscillatory motion, or anything else. Even the velocity, if you know a little bit of calculus, you can figure out the velocity at anytime, of the object. And that's pretty neat. So what can we do now? Well, let's try to figure out the period of this oscillating system. And just so you know-- I know I put the label harmonic motion on all of these-- this is simple harmonic motion. Simple harmonic motion is something that can be described by a trigonometric function like this. And it just oscillates back and forth, back and forth. And so, what we're doing is harmonic motion. And now, let's figure out what this period is. Remember we said that after T seconds, it gets back to its original position, and then after another T seconds, it gets back to its original position. Let's figure out with this T is. And that's essentially its period, right? What's the period of a function? It's how long it takes to get back to your starting point. Or how long it takes for the whole cycle to happen once. So what is this T? So let me ask you a question. What are all the points-- that if this is a cosine function, right? What are all of the points at which cosine is equal to 1? Or this function would be equal to A, right? Because whenever cosine is equal to 1, this whole function is equal to A. And it's these points. Well cosine is equal to 1 when-- so, theta-- let's say, when is cosine of theta equal to 1? So, at what angles is this true? Well it's true at theta is equal to 0, right? Cosine of 0 is 1. Cosine of 2 pi is also 1, right? We could just keep going around that unit circle. You should watch the unit circle video if this makes no sense to you. Or the graphing trig functions. It's also true at 4 pi. Really, any multiple of 2 pi, this is true. Right? Cosine of that angle is equal to 1. So the same thing is true. This function, x of t, is equal to A at what points? x of t is equal to A whenever this expression-- within the cosines-- whenever this expression is equal to 0, 2 pi, 4 pi, et cetera. And this first time that it cycles, right, from 0 to 2 pi-- from 0 to T, that'll be at 2 pi, right? So this whole expression will equal A, when k-- and that's these points, right? That's when this function is equal to A. It'll happen again over here someplace. When this little internal expression is equal to 2 pi, or really any multiple of 2 pi. So we could say, so x of t is equal to A when the square root of k over m times t, is equal to 2 pi. Or another way of thinking about it, is let's multiply both sides of this equation times the inverse of the square root of k over m. And you get, t is equal to 2 pi times the square root-- and it's going to be the inverse of this, right? Of m over k. And there we have the period of this function. This is going to be equal to 2 pi times the square root of m over k. So if someone tells you, well I have a spring that I'm going to pull from some-- I'm going to stretch it, or compress it a little bit, then I let go-- what is the period? How long does it take for the spring to go back to its It'll keep doing that, as we have no friction, or no gravity, or any air resistance, or Air resistance really is just a form of friction. You could immediately-- if you memorize this formula, although you should know where it comes from-- you could immediately say, well I know how long the period is. It's 2 pi times m over k. That's how long it's going to take the spring to get back-- to complete one cycle. And then what about the frequency? If you wanted to know cycles per second, well that's just the inverse of the period, right? So if I wanted to know the frequency, that equals 1 over the period, right? Period is given in seconds per cycle. So frequency is cycles per second, and this is seconds per cycle. So frequency is just going to be 1 over this. Which is 1 over 2 pi times the square root of k over m. That's the frequency. But I have always had trouble memorizing this, and this. k over m, and m over k, and all of that. All you have to really memorize is this. And even that, you might even have an intuition as to why it's true. You can even go to the differential equations if you want to reprove it to yourself. Because if you have this, you really can answer any question about the position of the mass, at any time. The velocity of the mass, at any time, just by taking the Or the period, or the frequency of the function. As long as you know how to take the period and frequency of trig functions. You can watch my videos, and watch my trig videos, to get a refresher on that. One thing that's pretty interesting about this, is notice that the frequency and the period, right? This is the period of the function, that's how long it takes do one cycle. This is how many cycles it does in one second-- both of them are independent of A. So it doesn't matter, I could stretch it only a little bit, like there, and it'll take the same amount of time to go back, and come back like that, as it would if I stretch it a lot. It would just do that. If I stretched it just a little bit, the function would look like this. Make sure I do this right. I'm not doing that right. Edit, undo. If I just do it a little bit, the amplitude is going to be less, but the function is going to essentially do the same thing. It's just going to do that. So it's going to take the same amount of time to complete the cycle, it'll just have a lower amplitude. So that's interesting to me, that how much I stretch it, it's not going to make it take longer or less time to complete one cycle. That's interesting. And so if I just told you, that I actually start having objects compressed, right? So in that case, let's say my A is minus 3. I have a spring constant of-- let's say k is, I don't know, 10. And I have a mass of 2 kilograms. Then I could immediately tell you what the equation of the position as a function of time at any point is. It's going to be x of t will equal-- I'm running out of space-- so x of t would equal-- this is just basic subsitution-- minus 3 cosine of 10 divided by 2, right? k over m, is 5. So square root of 5t. I know that's hard to read, but you get the point. I just substituted that. But the important thing to know is this-- this is, I think, the most important thing-- and then if given a trig function, you have trouble remembering how to figure out the period or frequency-- although I always just think about, when does this expression equal 1? And then you can figure out-- when does it equal 1, or when does it equal 0-- and you can figure out its period. If you don't have it, you can memorize this formula for period, and this formula for frequency, but I think that might be a waste of your brain space. Anyway, I'll see you in the next video." + }, + { + "Q": "At 2:05 Sal puts in the -9y^2x at the end of the simplified polynomial equation. Could he have put -9y^2x at the beginning? Why did he put the -9y^2x where he did?", + "A": "The terms in the polynomial can be listed in any order. For example - these are all the same: 4x^2y - 10xy + 45 - 9y^2x (this is Sal s version) 4x^2y - 10xy - 9y^2x + 45 45 - 9y^2x - 10xy + 4x^2y - 9y^2x + 45 + + 4x^2y - 10xy etc.", + "video_name": "AqMT_zB9rP8", + "transcript": "We've got 4x squared y minus 3xy plus 25 minus the entire expression 9y squared x plus 7xy minus 20. So when we're subtracting this entire expression, that's equivalent to subtracting each of these terms individually if we didn't have the parentheses. Or another way of thinking about it-- we could distribute this negative sign. Or you could view this as a negative 1 times this entire expression. And we can distribute it. So let's do that. So let me write this first expression here. I'm going to write it unchanged. So it is 4x squared y minus 3xy plus 25. And now let me distribute the negative 1, or the negative sign times all of this stuff. So negative 1 times 9y squared x is negative 9y squared x. Negative 1 times 7xy is negative 7xy. And then negative 1 times 20 is positive 20. And now we just have to add these terms. And we just want to group like terms. So let's see, is there another x squared y term anywhere? No, I don't see one. So I'll just rewrite this. So we have 4x squared y. Now, is there another xy term? Yeah, there is. So we can group negative 3xy and negative 7xy. Negative 3 of something minus another 7 of that something is going to be negative 10 of that something. So it's negative 10xy. And then we have a 25, which is just a constant term. Or an x to the 0 term. It's 25x to the 0. You could view it that way. And there's another constant term right over here. We can always add 25 to 20. That gives us 45. And then we have this term right over here, which clearly can't be merged with anything else. So minus 9y squared. Let me do that in that original color. Minus 9-- I'm having trouble shifting colors-- minus 9y squared x. And we are done." + }, + { + "Q": "How to balance Pb3O4 ---> PbO + O2? sorry i know the answer, but i don't understand why and how to get it. can someone explain step-by-step to me please? Thank you!", + "A": "Balancing Pb3O4 ---> PbO + O2 _ Pb3O4 --> _ PbO + __O2 Starting off. 1 Pb3O4 --> 3 PbO + __O2 . The Pb atoms are now balanced with 3 Pb atoms on the reactant side and 3 Pb atoms on the product side. 1 Pb3O4 --> 3 PbO + 1/2 O2 Balancing the oxygen atoms with 4 on each side. Since it s not possible to have 1/2 a O2 molecule, all the coefficients would need to be multiplied by 2. The final equation would be: 2 Pb3O4 --> 6 PbO + O2", + "video_name": "8KXWJCmshEE", + "transcript": "- [Voiceover] All right, let's see what's going on in this chemical reaction. On the reactant side, we have an iron oxide. This is ferric oxide right over here, reacting with sulfuric acid, and it's producing, this is ferric sulfate and water. We want to balance this chemical equation. I encourage you to pause this video and try to balance this. I'm assuming you've had a go at it, and you might have been able to successfully balance it. If you didn't, if you weren't able to successfully balance it, one theory of why you weren't able to is because this sulfate group made things really confusing. You have four oxygens, three oxygens here, you have seven on this side, then you have four oxygens in the sulfate group here, but you have three sulfate groups. This is 12 oxygens here, and then you have another oxygen here. This seems really, really, really, really confusing, especially with all this sulfate group business. The key here is to appreciate that the sulfate group is kind of staying together. You can kind of treat it ... Instead of just saying, \"Hey, let's try to balance \"all of the oxygens,\" you can say, \"Let's balance the oxygens \"that are outside of the sulfate groups separately, \"and let's balance the sulfate groups separately.\" To help our brains grapple with that, I'm going to rewrite this chemical equation with a substitution. I'm going to say, let's say that x is equal to a sulfate, a sulfate group. Let me rewrite all of this business. I have ... It's nice to have, I guess this is close to a rust color, which seems appropriate. Let's say I have some ferric oxide. I'm just rewriting what I have above right over here. Ferric oxide plus some sulfuric acid. But instead of writing H2, and then writing the sulfate group, I'm going to write H2 and then x. H2 and then x is going to yield ... Is going to yield ferric sulfate. Ferric sulfate has three sulfate groups. So x is a sulfate group. It's going to have three of them. Ferric sulfate plus molecular, plus molecular water. Now, even though x represents an entire group, let's treat it like an element and just ... make sure we have the same number of x's, or the same number of sulfate groups on both sides. Let's balance this chemical equation. Let's start with the iron. Over here, I have two irons. And over here, on the right-hand side, I have two irons. It doesn't seem like I have to tweak the irons at all. Now let's move on to the oxygens. I have three oxygens here, on the left-hand side. On the right-hand side, I only have one oxygen. But I can change that by saying, \"Let's have three water molecules.\" Then this is going to be three, right over here. Now let's focus on the hydrogens. Focus on the hydrogens, I have two hydrogens here, and I have six hydrogens right over here. If I have six hydrogens right over there, how do I get six hydrogens here? Well, I'll have three molecules of sulfuric acid. Each of them have two hydrogen atoms. Now I have six hydrogens. My hydrogens, my irons, and my oxygens that are not part of the sulfate group are all balanced. Now let's see if we can balance the sulfate groups. On the left-hand side, I have three sulfate groups. Let me do that in that magenta color. I have three sulfate groups. On the right-hand side, I also have three sulfate groups. I'm all balanced. And if we want to un-substitute, we just go back up here. Okay, we didn't change the coefficient on this molecule, on the ferric oxide. We did change the coefficient on the sulfuric acid. We say we have, for every molecule of ferric oxide, we have three molecules of sulfuric acid. We didn't change this. They're going to yield one for every one molecule of ferric oxide, and three molecules of sulfuric acid. It's going to yield, the product, are going to be one molecule of ferric sulfate and three molecules of water. And we are done." + }, + { + "Q": "I'm confused, when he divided 5 from 4 how did he go from getting 0.8 to 1.8?", + "A": "The original number before he divided as 1 4/5. He did the division of the 4/5 = 0.8 (temporarily ignoring the 1). Then add the 1 with the decimal to get the decimal equivalent of 1 4/5 = 1 + 0.8 = 1.8", + "video_name": "-lUEWEEpmIo", + "transcript": "Let's see if we can figure out what 30% of 6 is. So one way of thinking about 30%-- this literally means 30 per 100. So you could view this as 30/100 times 6 is the same thing as 30% of 6. Or you could view this as 30 hundredths times 6, so 0.30 times 6. Now we could solve both of these, and you'll see that we'll get the same answer. If you do this multiplication right over here, 30/100-- and you could view this times 6/1-- this is equal to 180/100. We can simplify. We can divide the numerator and the denominator by 10. And then we can divide the numerator and the denominator by 2. And we will get 9/5, which is the same thing as 1 and 4/5. And then if we wanted to write this as a decimal, 4/5 is 0.8. And if you want to verify that, you could verify that 5 goes into 4-- and there's going to be a decimal. So let's throw some decimals in there. It goes into 4 zero times. So we don't have to worry about that. It goes into 40 eight times. 8 times 5 is 40. Subtract. You have no remainder, and you just have 0's left here. So 4/5 is 0.8. You've got the 1 there. This is the same thing as 1.8, which you would have gotten if you divided 5 into 9. You would've gotten 1.8. So 30% of 6 is equal to 1.8. And we can verify it doing this way as well. So if we were to multiply 0.30 times 6-- let's do that. And I could just write that literally as 0.3 times 6. Well, 3 times 6 is 18. I have only one digit behind the decimal amongst both of these numbers that I'm multiplying. I only have the 3 to the right of the decimal. So I'm only going to have one number to the right of the decimal here. So I just count one number. It's going to be 1.8. So either way you think about it or calculate it, 30% of 6 is 1.8." + }, + { + "Q": "Is mean and arithmetic mean the same ?", + "A": "Usually yes, if left as mean the arithmetic mean is implied.", + "video_name": "h8EYEJ32oQ8", + "transcript": "We will now begin our journey into the world of statistics, which is really a way to understand or get our head around data. So statistics is all about data. And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments. And we'll start to do a lot of inferential statistics, make inferences. So with that out of the way, let's think about how we can describe data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants. And the heights are 4 inches, 3 inches, 1 inch, 6 inches, and another one's 1 inch, and another one is 7 inches. And let's say someone just said-- in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that-- maybe I want a typical number. Maybe I want some number that somehow represents the middle. Maybe I want the most frequent number. Maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. And in every day terminology, average has a very particular meaning, as we'll see. When many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, or give me a middle number, or-- and these are or's. And really it's an attempt to find a measure of central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number we'll call the average, that's somehow typical, or middle, or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one-- and people talk about hey, the average on this exam or the average height. And that's the arithmetic mean. Just let me write it in. I'll write in yellow, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic, arithmetic mean. And this is really just the sum of all the numbers divided by-- this is a human-constructed definition that we've found useful-- the sum of all these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be 4 plus 3 plus 1 plus 6 plus 1 plus 7 over the number of data points we have. So we have six data points. So we're going to divide by 6. And we get 4 plus 3 is 7, plus 1 is 8, plus 6 is 14, plus 1 is 15, plus 7. 15 plus 7 is 22. Let me do that one more time. You have 7, 8, 14, 15, 22, all of that over 6. And we could write this as a mixed number. 6 goes into 22 three times with a remainder of 4. So it's 3 and 4/6, which is the same thing as 3 and 2/3. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human-constructed. No one ever-- it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as, say, finding the circumference of the circle, which there really is-- that was kind of-- we studied the universe. And that just fell out of our study of the universe. It's a human-constructed definition that we found useful. Now there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median. I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have 1. Then we have another 1. Then we have a 3. Then we have a 4, a 6, and a 7. So all I did is I reordered this. And so what's the middle number? Well, you look here. Since we have an even number of numbers, we have six numbers, there's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the 3 and the 4. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. You're essentially taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in-between 3 and 4, which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, the-- essentially the arithmetic mean of the middle two, or halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set-- and I'll order it for us-- let's say our data set was 0, 7, 50, I don't know, 10,000, and 1 million. Let's say that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is the number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it. It sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all of the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one 4. We only have one 3. But we have two 1's. We have one 6 and one 7. So the number that shows up the most number of times here is our 1. So the mode, the most typical number, the most common number here is a 1. So, you see, these are all different ways of trying to get at a typical, or middle, or central tendency. But they do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently. Anyway, I'll leave you there. And we'll-- the next few videos, we will explore statistics even deeper." + }, + { + "Q": "Why did you multiply by 20 to get numerators?", + "A": "It s the LCM (Lowest Common Multiple)", + "video_name": "N8dIOmk_lHs", + "transcript": "What I want to do in this video is order these fractions from least to greatest. And the easiest way and the way that I think we can be sure we'll get the right answer here is to find a common denominator, because if we don't find a common denominator, these fractions are really 4/9 versus 3/4 versus 4/5, 11/12, 13/15. You can try to estimate them, but you'll be able to directly compare them if they all had the same denominator. So the trick here, or at least the first trick here, is to try to find that common denominator. And there's many ways to do it. You could just pick one of these numbers and keep taking its multiples and find that multiple that is divisible by all the rest. Another way to do it is look at the prime factorization of each of these numbers. And then the least common multiple of them will have to have at least all of those prime numbers in it. It has to be composed of all of these numbers. So let's do it that second way. And then let's verify that it definitely is divisible. So 9 is the same thing as 3 times 3. So our least common multiple is going to have at least one 3 times 3 in it. And then 4 is the same thing as 2 times 2. So we're going to also have to have a 2 times 2 in our prime factorization of our least common multiple. 5 is a prime number. So we're going to need to have a 5 in there. And then 12-- I'm going to do that in yellow. 12 is the same thing as 2 times 6, which is the same thing as 2 times 3. And so in our least common multiple, we have to have two 2's. But we already have two 2's right over here from our 4. And we already have one 3 right over here. Another way to think about it is something that is divisible by both 9 and 4 is going to be divisible by 12, because you're going to have the two 2's. And you're going to have that one 3 right over there. And then, finally, we need to be divisible by 15's prime factors. So let's look at 15's prime factors. 15 is the same thing as 3 times 5. So once again, this number right over here already has a 3 in it. And it already has a 5 in it. So we're cool for 15, for 12, and, obviously, for the rest of them. So this is our least common multiple. And we can just take this product. And so this is going to be equal to 3 times 3 is 9. 9 times 2 is 18. 18 times 2 is 36. 36 times 5, you could do that in your head if you're like. But I'll do it on the side just in case. 36 times 5, just so that we don't mess up. 6 times 5 is 30. 3 times 5 is 15 plus 3 is 180. So our least common multiple is 180. So we want to rewrite all of these fractions with 180 in the denominator. So this first fraction, 4/9, is what over 180? To go from 9 to 180, we have to multiply the denominator by 20. So let me do it this way. So if we do 4/9, to get the denominator of 9 to be 180, you have to multiply it by 20. And since we don't want to change the value of the fraction, we should also multiply the 4 by 20. So we're just really multiplying by 20/20. And so 4/9 is going to be the same thing as 80/180. Now, let's do the same thing for 3/4. Well, what do we have to multiply the denominator by to get us to 180? So it looks like 45. You could divide 4 into 180 to figure that out. But if you take 4 times 45, 4 times 40 is 160. 4 times 5 is 20. You get 180. So if you multiply the denominator by 45, you also have to multiply the numerator by 45. 3 times 45 is 120 plus 15. So it's 135. And the denominator here is 180. Now, let's do 4/5. To get our denominator to be 180, what do you have to multiply 5 by? Let's see. If you multiply 5 by 30, you'll get to 150. But then you have another 30. Actually, we know it right over here. You have to multiply it by 36. Well, then you have to multiply the numerator by 36 as well. And so our denominator is going to be 180. Our numerator, 4 times 30 is 120. 4 times 6 is 24. So it's 144/180. And then we have only two more to do. So we have our 11/12. So to get the denominator to be 180, we have to multiply 12 by-- so 12 times 10 is 120. Then you have 60 left. So you have to multiply it by 15, 15 In the denominator, and 15 in the numerator. And so the denominator gives us 180. And 11 times 15. So 10 times 15 is 150. And then you have one more 15. So it's going to be 165. And then, finally, we have 13/15. To get our denominator to be 180, have to multiply it by 12. We already figured out that 12 times 15 is 180. So you have to multiply it by 12. That will give us 180 in the denominator. And so you have to also multiply the numerator by 12, so that we don't change the value of the fraction. We know 12 times 12 is 144. You could put one more 12 in there. You get 156. Did I do that right? 12 plus 144 is going to be 156. So we've rewritten each of these fractions with that new common denominator of 180. And now, it's very easy to compare them. You really just have to look at the numerators. So the smallest of the numerators is this 80 right over here. So 4/9 is the smallest. 4/9 is the least of these numbers. So let me just write it over here. So this is our ordering. We have 4/9 comes first, which is the same thing as 80/180. Let me write it both ways-- 80/180. Then the next the smallest number looks like it's this 135 right over here. I want to do it in that same color. The next one is going to be that 135/180, which is the same thing as 3/4. And then the next one is going to be-- let's see, we have the 144/180. So this is going to be the 144/180, which is the same thing as 4/5. And then we have two more. The next is this 156/180. So then we have our 156/180, which is the same thing as 13/15. And then we have one left over, the 165/180, which is the same thing-- I want to do that in yellow. We have our 165/180, which is the same thing as 11/12. And we're done. We have finished our ordering. So if you're doing the Khan Academy module on this, this is what you would input into that little box there." + }, + { + "Q": "This video confused me. What does this video have to do with polynomials?", + "A": "It s basically one polynomial divided by another.", + "video_name": "S-XKGBesRzk", + "transcript": "Let's see if we can learn a thing or two about partial fraction expansion, or sometimes it's called partial fraction decomposition. The whole idea is to take rational functions-- and a rational function is just a function or expression where it's one expression divided by another-- and to essentially expand them or decompose them into simpler parts. And the first thing you've got to do, before you can even start the actual partial fraction expansion process, is to make sure that the numerator has a lower degree than the denominator. In the situation, the problem, that I've drawn right here, I've written right here, that's not the case. The numerator has the same degree as the denominator. So the first step we want to do to simplify this and to get it to the point where the numerator has a lower degree than the denominator is to do a little bit of algebraic division. And I've done a video on this, but it never hurts to get a review here, so to do that, we divide the denominator into the numerator to figure out the remainder, so we divide x squared minus 3x minus 40 into x squared minus 2x minus 37. So how many times? You look at the highest degree term, so x squared goes into x squared one time, one times this whole thing is x squared minus 3x minus 40, and now you want to subtract this from that to get the remainder. And see, if I'm subtracting, so I'm going to subtract, and then minus minus is a plus, a plus, and then you can add them. These cancel out. Minus 2x plus 3x, that's x. Minus 37 plus 40, that's plus 3. So this expression up here can be rewritten as-- let me scroll down a little bit-- as 1 plus x plus 3 over x squared minus 3x minus 40. This might seem like some type of magic thing I just did, but this is no different than what you did in the fourth or fifth grade, where you learned how to convert improper regular fractions into mixed numbers. Let me just do a little side example here. If I had 13 over 2, and I want to turn it into a mixed number, what you do-- you can probably do this in your head now-- but what you did is, you divide the denominator into the numerator, just like we did over here. 2 goes into 13. We see 2 goes into 13 six times, 6 times 2 is 12, you subtract that from that, you get a remainder of 1. 2 doesn't go into 1, so that's just the remainder. So if you wanted to rewrite this, it would be the number of times the denominator goes into the numerator, that's 6, plus the remainder over the denominator. Plus 6-- plus 1 over 2. And when you did it in elementary school, you would just write 6 1/2, but 6 1/2 is the same thing as 6 plus 1/2. That's exactly the same thing we did here. The denominator went to the numerator one time, and then there was a remainder of x plus 3 left over, so it's 1 plus x plus 3 over this expression. Now we see that that numerator in this rational expression does have a lower degree than the denominator. The highest degree here is 1, the highest degree here is 2, so we're ready to commence our partial fraction decomposition. And all that is, is taking this expression up here and turning it into two simpler expressions where the denominators are the factors of this lower term. So given that, let's factor this lower term. So let's see. What two numbers add up to minus 3, and when you multiply them, you get minus 40? So let's see. They have to be different signs, because when you multiply them you get a negative, so it has to be minus 8 and plus 5. So we can rewrite this up here as-- I'll switch colors-- 1 plus x plus 3 over x plus 5 times x minus 8. 5 times 8 is minus 40-- 5 times negative 8 is minus 40, plus 5 minus 8 is minus 3, so we're all set. Now I'll just focus on this part right now. We can just remember that that 1 is sitting out there out front. This is the expression we want to decompose or expand. And we're going to expand it into two simpler expressions where each of these are the denominator-- and I will make the claim, and if the numbers work out then the claim is true-- I'll make the claim that I can expand this, or decompose this, into two fractions where the first fraction is just some number a over the first factor, over x plus 5, plus some number b over the second factor, over x minus 8. That's my claim, and if I can solve for a and b in a way that it actually does add up to this, then I'm done and I will have fully decomposed this fraction. I guess is the way-- I don't know if that's the correct terminology. So let's try to do that. So if I were to add these two terms, what do I get? When you add anything, you find the common denominator, and the common denominator, the easiest common denominator, is to multiply the two denominators, so let me write this here. So a over x plus 5 plus b over x minus 8 is equal to-- well, let's get the common denominator-- it's equal to x plus 5 times x minus 8. And then the a term, we would-- a over x plus 5 is the same thing as a times x minus 8 over this whole thing. I mean, if I just wrote this right here, you would just cancel these two terms out and you would get a over x plus 5. And then you could add that to the common denominator, x plus 5 times x minus 8, and it would be b times x plus 5. Important to realize, that, look. This term is the exact same thing as this term if you just cancel the x minus 8 out, and this term is the exact same thing as this term if you just cancel the x plus 5 out. But now that we have an actual common denominator, we can add them together, so we get-- let me just write the left side here over-- a over x plus 5-- I'm sorry. I want to write this over here. I want to write x plus 3 over plus 5 times x minus 8 is equal to is equal to the sum of these two things on top. a times x minus 8 plus b times x plus 5, all of that over their common denominator, x plus 5 times x minus eight. So the denominators are the same, so we know that this, when you add this together, you have to get this. So if we want to solve for a and b, let's just set that equality. We can ignore the denominators. So we can say that x plus 3 is equal to a times x minus 8 plus b times x plus 5. Now, there's two ways to solve for a and b from this point going forward. One is the way that I was actually taught in the seventh or eighth grade, which tends to take a little longer, then there's a fast way to do it and it never hurts to do the fast way first. If you want to solve for a, let's pick an x that'll make this term disappear. So what x would make this term disappear? Well, if I say x is minus 5, then this becomes 0, and then the b disappears. So if we say x is minus 5-- I'm just picking an arbitrary x to be able to solve for this-- then this would become minus 5 plus 3-- let me just write it out, minus 5 plus 3-- is equal to a times minus 5 minus 8-- let me just write it out, minus 5 minus 8-- plus b times minus 5 plus 5. And I picked the minus 5 to make this expression 0. So then you get-- pick a brighter color-- minus 5 plus 3 is minus 2, is equal to-- what is this?-- minus 13a plus-- this is 0, right? That's 0. Minus 5 plus 5 is 0, 0 times b is 0, and then you divide both sides by minus 13, you get-- negatives cancel out-- you get 2 over 13 is equal to a, and now we can do the same thing up here and get rid of the a terms by making x is equal to 8. If x is equal to 8, you get x plus 3 is equal to 11, is equal to a times 0 plus b times-- what's 5-- 8 plus 5 is-- plus b times 13. Their b looks a bit like a 13. And then you get 11 is equal to 13b, divide both sides by 13, you get b is equal to 11 over 13. So we were able to solve for our a's and our b's. And so we can go back to our original equation and we could say, wow. This just has to be equal to 2 over 13, and this just has to be equal to 11 over 13. So our original, our very original thing we wrote up here, can be decomposed into 1, that's this 1 over here, plus this, which is 2 over 13-- I'll just write it like this for now-- 2 over 13, over x plus 5. You could bring the 13 down here if you want to write it so you don't have a fraction over a fraction. Plus 11 over 13 times-- over x minus 8. And once again, you could bring the 13 down so you don't have a fraction over a fraction. But we have just successfully decomposed this pretty-- I don't want to say that we necessarily simplified it, because you could say, oh, we only have one expression here, now I have three-- but I've reduced the degree of both the numerators and the denominators. And you might say, well, Sal, why would I ever have to do this? And you're right. In algebra you probably won't. But this is actually a really useful technique later on when you get to calculus, and actually, differential equations, because a lot of times it's much easier-- and I'll throw out a word here that you don't understand-- to take the integral or the antiderivative of something like this, then something like this. And later, when you do inverse Laplace transforms and differential equations, it's much easier take an inverse Laplace transform of something like this than something like that. So anyway, hopefully I've given you another tool kit in your-- or another tool in your tool kit, and I'll probably do a couple more videos because we haven't exhausted all of the examples that we could we could show for partial fraction decomposition." + }, + { + "Q": "e^(x+1)=25 please solve (for Nana)", + "A": "e^(x + 1) = 25 x + 1 = ln(25) x = ln(25) - 1 x \u00e2\u0089\u0088 2.2189", + "video_name": "sBhEi4L91Sg", + "transcript": "I would guess that you're reasonably familiar with linear scales. These are the scales that you would typically see in most of your math classes. And so just to make sure we know what we're talking about, and maybe thinking about in a slightly different way, let me draw a linear number line. Let me start with 0. And what we're going to do is, we're going to say, look, if I move this distance right over here, and if I move that distance to the right, that's equivalent to adding 10. So if you start at 0 and you add 10, that would obviously get you to 10. If you move that distance to the right again, you're going to add 10 again, that would get you to 20. And obviously we could keep doing it, and get to 30, 40, 50, so on and so forth. And also, just looking at what we did here, if we go the other direction. If we start here, and move that same distance to the left, we're clearly subtracting 10. 10 minus 10 is equal to 0. So if we move that distance to the left again, we would get to negative 10. And if we did it again, we would get to negative 20. So the general idea is, however many times we move that distance, we are essentially adding-- or however many times you move that distance to the right-- we are essentially adding that multiple of 10. If we do it twice, we're adding 2 times 10. And that not only works for whole numbers, it would work for fractions as well. Where would 5 be? Well to get to 5, we only have to multiply 10-- or I guess one way to think about it is 5 is half of 10. And so if we want to only go half of 10, we only have to go half this distance. So if we go half this distance, that will get us to 1/2 times 10. In this case, that would be 5. If we go to the left, that would get us to negative 5. And there's nothing-- let me draw that a little bit more centered, negative 5-- and there's nothing really new here. We're just kind of thinking about it in a slightly novel way that's going to be useful when we start thinking about logarithmic. But this is just the number line that you've always known. If we want to put 1 here, we would move 1/10 of the distance, because 1 is 1/10 of 10. So this would be 1, 2, 3, 4, I could just put, I could label frankly, any number right over here. Now this was a situation where we add 10 or subtract 10. But it's completely legitimate to have an alternate way of thinking of what you do when you move this distance. And let's think about that. So let's say I have another line over here. And you might guess this is going to be the logarithmic number line. Let me give ourselves some space. And let's start this logarithmic number line at 1. And I'll let you think about, after this video, why I didn't start it at 0. And if you start at 1, and instead of moving that, so I'm still going to define that same distance. But instead of saying that that same distance is adding 10 when I move to the right, I'm going to say when I move the right that distance on this new number line that I have created, that is the same thing as multiplying by 10. And so if I move that distance, I start at 1, I multiply by 10. That gets me to 10. And then if I multiply by 10 again, if I move by that distance again, I'm multiplying by 10 again. And so that would get me to 100. I think you can already see the difference that's happening. And what about moving to the left that distance? Well we already have kind of said what happens. Because if we start here, we start at 100 and move to the left of that distance, what happens? Well I divided by 10. 100 divided by 10 gets me 10. 10 divided by 10 get me 1. And so if I move that distance to the left again, I'll divide by 10 again. That would get me to 1/10. And if I move that distance to the left again, that would get me to 1/100. And so the general idea is, is however many times I move that distance to the right, I'm multiplying my starting point by 10 that many times. And so for example, when I move that distance twice, so this whole distance right over here, I went that distance twice. So this is times 10 times 10, which is the same thing as times 10 to the second power. And so really I'm raising 10 to what I'm multiplying it times 10 to whatever power, however many times I'm jumping to the right. Same thing if I go to the left. If I go to the left that distance twice-- let me do that in a new color-- this will be the same thing as dividing by 10 twice. Dividing by 10, dividing by 10, which is the same thing as multiplying by-- one way to think of it-- 1/10 squared. Or dividing by 10 squared is another way of thinking about it. And so that might make a little, that might be hopefully a little bit intuitive. And you can already see why this is valuable. We can already on this number line plot a much broader spectrum of things than we can on this number line. We can go all the way up to 100, and then we even get some nice granularity if we go down to 1/10 and 1/100. Here we don't get the granularity at small scales, and we also don't get to go to really large numbers. And if we go a little distance more, we get to 1,000, and then we get to 10,000, so on and so forth. So we can really cover a much broader spectrum on this line right over here. But what's also neat about this is that when you move a fixed distance, so when you move fixed distance on this linear number line, you're adding or subtracting that amount. So if you move that fixed distance you're adding 2 to the right. If you go to the left, you're subtracting 2. When you do the same thing on a logarithmic number line, this is true of any logarithmic number line, you will be scaling by a fixed factor. And one way to think about what that fixed factor is is this idea of exponents. So if you wanted to say, where would 2 sit on this number line? Then you would just think to yourself, well, if I ask myself where does 100 sit on that number line-- actually, that might be a better place to start. If I said, if I didn't already plot it and said where does 100 sit on that number line? I would say, how many times would we have to multiply 10 by itself to get 100? And that's how many times I need to move this distance. And so essentially I'll be asking 10 to the what power is equal to 100? And then I would get that question mark is equal to 2. And then I would move that many spaces to plot my 100. Or another way of stating this exact same thing is log base 10 of 100 is equal to question mark. And this question mark is clearly equal to 2. And that says, I need to plot the 100 2 of this distance to the right. And to figure out where do I plot the 2, I would do the exact same thing. I would say 10 to what power is equal to 2? Or log base 10 of 2 is equal to what? And we can get the trusty calculator out, and we can just say log-- and on most calculators if there's a log without the base specified, they're assuming base 10-- so log of 2 is equal to roughly 0.3. 0.301. So this is equal to 0.301. So what this tells us is we need to move this fraction of this distance to get to 2. If we move this whole distance, it's like multiplying times 10 to the first power. But since we only want to get 10 to the 0.301 power, we only want to do 0.301 of this distance. So it's going to be roughly a third of this. It's going to be roughly-- actually, a little less than a third. 0.3, not 0.33. So 2 is going to sit-- let me do it a little bit more to the right-- so 2 is going to sit right over here. Now what's really cool about it is this distance in general on this logarithmic number line means multiplying by 2. And so if you go that same distance again, you're going to get to 4. If you multiply that same distance again, you're going to multiply by 4. And you go that same distance again, you're going to get to 8. And so if you said where would I plot 5? Where would I plot 5 on this number line? Well, there's a couple ways to do it. You could literally figure out what the base 10 logarithm of 5 is, and figure out where it goes on the number line. Or you could say, look, if I start at 10 and if I move this distance to the left, I'm going to be dividing by 2. So if I move this distance to the left I will be dividing by 2. I know it's getting a little bit messy here. I'll maybe do another video where we learn how to draw a clean version of this. So if I start at 10 and I go that same distance I'm dividing by 2. And so this right here would be that right over there would be 5. Now the next question, you said well where do I plot 3? Well we could do the exact same thing that we did with 2. We ask ourselves, what power do we have to raise 10 to to get to 3? And to get that, we once again get our calculator out. log base 10 of 3 is equal to 0.477. So it's almost halfway. So it's almost going to be half of this distance. So half of that distance is going to look something like right over there. So 3 is going to go right over here. And you could do the logarithm-- let's see, we're missing 6, 7, and 8. Oh, we have 8. We're missing 9. So to get 9, we just have to multiply by 3 again. So this is 3, and if we go that same distance, we multiply by 3 again, 9 is going to be squeezed in right over here. 9 is going to be squeezed in right over there. And if we want to get to 6, we just have to multiply by 2. And we already know the distance to multiply by 2, it's this thing right over here. So you multiply that by 2, you do that same distance, and you're going to get to 6. And if you wanted to figure out where 7 is, once again you could take the log base-- let me do it right over here-- so you'll take the log of 7 is going to be 0.8, roughly 0.85. So 7 is just going to be squeezed in roughly right over there. And so a couple of neat things you already appreciated. One, we can fit more on this logarithmic scale. And, as I did with the video with Vi Hart, where she talked about how we perceive many things with logarithmic scales. So it actually is a good way to even understand some of human perception. But the other really cool thing is when we move a fixed distance on this logarithmic scale, we're multiplying by a fixed constant. Now the one kind of strange thing about this, and you might have already noticed here, is that we don't see the numbers lined up the way we normally see them. There's a big jump from 1 to 2, then a smaller jump from 3 to 4, then a smaller jump from that from 3 to 4, then even smaller from 4 to 5, then even smaller 5 to 6 it gets. And then 7, 8, 9, you know 7's going to be right in there. They get squeeze, squeeze, squeezed in, tighter and tighter and tighter, and then you get 10. And then you get another big jump. Because once again, if you want to get to 20, you just have to multiply by 2. So this distance again gets us to 20. If you go this distance over here that will get you to 30, because you're multiplying by 3. So this right over here is a times 3 distance. So if you do that again, if you do that distance, then that gets you to 30. You're multiplying by 3. And then you can plot the whole same thing over here again. But hopefully this gives you a little bit more intuition of why logarithmic number lines look the way they do. Or why logarithmic scale looks the way it does. And also, it gives you a little bit of appreciation for why it might be useful." + }, + { + "Q": "I'm stuck on a problem.\nHow would you simplify the following: (x^3)^(2/3)\n\nMy first thought would be to multiply the exponents: 3/1 * 2/3 which would leave me with an exponent of 2. Can anyone confirm this answer for me?", + "A": "If my brain does not fail me I think that s correct. The answer is x^2.", + "video_name": "S34NM0Po0eA", + "transcript": "We've already seen how to think about something like 64 to the 1/3 power. We saw that this is the exact same thing as taking the cube root of 64. And because we know that 4 times 4 times 4, or 4 to the third power, is equal to 64, if we're looking for the cube root of 64, we're looking for a number that that number times that number times that same number is going to be equal to 64. Well, we know that number is 4, so this thing right over here is going to be 4. Now we're going to think of slightly more complex fractional exponents. The one we see here has a 1 in the numerator. Now we're going to see something different. So what I want to do is think about what 64 to the 2/3 power is. And here I'm going to use a property of exponents that we'll study more later on. But this property of exponents is the idea that-- let's say with a simple number-- if I raise something to the third power and then I were to raise that to, say, the fourth power, this is going to be the same thing as raising it to the 2 to the 3 times 4 power, or 2 to the 12th power, which you could also write as raising it to the fourth power and then the third power. All this is saying is, if I raise something to a power and then raise that whole thing to a power, it's the same thing as multiplying the two exponents. This is the same thing as 2 to the 12th. So we could use that property here to say, well, 2/3 is the same thing as 1/3 times 2. So we could go in the other direction. We could say, hey look, well this is going to be the same thing as 64 to the 1/3 power and then that thing squared. Notice, I'm raising something to a power and then raising that to a power. If I were to multiply these two things, I would get 64 to the 2/3 power. Now, why did I do this? Well, we already know what 64 to the 1/3 power is. We just calculated it. That's equal to 4. So we could say that-- and I'll write it in that same yellow color-- this is equal to 4 squared, which is equal to 16. So 64 to the 2/3 power is equal to 16. The way I think of it, let me find the cube root of 64, which is 4. And then let me square it. And that is going to get me to 16. Now I'll give you in even hairier problem. And I encourage you to try this one on your own before I work through it. So we're going to work with 8/27. And we're going to raise this thing to the-- and I'll try to color code it-- negative 2 over 3 power, to the negative 2/3 power. I encourage you to pause and try this on your own. Well the first thing I do whenever I see a negative exponent is to say, well, how can I get rid of that negative exponent? And I just remind myself, well, the negative exponent really just says, take the reciprocal of this to the positive exponent. I'm using a different color. I'm going to use that light mauve color. So this is going to be equal to 27/8. I just took the reciprocal of this right over here. It's equal to 27/8 to the positive 2/3 power. So notice, all I did, I got rid of the exponent and took the reciprocal of the base right over here. 8/27 is the base. Negative 2/3 is the exponent. Now, how can we handle this? Well, we've already seen that if I have a numerator to some power over a denominator to some power-- and this is another very powerful exponent property-- this is going to be the exact same thing as raising 27 to the 2/3 power-- to the 2 over 3 power-- over 8 to the 2/3 power. This is another very powerful exponent property. Notice, if I have something divided by something and I'm raising the whole thing to a power, I can essentially raise the numerator to that power over the denominator raised to that power. Now, let's think about what this is. Well just like we saw before, this is going to be the same thing. This is going to be the same thing as 27 to the 1/3 power and then that squared because 1/3 times 2 is 2/3. So I'm going to raise 27 to the 1/3 power and then square whatever that is. All this color coding is making this have to switch a lot of colors. This is going to be over 8 to the 1/3 power. And then that's going to be raised to the second power. Same thing we were doing in the denominator-- we raise 8 to the 1/3 and then square that. So what's this going to be? Well, 27 to the 1/3 power is the cube root of 27. It's some number-- that number times that same number times that same number is going to be equal to 27. Well, it might jump out at you already that 3 to the third is equal to 27 or that 27 to the 1/3 is equal to 3. So the numerator, we're going to end up with 3 squared. And then in the denominator, we are going to end up with-- well, what's 8 to the 1/3 power? Well, 2 times 2 times 2 is 8. So this is 8 to the 1/3 third is 2. Let me do that same orange color. 8 to the 1/3 is 2, and then we're going to square that. So this is going to simplify to 3 squared over 2 squared, which is just going to be equal to 9/4. So if you just break it down step by step, it actually is not too daunting." + }, + { + "Q": "What does it mean when a slope is undefined?", + "A": "If a slope is undefined that means the line is perfectly vertical.", + "video_name": "CFSHq099Mx0", + "transcript": "Find the slope of the line that goes through the ordered pairs 7, negative 1 and negative 3, negative 1. Let me just do a quick graph of these just so we can visualize what they look like. So let me draw a quick graph over here. So our first point is 7, negative 1. So 1, 2, 3, 4, 5, 6, 7. This is the x-axis. 7, negative 1. So it's 7, negative 1 is right over there. 7, negative 1. This, of course, is the y-axis. And then the next point is negative 3, negative 1. So we go back 3 in the horizontal direction. Negative 3 for the y-coordinate is still negative 1. So the line that connects these two points will look like this. It will look like that. Now, they're asking us to find the slope of the line that goes through the ordered pairs. Find the slope of this line. And just to give a little bit of intuition here, slope is a measure of a line's inclination. And the way that it's defined-- slope is defined as rise over run, or change in y over change in x, or sometimes you'll see it defined as the variable m. And then they'll define change in y as just being the second y-coordinate minus the first y-coordinate and then the change in x as the second x-coordinate minus the first x-coordinate. These are all different variations in slope, but hopefully you'll appreciate that these are measuring inclination. If I rise a ton when I run a little bit, if I move a little bit in the x direction, and I rise a bunch, then I have a very steep line. I have a very steep upward-sloping line. If I don't change at all when I run a bit, then I have a very low slope. And that's actually what's happening here. I'm going from-- you could either view this as the starting point or view this as the starting point. But let's view this as the starting point. So this negative 3, 1. If I go from negative 3, negative 1 to 7, negative 1, I'm running a good bit. I'm going from negative 3. My x value is negative 3 here, and it goes all the way to 7. So my change in x here is 10. To go from negative 3 to 7, I changed my x value by 10. But what's my change in y? Well, my y value here is negative 1, and my y value over here is still negative 1. So my change in y is a 0. My change in y is going to be 0. My y value does not change no matter how much I change my x value. So the slope here is going to be-- when we run 10, what was our rise? How much did we change in y? Well, we didn't rise at all. We didn't go up or down. So the slope here is 0. Or another way to think about is this line has no inclination. It's a completely flat-- it's a completely horizontal line. So this should make sense. This is a 0. The slope here is 0. And just to make sure that this gels with all of these other formulas that you might know-- but I want to make it very clear. These are all just telling you rise over run or change in y over change in x, a way to measure inclination. But let's just apply them just so, hopefully, it all makes sense to you. So we could also say slope is change in y over change in x. If we take this to be our start and if we take this to be our end point, then we would call this over here x1. And then this is over here. This is y1. And then we would call this x2 and we would call this y2, if this is our start point and that is our end point. And so the slope here, the change in y, y2 minus y1. So it's negative 1 minus negative 1, all of that over x2, negative 3, minus x1, minus 7. So the numerator, negative 1 minus negative 1, that's the same thing as negative 1 plus 1. And our denominator is negative 3 minus 7, which is negative 10. So once again, negative 1 plus 1 is 0 over negative 10. And this is still going to be 0. And the only reason why we got a negative 10 here and a positive 10 there is because we swapped the starting and the ending point. In this example right over here, we took this as the start point and made this coordinate over here as the end point. Over here, we swapped them around. 7, negative 1 was our start point, and negative 3, negative 1 is our end point. So if we start over here, our change in x is going to be negative 10. But our change in y is still going to be 0. So regardless of how you do it, the slope of this line is 0. It's a horizontal line." + }, + { + "Q": "what would we be if non of that hapend", + "A": "nothing we wouldn t be here", + "video_name": "VbNXh0GaLYo", + "transcript": "What I'm going to attempt to do in the next two videos is really just give an overview of everything that's happened to Earth since it came into existence. We're going start really at the formation of Earth or the formation of our Solar system or the formation of the Sun, and our best sense of what actually happened is that there was a supernova in our vicinity of the galaxy, and this right here is a picture of a supernova remnant, actually, the remnant for Kepler's supernova. The supernova in this picture actually happened four hundred years ago in 1604, so right at the center a star essentially exploded and for a few weeks was the brightest object in the night sky, and it was observed by Kepler and other people in 1604, and this is what it looks like now. What we see is kinda the shockwave that's been traveling out for the past 400 years, so now it must be many light years across. It wasn't, obviously, matter wasn't traveling at the speed of light, but it must've been traveling pretty, pretty fast, at least relativistic speeds, a reasonable fraction of the speed of light. This has traveled a good bit out now, but what you can imagine is when you have the shockwave traveling out from a supernova, let's say you had a cloud of molecules, a cloud of gas, that before the shockwave came by just wasn't dense enough for gravity to take over, and for it to accrete, essentially, into a solar system. When the shockwave passes by it compresses all of this gas and all of this material and all of these molecules, so it now does have that critical density to form, to accrete into a star and a solar system. We think that's what's happened, and the reason why we feel pretty strongly that it must've been caused by a supernova is that the only way that the really heavy elements can form, or the only way we know that they can form is in kind of the heat of a supernova, and our uranium, the uranium that seems to be in our solar system on Earth, seems to have formed roughly at the time of the formation of Earth, at about four and a half billion years ago, and we'll talk in a little bit more depth in future videos on exactly how people figure that out, but since the uranium seems about the same age as our solar system, it must've been formed at around the same time, and it must've been formed by a supernova, and it must be coming from a supernova, so a supernova shockwave must've passed through our part of the universe, and that's a good reason for gas to get compressed and begin to accrete. So you fast-forward a few million years. That gas would've accreted into something like this. It would've reached the critical temperature, critical density and pressure at the center for ignition to occur, for fusion to start to happen, for hydrogen to start fusing into helium, and this right here is our early sun. Around the sun you have all of the gases and particles and molecules that had enough angular velocity to not fall into the sun, to go into orbit around the sun. They were actually supported by a little bit of pressure, too, because you can kinda view this as kind of a big cloud of gas, so they're always bumping into each other, but for the most part it was their angular velocity, and over the next tens of millions of years they'll slowly bump into each other and clump into each other. Even small particles have gravity, and they're gonna slowly become rocks and asteroids and, eventually, what we would call \"planetesimals,\" which are, kinda view them as seeds of planets or early planets, and then those would have a reasonable amount of gravity and other things would be attracted to them and slowly clump up to them. This wasn't like a simple process, you know, you could imagine you might have one planetesimal form, and then there's another planetesimal formed, and instead of having a nice, gentle those two guys accreting into each other, they might have huge relative velocities and ram into each other, and then just, you know, shatter, so this wasn't just a nice, gentle process of constant accretion. It would actually have been a very violent process, actually happened early in Earth's history, and we actually think this is why the Moon formed, so at some point you fast-forward a little bit from this, Earth would have formed, I should say, the mass that eventually becomes our modern Earth would have been forming. Let me draw it over here. So, let's say that that is our modern Earth, and what we think happened is that another proto-planet or another, it was actually a planet because it was roughly the size of Mars, ran into our, what it is eventually going to become our Earth. This is actually a picture of it. This is an artist's depiction of that collision, where this planet right here is the size of Mars, and it ran into what would eventually become Earth. This we call Theia. This is Theia, and what we believe happened, and if you look up, if you go onto the Internet, you'll see some simulations that talk about this, is that we think it was a glancing blow. It wasn't a direct hit that would've just kinda shattered each of them and turned into one big molten ball. We think it was a glancing blow, something like this. This was essentially Earth. Obviously, Earth got changed dramatically once Theia ran into it, but Theia is right over here, and we think it was a glancing blow. It came and it hit Earth at kind of an angle, and then it obviously the combined energies from that interaction would've made both of them molten, and frankly they probably already were molten because you had a bunch of smaller collisions and accretion events and little things hitting the surface, so probably both of them during this entire period, but this would've had a glancing blow on Earth and essentially splashed a bunch of molten material out into orbit. It would've just come in, had a glancing blow on Earth, and then splashed a bunch of molten material, some of it would've been captured by Earth, so this is the before and the after, you can imagine, Earth is kind of this molten, super hot ball, and some of it just gets splashed into orbit from the collision. Let me just see if I can draw Theia here, so Theia has collided, and it is also molten now because huge energies, and it splashes some of it into orbit. If we fast-forward a little bit, this stuff that got splashed into orbit, it's going in that direction, that becomes our Moon, and then the rest of this material eventually kind of condenses back into a spherical shape and is what we now call our Earth. So that's how we actually think right now that the Moon actually formed. Even after this happened, the Earth still had a lot more, I guess, violence to experience. Just to get a sense of where we are in the history of Earth, we're going to refer to this time clock a lot over the next few videos, this time clock starts right here at the formation of our solar system, 4.6 billion years ago, probably coinciding with some type of supernova, and as we go clockwise on this diagram, we're moving forward in time, and we're gonna go all the way forward to the present period, and just so you understand some of the terminology, \"Ga\" means \"billions of years ago\" 'G' for \"Giga-\" \"Ma\" means \"millions of years ago\" 'M' for \"Mega-\" So where we are right now, the Moon has formed, and we're in what we call the Hadean period or actually I shouldn't say \"period.\" It's the Hadean eon of Earth. \"Period\" is actually another time period, so let me make this very clear. It's the Hadean, we are in the Hadean eon, and an eon is kind of the largest period of time that we talk about, especially relative to Earth, and it's roughly 500 million to a billion years is an eon, and what makes the Hadean eon distinctive, well, from a geological point of view what makes it distinctive is really we don't have any rocks from the Hadean period. We don't have any kind of macroscopic-scale rocks from the Hadean period, and that's because at that time, we believe, the Earth was just this molten ball of kind of magma and lava, and it was molten because it was a product of all of these accretion events and all of these collisions and all this kinetic energy turning into heat. If you were to look at the surface of the Earth, if you were to be on the surface of the Earth during the Hadean eon, which you probably wouldn't want to be because you might get hit by a falling meteorite or probably burned by some magma, whatever, it would look like this, and you wouldn't be able to breathe anyway; this is what the surface of the Earth would look like. It would look like a big magma pool, and that's why we don't have any rocks from there because the rocks were just constantly being recycled, being dissolved and churned inside of this giant molten ball, and frankly the Earth still is a giant molten ball, it's just we live on the super-thin, cooled crust of that molten ball. If you go right below that crust, and we'll talk a little bit more about that in future videos, you will get magma, and if you go dig deeper, you'll have liquid iron. I mean, it still is a molten ball. And this whole period is just a violent, not only was Earth itself a volcanic, molten ball, it began to harden as you get into the late Hadean eon, but we also had stuff falling from the sky and constantly colliding with Earth, and really just continuing to add to the heat of this molten ball. Anyway, I'll leave you there, and, as you can imagine, at this point there was no, as far as we can tell, there was no life on Earth. Some people believe that maybe some life could've formed in the late Hadean eon, but for the most part this was just completely inhospitable for any life forming. I'll leave you there, and where we take up the next video, we'll talk a little bit about the Archean eon." + }, + { + "Q": "Sal, aren't we meant to name a molecule on the basis of having the smallest number of branches on the main chain? (I'm confused because my chemistry book says that :s) So for the first example, isn't it meant to be 4-(1-methylpropyl) tridecane instead of 3-methyl-4-propyltridecane? because the first name would mean that there's only one branch instead of two.", + "A": "As you noted, there is more than one possible choice of main chains. When this happens, you do not name the molecule based on the smallest number of branches on the main chain. Instead, you choose the main chain whose substituents are least substituted. Both methyl and propyl are simpler than methylpropyl, so the name in the video is correct. In effect, it boils down to choosing the main chain based on the largest number of substituents.", + "video_name": "Se-ekDNhCDk", + "transcript": "We've got a few more molecular structures to name, so let's look at this first one right here. The first thing you always want to do is identify the longest chain. If we start over here, we have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen carbons, looks pretty long. Now what if we start over here? This looks like it could also be a long chain: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen. So we have two different chains, depending on whether we want to go up here or whether we go over here, that have a length of thirteen. So you're probably asking, which chain do you choose? You should always, if we have two chains of equal lengths and they're the longest chains, you pick the one that will have more branches or more alkyl groups on it. So this group right here, if we pick this from here to here as our chain, we only have one group on it, that group up there. If we pick this chain, starting over here and then going over here, we have two groups on it. We would have this group over here and then we would have this group over here. This is the better chain to use because it has more groups on it. It has more groups, but the groups are smaller and simpler. So let's start counting. And the direction we want to count, we always want to start on the side of the chain where we're going to encounter something first, and everything is closer this end of the chain so we'll start counting here. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, so we have thirteen carbons on our main chain. Let me draw our main chain. So our main chain is this thing in orange I'm drawing right here. That is our main chain. Thirteen, that's three and ten. The prefix is tridec- and it's tridecane because we have all single bonds here, so it's tridecane. Tridecane is the main chain. Then we have two groups over here. This one in green, this only has one the carbon branching off of the main chain, so it's prefix will be meth- and it'll be a methyl group, so that is methyl. That is a methyl group right there. Then this one down here, we have one, two, three carbons. the prefix is prop-, so this is a propyl group. And the methyl is sitting on the three carbon of our main chain and the propyl group is sitting on the four carbon: one, two, three, four. Now when we figure out what order to list them in when we actually write out the name, M, we just do it in alphabetical order. M comes before P, so we right 3-methyl before 4-propyl. The entire compound here is or the entire molecule is 3-methyl-4-propyltridecane And this is actually all going to be one word. You use dashes to separate when you have numbers, but if you have a word followed by a word, it just becomes propyltridecane, so 3-methyl-4-propyltridecane. And we're done. Let's do this one down here. Now this one seems a little bit more complex. The first thing to see is what is the largest chain or the largest ring that we have in our structure? The two candidates, we have this chain over here. This has one, two, three, four, five, six, seven carbons. Let's see how many carbons our ring has. Our ring has one, two, three, four, five, six, seven, eight, nine carbons. So the ring is the largest core structure in this molecule, so that will be our core structure. We have nine carbons-- let me highlight it-- so the ring is this right here. It has nine carbons. The prefix for nine is non-. It's all single bonds, so it's nonane, and it is in a cycle. It's a ring, so it's cyclononane. This part right here, that part right there, is cyclononane. We have several things that branch off of the cyclononane, so let's look at them one at a time, and then we'll think about how we're going to number them on the ring. We looked already at this at this chain that has seven carbons: one, two, three, four, five, six, seven. So that is a heptyl group. This over here, let's see what we're dealing with. We have one, two, three carbons, so that is just a standard propyl group. Then here, we have one, two, three, four carbons, so we could say this is a butyl group. But this isn't just any butyl group. If we use the common naming, the carbon we immediately touch on or that we immediately get to when we go off of our main ring, that branches off into three other carbons, so this is tert-, tert- for three. The tert- is usually written in italics. It's hard to differentiate that when you see it. I'll write it in cursive. This is tert. This is a tertbutyl group. Now, the next question is how do we specify where these different groups sit on this main ring? If you just had one group, you wouldn't have to specify it, but when you have more than one, what you actually do is you figure out which one would be alphabetically first, and that would be number one. Now, we have an H in heptyl, a P in propyl, and tert-butyl, you might say, well, do I use the T or do I use the B? And this is just the convention, you use the B. If you have sec-butyl or tert-butyl, you ignore the sec- or the tert-. If this was an isobutyl or an isopropyl, you actually would use the I, so it's a little bit-- I guess the best way to think about it is there's a dash here so you can kind of ignore it, but if this was isobutyl it would all be one word, so you would consider the I. So in this situation, we would consider the B, and B comes before P or an H, so that is where we will start numbering one. Then to figure out which direction to keep numbering in, we just go in the direction where we're likely to encounter the first or where we will encounter the first side chain, so we'll go in this direction because we get right to the propyl group. One, two, three, four, five, six, seven, eight, nine. So this compound, we're going to start with the alphabetically first side chain, so it's 1-tert-butyl. I'll write this in cursive. 1-tert-butyl. Then the next one alphabetically is the heptyl group. That's H for heptyl, so then it is 5-heptyl. And then we have the propyl, and then it is 2-propyl, and then finally cyclononane, and we're done!" + }, + { + "Q": "Hi,\n\nI have a question about commas and the word 'AND'. If my memory serves me correctly, the rule was that the comma would replace the word 'and' in a list etc. E.g. If the sentence is -: I have an apple and a banana and a mango. We would replace the \"and's\" with commas and it would now read, 'I have an apple, a banana and a mango'. How is it that in the example above, you guys are placing a comma before the word 'and'? Isn't that like saying \"and and\"?", + "A": "The comma you re questioning is called the Oxford comma. For more information, check out the bonus video titled The Oxford Comma.", + "video_name": "DBMQOK64VQY", + "transcript": "- [Voiceover] Hey Paige! - [Voiceover] What's up David? Okay, so I'm about to go to the grocery store, and it looks like it says, \"I need to get squid pickles and chocolate at the grocery store.\" - [Voiceover] Yeah. - [Voiceover] Did you want squid pickles? - [Voiceover] No, I wanted squid and pickles. - [Voiceover] I must have written it down wrong, okay. So I think what we need to do in order to fix this list and avoid confusion like this in the future is using commas to punctuate a list. 'Cause right now this just looks like squid pickles, which, I mean probably delicious, pickled squid. - [Voiceover] Yeah. - [Voiceover] Sure. - [Voiceover] But not what we were looking to get today. - [Voiceover] Right. - [Voiceover] If we don't want to get pickled squid today, then we have to put commas in between the elements of the list. - [Voiceover] Right. - [Voiceover] Right, because this is what commas do. The separate elements of every everything. So let's put in those dividers. I need to get squid, comma, pickles, comma, and chocolate at the grocery store. - [Voiceover] Exactly. - [Voiceover] Okay, so we can punctuate a list by separating out nouns, and I see from the second sentence here, Paige could you give me a read for that? - [Voiceover] I'm going to go for a run, read a chapter of my book and go see the new Colonel Justice movie. - [Voiceover] Oh I hear that's good. So right now it says that, but it could also just be \"I'm going to go for a run read a chapter of my book, and go see...\" you know there's like no-- - [Voiceover] It's a little confusing. - [Voiceover] It's a little confusing, right? There could be such a thing as a run read. - [Voiceover] There probably is. - [Voiceover] You know, like where you go for a jog while holding a book. - [Voiceover] Sounds difficult. - [Voiceover] And so we can also use commas in lists to separate not just nouns like in this first one, but also verb phrases. So I'm going for a run, comma, read a chapter of my book, comma, and go see the new Colonel Justice movie. - [Voiceover] Perfect. - [Voiceover] Cool. So that's how you punctuate a list with commas. - [Voiceover] Yeah, you got it. - [Voiceover] You can learn anything. David out. - [Voiceover] Paige out." + }, + { + "Q": "Would this apply to an equation like f(g(x))? Would that be the same as f(x) * g(x)? or would it be adding? I was told I would add but that just doesn't make sense. I would love a clarification.", + "A": "No, when finding f(g(x)), you first find what g(x) equals, then using the answer to g(x) as x, you find f(x). For example, f(x)=2x g(x)=4x-1 Find f(g(1)) You would first find g(1), which would be 4(1)-1=3. Then, you would use the 3 to find f(x), which would be 2(3)=6. So, f(g(1))=6. Hope that clarifies things :)", + "video_name": "JKvmAexeMgY", + "transcript": "f of x is equal to 7x minus 5. g of x is equal to x to the third power plus 4x. And then they ask us to find f times g of x So the first thing to realize is that this notation f times g of x is just referring to a function that is a product of f of x and g of x. So by definition, this notation just means f of x times g of x. And then we just have to substitute f of x with this definition, g of x with this definition, and then multiply out these algebraic expressions. f of x is right over there. And g of x, is right over there. So let's do it. So this is going to be equal to-- switch back to the orange color. It's going to be equal to f of x, which is 7x minus 5 times g of x, and g of x is x to the third power plus 4x. And you could-- we're multiplying two expressions that each have two terms. You could use FOIL if you like. I don't like using FOIL because you might forget what it's even about. Foil is really just using the distributive property twice. So for example, you take this expression. Whatever you have out here, if you had a 9 out here, or an a, or an x, or anything. Now you have 7x minus 5. If you're multiplying it times this expression, you would multiply this times each term over here. So when you multiply 7x minus 5 times x to the third, you get-- I'll write it this way. You get x to the third times-- actually, let me write it the other way. You get 7x minus 5 times x to the third. And then you have plus 7x minus 5 times 4x. And now we can do the distributive property again. We're not normally used to seeing the things we distribute on the right hand side. It's the same exact idea. We could put the x to the third here as well. And when we distribute, you multiply x to the third times 7x and times negative 5. x to the third times 7x is 7x to the fourth power. X to the third times negative 5 is minus 5x to the third. And then you do it over here. You distribute the 4x over the 7x. 4x times 7x is plus 28x squared. 4x times negative 5 is minus 20x. And let's see if we can simplify this. We only have one fourth degree term, one third degree term, one second degree term, and one first degree term. Actually, we can't simplify this anymore. And we're done. This is the product of those two function definitions. This is f times g of x. It is a new function created by multiplying the other two functions." + }, + { + "Q": "What if you use the current year as the base year, and calculate real GDP for the previous years using the current year's prices? Would that give you any usable information?", + "A": "When you compare the result of this calculation with that obtained from a normal kind of calculation for the actual year involved (that is as a percentage), it will show you how much the prices of goods taken all together have changed.", + "video_name": "lBDT2w5Wl84", + "transcript": "Let's say we're studying a very small and oversimplified country that only sells apples, and we measure the GDP in year one. And we measure that GDP as $1,000. And all of that is due to apples. And we also know that the price of apples in year one was $0.50 a pound. So I'll write it as $0.50 per pound. And let's say that now year one has gone by and even year two has gone by, and we're able to measure the GDP in year two. So the GDP in year two is $1,200. And the price of apples in year two, let's just say it is $0.55 a pound. So my question to you is, GDP, the whole point of measuring GDP, is measuring the productivity of a country. I mean we are measuring in terms of dollars, but we care more about just the dollar amount. We really care is, was this country more productive? And if it was more productive, how much more productive was it? And if we just look at these GDP numbers right over here, this $1,000 versus this $1,200, it gives you the sense that-- well, at least if you just look at the numbers-- $1,200 is 20% larger than $1,000. So if you just look at those numbers right over there, it looks like the GDP grew by 20%. But is that an accurate representation of the productivity of this country? Did it actually produce 20% more goods? And a big clue is looking at this price here. Because some of this GDP actually might have increased just due to price. But that doesn't actually make the country more productive. The quantity, the extra quantity of apples that the country produces, is actually what adds to the total productivity. One way to think about it-- Let me draw a little diagram over here. On this axis, I'll do quantity. On this axis, I will do price. And P1, so if I want to figure out the GDP in year one, I would have the price of apples in year one-- that's the only good or service, just to simplify things-- times the quantity of apples in year one. And then this right over here, the area of this green rectangle, would GDP in year one. And then GDP in year two would be the price in year two. So we're going to go from $0.50 to $0.55. The price in year two times the quantity in year two-- we'll assume some growth as occurred-- times the quantity in year two. And so GDP in year two would be the area of this entire rectangle. And if we want to find the difference between GDP in year two and GDP in year one, it would be the difference in area. So it would be what I am shading in, in blue right over here. And based on the numbers that we went over right over here, the area that I'm shading in, in blue-- so the difference between GDP in year two and GDP in year one, the area I'm shading in blue-- would be this 200, the 200 increment. So this area right over here would be that 200. Now when you look at it over here, you see that that 200, some of it is due to an increase in quantity. But a lot of it is also due to an increase in price. So if we really wanted to figure out how much more productive the country got, and we still want to measure GDP in dollars, maybe we can take a measure of GDP that measures year two's GDP, but it does it in year one's prices. So if we could somehow multiply-- if we could multiply year two's quantity by year one's prices, then we would get this rectangle right over here. And then the difference between that and year one, would give us the incremental GDP in year one prices due to quantity. And that's what we care about. We care about total productivity. When we're thinking about GDP one, we say how much more productive did the country get? So let's try to do it with these numbers right over here. So we can figure out quantity two, we could figure out the quantity in year two just by dividing the GDP by the price. Just by dividing this area of the entire blue rectangle and dividing it by the price, that will give us the quantity. So if we divide 1,200 divided by $0.55-- let me get my calculator out. So if I do 1,200 divided by $0.55, this is my quantity of apples and in pounds in year two. And I'll just round it, 2,182. So this is 2,182. So the quantity in year two is 2,182 pounds. So this is equal to that. And then I could multiply this times the price. So this is this quantity. It's 2,182 pounds. And then I could multiply it times the price in year one at year one's price. So I'm going to multiply it times-- P1 is equal to $0.50 a pound, $0.50 per pound. And this will give me-- so let me just get my calculator out. I should be able to do that one in my head. But let's see 0.5. And I get 1,090. Obviously, I'll round it to 1,091. So this is equal to 1,091. And this is an interesting number. So this is-- you could view this as year two's GDP. In year-- or adjusted for-- I'll write it, adjusted for prices, or adjusted for price increases. Or you could say in year one prices. And what's useful about this is, this says, look, if prices had remained constant, this is what our GDP would have gotten to. If prices did not increase, our GDP would have gotten to this 1,091. 1,091 is this area that I drew in pink here. And so now, you could say if prices were held constant, the growth in GDP would have been $91 not $200. So this area right over here that I'm-- actually, let me do it in a color. Let me do it in orange, maybe. This area right over here, the actual growth, if prices were held constant, would have been $91. We would have gone from $1,000 of GDP to $1,091. So this right over here, that area, is $91 of-- and we could even call it real growth. It really measures the productivity. Now this gives us an interesting, I guess, set of ideas. One idea is to just measure your GDP in the current year's dollars. So this was GDP measured in year two's dollars. It was year two GDP measured in year two dollars, year two prices. So we could call that year two's nominal GDP. Nominal, in name. So it's GDP in name, in that year's prices. But this right over here, where we measured year two's GDP, in some base year's prices-- so it allows a real comparison of how much did our productivity actually increase. Our productivity actually increased by 9%. We produced 9% more apples. This, we call real GDP. Because it gives you a measure of real productivity. It tries to take out price increases. What we'll see in the future, or we might not do it in an introductory course, but in practice, it's kind of hard to really measure what the absolute-- this was a simple economy, where we only had one product. But if you have many, many, many products-- actually gazillions of products in a real economy and the prices are adjusting and the quantities are adjusting, it's not so easy to figure out how to adjust for price. But the folks running the national income accounts do try to do this. So they get a sense of how much was the actual real growth." + }, + { + "Q": "4:05 12 is a number that is in the problem!\n\nAll numbers divisible by both 12 and 20 are also divisible by 12?", + "A": "Yup, and all numbers divisible by both 12 and 20 are also divisible by 20.", + "video_name": "zWcfVC-oCNw", + "transcript": "- [Voiceover] In this video I wanna do a bunch of example problems that show up on standardized exams and definitely will help you with our divisibility module because it's asking you questions like this. And this is just one of the examples. All numbers divisible by both 12 and 20 are also divisible by: And the trick here is to realize that if a number is divisible by both 12 and 20, it has to be divisible by each of these guy's prime factors. So let's take the prime factorization, the prime factorization of 12, let's see, 12 is 2 times 6. 6 isn't prime yet so 6 is 2 times 3. So that is prime. So any number divisible by 12 needs to be divisible by 2 times 2 times 3. So its prime factorization needs to have a 2 times a 2 times a 3 in it, any number that's divisible by 12. Now any number that's divisible by 20 needs to be divisible by, let's take it's prime factorization. 2 times 10 10 is 2 times 5. So any number divisible by 20 needs to also be divisible by 2 times 2 times 5. Or another way of thinking about it, it needs to have two 2's and a 5 in its prime factorization. If you're divisible by both, you have to have two 2's, a 3, and a 5, two 2's and a 3 for 12, and then two 2's and a 5 for 20, and you can verify this for yourself that this is divisible by both. Obviously if you divide it by 20, let me do it this way. Dividing it by 20 is the same thing as dividing by 2 times 2 times 5, so you're going to have the 2's are going to cancel out, the 5's are going to cancel out. You're just going to have a 3 left over. So it's clearly divisible by 20, and if you were to divide it by 12, you'd divide it by 2 times 2 times 3. This is the same thing as 12. And so these guys would cancel out and you would just have a 5 left over so it's clearly divisible by both, and this number right here is 60. It's 4 times 3, which is 12, times 5, it's 60. This right here is actually the least common multiple of 12 and 20. Now this isn't the only number that's divisible by both 12 and 20. You could multiply this number right here by a whole bunch of other factors. I could call them a, b, and c, but this is kind of the smallest number that's divisible by 12 and 20. Any larger number will also be divisible by the same things as this smaller number. Now with that said, let's answer the questions. All numbers divisible by both 12 and 20 are also divisible by: Well we don't know what these numbers are so we can't really address it. They might just be 1's or they might not exist because the number might be 60. It might be 120. Who knows what this number is? So the only numbers that we know can be divided into this number, well we know 2 can be, we know that 2 is a legitimate answer. 2 is obviously divisible into 2 times 2 times 3 times 5. We know that 2 times 2 is divisible into it, cuz we have the 2 times 2 over there. We know that 3 is divisible into it. We know that 2 times 3 is divisible into it. So that's 6. Let me write these. This is 4. This is 6. We know that 2 times 2 times 3 is divisible into it. I could go through every combination of these numbers right here. We know that 3 times 5 is divisible into it. We know that 2 times 3 times 5 is divisible into it. So in general you can look at these prime factors and any combination of these prime factors is divisible into any number that's divisible by both 12 and 20, so if this was a multiple choice question, and the choices were 7 and 9 and 12 and 8. You would say, well let's see, 7 is not one of these prime factors over here. 9 is 3 times 3 so I need to have two 3's here. I only have one 3 here so 9 doesn't work. 7 doesn't work, 9 doesn't work. 12 is 4 times 3 or another way to think about it, 12 is 2 times 2 times 3. Well there is a 2 times 2 times 3 in the prime factorization, of this least common multiple of these two numbers, so this is a 12 so 12 would work. 8 is 2 times 2 times 2. You would need three 2's in the prime factorization. We don't have three 2's, so this doesn't work. Let's try another example just so that we make sure that we understand this fairly well. So let's say we wanna know, we ask the same question. All numbers divisible by and let me think of two interesting numbers, all numbers divisible by 12 and let's say 9, and I don't know, let's make it more interesting, 9 and 24 are also divisible by are also divisible by And once again we just do the prime factorization. We essentially think about the least common multiple of 9 and 24. You take the prime factorization of 9, it's 3 times 3 and we're done. Prime factorization of 24 is 2 times 12. 12 is 2 times 6, 6 is 2 times 3. So anything that's divisible by 9 has to have a 9 in it's factorization or if you did its prime factorization would have to be a 3 times 3, anything divisible by 24 has got to have three 2's in it, so it's gotta have a 2 times a 2 times a 2, and it's got to have at lease one 3 here. And we already have at least one 3 from the 9, so we have that. So this number right here is divisible by both 9 and 24. And this number right here is actually 72. This is 8 times 9, which is 72. So if the choices for this question, let's assume that it was multiple choice. Let's say the choices here were 16 27 5 11 and 9. So 16, if you were to do its prime factorization, is 2 times 2 times 2 times 2. It's 2 to the 4th power. So you would need four 2's here. We don't have four 2's over here. I mean there could be some other numbers here but we don't know what they are. These are the only numbers that we can assume are in the prime factorization of something divisible by both 9 and 24. So we can rule out 16. We don't have four 2's here. 27 is equal to 3 times 3 times 3, so you need three 3's in the prime factorization. We don't have three 3's. So once again, cancel that out. 5's a prime number. There are no 5's here. Rule that out. 11, once again, prime number. No 11's here. Rule that out. 9 is equal to 3 times 3. And actually I just realized that this is a silly answer because obviously all numbers divisible by 9 and 24 are also divisible by 9. So obviously 9 is going to work but I shouldn't have made that a choice cuz that's in the problem, but 9 would work, and what also would work if we had a, if 8 was one of the choices, because 8 is equal to 2 times 2 times 2, and we have a 2 times 2 times 2 here. 4 would also work. That's 2 times 2. That's 2 times 2. 6 would work since that's 2 times 3. 18 would work cuz that's 2 times 3 times 3. So anything that's made up of a combination of these prime factors will be divisible into something divisible by both 9 and 24. Hopefully that doesn't confuse you too much." + }, + { + "Q": "I know that tan(theta) = opp/adj, so, if I want to calculate the opposite side (the length traveled by the globe) at the given time, and at the given time+1 min, shouldn't I be able to:\ntan( pi/4+.2 ) * 500 - tan( pi/4 ) * 500 = 254.25 meters/min\n\n\nThat all divided by 1 minute of course. However, that clearly doesn't give the expected result, which is 200 meters/min... where am I wrong?", + "A": "To calculate [tan(\u00cf\u0080/4 + 0.2) - tan(\u00cf\u0080/4)] * 500 will give you the result whose unit is meters, not meters/min. We need to find the rate of change.", + "video_name": "_kbd6troMgA", + "transcript": "You're watching some type of hot air balloon show, and you're curious about how quickly one hot air balloon in particular is rising. And you have some information at your disposal. You know the spot on the ground that is directly below the hot air balloon. Let's say it took off from that point, it's just been going straight up ever since. And you know, you've measured it out, that you're 500 meters away from there. So you know that you are 500 meters away from that. And you're also able to measure the angle between the horizontal and the hot air balloon. You could do that with, I don't know, I'm not exactly a surveyor, but I guess a viewfinder or something like that. So you're able to-- and I'm not sure if that's the right tool, but there are tools that you can measure the angles between the horizontal and something that's not on the horizontal. So you know that this angle right over here is pi over 4 radians, or 45 degrees. We're going to keep it pi over 4, because when you take derivatives of trig functions, you assume that you're dealing with radians. So right over here, this is pi over 4 radians. And you also are able to measure the rate at which this angle is changing. So this is changing at 0.2 radians per minute. Now my question to you, or the question that you're trying to figure out as you watch this hot air balloon, is how fast is it rising right now? How fast is it rising just as the angle between the horizontal and kind of the line between you and the hot air balloon is pi over 4 radians, and that angle is changing at 0.2 radians per minute? So let's think about what we know and what we're trying to figure out. So we know a couple of things. We know that theta is equal to pi over 4 if we call theta the angle right over here. So this is theta. We also know the rate at which data is changing. We know d theta. Let me do this in yellow. We know d theta dt is equal to 0.2 radians per minute. Now what are we trying to figure out? Well, we're trying to figure out the rate at which the height of the balloon is changing. So if you call this distance right over here h, what we want to figure out is dh dt. That's what we don't know. So what we'd want to come up with is a relationship between dh dt, d theta dt, and maybe theta, if we need it. Or another way to think about it, if we can come up with the relationship between h and theta, then we could take the derivative with respect to t, and we'll probably get a relationship between all of this stuff. So what's the relationship between theta and h? Well, it's a little bit of trigonometry. We know we're trying to figure out h. We already know what this length is right over here. We know opposite over adjacent. That's the definition of tangent. So let's write that down. So we know that the tangent of theta is equal to the opposite side-- the opposite side is equal to h-- over the adjacent side, which we know is going to be a fixed 500. So there you have a relationship between theta and h. And then to figure out a relationship between d theta dt, the rate at which theta changes with respect to t, and the rate at which h changes with respect to t, we just have to take the derivative of both sides of this with respect to t implicitly. So let's do that. And actually, let me move over this h over 500 a little bit. So let me move it over a little bit so I have space to show the derivative operator. So let's write it like that. And now let's take the derivative with respect to t. So d dt. I'm going to take the derivative with respect to t on the left. We're going to take the derivative with respect to t on the right. So what's the derivative with respect to t of tangent of theta? Well, we're just going to apply the chain rule here. It's going to be first the derivative of the tangent of theta with respect to theta, which is just secant squared of theta, times the derivative of theta with respect to t, times d theta dt. Once again, this is just the derivative of the tangent, the tangent of something with respect to that something times the derivative of the something with respect to t. Derivative of tangent theta with respect to theta times the derivative theta with respect to t gives us the derivative of tangent of theta with respect to t, which is what we want when we use this type of a derivative operator. We're taking the derivative with respect to t, not just the derivative with respect to theta. So this is the left hand side. And then the right hand side becomes, well, it's just going to be 1 over 500 dh dt. So 1 over 500 dh dt. We're literally saying it's just 1 over 500 times the derivative of h with respect to t. But now we have our relationship. We have the relationship that we actually care about. We have a relationship between the rate at which the height is changing with respect to time and the rate at which the angle is changing with respect to time and our angle at any moment. So we can just take these values up here, throw it in here, and then solve for the unknown. So let's do that. Let's do that right over here. So we get secant squared of theta. So we get secant squared. Right now our theta is pi over 4. Secant squared of pi over 4. Let me write those colors in to show you that I'm putting these values in. Secant squared of pi over 4. Secant times d theta dt. Well, that is just 0.2. So times 0.2. And then this is going to be equal to 1 over 500. And we want to make sure. Since this is in radians per minute, we're going to get meters per. And this is meters right over here. We're going to get meters per minute right over here. We just want to make sure we know what our units are doing. I haven't written the units here to save some space. But we get 1 over 500 times dh dt. So if we want to solve for dh dt, we can multiply both sides by 500, and you get the rate at which our height is changing is equal to 500 times, let's see, secant squared of pi over 4. That is 1 over cosine squared of pi over 4. Let me write this over here. Cosine of pi over 4 is square root of 2 over 2. Cosine squared of pi over 4 is going to be equal to 2 over 4, which is equal to 1/2. And so secant squared of pi over 4 is just 1 over that, is equal to 2. So this is going to be equal to-- let me rewrite this instead of-- so the secant squared of pi over 4-- let me erase this right over here. Secant squared of pi over 4, all of this business right over here simplifies to 2. So times 2 times 0.2 times 0.2. So what is this going to be? This is going to be 500 times 0.4. So this is equal to 500 times-- let me just write a dot instead-- times 0.4, which is equal to-- let me make sure I get this right. This would be with two 0's and one behind the decimal. Yep, there you go. It would be 200. So the rate at which our height is changing with respect to time right at that moment is 200 meters per minute. dh dt is equal to 200 meters per minute." + }, + { + "Q": "If the plates are still moving, will there be a super continent in the future?", + "A": "yes.", + "video_name": "axB6uhEx628", + "transcript": "We know that new plate material is being formed, and these lithosphere plates on the surface of the Earth are moving around. And that might raise the question in your brain-- what happens if we kind of reverse things? We know the direction they're moving in. What does that tell us about where they came from? So let's just do the thought experiment. Right now, South America and Africa are moving away from each other, because of new plate material being created at the mid-Atlantic rift. Let's rewind it. Let's bring them back together. We know that India is jamming into the Eurasian Plate right now, causing the Himalayas to get higher and higher. What if we rewind that? Let's bring India back down towards Antarctica. Same thing with Australia. We have new plate material being formed between Australia and Antarctica that's making the continents move apart. Let's bring them back together. Let's rewind the clock. Even North America-- it's not as obvious from this diagram, but if you actually look at the GPS data, it becomes pretty obvious that North America, right now is moving in a counterclockwise rotation. So let's rewind it into a-- let's go back, moving it in a clockwise direction. Let's, instead of Eurasia going further away from North America, let's bring it back together. And so what you could imagine is a reality where India, Australia are jammed down into South America-- sorry, into Antarctica. South America and Africa are jammed together. North America is jammed in there. And essentially, Eurasia is also jammed in there. So it looks like they all would clump together if you go back a few hundred million years. And based on, literally-- based on just that thought experiment, you could imagine at one point, all of the continents on the world were merged into one supercontinent. And that supercontinent is called Pangaea-- pan for entire, or whole, and gaea, coming from Gaia, for the world. And it turns out that all of the evidence we've seen actually does make us believe that there was a supercontinent called-- well, we call it Pangaea, now. Obviously, there probably weren't things on the planet calling it anything back then. Or, there were things back then, but not things that would actually go and try to label continents that we know of. But all of the evidence tells us that Pangaea existed about 200 to 300 million years ago, roughly maybe 250 million, give or take, years ago. And I want to be clear. This was not the first supercontinent. To a large degree, it's kind of the most recent supercontinent. And it's easiest for us to construct because it was the most recent one. But we believe that there were other supercontinents before this. That if you rewind even more that you would have to break up Pangaea and it would reform. But we're now going back in time. Or that there were several supercontinents in the past that broke up, reformed, broke up, reformed. And the last time we had a supercontinent was Pangaea, about 250 million years ago. And now it's broken up into our current day geography. Now, I won't go into all of the detail why we believe that there was a Pangaea about 250 million years ago-- or, this diagram tells us, about 225 million years ago, give or take. But I'll go into some of the interesting evidence. On a very high level, you have a lot of rock commonalities between things that would have had to combine during Pangaea. And probably the most interesting thing is the fossil evidence. There are a whole bunch of fossils. And here are examples of it, from species that were around between 200 and 300 million years ago. And their fossils are found in a very specific place. This animal right here, cynognathus-- I hope I'm pronouncing that right-- cynognathus. This animal's fossils are only found in this area of South America on a nice clean band here, and in this part of Africa. So not only does South America look like it fits very nicely into Africa. But the fossil evidence also makes it look like there was a nice clean band where this animal lived and where we find the fossils. So it really makes it seem like these were connected, at least when this animal lived, maybe on the order of 250 million years ago. This species right over here, its fossils are found in this area-- let me do it in a color that has more contrast-- in this area right over here. This plant, its fossils-- now, this starts to connect to a lot of dots between a lot of cont-- its fossils are found in this entire area, across South America, Africa, Antarctica, India, and Australia. And so not only does it look like the continents fit together in a puzzle piece, not only do we get it to a configuration like this if we essentially just rewind to the movement that we're seeing now-- but the fossil evidence also kind of confirms that they fit together in this way, This animal right here, we find fossils on this nice stripe that goes from Africa through India, all the way to Antarctica. Now, this only gives us evidence of the Southern Hemisphere of Pangaea. But there is other evidence. We find kind of continuing mountain chains between North America and Europe. We find rock evidence, where just the way we see the fossils line up nicely. We see common rock that lines up nicely between South America and Africa and other continents that were at once connected. So all the evidence, as far as we can tell now, does make us think that there at one time was a Pangaea. And, for all we know, all the continents are going to keep moving. And maybe in a few hundred million years, we'll have another supercontinent. Who knows?" + }, + { + "Q": "why we only took 5 from 20 what about 2,2?", + "A": "we already have the 2 and 2 from prime factorization of 12. We don t want to repeat them.", + "video_name": "zWcfVC-oCNw", + "transcript": "- [Voiceover] In this video I wanna do a bunch of example problems that show up on standardized exams and definitely will help you with our divisibility module because it's asking you questions like this. And this is just one of the examples. All numbers divisible by both 12 and 20 are also divisible by: And the trick here is to realize that if a number is divisible by both 12 and 20, it has to be divisible by each of these guy's prime factors. So let's take the prime factorization, the prime factorization of 12, let's see, 12 is 2 times 6. 6 isn't prime yet so 6 is 2 times 3. So that is prime. So any number divisible by 12 needs to be divisible by 2 times 2 times 3. So its prime factorization needs to have a 2 times a 2 times a 3 in it, any number that's divisible by 12. Now any number that's divisible by 20 needs to be divisible by, let's take it's prime factorization. 2 times 10 10 is 2 times 5. So any number divisible by 20 needs to also be divisible by 2 times 2 times 5. Or another way of thinking about it, it needs to have two 2's and a 5 in its prime factorization. If you're divisible by both, you have to have two 2's, a 3, and a 5, two 2's and a 3 for 12, and then two 2's and a 5 for 20, and you can verify this for yourself that this is divisible by both. Obviously if you divide it by 20, let me do it this way. Dividing it by 20 is the same thing as dividing by 2 times 2 times 5, so you're going to have the 2's are going to cancel out, the 5's are going to cancel out. You're just going to have a 3 left over. So it's clearly divisible by 20, and if you were to divide it by 12, you'd divide it by 2 times 2 times 3. This is the same thing as 12. And so these guys would cancel out and you would just have a 5 left over so it's clearly divisible by both, and this number right here is 60. It's 4 times 3, which is 12, times 5, it's 60. This right here is actually the least common multiple of 12 and 20. Now this isn't the only number that's divisible by both 12 and 20. You could multiply this number right here by a whole bunch of other factors. I could call them a, b, and c, but this is kind of the smallest number that's divisible by 12 and 20. Any larger number will also be divisible by the same things as this smaller number. Now with that said, let's answer the questions. All numbers divisible by both 12 and 20 are also divisible by: Well we don't know what these numbers are so we can't really address it. They might just be 1's or they might not exist because the number might be 60. It might be 120. Who knows what this number is? So the only numbers that we know can be divided into this number, well we know 2 can be, we know that 2 is a legitimate answer. 2 is obviously divisible into 2 times 2 times 3 times 5. We know that 2 times 2 is divisible into it, cuz we have the 2 times 2 over there. We know that 3 is divisible into it. We know that 2 times 3 is divisible into it. So that's 6. Let me write these. This is 4. This is 6. We know that 2 times 2 times 3 is divisible into it. I could go through every combination of these numbers right here. We know that 3 times 5 is divisible into it. We know that 2 times 3 times 5 is divisible into it. So in general you can look at these prime factors and any combination of these prime factors is divisible into any number that's divisible by both 12 and 20, so if this was a multiple choice question, and the choices were 7 and 9 and 12 and 8. You would say, well let's see, 7 is not one of these prime factors over here. 9 is 3 times 3 so I need to have two 3's here. I only have one 3 here so 9 doesn't work. 7 doesn't work, 9 doesn't work. 12 is 4 times 3 or another way to think about it, 12 is 2 times 2 times 3. Well there is a 2 times 2 times 3 in the prime factorization, of this least common multiple of these two numbers, so this is a 12 so 12 would work. 8 is 2 times 2 times 2. You would need three 2's in the prime factorization. We don't have three 2's, so this doesn't work. Let's try another example just so that we make sure that we understand this fairly well. So let's say we wanna know, we ask the same question. All numbers divisible by and let me think of two interesting numbers, all numbers divisible by 12 and let's say 9, and I don't know, let's make it more interesting, 9 and 24 are also divisible by are also divisible by And once again we just do the prime factorization. We essentially think about the least common multiple of 9 and 24. You take the prime factorization of 9, it's 3 times 3 and we're done. Prime factorization of 24 is 2 times 12. 12 is 2 times 6, 6 is 2 times 3. So anything that's divisible by 9 has to have a 9 in it's factorization or if you did its prime factorization would have to be a 3 times 3, anything divisible by 24 has got to have three 2's in it, so it's gotta have a 2 times a 2 times a 2, and it's got to have at lease one 3 here. And we already have at least one 3 from the 9, so we have that. So this number right here is divisible by both 9 and 24. And this number right here is actually 72. This is 8 times 9, which is 72. So if the choices for this question, let's assume that it was multiple choice. Let's say the choices here were 16 27 5 11 and 9. So 16, if you were to do its prime factorization, is 2 times 2 times 2 times 2. It's 2 to the 4th power. So you would need four 2's here. We don't have four 2's over here. I mean there could be some other numbers here but we don't know what they are. These are the only numbers that we can assume are in the prime factorization of something divisible by both 9 and 24. So we can rule out 16. We don't have four 2's here. 27 is equal to 3 times 3 times 3, so you need three 3's in the prime factorization. We don't have three 3's. So once again, cancel that out. 5's a prime number. There are no 5's here. Rule that out. 11, once again, prime number. No 11's here. Rule that out. 9 is equal to 3 times 3. And actually I just realized that this is a silly answer because obviously all numbers divisible by 9 and 24 are also divisible by 9. So obviously 9 is going to work but I shouldn't have made that a choice cuz that's in the problem, but 9 would work, and what also would work if we had a, if 8 was one of the choices, because 8 is equal to 2 times 2 times 2, and we have a 2 times 2 times 2 here. 4 would also work. That's 2 times 2. That's 2 times 2. 6 would work since that's 2 times 3. 18 would work cuz that's 2 times 3 times 3. So anything that's made up of a combination of these prime factors will be divisible into something divisible by both 9 and 24. Hopefully that doesn't confuse you too much." + }, + { + "Q": "At about 7:12 sal says \"for every one molecule of this, we need one molecule of that.\" I thought Fe2O3 would have an ionic bond and therefore technically wouldn't be a molecule. Am I mistaken?", + "A": "You are not mistaken, but we are often a little sloppy in terminology. We technically should be saying formula unit when speaking of Fe\u00e2\u0082\u0082O\u00e2\u0082\u0083 or any other ionic compound instead of molecule .", + "video_name": "SjQG3rKSZUQ", + "transcript": "We know what a chemical equation is and we've learned how to balance it. Now, we're ready to learn about stoichiometry. And this is an ultra fancy word that often makes people think it's difficult. But it's really just the study or the calculation of the relationships between the different molecules in a reaction. This is the actual definition that Wikipedia gives, stoichiometry is the calculation of quantitative, or measurable, relationships of the reactants and the products. And you're going to see in chemistry, sometimes people use the word reagents. For most of our purposes you can use the word reagents and reactants interchangeably. They're both the reactants in a reaction. The reagents are sometimes for special types of reactions where you want to throw a reagent in and see if something happens. And see if your belief about that substance is true or things like that. But for our purposes a reagent and reactant is the same thing. So it's a relationship between the reactants and the products in a balanced chemical equation. So if we're given an unbalanced one, we know how to get to the balanced point. A balanced chemical equation. So let's do some stoichiometry. Just so we get practice balancing equations, I'm always going to start with unbalanced equations. Let's say we have iron three oxide. Two iron atoms with three oxygen atoms. Plus aluminum, Al. And it yields Al2 O3 plus iron. So remember when we're doing stoichiometry, first of all we want to deal with balanced equations. A lot of stoichiometry problems will give you a balanced equation. But I think it's good practice to actually balance the equations ourselves. So let's try to balance this one. We have two iron atoms here in this iron three oxide. How many iron atoms do we have on the right hand side? We only have one. So let's multiply this by 2 right here. All right, oxygen, we have three on this side. We have three oxygens on that side. That looks good. Aluminum, on the left hand side we only have one aluminum atom. On the right hand side we have two aluminum atoms. So we have to put a 2 here. And we have balanced this equation. So now we're ready to do some stoichiometry. So the stoichiometry essentially ... If I give you... There's not just one type of stoichiometry problem, but they're all along the lines of, if I give you x grams of this how many grams of aluminum do I need to make this reaction happen? Or if I give you y grams of this molecule and z grams of this molecule which one's going to run out first? That's all stoichiometry. And we'll actually do those exact two types of problems in this video. So let's say that we were given 85 grams of the iron three oxide. So 85 grams. So my question to you is how many grams of aluminum do we need? How many grams of aluminum? Well you look at the equation, you immediately see the mole ratio. So for every mole of this, so for every one atom we use of iron three oxide we need two aluminums. So what we need to do is figure out how many moles of this molecule there are in 85 grams. And then we need to have twice as many moles of aluminum. Because for every mole of the iron three oxide, we have two moles of aluminum. And we're just looking at the coefficients, we're just looking at the numbers. One molecule of iron three oxide combines with two molecule of aluminum to make this reaction happen. So lets first figure out how many moles 85 grams are. So what's the atomic mass or the mass number of this entire molecule? Let me do it down here. So we have two irons and three oxygens. So let me go down and figure out the atomic masses of iron and oxygen. So iron is right here, 55.85. I think it's fair enough to round to 56. Let's say we're dealing with the version of iron, the isotope of iron, that has 30 neutrons. So it has an atomic mass number of 56. So iron has 56 atomic mass number. And then oxygen, we already know, is 16. Iron was 56. This mass is going to be 2 times 56 plus 3 times 16. We can do that in our heads. But this isn't a math video, so I'll get the calculator out. Is that right? That's 48 plus 112, right, 160. So one molecule of iron three oxide is going to be 160 atomic mass units. So one mole or 6.02 times 10 to the 23 molecules of iron oxide is going to have a mass of 160 grams. So in our reaction we said we're starting off with 85 grams of iron oxide. How many moles is that? Well 85 grams of iron three oxide is equal to 85 over 160 moles. So that's equal to, 85 divided by 160 equals 0.53125. Equals 0.53 moles. So everything we've done so far in this green and light blue, we figured out how many moles 85 grams of iron three oxide is. And we figured out it's 0.53 moles. Because a full mole would have been 160 grams. But we only have 85. So it's .53 moles. And we know from this balanced equation, that for every mole of iron three oxide we have, we need to have two moles of aluminum. So if we have 0.53 moles of the iron molecule, iron three oxide, then we're going to need twice as many aluminum. So we're going to need 1.06 moles of aluminum. I just took 0.53 times 2. Because the ratio is 1:2. For every molecule of this, we need two molecules of that. So for every mole of this, we need two moles of this. If we have 0.53 moles, you multiply that by 2, and you have 1.06 moles of aluminum. All right, so we just have to figure out how many grams is a mole of aluminum and then multiply that times 1.06 and we're done. So aluminum, or aluminium as some of our friends across the pond might say. Aluminium, actually I enjoy that more. Aluminium has the atomic weight or the weighted average is 26.98. But let's just say that the aluminium that we're dealing with has a mass of 27 atomic mass units. So one aluminum is 27 atomic mass units. So one mole of aluminium is going to be 27 grams. Or 6.02 times 10 to 23 aluminium atoms is going to be 27 grams. So if we need 1.06 moles, how many is that going to be? So 1.06 moles of aluminium is equal to 1.06 times 27 grams. And what is that? What is that? Equals 28.62. So we need 28.62 grams of aluminium, I won't write the whole thing there, in order to essentially use up our 85 grams of the iron three oxide. And if we had more than 28.62 grams of aluminium, then they'll be left over after this reaction happens. Assuming we keep mixing it nicely and the whole reaction happens all the way. And we'll talk more about that in the future. And in that situation where we have more than 28.63 grams of aluminium, then this molecule will be the limiting reagent. Because we had more than enough of this, so this is what's going to limit the amount of this process from happening. If we have less than 28.63 grams of, I'll start saying aluminum, then the aluminum will be the limiting reagent, because then we wouldn't be able to use all the 85 grams of our iron molecule, or our iron three oxide molecule. Anyway, I don't want to confuse you in the end with that limiting reagents. In the next video, we'll do a whole problem devoted to limiting reagents." + }, + { + "Q": "What about 'investing' in a ponzi scheme and withdrawing in the early days before it busts?", + "A": "Lots of people think they are going to be the ones to get out in time! Surprise! A typical characteristic of Ponzi schemes is that when they go, they go fast. If you suspect something is a Ponzi scheme, how will you know whether it is early days or not? You won t. Don t do it.", + "video_name": "5UVpLPtgdF4", + "transcript": "You've probably heard the term Ponzi scheme before, let me write it down, and what we're going to do in this video is explain what it really is. You might have a sense it's some type of scam, that you're taking one person's money and giving it to another, but we're going to do a tangible example of how it actually works. And right here, I have two pictures of the probably the two most famous perpetrators of Ponzi schemes, this is Charles Ponzi right here. It was obviously named after him and then, more recently, this is Bernie Madoff, who pulled off probably the longest-lasting and largest Ponzi scheme of all time. Who knows, maybe there's a longer-lasting and larger one out there that we have still haven't figured out yet, but this is the largest one to date. And Ponzi wasn't the first person to come up with the Ponzi scheme, but they decided to name it after him because he was the first person to really make it famous. This mugshot was taken in the early 1900s when he was finally caught for perpetrating his scheme. So how does it work? So let's start with some investors here. We could get rid of pictures of these two gentlemen. So let's say that I've got a scheme, and what I'm going to do is I'm going to set up my investors, so these are my investors. And then, let's say, we have several time periods. So we can see why the investors think, at least initially, that my scheme is legitimate. So, let's say, that this is the time period, so we have period one, maybe this is years, year one, year two, year three, year four, year five. What I'm going to do is I'm going to write each investor-- I'm going to write down how much money they think they have with me, the person operating the Ponzi scheme and then I'll show you exactly how much money I have and how I can even get away with having the first investors think that I'm legitimate. So let's say I have investor A. So this is going to be their investor's perceived-- let me do this in a different color-- investor's perceived holdings or perceived value. And over here, I'm going to write total actual value. So you can imagine, this is the actual amount of cash that I have. So we have the investor A in blue, and let's say in year one, he gives me $10,000. And I say I I've got a surefire way of doubling his money in the second year. So in the second year, I actually do nothing with the money and you know, I might actually be spending it on my own yachts and you know, fancy suits and whatnot. But let's just say I'm just keeping it in a bank account so you know, he gives it to me, $10,000 and I do nothing with that money. I don't even get interest on it. It's not even in a bank account, I stuff it into my mattress. So the reality is, after a year, it's still only $10,000, but I promised him that I had some type of a genius scheme that could double his money in a year, so I send him a statement that says that his $10,000 is now $20,000. And I feel good about it because I know that he's going to be so excited that his money doubled that he's going to want to keep his money with me, because he'll hope that it can double again. And not only is he going to do that, but he's going to go to the country club and show off to all of his other friends how he was able to do way better than they did with their investments. So he's going to essentially convince other people to join in. So let's say he convinces investor B to join in. Investor B looks at the statement says, hey, this guy running this-- well he doesn't know it's a scheme, let's say he says , Sal seems to know what he's doing, he doubled investor A's money in a year. I'm going to give him a bunch of money, let's say I'm going to give him $15,000 in year two. So how much total actual value do I have in year two now? I have the $10,000 from investor A plus this $15,000, so I have a total of $25,000. This is the actual amount that I have in my bank account, assuming that I'm not spending it on my yacht or my fancy suits. But the total perceived value, let me write that down another line, if everyone actually wanted the amount of money that they thought they had back, I would have to pay out $35,000. But we know that's not going to happen, because people think I'm such a good investor, they want their money to ride as long as possible. So you already see this discrepancy. There's only $25,000 in this little pile of money that I'm collecting, but people think that there should be $35,000. This is their perceived value, because the this guy thought his money doubled, although it didn't, it just sat there. Now, let's say that the next year I sent them statements that say, look, I made super-awesome investments again, the money doubled again. So this guy's money, his perceived value, he gets a statement that says you now have $40,000. This guy down here, investor B, gets a statement that says you now have $30,000. And then they go back to the country club, and they get investor C onboard. They're like, look, both of us have tried out this guy, he's doubled our money two years in a row for both of us, you probably want in on this as well. And investor C is like yeah, well you know, my two buddies, they look like legitimate guys, let me put my money there. Let's say it gets bigger every time because that's usually how these things go, you know, you normally don't don't have just three investors, you'll have hundreds of investors. And the more fake positive returns you get, the more people that want to put their money in. So investor C, let's say he comes in and he puts in $20,000. So what's the reality? Let's focus on it. So there's a perception on year 3, this guy just put his $20,000. A think he's got $40,000, B thinks he has $30,000, so the perceived value here's $40,000 plus $30,000, $70,000 plus $20,000 is $90,000. That's the perceived value. But the actual value is just going to be this $25,000 we had in period two, assuming we didn't spend the money or do anything with it plus the $20,000 that this guy just deposited. So the reality is $45,000. Now let's say that as we go from period three to period four, or let's say right when this guy gets his statement for $40,000, and it's the exact same time period that this guy had put in his $20,000. Let's say person A, he says, you know what, I felt like my ride has gone long enough, I don't want to test fate, let me take my money out and you might say, oh you the scheme will be ruined, but it's going to work because enough money is coming in from new investors to pay this guy off. We now have $45,000 actual value, even though the people think there's $90,000. So if this guy withdraws all of his money, so if he withdraws all of his money so it goes to zero, I have the cash to pay them. I have $40,000 even though people think there's $90,000. So I subtract out $40,000 right here, and there's only $5,000 dollars left in the bank account and since this guy withdrew his money, the perceived value-- this $40,000 is no longer there, I gave the guy the cash-- the perceived value now is $50,000. So I essentially owe people $50,000, investors B and C think that they have $50,000 invested with me, but the reality is that I only have $5,000 in my bank account. And probably a more realistic reality is I was probably spending a lot of this money on my own little luxuries the whole time. But let's continue another way, once again this guy, not only did he double his money for two years, they're all the same country club. This guy doubled his money again, this guy just invested his money. And now this guy says, look, the scheme is legitimate. This legitimized the scheme This is legit. Because, look, I doubled my money for two years and I was able to withdraw the money. So he was able to withdraw his money, so when he goes back to the country club, he gives this guy and that guy more conviction that this is all on the up and up. And then investor D will probably jump in too and say, wow, now that he's doubled money two years in a row, I see it's legitimate. Investor A was able to withdraw his money, I'll give even more money, I'll give $100,000. Maybe this is a ton of people who are now going to put in a $100,000. And then the next year, I double the money again. And obviously, I won't make it exactly double, I'll make it, you know 40% one year and 30% percent the next year so doesn't look too suspicious. I want to make it look like real returns, but for the sake of our math, let's say I double it again. So now and we're in year four. And this guy withdrew all of his money, but investor B now thinks he has $60,000. Investor C thinks he has $40,000. And investor D thinks he has $200,000. Oh, and I forgot to put investor D's deposit here. So when he put a $100,000, I only had $5,000 in my bank account, but then if I add $100,000 I'll now have $105,000 in my bank account after this guy comes in at, you know, at the end of period three, we can imagine. And, with the perceived amount, I owe is $150,000. So the green is what happens after D comes in, so these are no longer valid. But you can see that as more money comes in, I have more and more money to pay out, even though I'm not doing anything. Even though all of these returns are fake. So now I actually have $105,000 even though people think, well that's at the end of period three, at the end of period four, what do people think? People think that I have $300,000 of holdings. Let me write that down. But the reality is I still only have $105,000 of holdings. This is the total actual value. But notice, if this guy or this guy, some of the early investors, wanted to pull out some of their money, although they probably don't want to, because where else can you double your money every year and this guy already showed that I'm good for paying back the money. But even if this guy or this guy wanted to pull out their money, I would be able to give it to them because I have at least enough for those withdrawals. Now, everything would be ruined if everyone gets freaked out or scared and if I have mass withdrawals or if more people withdraw money than there is in the bank plus the amount of money that comes in. So in order for a Ponzi scheme to keep going, and Bernie Madoff was able to do this for very long time, you have to have good, believable, legitimate returns. Although they're not legitimate, they just need to look legitimate. So that you have more money coming in that out. And the whole point of the doing this Ponzi scheme, if you're a stylish criminal, isn't just to keep the cash there, you know, the $10,000 from one period to the next. The whole point of it is to take a lot of that for yourself, for you to live off of and put into some Swiss bank account to be able to escape the country at some point. Anyway, hopefully you found that enjoyable." + }, + { + "Q": "At 2:20 sir, you said that Concrete nouns are those which we can see, count, or measure. We can measure the velocity of air, do we? Is it Concrete or Abstract Noun sir? Thanks!", + "A": "Good question, Jibran! Let s break it down: you can measure the velocity of wind. But is velocity abstract or concrete? I d say it s abstract, because it is about the idea of directional speed. Is air abstract or concrete? Well, I d say it s concrete, because it s nitrogen and oxygen that we can feel and inhale and touch.", + "video_name": "3AF_rN-yN-Y", + "transcript": "- [Voiceover] Hello grammarians. So today I'd like to talk to you about the idea of concrete and abstract nouns, and before we do that, I'd like to get into some word origins or etymology. So let's take each of these words in turn, because I think by digging into what these words mean, literally what they mean and where they come from, we'll get a better understanding of this concept. So both of these words come to us from Latin. Concrete comes to us from the Latin concretus, which means to grow together. So this part of it means grown. And this part means together. It refers to something that, you know, has grown together and become thick and kind of hard to get through and physical. The connotation here is that this is a physical thing. Something that is concrete is physical. Abstract, on the other hand, means to draw something away. So something that is abstract is drawn away from the real, from the concrete, from the physical. So this is not physical. And we make this distinction in English when we're talking about nouns. Is it something that is concrete, is it something you can look at or pick up or smell or sense or something that is abstract, something that isn't physical, but can still be talked about. So for example, the word sadness... Is a noun, right? This is definitely a noun. It's got this noun-making ending, this noun-forming suffix, ness. You know, we take the adjective sad and we toss this ness part onto it, we've got a noun. But can you see sadness? Is it something you can pick up? Sure, you can tell by being, you know observant and empathetic that your friend is sad, but it's not something you can pick up. You can't be like a measurable degree of sad. You couldn't take someone's sadness, put it under a microscope and say \"Oh, Roberta, you are 32 degrees microsad.\" You know, it's not something physical. Concrete things, on the other hand, are things that we can see or count or measure. Just parts of the physical world. So anything you look at, like a dog is concrete, a ball is concrete, a cliff is concrete. Happiness... Is abstract. The idea of freedom... Is abstract. Though the presence of freedom in your life may manifest in physical objects, like \"Oh, my parents let me have the freedom to eat ice cream.\" Ice cream is, you know, a concrete noun. But freedom, the thing that allows you, you know, the permission that you get from your parents to have ice cream. That's not a physical object. So that's basically the difference. So a concrete noun is a physical object and an abstract noun is not. This is why I really wanted to hit the idea that a noun can be a person, place, thing or idea, because nouns can be ideas, and those ideas tend to be abstract. Sadness, happiness, freedom, permission, liberty, injustice. All of these are abstract ideas. That's the difference. You can learn anything. David out." + }, + { + "Q": "at 3:46 , Can you really just square the 1/3 and then plug it into the radical? I have never seen that before.", + "A": "Yes as you re not changing the equation in any way. You re basically squaring a positive multiple then when you place it inside the radical you re square rooting it again so it s exactly the same multiple. (1/3)^2 = 1/9 sqrt(1/9) = 1/3", + "video_name": "WAoaBTWKLoI", + "transcript": "In the last video, in order to evaluate this indefinite integral, we first made the substitution that x is equal to 3 sine theta. And then this got us to an integral of this form. Then we were able to break up these sines and cosines and use a little bit of our trig identities. To get it into the form where we could do u substitution, we did another substitution where we said that u is equal to cosine of theta. And then finally, we were able to get it into a form using that second round of substitution. And this time, it was u substitution. We were able to get it into a form that we could actually take the antiderivative. And we got the final answer here in terms of u. But now we've got to go and undo everything. We have to undo the substitutions. So the last substitution we had done-- we're now going to go in reverse order-- was that u was equal to cosine theta. So you might just want to substitute u with cosine theta here. But then we're going to have everything in terms of cosine theta, which still doesn't get us to x. So the ideal is if I can somehow express u in terms of x. So let's think how we can do it. We know that u is equal to cosine theta. We know the relationship between x and theta is right over here. x is equal to 3 sine theta. So let's write that over here. So we know that x-- let me write it over here-- we know that x is equal to 3 sine theta. So if we could somehow write cosine-- let me rewrite this a little bit differently. Or we could also say that x over 3 is equal to sine theta. I just divided both sides by 3. So if we could somehow re-express this in terms of sine theta, then we can replace all the sine thetas with x over 3's, and we are done. So how can we do that? And I'll actually show you two techniques for doing it. So the first one is to make the realization, OK, u is equal to cosine of theta. If I want to write this in terms of sine of theta, I can just say that this is equal to-- straight up, this is the most fundamental trigonometric identity. Cosine theta is the square root of 1 minus sine squared theta. And we see sine of theta is equal to x over 3. So this is the square root of 1 minus x over 3 squared. So this is u in terms of x. So everywhere we see a u up here we can replace it with this expression. And we are essentially done. We would have written this in terms of x. Now, there's another technique you might sometimes see in a calculus class where someone says, OK, we know that u is equal to cosine theta. We know this relationship. How can we express u in terms of x? And we'll say, let's draw a right triangle. They'll draw a right triangle like this. They'll draw a right triangle, and they'll say, OK, look, sine of theta is x over 3. So if we say that this is theta right over here, sine of theta is the same thing as opposite over hypotenuse. Opposite over hypotenuse is equal to x over 3. So let's say that this is x and then this right over here is 3. Then the sine of theta will be x over 3. So we look at that first substitution right over here. But in order to figure out what u is in terms of x, we need to figure out what cosine of theta is. Well, cosine is adjacent over hypotenuse. So we have to figure out what this adjacent side is. Well, we can just use the Pythagorean theorem for that. Pythagorean theorem would tell us that this is going to be the square root of the hypotenuse squared, which is 9, minus the other side squared, minus x squared. So from this, we fully solved the right triangle in terms of x. We can realize that cosine of theta is going to be equal to the adjacent side, square root of 9 minus x squared, over the hypotenuse, over 3, which is the same thing as 1/3 times the square root of 9 minus x squared, which is the same thing if we square 1/3 and put it into the radical. So we're essentially going to take the square-- 1/3 is the same thing as the square root of 1/9. So can rewrite this as the square root of 1/9 times 9 minus x squared. Essentially, we just brought the 1/3 third into the radical. Now it's 1/9. And so now this is going to be the same thing as the square root of 1 minus x squared over 9, which is exactly this thing right over here. x squared over 9 is the same thing as x over 3 squared. So either way, you get the same result. I find using the trig identity right over here to express cosine of theta in terms of sine theta and then just do the substitution to be a little bit more straightforward. But now we can just substitute into the original thing. So either of these-- I can write it as either way-- this thing right over here, this is the same thing as 1 minus x squared over 9 to the 1/2 power. That's what u is equal to. And everywhere we see u, we just substitute it with this thing. So our final answer in terms of x is going to be equal to 243 times u to the fifth, this to the fifth power over 5. This to the fifth power is 1 minus x squared over 9. It was to the 1/2, but if we raise that to the fifth power, it's now going to be to the 5/2 power over 5 minus this to the third power, 1 minus x squared over 9 to the 3/2 raising this to the third power-- that's this right over here-- over 3, and then all of that plus c. And we're done. It's messy, but using first trig substitution then u substitution, or trig substitution then rearranging using a couple of our techniques for manipulating these powers of trig functions, we're able to get into a form where we could use u substitution, and then we were able to unwind all the substitutions and actually evaluate the indefinite integral." + }, + { + "Q": "I couldn't seem to find any videos explaining triangle altitudes and how to use them. I have a math problem that I can't seem to figure out:\n\n~An equilateral triangle has an altitude length of 10 feet. Determine the length of a side of a triangle.\n\nWould the sides be 10 feet as well?", + "A": "They would not, because the height of the equilateral triangle is not the length of a side. Before giving you the answer, let me give you a few hints, and you try it out. 1. Draw an equilateral triangle and mark the altitude as a dashed line on the triangle. 2. Let each side be of length l Can you solve for the side length? Hint: Think Pythagoras", + "video_name": "aGwT2-RERXY", + "transcript": "What I want to do in this video is to show that if we start with any arbitrary triangle-- and this will be the arbitrary triangle that we're starting with-- that we can always make this the medial triangle of a larger triangle. And when we say the medial triangle, we mean that each of the vertices of this triangle will be the midpoint of the sides of a larger triangle. And I wanted to show that you can always construct that. If you start with this triangle, you can always have this be the medial triangle of a larger triangle. So to do that, let's draw a line that goes through this point right over here, but that's parallel to this line down here. So this line and this line up here are going to be parallel. So just like that. And immediately we can start to say some interesting things about the angles. So if we have a transversal right over here, we could view this side as a transversal of these two parallel lines, or of this line in the segment. We know that alternate interior angles are congruent. So that angle is going to be congruent to that angle. And we also know that this angle in blue, is going to be congruent to that angle right over there. Now, let's do that for the other two sides. So let's create a line that is parallel to this side of the triangle, but that goes through this point right over here. So let me draw it as well as possible. And so these two characters are going to be parallel, and you could always construct a line that's parallel to another line that goes to a point that's not on that line. And so once again, we can use alternate interior angles. We know that if this angle right over here-- let's say we have this orange angle-- it's alternate interior angle is this angle right over there. We also have corresponding angles. This blue angle corresponds to this angle right over here. So it will correspond to that angle right over there. And now let's draw another line that is parallel to this line right over here, but it goes through this vertex. It goes through the vertex that's opposite that line. And so let me just draw it. And you can always construct these parallel lines just like that. And let's see what happens. So once again, these two lines are parallel. So you could view this green line as a transversal. If this green line is a transversal, this corresponding angle is this angle right over here. If we view this green line as a transversal of both of these pink lines, then this angle corresponds to this angle right over here. If we view this yellow line as a transversal of both of these pink lines-- actually, let's look at it this way. View the pink line as a transversal of these two yellow lines, then we know that this angle corresponds to this angle right over here. And if you view this yellow line as a transversal of these two pink lines, then this angle corresponds to this angle right over here. And then the last thing we need to think about is if we think about the two green parallel lines and you view this yellow line as a transversal, then this corresponding angle in orange is right over here. This corresponds to that angle, because this yellow line is a transversal on both of these green lines. So what I've just shown starting with this inner triangle right over here is that if I construct these parallel lines in this way, that I now have four triangles if I include the original one, and they're all going to be similar to each other. And we know that they're all similar because they all have the exact same angles. You just need two angles to prove similarity. But all four of these triangles have the exact three angles. Now, the other thing we can show is that they're congruent. So all of these four are similar. And we also know they're congruent. For example, this side right over here in yellow is the side in this triangle, between the orange and the green side, is the side between the orange and the green side on this triangle right over here. So these two-- we have an angle, a side, and an angle. Angle-side-angle congruency. So these two are going to be congruent to each other. Then over here, on this inner triangle, our original triangle, the side that's between the orange and the blue side is going to be congruent to the side between the orange and the blue side on that triangle. Once again, we have angle-side-angle congruency. So this is congruent to this, which is congruent to that. All of these are going to be congruent. And by the same exact argument, this middle triangle is going to be congruent to this bottom triangle. You have an angle, blue angle, purple side, green angle. Blue angle, purple side, green angle. They're congruent to each other. So you have all of these triangles are congruent to each other. So their corresponding sides are equal. So if you look at this triangle over here, we know that the side between the blue angle and the green angle is going to be equal to this angle right over here. Sorry, equal to this length. So it's going to be equal to this length. Between the blue and the green we have this length, between the blue and the green we have that length, between the blue and the green we have that length right over there. So you immediately see that this point-- and let me label it now, maybe I should've labeled it before. If we call that point A, we see that A is the midpoint of-- let's call this point B, and call this point C right over here. So A is the midpoint of BC. So that's fair enough. So I was able to construct it in that way. Now let's look at the other sides. So this green side on all the triangles is the side between the blue and the orange angle. So between the blue and the orange angle, you have the green side, between the blue and the orange angle you have the green side. So once again, this length is equal to this length. And so if we call this point over here D, and maybe this point over here E, you see that D is the midpoint of BE. And then finally, the yellow side is between the green and the orange. So between the green and the orange, we have a yellow side. Between the green and the orange you have a yellow side. All of these triangles are congruent. So once again, let me call this F. We see that F is the midpoint of EC. So we've done what we wanted to do. We've shown that if you start with any arbitrary triangle, triangle ADF, we can construct a triangle BCE so that ADF is triangle BCE's medial triangle. And all that means is that the vertices of ADF sit on the midpoints of BCE. So you might say Sal, that by itself is interesting, but what's the whole point of this? The whole point of this is actually, I wanted to use this fact that if you give me any triangle, I can make it the medial triangle of the larger one to prove that the altitudes of this triangle are concurrent. And to see that, let me first draw the altitudes. So an altitude from vertex A looks like this. It starts at the vertex, goes to the opposite side, and is perpendicular to the opposite side. If I draw an altitude from vertex D, it would look like this. And if I draw an altitude from vertex F, it will look like this. And what I did, this whole set up of this video is to show, to prove that these will always be concurrent. And you might say, wait how do we know that they are concurrent? Well all you have to do is think about how they interact with the larger triangle. What are these altitudes to the larger triangle? Well, this yellow altitude to the larger triangle. Remember, these two yellow lines, line AD and line CE are parallel. So if this is a 90-degree angle, so its alternate interior angle is also going to be 90 degrees. So this right over here is perpendicular to CE, and it bisects CE, because we know that ADE is the medial triangle. This is the midpoint. So this right over here is perpendicular bisector. This is a perpendicular bisector for the larger triangle, for triangle BCE. So this altitude for the smaller one is a perpendicular bisector for the larger one. We can do that for all of them. If this angle right over here is 90 degrees, then this angle right over there is going to be 90 degrees, because this line is parallel to this, this is a transversal, alternate interior angles are the same. So this line right over here, this altitude of the smaller triangle, it bisects right at the midpoint of the larger one, on this side, and it's also a perpendicular bisector. So it's a perpendicular bisector of the larger triangle. And then finally, the same thing is true of this altitude right over here. It bisects this side of the larger triangle at a 90-degree angle. We know that because these two magenta lines the way we constructed the larger triangle, they're going to be parallel. So once again, this is a perpendicular bisector. So this whole reason, if you just give me any triangle, I can take its altitudes and I know that its altitude are going to intersect in one point. They're going to be concurrent. Because for any triangle, I can make it the medial triangle of a larger one, and then it's altitudes will be the perpendicular bisector for the larger And we already know that the perpendicular bisectors for any triangle are concurrent. They do intersect in exactly one point." + }, + { + "Q": "Did she ever have arms or a head or did she lose them ?", + "A": "Marble is very fragile material. Sculptors often carved torsos separately from heads (sometimes) and from arms, particularly arms held out and away from the body. The elements can become detached. They also can break off.", + "video_name": "TPM1LuW3Y5w", + "transcript": "STEVEN ZUCKER: We're in the Louvre at the top of one of the grand staircases. And we're looking at the \"Nike of Samothrace,\" that dates to the second century CE, or after Christ. BETH HARRIS: So we're in the Hellenistic period. And the sculpture is nine feet high, so it's really large. STEVEN ZUCKER: It's called the \"Nike of Samothrace\" because it was found on the island in the north of the Aegean which is called Samothrace. It was found in a sanctuary in the harbor that actually faces in such a way the predominant wind that blows off the coast actually seems to be enlivening her drapery. BETH HARRIS: So she never stood on the prow of a real boat. STEVEN ZUCKER: No, she stood on the prow of a stone ship that was within a temple environment. BETH HARRIS: So she's the goddess of victory. She's a messenger goddess who spreads the news of victory. STEVEN ZUCKER: In fact, there are some reconstructions of what the sculpture would've originally looked like that show her as literally a herald with a horn. This is an image that will have an enormous impact on Western art. But you had mentioned the Hellenistic before. And so gone is all of that very reserved, high classical style. And in its place is a kind of voluptuousness. is a kind of windswept energy that is full of motion and full of emotion. BETH HARRIS: I feel as though she moves in several directions at the same time. She's grounded by her legs but strides forward. Her torso lifts up. Her abdomen twists. Her wings move back. One can almost feel the wind around her, whipping her, pulling back that drapery that flows out behind her, swirling around her abdomen, where it really reminds us of, actually, the sculptures of hundreds of years earlier on the Parthenon frieze. STEVEN ZUCKER: Yes, exactly. But instead of the quiet, relaxed attitude of the gods on Mount Olympus, you have instead this sense of energy and a goddess that's responding, in this case, to actually natural forces. BETH HARRIS: The environment. STEVEN ZUCKER: Absolutely, just as we would stand there, very likely having the wind whip around us. BETH HARRIS: And that drapery that clings to her body and creates so many creases and folds that play against the light, and the different texture of her wings-- the marble is really made to do so many different things in terms of texture. STEVEN ZUCKER: So here is a culture that has studied the body, celebrated the body, and then is willing then to use the body for tremendous expressive force." + }, + { + "Q": "how dose it make oxygen is only trees do?", + "A": "Bacteria also make oxygen.", + "video_name": "pBZ-RiT5nEE", + "transcript": "In the video on electronegativity, we learned how to determine whether a covalent bond is polar or nonpolar. In this video, we're going to see how we figure out whether molecules are polar or nonpolar and also how to apply that polarity to what we call intermolecular forces. Intermolecular forces are the forces that are between molecules. And so that's different from an intramolecular force, which is the force within a molecule. So a force within a molecule would be something like the covalent bond. And an intermolecular force would be the force that are between molecules. And so let's look at the first intermolecular force. It's called a dipole-dipole interaction. And let's analyze why it has that name. If I look at one of these molecules of acetone here and I focus in on the carbon that's double bonded to the oxygen, I know that oxygen is more electronegative than carbon. And so we have four electrons in this double bond between the carbon and the oxygen. So I'll try to highlight them right here. And since oxygen is more electronegative, oxygen is going to pull those electrons closer to it, therefore giving oxygen a partial negative charge. Those electrons in yellow are moving away from this carbon. So the carbon's losing a little bit of electron density, and this carbon is becoming partially positive like that. And so for this molecule, we're going to get a separation of charge, a positive and a negative charge. So we have a polarized double bond situation here. We also have a polarized molecule. And so there's two different poles, a negative and a positive pole here. And so we say that this is a polar molecule. So acetone is a relatively polar molecule. The same thing happens to this acetone molecule down here. So we get a partial negative, and we get a partial positive. So this is a polar molecule as well. It has two poles. So we call this a dipole. So each molecule has a dipole moment. And because each molecule is polar and has a separation of positive and negative charge, in organic chemistry we know that opposite charges attract, So this negatively charged oxygen is going to be attracted to this positively charged carbon. And so there's going to be an electrostatic attraction between those two molecules. And that's what's going to hold these two molecules together. And you would therefore need energy if you were to try to pull them apart. And so the boiling point of acetone turns out to be approximately 56 degrees Celsius. And since room temperature is between 20 and 25, at room temperature we have not reached the boiling point of acetone. And therefore, acetone is still a liquid. So at room temperature and pressure, acetone is a liquid. And it has to do with the intermolecular force of dipole-dipole interactions holding those molecules together. And the intermolecular force, in turn, depends on the electronegativity. Let's look at another intermolecular force, and this one's called hydrogen bonding. So here we have two water molecules. And once again, if I think about these electrons here, which are between the oxygen and the hydrogen, I know oxygen's more electronegative than hydrogen. So oxygen's going to pull those electrons closer to it, giving the oxygen a partial negative charge like that. The hydrogen is losing a little bit of electron density, therefore becoming partially positive. The same situation exists in the water molecule down here. So we have a partial negative, and we have a partial positive. And so like the last example, we can see there's going to be some sort of electrostatic attraction between those opposite charges, between the negatively partially charged oxygen, and the partially positive hydrogen like that. And so this is a polar molecule. Of course, water is a polar molecule. And so you would think that this would be an example of dipole-dipole interaction. And it is, except in this case it's an even stronger version of dipole-dipole interaction that we call hydrogen bonding. So at one time it was thought that it was possible for hydrogen to form an extra bond. And that's where the term originally comes from. But of course, it's not an actual intramolecular force. We're talking about an intermolecular force. But it is the strongest intermolecular force. The way to recognize when hydrogen bonding is present as opposed to just dipole-dipole is to see what the hydrogen is bonded to. And so in this case, we have a very electronegative atom, hydrogen, bonded-- oxygen, I should say-- bonded to hydrogen. And then that hydrogen is interacting with another electronegative atom like that. So we have a partial negative, and we have a partial positive, and then we have another partial negative over here. And this is the situation that you need to have when you have hydrogen bonding. Here's your hydrogen showing intermolecular force here. And what some students forget is that this hydrogen actually has to be bonded to another electronegative atom in order for there to be a big enough difference in electronegativity for there to be a little bit extra attraction. And so the three electronegative elements that you should remember for hydrogen bonding are fluorine, oxygen, and nitrogen. And so the mnemonics that students use is FON. So if you remember FON as the electronegative atoms that can participate in hydrogen bonding, you should be able to remember this intermolecular force. The boiling point of water is, of course, about 100 degrees Celsius, so higher than what we saw for acetone. And this just is due to the fact that hydrogen bonding is a stronger version of dipole-dipole interaction, and therefore, it takes more energy or more heat to pull these water molecules apart in order to turn them into a gas. And so, of course, water is a liquid at room temperature. Let's look at another intermolecular force. And this one is called London dispersion forces. So these are the weakest intermolecular forces, and they have to do with the electrons that are always moving around in orbitals. And even though the methane molecule here, if we look at it, we have a carbon surrounded by four hydrogens for methane. And it's hard to tell in how I've drawn the structure here, but if you go back and you look at the video for the tetrahedral bond angle proof, you can see that in three dimensions, these hydrogens are coming off of the carbon, and they're equivalent in all directions. And there's a very small difference in electronegativity between the carbon and the hydrogen. And that small difference is canceled out in three dimensions. So the methane molecule becomes nonpolar as a result of that. So this one's nonpolar, and, of course, this one's nonpolar. And so there's no dipole-dipole interaction. There's no hydrogen bonding. The only intermolecular force that's holding two methane molecules together would be London dispersion forces. And so once again, you could think about the electrons that are in these bonds moving in those orbitals. And let's say for the molecule on the left, if for a brief transient moment in time you get a little bit of negative charge on this side of the molecule, so it might turn out to be those electrons have a net negative charge on this side. And then for this molecule, the electrons could be moving the opposite direction, giving this a partial positive. And so there could be a very, very small bit of attraction between these two methane molecules. It's very weak, which is why London dispersion forces are the weakest intermolecular forces. But it is there. And that's the only thing that's holding together these methane molecules. And since it's weak, we would expect the boiling point for methane to be extremely low. And, of course, it is. So the boiling point for methane is somewhere around negative 164 degrees Celsius. And so since room temperature is somewhere around 20 to 25, obviously methane has already boiled, if you will, and turned into a gas. So methane is obviously a gas at room temperature and pressure. Now, if you increase the number of carbons, you're going to increase the number of attractive forces that are possible. And if you do that, you can actually increase the boiling point of other hydrocarbons dramatically. And so even though London dispersion forces are the weakest, if you have larger molecules and you sum up all those extra forces, it can actually turn out to be rather significant when you're working with larger molecules. And so this is just a quick summary of some of the intermolecular forces to show you the application of electronegativity and how important it is." + }, + { + "Q": "what is a propaganda", + "A": "Information, especially of a biased or misleading nature, used to promote a political cause or point of view.", + "video_name": "dHXzusNSF60", + "transcript": "Despite the fact that Wilson had just won reelection in 1916 based on a platform of keeping the United States out of war, by April of 1917, the administration had decided that Germany had gone too far. And in particular, had gone too far with the unrestricted submarine warfare. So this right over here is a picture of President Wilson on April 2, 1917, giving a war message to Congress as to why the US needs to declare war on Germany. And April 4, Congress passes the resolution to declare war. And then the President approves it on April 6. So by early April, the United States was at war with Germany. Which is a good time to start thinking about, why did all of this happen. Now, the things that are typically cited, and these are the things that are inflamed public opinion in the US and that many of which were cited by President Woodrow Wilson. And in this tutorial that this is part of on khanacademy.org, I put the entire text of his speech, which I highly recommend reading to see all of the things the President Wilson cited in his speech. But just as a summary of that, the things that tend to get cited most often are the unrestricted submarine warfare on the part of Germany. And particular cases or the most cited example of that is the sinking of the Lusitania. The Germans had stopped doing that for a little under two years. But then, as we enter into 1917, they began doing it again. And it also made the Americans quite angry to realize that the Germans were trying to incite the Mexicans against them. So you have the Zimmerman telegram. Zimmerman telegram is also a reason that the Wilson administration, and why people in general, were fairly angry about things. Now, on top of that, there were atrocities committed by the Germans in their march through Belgium as they were trying to execute on the Schlieffen Plan. So Belgian atrocities. And these were earlier in the war in 1914, which immediately made many Americans not like what's going on. Belgian atrocities. And to put on top of that, the British were able to leverage the Belgian atrocities to fairly, to execute a fairly effective propaganda campaign in America. Now on top of that-- and this is something that Wilson speaks very strongly about in his speech-- is the notion of fighting for democracy. And what you have here, in the First World War, the Central Powers. You're talking about the German Empire, you're talking about the Austro-Hungarians. These are monarchies. These are emperors who are controlling it. And even though the UK, the United Kingdom, was nominally a kingdom, it was really a democracy. At least for those who could vote. We're not talking about the entire British Empire. So UK is functionally a democracy, democratic. And so was the Third French Republic. And so was France. So there's this argument that the US is fighting for the representation of people. Now, there is a more cynical argument that some people have made. And I think it's reasonable to give that to due time. And one of the cynical arguments, or more cynical arguments, is that the US had close financial and trade ties to Britain, not to mention cultural ties. Financial ties to the British. On top of that, you had very successful British propaganda. One, talking about the atrocities in Belgium, which did actually happen. But the British were able to exploit this as a propaganda machine. Successful propaganda. But they also spread rumors that after the sinking of the Lusitania that the Germans had their school children celebrating. And these were all made up propaganda. And then, more cynical view of why the US entered the war-- and this is true of probably most wars-- is that there was a lot of lobbying on the part of war profiteers. In fact, in \"Little Orphan Annie,\" Daddy Warbucks, the name, the reason why his last name is Warbucks is because he made his fortune as a war profiteer during World War I. And war profiteers, these are people who might be selling arms to the Allies. Or who might sell arms to the US government if the US were to get into a war that might somehow supply the troops. And it includes, potentially, folks on Wall Street. There were significant lending to the Allies, and mainly the Allies, not the Central Power. And so the view is if the Allies win, those loans are going to be made good. And I had the entire text of the speech from Senator George Norris who was one of five senators, or sorry, one of six senators to vote against the resolution There were 50 representatives who also voted against it. This is a little excerpt but also in this tutorial, I have the full text of his speech. And I highly, highly, highly recommend reading that along with Wilson's text of his speech to Congress in his war message. But I'll just read this part because it does, I think, point out that the US, from the beginning, did have biases that were more pro-British. And so this is part of his speech. \"The reason given by the President in asking Congress to declare war against Germany is that the German government has declared certain war zones, within which by the use of submarines, she sinks, without notice, American ships and destroys American lives. The first war zone was declared by Great Britain. She gave us and the world notice of it on the 4th day of November 1914. The zone became effective November 5, 1914. This zone, so declared by Great Britain, covered the whole of the North Sea. The first German war zone was declared on the 4th day of February, 1915, just three months after the British war zone was declared. Germany gave 15 days notice of the establishment of her zone, which became effective on the 18th day of February, 1915. The German war zone cover the English Channel and the high seawaters around the British Isles. It is unnecessary to cite authority to show that both of these orders declaring military zones were illegal and contrary to international law. It is sufficient to say that our government has officially declared both of them to be illegal and has officially protested against both of them. The only difference is that, in the case of Germany we have persisted in our protest, while in the case of England, we have submitted.\" And I encourage you, once again, to read the text of both Wilson's speech and Senator Norris' speech and come up to your, with your own decisions. And it might be a little bit of both." + }, + { + "Q": "on minute 2:50, Sal says US government buys treasury securities, or other kind of securities...\nthis means treasury bonds, that they issued themselves right?\nwhat other kind of securities is Sal referring to?\nAlso, who are those people government buys securities from?", + "A": "Under normal operations, the Fed (not the US Government) buys treasury securities. In the last few years, in the wake of the credit crisis, the Fed has taken to purchasing other types of securities as well (MBS, for example). They buy these on the open market, through broker/dealers.", + "video_name": "wDuCOxDxMzY", + "transcript": "Let's say we have two banks, bank A and bank B, and you might already know that banks, all banks, lend out the great majority of the money that they get in as deposits, but they keep some of the money as reserves. One, just in case their depositor comes and hey, can I have some of my money back and two because the central bank, the Federal Reserve, says you have to keep a certain amount of your deposits in reserve. There is a reserve requirement, but you can imagine over the course of doing transactions, thousands of transactions a day, millions maybe, maybe bank B more of its depositors come by and say hey, give me some of my deposits back. Obviously he's lent out a lot of that money and so he starts running low on reserves. Maybe bank A, the depositors haven't asked for the money or for whatever reason Bank A is sitting on a lot of cash. In this situation, what you're going to have happen is bank A is going to lend some reserves, is going to lend some cash to bank B. This is lending some cash and they'll charge an interest rate for lending that cash. Maybe it will be 5% interest and that won't be 5% per day and usually these loans are on a per day basis and then the next day it'll be renegotiated on a per day basis, but it's not 5% per day. It'll be 5% per year, so it will be a much smaller fraction, but usually as I mentioned this lending takes place on a per day basis. We'll say hey, this is your cash for just tonight. If you need it for the next night, we'll talk again and maybe it will be another 5% or maybe the interest rate can change again. Let's say the Federal Reserve is sitting over here and for whatever reason wants to stimulate the economy. This is the Fed and they want to stimulate the economy so they start printing some money. I should do money in green. The Federal Reserve here, they're starting to print some money and they want to do two things. They want to inject this money into the banking system which essentially, hopefully, will find its way into the economy and they also want to lower the interest rate, especially the short term interest rate. This overnight borrowing. Remember this is the annual interest rate, but this is an overnight loan. Overnight loan. When I talk about the short term interest rate, I'm talking about the interest rate on loans that are made over very short periods of time. What the Federal Reserve will do is what's called open market operations. They will go to the market and maybe directly to these banks or some other banks and they will buy treasuries. They will give this money to the market and in exchange, they will usually buy treasury securities. Sometimes something slightly different, but usually very safe securities and maybe it's temporarily buy. I'll take about repurchase agreements in the future. What happens is that this cash goes in the hands of the people who just sold the treasury securities and they have to deposit it in banks. They might deposit it in this bank over here. They might deposit it in this bank over here or other banks, but the net-net effect is that there's more cash now in the banking system. If there's more cash in the banking system, this guy right over here needs less. This guy needs less cash, so it lowers demand. It lowers the demand for cash and then this guy has more to give, so it raises supply. Raises the supply because some people maybe just took some of this cash and deposited with them. If it raises the supply of cash and this guy needs it less, then the rate to borrow this cash is going to go down. Maybe instead of 5%, it's goes down to 4%. What that would do is it would lower the short term of the yield curve, the short end of the yield curve. Let me draw a yield curve right over here. This is maturity on this axis, maturity, and this is yield. Let's say the yield curve before looked like this. Let's say it looked like this where this right over here is 5% and this overnight. Overnight. This might be yield on, I don't know, one year debt. This might be yield on, I don't know, maybe it's five year debt. Whatever, I could keep going, but by doing this open market operation, the Fed was able to do both of its goals. It was able to inject cash, printed cash, into the economy and it's also able to lower the interest rate. It took it from being 5% to down to 4%. Now because of this open market operation, the Fed, the yield curve might start to look something like that." + }, + { + "Q": "why is velocity zero when no external forces are acting on body?", + "A": "Velocity may or may not be zero. Acceleration is zero because F = ma (Newton s second law)", + "video_name": "VrflZifKIuw", + "transcript": "I will now do a presentation on the center of mass. And the center mass, hopefully, is something that will be a little bit intuitive to you, and it actually has some very neat applications. So in very simple terms, the center of mass is a point. Let me draw an object. Let's say that this is my object. Let's say it's a ruler. This ruler, it exists, so it has some mass. And my question to you is what is the center mass? And you say, Sal, well, in order to know figure out the center mass, you have to tell me what the center of mass is. And what I tell you is the center mass is a point, and it actually doesn't have to even be a point in the object. I'll do an example soon where it won't be. But it's a point. And at that point, for dealing with this object as a whole or the mass of the object as a whole, we can pretend that the entire mass exists at that point. And what do I mean by saying that? Well, let's say that the center of mass is here. And I'll tell you why I picked this point. Because that is pretty close to where the center of mass will be. If the center of mass is there, and let's say the mass of this entire ruler is, I don't know, 10 kilograms. This ruler, if a force is applied at the center of mass, let's say 10 Newtons, so the mass of the whole ruler is 10 kilograms. If a force is applied at the center of mass, this ruler will accelerate the same exact way as would a point mass. Let's say that we just had a little dot, but that little dot had the same mass, 10 kilograms, and we were to push on that dot with 10 Newtons. In either case, in the case of the ruler, we would accelerate upwards at what? Force divided by mass, so we would accelerate upwards at 1 meter per second squared. And in this case of this point mass, we would accelerate that point. When I say point mass, I'm just saying something really, really small, but it has a mass of 10 kilograms, so it's much smaller, but it has the same mass as this ruler. This would also accelerate upwards with a magnitude of 1 meters per second squared. So why is this useful to us? Well, sometimes we have some really crazy objects and we want to figure out exactly what it does. If we know its center of mass first, we can know how that object will behave without having to worry about the shape of that object. And I'll give you a really easy way of realizing where the center of mass is. If the object has a uniform distribution-- when I say that, it means, for simple purposes, if it's made out of the same thing and that thing that it's made out of, its density, doesn't really change throughout the object, the center of mass will be the object's geometric center. So in this case, this ruler's almost a one-dimensional object. We just went halfway. The distance from here to here and the distance from here to here are the same. This is the center of mass. If we had a two-dimensional object, let's say we had this triangle and we want to figure out its center of mass, it'll be the center in two dimensions. So it'll be something like that. Now, if I had another situation, let's say I have this square. I don't know if that's big enough for you to see. I need to draw it a little thicker. Let's say I have this square, but let's say that half of this square is made from lead. And let's say the other half of the square is made from something lighter than lead. It's made of styrofoam. That is lighter than lead. So in this situation, the center of mass isn't going to be the geographic center. I don't know how much denser lead is than styrofoam, but the center of mass is going to be someplace closer to the right because this object does not have a uniform density. It'll actually depend on how much denser the lead is than the styrofoam, which I don't know. But hopefully, that gives you a little intuition of what the center of mass is. And now I'll tell you something a little more interesting. Every problem we have done so far, we actually made the simplifying assumption that the force acts on the center of mass. So if I have an object, let's say the object that looks like a horse. Let's say that object. If this is the object's center of mass, I don't know where the horse's center of mass normally is, but let's say a horse's center of mass is here. If I apply a force directly on that center of mass, then the object will move in the direction of that force with the appropriate acceleration. We could divide the force by the mass of the entire horse acceleration in that direction. But now I will throw in a twist. And actually, every problem we did, all of these Newton's Law's problems, we assumed that the force acted at the center of mass. But something more interesting happens if the force acts away from the center of mass. Let me actually take that ruler example. I don't know why I even drew the horse. If I have this ruler again and this is the center of mass, as we said, any force that we act on the center of mass, the whole object will move in the direction of the force. It'll be shifted by the force, essentially. Now, this is what's interesting. If that's the center of mass and if I were to apply a force someplace else away from the center of mass, let' say I apply a force here, I want you to think about for a second what will probably happen to the object. Well, it turns out that the object will rotate. And so think about if we're on the space shuttle or we're in deep space or something, and if I have a ruler, and if I just push at one end of the ruler, what's going to happen? Am I just going to push the whole ruler or is the whole ruler is going to rotate? And hopefully, your intuition is correct. The whole ruler will rotate around the center of mass. And in general, if you were to throw a monkey wrench at someone, and I don't recommend that you do, but if you did, and while the monkey wrench is spinning in the air, it's spinning around its center of mass. Same for a knife. If you're a knife catcher, that's something you should think about, that the object, when it's free, when it's not fixed to any point, it rotates around its center of mass, and that's very interesting. So you can actually throw random objects, and that point at which it rotates around, that's the object's center of mass. That's an experiment that you should do in an open field around no one else. Now, with all of this, and I'll actually in the next video tell you what this is. When you have a force that causes rotational motion as opposed to a shifting motion, that's torque, but we'll do that in the next video. But now I'll show you just a cool example of how the center of mass is relevant in everyday applications, like high jumping. So in general, let's say that this is a bar. This is a side view of a bar, and this is the thing holding the bar. And a guy wants to jump over the bar. His center of mass is-- most people's center of mass is around their gut. I think evolutionarily that's why our gut is there, because it's close to our center of mass. So there's two ways to jump. You could just jump straight over the bar, like a hurdle jump, in which case your center of mass would have to cross over the bar. And we could figure out this mass, and we can figure out how much energy and how much force is required to propel a mass that high because we know projectile motion and we know all of Newton's laws. But what you see a lot in the Olympics is people doing a very strange type of jump, where, when they're going over the bar, they look something like this. Their backs are arched over the bar. Not a good picture. But what happens when someone arches their back over the bar like this? I hope you get the point. This is the bar right here. Well, it's interesting. If you took the average of this person's density and figured out his geometric center and all of that, the center of mass in this situation, if someone jumps like that, actually travels below the bar. Because the person arches their back so much, if you took the average of the total mass of where the person is, their center of mass actually goes below the bar. And because of that, you can clear a bar without having your center of mass go as high as the bar and so you need less force to do it. Or another way to say it, with the same force, you could clear a higher bar. , Hopefully, I didn't confuse you, but that's exactly why these high jumpers arch their back, so that their center of mass is actually below the bar and they don't have to exert as much force. Anyway, hopefully you found that to be a vaguely useful introduction to the center of mass, and I'll see you in the next video on torque." + }, + { + "Q": "This might be a stupid question but, I'm confused on how sadness is a noun? when I look at sadness in a sentence to me its more of a verb word. Can you help me understand why its a noun and maybe help me understand how I can find nouns in a sentence that aren't as normal as person, place, or thing.", + "A": "Sadness isn t an action or verb. You can t say He sadness today because he failed an exam or The dog sadness because he can t have a treat. Sadness is an emotion, a thing. Not something you do . There are plenty of things you can t touch, but they are still considered nouns. These are called abstract nouns. There was sadness in her voice . The verb here is was and the noun here is sadness . Just like the sentence There was water everywhere . The verb is was and the noun is water .", + "video_name": "3AF_rN-yN-Y", + "transcript": "- [Voiceover] Hello grammarians. So today I'd like to talk to you about the idea of concrete and abstract nouns, and before we do that, I'd like to get into some word origins or etymology. So let's take each of these words in turn, because I think by digging into what these words mean, literally what they mean and where they come from, we'll get a better understanding of this concept. So both of these words come to us from Latin. Concrete comes to us from the Latin concretus, which means to grow together. So this part of it means grown. And this part means together. It refers to something that, you know, has grown together and become thick and kind of hard to get through and physical. The connotation here is that this is a physical thing. Something that is concrete is physical. Abstract, on the other hand, means to draw something away. So something that is abstract is drawn away from the real, from the concrete, from the physical. So this is not physical. And we make this distinction in English when we're talking about nouns. Is it something that is concrete, is it something you can look at or pick up or smell or sense or something that is abstract, something that isn't physical, but can still be talked about. So for example, the word sadness... Is a noun, right? This is definitely a noun. It's got this noun-making ending, this noun-forming suffix, ness. You know, we take the adjective sad and we toss this ness part onto it, we've got a noun. But can you see sadness? Is it something you can pick up? Sure, you can tell by being, you know observant and empathetic that your friend is sad, but it's not something you can pick up. You can't be like a measurable degree of sad. You couldn't take someone's sadness, put it under a microscope and say \"Oh, Roberta, you are 32 degrees microsad.\" You know, it's not something physical. Concrete things, on the other hand, are things that we can see or count or measure. Just parts of the physical world. So anything you look at, like a dog is concrete, a ball is concrete, a cliff is concrete. Happiness... Is abstract. The idea of freedom... Is abstract. Though the presence of freedom in your life may manifest in physical objects, like \"Oh, my parents let me have the freedom to eat ice cream.\" Ice cream is, you know, a concrete noun. But freedom, the thing that allows you, you know, the permission that you get from your parents to have ice cream. That's not a physical object. So that's basically the difference. So a concrete noun is a physical object and an abstract noun is not. This is why I really wanted to hit the idea that a noun can be a person, place, thing or idea, because nouns can be ideas, and those ideas tend to be abstract. Sadness, happiness, freedom, permission, liberty, injustice. All of these are abstract ideas. That's the difference. You can learn anything. David out." + }, + { + "Q": "does a euro pound count as a pound in England", + "A": "No, at the current exchange rate, a british pound is worth about 1.3 euros.", + "video_name": "s6NOa1KTCxQ", + "transcript": "- [Voiceover] Zhang Tao has two one dollar bills. So two one dollar bills. So let's just write that as one plus one, because each one dollar bill represents one dollar. So, one dollar plus one dollar. That's the two one dollar bills. One five dollar bill. One five dollar bill. So that represents five dollars. And three 10 dollar bills. Three 10 dollar bills. So that's plus ten plus ten plus ten. How much money does he have in all? Well, let's see. One plus, let's see if we can calculate it. One plus one is two. Two plus five is seven. So, these are going to be seven dollars. And then 10 plus 10 plus 10, that's three 10's. So that's going to be 30 dollars. Now what's seven ones plus three 10's? Or what's seven plus 30? Well, that's going to be 37 dollars. So Zhang Tao has 37 dollars in all. Let's do another one of these. Diya has six one dollar bills. And actually, let's draw it out. Diya has six one dollar bills. So that's one, two, three, four, five and six. And these are one dollar bills. So, one dollar bills. This is my rough drawing of one dollar bills. Have to draw a lot of one dollar bills here. So, she has six one dollar bills. She has three five dollar bills. Can we do that in a different color? So, three five dollar bills. So one, so that's a five dollar bill. Two, that's a five dollar bill. And three five dollar bills. That little circle in .... That's supposed to be the picture of someone. Let me make it clear these are pictures of folks. Of famous historical figures. So, these are pictures of famous historical figures here. And then finally she has one 10 dollar bill. One 10 dollar bill. So one 10 dollar bill. So it's worth 10 dollars. A picture of a famous historical figure right over there. So, how much money does she have in all? Well, the six one dollar bills that's going to be six dollars. The three five dollar bills that's going to be worth five, 10, 15 dollars. So plus 15 dollars. And then the one 10 dollar bill. That's going to be worth 10 dollars. 10 dollars. So, what's six plus 15 plus 10? Well, let's see, six plus fifteen is going to be 21. 21 plus 10 is going to be equal to 31 dollars. Is equal to 31 dollars. And you could have also added it this way, You could have said 15 plus 10 plus 10 plus six plus six. Add them together and we would have gotten five plus zero plus six is 11. That's one one and one 10. And then you have three 10's here. So, 31 dollars." + }, + { + "Q": "why is the electric field of a point charge is not uniform ?\nPlease explain.\nThank you.", + "A": "Because the electric field generated by a point charge extends radially from the point iteself. This means that the field decreases with the square of theradius of distance from the point.", + "video_name": "0YOGrTNgGhE", + "transcript": "Let's imagine that instead of having two charges, we just have one charge by itself, sitting in a vacuum, sitting in space. So that's this charge here, and let's say its charge is Q. That's some number, whatever it is. That's it's charge. And I want to know, if I were to place another charge close to this Q, within its sphere of influence, what's going to happen to that other charge? What's going to be the net impact on it? And we know if this has some charge, if we put another charge here, if this is 1 coulomb and we put another charge here that's 1 coulomb, that they're both positive, they're going to repel each other, so there will be some force that pushes the next charge away. If it's a negative charge and I put it here, it'll be even a stronger force that pulls it in because it'll be closer. So in general, there's this notion of what we can call an electric field around this charge. And what's an electric field? We can debate whether it really exists, but what it allows us to do is imagine that somehow this charge is affecting the space around it in some way that whenever I put-- it's creating a field that whenever I put another charge in that field, I can predict how the field will affect that charge. So let's put it in a little more quantitative term so I stop confusing you. So Coulomb's Law told us that the force between two charges is going to be equal to Coulomb's constant times-- and in this case, the first charge is big Q. And let's say that the second notional charge that I eventually put in this field is small q, and then you divide by the distance between them. Sometimes it's called r because you can kind of view the distance as the radial distance between the two charges. So sometimes it says r squared, but it's the distance between them. So what we want to do if we want to calculate the field, we want to figure out how much force is there placed per charge at any point around this Q, so, say, at a given distance out here. At this distance, we want to know, for a given Q, what is the force going to be? So what we can do is we could take this equation up here and divide both sides by this small 1, and say, OK, the force-- and I will arbitrarily switch colors. The force per charge at this point-- let's call that d1-- is equal to Coulomb's constant times the charge of the particle that's creating the field divided by-- well, in this case, it's d1-- d1 squared, right? Or we could say, in general-- and this is the definition of the electric field, right? Well, this is the electric field at the point d1, and if we wanted a more general definition of the electric field, we'll just make this a general variable, so instead of having a particular distance, we'll define the field for all distances away from the point Q. So the electric field could be defined as Coulomb's constant times the charge creating the field divided by the distance squared, the distance we are away from the charge. So essentially, we've defined-- if you give me a force and a point around this charge anywhere, I can now tell you the exact force. For example, if I told you that I have a minus 1 coulomb charge and the distance is equal to-- oh, I don't know. The distance is equal to let's say-- let's make it easy. Let's say 2 meters. So first of all, we can say, in general, what is the electric field 2 meters away from? So what is the electric field out here? This is 2, right? And it's going to be 2 meters away. It's radial so it's actually along this whole circle. What is the electric field there? Well, the electric field at that point is going to be equal to what? And it's also a vector quantity, right? Because we're dividing a vector quantity by a scalar quantity charge. So the electric field at that point is going to be k times whatever charge it is divided by 2 meters, so divided by 2 meters squared, so that's 4, right, distance squared. And so if I know the electric field at any given point and then I say, well, what happens if I put a negative 1 coulomb charge there, all I have to do is say, well, the force is going to be equal to the charge that I place there times the electric field at that point, right? So in this case, we said the electric field at this point is equal to-- and the units for electric field are newtons per coulomb, and that makes sense, right? Because it's force divided by charge, so newtons per coulomb. So if we know that the electric charge-- well, let me put some real numbers here. Let's say that this is-- I don't know. It's going to be a really large number, but let's say this-- let me pick a smaller number. Let's say this is 1 times 10 to the minus 6 coulombs, right? If that's 1 times 10 to the minus 6 coulombs, what is the electric field at that point? Let me switch colors again. What's the electric field at that point? Well, the electric field at that point is going to be equal to Coulomb's constant, which is 9 times 10 to the ninth-- times the charge generating the field-- times 1 times 10 to the minus 6 coulombs. And then we are 2 meters away, so 2 squared. So that equals 9 times 10 to the third divided by 4. So I don't know, what is that? 2.5 times 10 to the third or 2,500 newtons per coulomb. So we know that this is generating a field that when we're 2 meters away, at a radius of 2 meters, so roughly that circle around it, this is generating a field that if I were to put-- let's say I were to place a 1 coulomb charge here, the force exerted on that 1 coulomb charge is going to be equal to 1 coulomb times the electric fields, times 2,500 newtons per coulomb. So the coulombs cancel out, and you'll have 2,500 newtons, which is a lot, and that's because 1 coulomb is a very, very large charge. And then a question you should ask yourself: If this is 1 times 10 to the negative 6 coulombs and this is 1 coulomb, in which direction will the force be? Well, they're both positive, so the force is going to be outwards, right? So let's take this notion and see if we can somehow draw an electric field around a particle, just to get an intuition of what happens when we later put a charge anywhere near the particle. So there's a couple of ways to visualize an electric field. One way to visualize it is if I have a-- let's say I have a point charge here Q. What would be the path of a positive charge if I placed it someplace on this Q? Well, if I put a positive charge here and this Q is positive, that positive charge is just going to accelerate outward, right? It's just going to go straight out, but it's going to accelerate at an ever-slowing rate, right? Because here, when you're really close, the outward force is very strong, and then as you get further and further away, the electrostatic force from this charge becomes weaker and weaker, or you could say the field becomes weaker and weaker. But that's the path of a-- it'll just be radially outward-- of a positive test charge. And then if I put it here, well, it would be radially outward that way. It wouldn't curve the way I drew it. It would be a straight line. I should actually use the line tool. If I did it here, it would be like that, but then I can't draw the arrows. If I was here, it would out like that. I think you get the picture. At any point, a positive test charge would just go straight out away from our charge Q. And to some degree, one measure of-- and these are called electric field lines. And one measure of how strong the field is, is if you actually took a unit area and you saw how dense So here, they're relatively sparse, while if I did that same area up here-- I know it's not that obvious. I'm getting more field lines in. But actually, that's not a good way to view it because I'm covering so much area. Let me undo both of them. You can imagine if I had a lot more lines, if I did this area, for example, in that area, I'm capturing two of these field lines. Well, if I did that exact same area out here, I'm only capturing one of the field lines, although you could have a bunch more in between here. And that makes sense, right? Because as you get closer and closer to the source of the electric field, the charge gets stronger. Another way that you could have done this, and this would have actually more clearly shown the magnitude of the field at any point, is you could have-- you could say, OK, if that's my charge Q, you could say, well, really close, the field is strong. So at this point, the vector, the newtons per coulomb, is that strong, that strong, that strong, that strong. We're just taking sample points. You can't possibly draw them at every single point. So at that point, that's the vector. That's the electric field vector. But then if we go a little bit further out, the vector is going to be-- it falls off. This one should be shorter, then this one should be even shorter, right? You could pick any point and you could actually calculate the electric field vector, and the further you go out, the shorter and shorter the electric field vectors get. And so, in general, there's all sorts of things you can draw the electric fields for. Let's say that this is a positive charge and that this is a negative charge. Let me switch colors so I don't have to erase things. If I have to draw the path of a positive test charge, it would go out radially from this charge, right? But then as it goes out, it'll start being attracted to this one the closer it gets to the negative, and then it'll curve in to the negative charge and these arrows go like this. And if I went from here, the positive one will be repelled really strong, really strong, it'll accelerate fast and it's rate of acceleration will slow down, but then as it gets closer to the negative one, it'll speed up again, and then that would be its path. Similarly, if there was a positive test charge here, its path would be like that, right? If it was here, its path would be like that. If it was here, it's path would be like that. If it was there, maybe its path is like that, and at some point, its path might never get to that-- this out here might just go straight out that way. That one would just go straight out, and here, the field lines would just come in, right? A positive test charge would just be naturally attracted to that negative charge. So that's, in general, what electric field lines show, and we could use our little area method and see that over here, if we picked a given area, the electric field is much weaker than if we picked that same area right here. We're getting more field lines in than we do right there. So that hopefully gives you a little sense for what an electric field is. It's really just a way of visualizing what the impact would be on a test charge if you bring it close to another charge. And hopefully, you know a little bit about Coulomb's constant. And let's just do a very simple-- I'm getting this out of the AP Physics book, but they say-- let's do a little simple problem: Calculate the static electric force between a 6 times 10 to the negative sixth coulomb charge. So 6 times-- oh, no, that's not on an electric field. Oh, here it says: What is the force acting on an electron placed in an external electric field where the electric field is-- they're saying it is 100 newtons per coulomb at that point, wherever the electron is. So the force on that, the force in general, is just going to be the charge times the electric field, and they say it's an electron, so what's the Well, we know it's negative, and then in the first video, we learned that its charge is 1.6 times 10 to the negative nineteenth coulombs times 100 newtons per coulomb. The coulombs cancel out. And this is 10 squared, right? This is 10 to the positive 2, so it'll be 10 to the minus 19 times 10 to the positive 2. The force will be minus 1.6 times 10 to the minus 17 newtons. So the problems are pretty simple. I think the more important thing with electric fields is to really understand intuitively what's going on, and kind of how it's stronger near the point charges, and how it gets weaker as it goes away, and what the field lines depict, and how they can be used to at least approximate the strength of the field. I will see you in the next video." + }, + { + "Q": "if we compress H2O will it turn in to ice ?", + "A": "If you mean normal ice, then the answer is no. But there some other kinds of ice that you can form by exposing water to EXTREME pressures. If you crush the water with over 10000 times normal atmospheric pressure (and there are some conditions that have to be just right for this to work), then it is possible to convert the water to one of the other kinds of ice such as Ice VI or Ice VII. These other kinds of ice are structurally different than normal ice.", + "video_name": "pKvo0XWZtjo", + "transcript": "I think we're all reasonably familiar with the three states of matter in our everyday world. At very high temperatures you get a fourth. But the three ones that we normally deal with are, things could be a solid, a liquid, or it could be a gas. And we have this general notion, and I think water is the example that always comes to at least my mind. Is that solid happens when things are colder, relatively colder. And then as you warm up, you go into a liquid state. And as your warm up even more you go into a gaseous state. So you go from colder to hotter. And in the case of water, when you're a solid, you're ice. When you're a liquid, some people would call ice water, but let's call it liquid water. I think we know what that is. And then when it's in the gas state, you're essentially vapor or steam. So let's think a little bit about what, at least in the case of water, and the analogy will extend to other types of molecules. But what is it about water that makes it solid, and when it's colder, what allows it to be liquid. And I'll be frank, liquids are kind of fascinating because you can never nail them down, I guess is the best way to view them. Or a gas. So let's just draw a water molecule. So you have oxygen there. You have some bonds to hydrogen. And then you have two extra pairs of valence electrons in the oxygen. And a couple of videos ago, we said oxygen is a lot more electronegative than the hydrogen. It likes to hog the electrons. So even though this shows that they're sharing electrons here and here. At both sides of those lines, you can kind of view that hydrogen is contributing an electron and oxygen is contributing an electron on both sides of that line. But we know because of the electronegativity, or the relative electronegativity of oxygen, that it's hogging these electrons. And so the electrons spend a lot more time around the oxygen than they do around the hydrogen. And what that results is that on the oxygen side of the molecule, you end up with a partial negative charge. And we talked about that a little bit. And on the hydrogen side of the molecules, you end up with a slightly positive charge. Now, if these molecules have very little kinetic energy, they're not moving around a whole lot, then the positive sides of the hydrogens are very attracted to the negative sides of oxygen in other molecules. Let me draw some more molecules. When we talk about the whole state of the whole matter, we actually think about how the molecules are interacting with Not just how the atoms are interacting with each other within a molecule. I just drew one oxygen, let me copy and paste that. But I could do multiple oxygens. And let's say that that hydrogen is going to want to be near this oxygen. Because this has partial negative charge, this has a partial positive charge. And then I could do another one right there. And then maybe we'll have, and just to make the point clear, you have two hydrogens here, maybe an oxygen wants to hang out there. So maybe you have an oxygen that wants to be here because it's got its partial negative here. And it's connected to two hydrogens right there that have their partial positives. But you can kind of see a lattice structure. Let me draw these bonds, these polar bonds that start forming between the particles. These bonds, they're called polar bonds because the molecules themselves are polar. And you can see it forms this lattice structure. And if each of these molecules don't have a lot of kinetic energy. Or we could say the average kinetic energy of this matter is fairly low. And what do we know is average kinetic energy? Well, that's temperature. Then this lattice structure will be solid. These molecules will not move relative to each other. I could draw a gazillion more, but I think you get the point that we're forming this kind of fixed structure. And while we're in the solid state, as we add kinetic energy, as we add heat, what it does to molecules is, it just makes them vibrate around a little bit. If I was a cartoonist, they way you'd draw a vibration is to put quotation marks there. That's not very scientific. But they would vibrate around, they would buzz around a little bit. I'm drawing arrows to show that they are vibrating. It doesn't have to be just left-right it could be up-down. But as you add more and more heat in a solid, these molecules are going to keep their structure. So they're not going to move around relative to each other. But they will convert that heat, and heat is just a form of energy, into kinetic energy which is expressed as the vibration of these molecules. Now, if you make these molecules start to vibrate enough, and if you put enough kinetic energy into these molecules, what do you think is going to happen? Well this guy is vibrating pretty hard, and he's vibrating harder and harder as you add more and more heat. This guy is doing the same thing. At some point, these polar bonds that they have to each other are going to start not being strong enough to contain the vibrations. And once that happens, the molecules-- let me draw a couple more. Once that happens, the molecules are going to start moving past each other. So now all of a sudden, the molecule will start shifting. But they're still attracted. Maybe this side is moving here, that's moving there. You have other molecules moving around that way. But they're still attracted to each other. Even though we've gotten the kinetic energy to the point that the vibrations can kind of break the bonds between the polar sides of the molecules. Our vibration, or our kinetic energy for each molecule, still isn't strong enough to completely separate them. They're starting to slide past each other. And this is essentially what happens when you're in a liquid state. You have a lot of atoms that want be touching each other but they're sliding. They have enough kinetic energy to slide past each other and break that solid lattice structure here. And then if you add even more kinetic energy, even more heat, at this point it's a solution now. They're not even going to be able to stay together. They're not going to be able to stay near each other. If you add enough kinetic energy they're going to start looking like this. They're going to completely separate and then kind of bounce around independently. Especially independently if they're an ideal gas. But in general, in gases, they're no longer touching They might bump into each other. But they have so much kinetic energy on their own that they're all doing their own thing and they're not touching. I think that makes intuitive sense if you just think about what a gas is. For example, it's hard to see a gas. Why is it hard to see a gas? Because the molecules are much further apart. So they're not acting on the light in the way that a liquid or a solid would. And if we keep making that extended further, a solid-- well, I probably shouldn't use the example with ice. Because ice or water is one of the few situations where the solid is less dense than the liquid. That's why ice floats. And that's why icebergs don't just all fall to the bottom of the ocean. And ponds don't completely freeze solid. But you can imagine that, because a liquid is in most cases other than water, less dense. That's another reason why you can see through it a little Or it's not diffracting-- well I won't go into that too much, than maybe even a solid. But the gas is the most obvious. And it is true with water. The liquid form is definitely more dense than the gas form. In the gas form, the molecules are going to jump around, not touch each other. And because of that, more light can get through the substance. Now the question is, how do we measure the amount of heat that it takes to do this to water? And to explain that, I'll actually draw a phase change diagram. Which is a fancy way of describing something fairly straightforward. Let me say that this is the amount of heat I'm adding. And this is the temperature. We'll talk about the states of matter in a second. So heat is often denoted by q. Sometimes people will talk about change in heat. They'll use H, lowercase and uppercase H. They'll put a delta in front of the H. Delta just means change in. And sometimes you'll hear the word enthalpy. Let me write that. Because I used to say what is enthalpy? It sounds like empathy, but it's quite a different concept. At least, as far as my neural connections could make it. But enthalpy is closely related to heat. It's heat content. For our purposes, when you hear someone say change in enthalpy, you should really just be thinking of change in heat. I think this word was really just introduced to confuse chemistry students and introduce a non-intuitive word into their vocabulary. The best way to think about it is heat content. Change in enthalpy is really just change in heat. And just remember, all of these things, whether we're talking about heat, kinetic energy, potential energy, enthalpy. You'll hear them in different contexts, and you're like, I thought I should be using heat and they're talking about enthalpy. These are all forms of energy. And these are all measured in joules. And they might be measured in other ways, but the traditional way is in joules. And energy is the ability to do work. And what's the unit for work? Well, it's joules. Force times distance. But anyway, that's a side-note. But it's good to know this word enthalpy. Especially in a chemistry context, because it's used all the time and it can be very confusing and non-intuitive. Because you're like, I don't know what enthalpy is in my everyday life. Just think of it as heat contact, because that's really But anyway, on this axis, I have heat. So this is when I have very little heat and I'm increasing my heat. And this is temperature. Now let's say at low temperatures I'm here and as I add heat my temperature will go up. Temperature is average kinetic energy. Let's say I'm in the solid state here. And I'll do the solid state in purple. No I already was using purple. I'll use magenta. So as I add heat, my temperature will go up. Heat is a form of energy. And when I add it to these molecules, as I did in this example, what did it do? It made them vibrate more. Or it made them have higher kinetic energy, or higher average kinetic engery, and that's what temperature is a measure of; average kinetic energy. So as I add heat in the solid phase, my average kinetic energy will go up. And let me write this down. This is in the solid phase, or the solid state of matter. Now something very interesting happens. Let's say this is water. So what happens at zero degrees? Which is also 273.15 Kelvin. Let's say that's that line. What happens to a solid? Well, it turns into a liquid. Ice melts. Not all solids, we're talking in particular about water, about H2O. So this is ice in our example. All solids aren't ice. Although, you could think of a rock as solid magma. Because that's what it is. I could take that analogy a bunch of different ways. But the interesting thing that happens at zero degrees. Depending on what direction you're going, either the freezing point of water or the melting point of ice, something interesting happens. As I add more heat, the temperature does not to go up. As I add more heat, the temperature does not go up for a little period. Let me draw that. For a little period, the temperature stays constant. And then while the temperature is constant, it stays a solid. We're still a solid. And then, we finally turn into a liquid. Let's say right there. So we added a certain amount of heat and it just stayed a solid. But it got us to the point that the ice turned into a liquid. It was kind of melting the entire time. That's the best way to think about it. And then, once we keep adding more and more heat, then the liquid warms up too. Now, we get to, what temperature becomes interesting again for water? Well, obviously 100 degrees Celsius or 373 degrees Kelvin. I'll do it in Celsius because that's what we're familiar with. That's the temperature at which water will vaporize or which water will boil. But something happens. And they're really getting kinetically active. But just like when you went from solid to liquid, there's a certain amount of energy that you have to contribute to the system. And actually, it's a good amount at this point. Where the water is turning into vapor, but it's not getting any hotter. So we have to keep adding heat, but notice that the temperature didn't go up. We'll talk about it in a second what was happening then. And then finally, after that point, we're completely vaporized, or we're completely steam. Then we can start getting hot, the steam can then get hotter as we add more and more heat to the system. So the interesting question, I think it's intuitive, that as you add heat here, our temperature is going to go up. But the interesting thing is, what was going on here? We were adding heat. So over here we were turning our heat into kinetic energy. Temperature is average kinetic energy. But over here, what was our heat doing? Well, our heat was was not adding kinetic energy to the system. The temperature was not increasing. But the ice was going from ice to water. So what was happening at that state, is that the kinetic energy, the heat, was being used to essentially break these bonds. And essentially bring the molecules into a higher energy state. So you're saying, Sal, what does that mean, higher energy state? Well, if there wasn't all of this heat and all this kinetic energy, these molecules want to be very close to each other. For example, I want to be close to the surface of the earth. When you put me in a plane you have put me in a higher energy state. I have a lot more potential energy. I have the potential to fall towards the earth. Likewise, when you move these molecules apart, and you go from a solid to a liquid, they want to fall towards each other. But because they have so much kinetic energy, they never quite are able to do it. But their energy goes up. Their potential energy is higher because they want to fall towards each other. By falling towards each other, in theory, they could do some work. So what's happening here is, when we're contributing heat-- and this amount of heat we're contributing, it's called the heat of fusion. Because it's the same amount of heat regardless how much direction we go in. When we go from solid to liquid, you view it as the heat of melting. It's the head that you need to put in to melt the ice into liquid. When you're going in this direction, it's the heat you have to take out of the zero degree water to turn it into ice. So you're taking that potential energy and you're bringing the molecules closer and closer to each other. So the way to think about it is, right here this heat is being converted to kinetic energy. Then, when we're at this phase change from solid to liquid, that heat is being used to add potential energy into the system. To pull the molecules apart, to give them more potential energy. If you pull me apart from the earth, you're giving me potential energy. Because gravity wants to pull me back to the earth. And I could do work when I'm falling back to the earth. A waterfall does work. It can move a turbine. You could have a bunch of falling Sals move a turbine as well. And then, once you are fully a liquid, then you just become a warmer and warmer liquid. Now the heat is, once again, being used for kinetic energy. You're making the water molecules move past each other faster, and faster, and faster. To some point where they want to completely disassociate from each other. They want to not even slide past each other, just completely jump away from each other. And that's right here. This is the heat of vaporization. And the same idea is happening. Before we were sliding next to each other, now we're pulling apart altogether. So they could definitely fall closer together. And then once we've added this much heat, now we're just heating up the steam. We're just heating up the gaseous water. And it's just getting hotter and hotter and hotter. But the interesting thing there, and I mean at least the interesting thing to me when I first learned this, whenever I think of zero degrees water I'll say, oh it must be ice. But that's not necessarily the case. If you start with water and you make it colder and colder and colder to zero degrees, you're essentially taking heat out of the water. You can have zero degree water and it hasn't turned into ice yet. And likewise, you could have 100 degree water that hasn't turned into steam yeat. You have to add more energy. You can also have 100 degree steam. You can also have zero degree water. Anyway, hopefully that gives you a little bit of intuition of what the different states of matter are. And in the next problem, we'll talk about how much heat exactly it does take to move along this line. And maybe we can solve some problems on how much ice we might need to make our drink cool." + }, + { + "Q": "how to know the quadratic formula does not have answer?", + "A": "I think you mean how to know if the quadratic equation has no real solutions, which just means it does not touch the x-axis. You have to look at the discriminant which is the b^2-4ac part. If it is negative then there is no real solutions and the parabola does not touch the x-axis, since there is no real number that gives you the square root of a negative.", + "video_name": "BGz3pkoGPag", + "transcript": "- [Instructor] In this video, we are going to talk about one of the most common types of curves you will see in mathematics, and that is the parabola. The word parabola sounds quite fancy, but we'll see it's describing something that is fairly straightforward. Now in terms of why it is called the parabola, I've seen multiple explanations for it. It comes from Greek para, that root word, similar to parable. You could view of something beside, alongside, something in parallel. Bola, same root as when we're talking about ballistics, throwing something. So you could interpret it as beside, alongside, something that is being thrown. Now how does that relate to curves like this? Well my brain immediately imagines this is the trajectory, this is the path that is a pretty good approximation for the path of things that are actually thrown. When you study physics, you will see the path, you'll approximate, the paths of objects being thrown, as parabolas, so maybe that's where it comes from, but there are other potential explanations for why it is actually called the parabola. It has been lost to history. But what exactly is a parabola? In future videos, we're gonna describe it a little bit more algebraically. In this one, we just wanna get a sense for what parabolas look like and introduce ourselves to some terminology around a parabola. These three curves, they are all hand-drawn versions of a parabola, and so you immediately notice some interesting things about them. Some of them are opened upwards like this yellow one and this pink one, and some of them are open downwards. You will hear people say things like open, opened down, open downwards or open down or open upwards, so it's good to know what they are talking about, and it's, hopefully, fairly self-explanatory. Open upwards, the parabola is open towards the top of our graph paper. Here it's open towards the bottom of our graph paper. This looks like a right-side up U. This looks like an upside down U right over here. This pink one would be open upwards. Now another term that you'll see associated with the parabola, and once again, in the future, we'll learn how to calculate these things and find them precisely, is the vertex. The vertex you should view as the maximum or minimum point on a parabola. So if a parabola opens upwards like these two on the right, the vertex is the minimum point. The vertex is the minimum point right over there, and so if someone said what is the vertex of this yellow parabola? Well it looks like the x, looks like the x coordinate is three and a half, so it is three and a half. It looks like the y coordinate, it looks like it is about negative three and a half. Once again, once we start representing these things with equations, we'll have techniques for calculating them more precisely, but the vertex of this other upward-opening parabola, it is the minimum point. It is the low point. There is no maximum point on an upward-opening parabola. It just keeps increasing as x gets larger in the positive or the negative direction. Now if your parabola opens downward, then your vertex is going to be your maximum point. Now related to the idea of a vertex is the idea of an axis of symmetry. In general when we're talking about, well not just three, two dimensions but even three dimensions, but especially in two dimensions, you can imagine a line over which you can flip the graph, and so it meets, it folds onto itself. The axis of symmetry for this yellow graph right over here, for this yellow parabola, it would be this line. I'm gonna have to draw it a little bit better. It would be that line right over there. You could fold the parabola over that line, and it would meet itself. And that line, I didn't draw it as neat as I should, that should go directly through the vertex, so to describe that line you'd say that line is x is equal to 3.5. Similarly the axis of symmetry for this pink parabola, it should go through the line x equals negative one, so let me do that. That's the axis of symmetry. It goes through the vertex, and if you were to fold the parabola over it, it would meet itself. The axis of symmetry for this green one? It should, once again, go through the vertex. It looks like it is x is equal to negative six. This is, let me write that down, that is the axis of symmetry. Now another concept that isn't unique to parabolas, but we'll talk a lot about it in the context of parabolas, are intercepts, so when people say y-intercept, and you saw this when you first graphed lines, they're saying where is the graph, where does the curve intercept or intersect the y-axis? So the y-intercept of this yellow line would be right there. It looks like it's the point zero comma three, zero comma three. The y-intercept for the pink one is right over there. At least on this graph paper, we don't see the y-intercept, but it eventually will intersect the y-axis. It just will be way off of this screen. You might also be familiar with the term x-intercept, and that's especially interesting with parabolas as we'll see in the future. X-intercept is where do you intercept or intersect the x-axis? Here this yellow one you see it does it two places, and this is where it gets interesting. Lines will only intersect the x-axis once at most, but here we see that a parabola can intersect the x-axis twice, because it curves back around to intersect it again, and so for here the x-intercepts are going to be the point one comma zero and six comma zero. You might already notice something interesting. The x-intercepts are symmetric around the axis of symmetry, so they should be equal distant from that axis of symmetry, and you can see they indeed are. They are both exactly two and a half away from that axis of symmetry, and so if you know where the intercepts are, you just take, you could say, the midpoint of the x coordinates, and then you're going to have the axis of symmetry, the x coordinate of the axis of symmetry and the x coordinate of the actual vertex. Similarly the x-intercept here looks like it's negative, the points are negative seven comma zero and negative five comma zero, and the x coordinate of the vertex, or the line of symmetry, is right in between those two points. It's worth noting not every parabola is going to intersect the x-axis. Notice this pink upward-opening parabola, it's low point is above the x-axis, so it's never going to intersect the actual x-axis, so this is actually not going to have any x-intercepts. I'll leave you there. Those are actually the core ideas or the core visual themes around parabolas, and we're going to discuss them in a lot more detail when we represent them with equations. As you'll see, these equations are going to involve second-degree terms. So the most simple parabola is going to be y is equal to x squared, but then you can complicate it a little bit. You could have things like y is equal to two x squared minus five x plus seven. These types that we'll talk about in more general terms, these types of equations sometimes called quadratics, they are represented, generally, by parabolas." + }, + { + "Q": "Even though I do everything right in the practice, I still get it wrong! What can I do to fix that?", + "A": "Study and watch more videos on the topic and pause them and try to do the questions before Sal does.", + "video_name": "xXIG8ouHcsc", + "transcript": "Let's divide 7,182 by 42. And what's different here is we're now dividing by a two-digit number, not a one-digit number, but the same idea holds. So we say, hey, how many times does 42 go into 7? Well, it doesn't really go into 7 at all, so let's add one more place value. How many times does 42 go into 71? Well, it goes into 71 one time. Just a reminder, whoever's doing the process where you say, hey, 42 goes into 71 one time. But what we're really saying, 42 goes into 7,100 100 times because we're putting this one in the hundreds place. But let's put that on the side for a little bit and focus on the process. So 1 times 42 is 42, and now we subtract. Now, you might be able to do 71 minus 42 in your head, knowing, hey, 72 minus 42 would be 30. So 71 minus 42 would be 29, but we could also do it by regrouping. To regroup, you want to subtract a 2 from a 1. You can't really do that in any traditional way. So let's take a 10 from the 70, so that it becomes a 60, and give that 10 to the ones place, and then that becomes an 11. And so 11 minus 2 is 9, and 6 minus 4 is 2. So you get 29. And we can bring down the next place value. Bring down an 8. And now, this is where the art happens when we're dividing by a multi-digit number right over here. We have to estimate how many times does 42 go into 298. And sometimes it might involve a little bit of trial and error. So you really just kind of have to eyeball it. If you make a mistake, try again. The way you know you make a mistake is, if say it goes into it 9 times, and you do 9 times 42 and you get a number larger than 298, then you overestimated. If you say it goes into it three times, you do 3 times 42, you get some number here. When you subtract, you get something larger than 42, then you also made a mistake, and you have to adjust upwards. Well, let's see if we can eyeball it. So this is roughly 40. This is roughly 300. 40 goes into 300 the same times as 4 goes into the 30, so it's going to be roughly 7. Let's see if that's right. 7 times 2 is 14. 7 times 1 is 28, plus 1 is 29. So I got pretty close. My remainder here-- notice 294 is less than 298. So I'm cool there. And my remainder is less than 42, so I'm cool as well. So now let's add another place value. Let's bring this 2 down. And here we're just asking ourselves, how many times does 42 go into 42? Well, 42 goes into 42 exactly one time. 1 times 42 is 42, and we have no remainder. So this one luckily divided exactly. 42 goes into 7,182 exactly 171 times." + }, + { + "Q": "why does a magnet always points towards south and north pole??", + "A": "The earth has a magnetic field, as though it has a giant bar magnet in it, with one pole near Earth s North Pole and the other pole near what we call the South Pole. The north end of a magnet attracts the south end of another magnet, and south attracts north. So a magnet that is free to spin in earth s magnetic field points its north pole toward the earth s North Pole. (this tells you that the magnetic pole that is near the North Pole is really a south magnetic pole)", + "video_name": "8Y4JSp5U82I", + "transcript": "We've learned a little bit about gravity. We've learned a little bit about electrostatic. So, time to learn about a new fundamental force of the universe. And this one is probably second most familiar to us, And that's magnetism. Where does the word come from? Well, I think several civilizations-- I'm no historian-- found these lodestones, these objects that would attract other objects like it, other magnets. Or would even attract metallic objects like iron. Ferrous objects. And they're called lodestones. That's, I guess, the Western term for it. And the reason why they're called magnets is because they're named after lodestones that were found near the Greek province of Magnesia. And I actually think the people who lived there were called Magnetes. But anyway, you could Wikipedia that and learn more about it than I know. But anyway let's focus on what magnetism is. And I think most of us have at least a working knowledge of what it is; we've all played with magnets and we've dealt with compasses. But I'll tell you this right now, what it really is, is pretty deep. And I think it's fairly-- I don't think anyone has-- we can mathematically understand it and manipulate it and see how it relates to electricity. We actually will show you the electrostatic force and the magnetic force are actually the same thing, just viewed from different frames of reference. I know that all of that sounds very complicated and all of that. But in our classical Newtonian world we treat them as two different forces. But what I'm saying is although we're kind of used to a magnet just like we're used to gravity, just like gravity is also fairly mysterious when you really think about what it is, so is magnetism. So with that said, let's at least try to get some working knowledge of how we can deal with magnetism. So we're all familiar with a magnet. I didn't want it to be yellow. I could make the boundary yellow. No, I didn't want it to be like that either. So if this is a magnet, we know that a magnet always has two poles. It has a north pole and a south pole. And these were just labeled by convention. Because when people first discovered these lodestones, or they took a lodestone and they magnetized a needle with that lodestone, and then that needle they put on a cork in a bucket of water, and that needle would point to the Earth's north pole. They said, oh, well the side of the needle that is pointing to the Earth's north, let's call that the north pole. And the point of the needle that's pointing to the south pole-- sorry, the point of the needle that's pointing to the Earth's geographic south, we'll call that the south pole. Or another way to put it, if we have a magnet, the direction of the magnet or the side of the magnet that orients itself-- if it's allowed to orient freely without friction-- towards our geographic north, we call that the north pole. And the other side is the south pole. And this is actually a little bit-- obviously we call the top of the Earth the north pole. You know, this is the north pole. And we call this the south pole. And there's another notion of magnetic north. And that's where-- I guess, you could kind of say-- that is where a compass, the north point of a compass, will point to. And actually, magnetic north moves around because we have all of this moving fluid inside of the earth. And a bunch of other interactions. It's a very complex interaction. But magnetic north is actually roughly in northern Canada. So magnetic north might be here. So that might be magnetic north. And magnetic south, I don't know exactly where that is. But it can kind of move around a little bit. It's not in the same place. So it's a little bit off the axis of the geographic north pole and the south pole. And this is another slightly confusing thing. Magnetic north is the geographic location, where the north pole of a magnet will point to. But that would actually be the south pole, if you viewed the Earth as a magnet. So if the Earth was a big magnet, you would actually view that as a south pole of the magnet. And the geographic south pole is the north pole of the magnet. You could read more about that on Wikipedia, I know it's a little bit confusing. But in general, when most people refer to magnetic north, or the north pole, they're talking about the geographic north area. And the south pole is the geographic south area. But the reason why I make this distinction is because we know when we deal with magnets, just like electricity, or electrostatics-- but I'll show a key difference very shortly-- is that opposite poles attract. So if this side of my magnet is attracted to Earth's north pole then Earth's north pole-- or Earth's magnetic north-- actually must be the south pole of that magnet. And vice versa. The south pole of my magnet here is going to be attracted to Earth's magnetic south. Which is actually the north pole of the magnet we call Earth. Anyway, I'll take Earth out of the equation because it gets a little bit confusing. And we'll just stick to bars because that tends to be a little bit more consistent. Let me erase this. There you go. I'll erase my Magnesia. I wonder if the element magnesium was first discovered in Magnesia, as well. Probably. And I actually looked up Milk of Magnesia, which is a laxative. And it was not discovered in Magnesia, but it has magnesium in it. So I guess its roots could be in Magnesia if magnesium was discovered in Magnesia. Anyway, enough about Magnesia. Back to the magnets. So if this is a magnet, and let me draw another magnet. Actually, let me erase all of this. All right. So let me draw two more magnets. We know from experimentation when we were all kids, this is the north pole, this is the south pole. That the north pole is going to be attracted to the south pole of another magnet. And that if I were to flip this magnet around, it would actually repel north-- two north facing magnets would repel each other. And so we have this notion, just like we had in electrostatics, that a magnet generates a field. It generates these vectors around it, that if you put something in that field that can be affected by it, it'll be some net force acting on it. So actually, before I go into magnetic field, I actually want to make one huge distinction between magnetism and electrostatics. Magnetism always comes in the form of a dipole. It means that we have two poles. A north and a south. In electrostatics, you do have two charges. You have a positive charge and a negative charge. So you do have two charges. But they could be by themselves. You could just have a proton. You don't have to have an electron there right next to it. You could just have a proton and it would create a positive electrostatic field. And our field lines are what a positive point charge would do. And it would be repelled. So you don't always have to have a negative charge there. Similarly you could just have an electron. And you don't have to have a proton there. So you could have monopoles. These are called monopoles, when you just have one charge when you're talking about electrostatics. But with magnetism you always have a dipole. If I were to take this magnet, this one right here, and if I were to cut it in half, somehow miraculously each of those halves of that magnet will turn into two more magnets. Where this will be the south, this'll be the north, this'll be the south, this will be the north. And actually, theoretically, I've read-- my own abilities don't go this far-- there could be such a thing as a magnetic monopole, although it has not been observed yet in nature. So everything we've seen in nature has been a dipole. So you could just keep cutting this up, all the way down to if it's just one electron left. And it actually turns out that even one electron is still a magnetic dipole. It still is generating, it still has a north pole and a south pole. And actually it turns out, all magnets, the magnetic field is actually generated by the electrons within it. By the spin of electrons and that-- you know, when we talk about electron spin we imagine some little ball of charge spinning. But electrons are-- you know, it's hard to-- they do have mass. But it starts to get fuzzy whether they are energy or mass. And then how does a ball of energy spin? Et cetera, et cetera. So it gets very almost metaphysical. So I don't want to go too far into it. And frankly, I don't think you really can get an intuition. It is almost-- it is a realm that we don't normally operate in. But even these large magnets you deal with, the magnetic field is generated by the electron spins inside of it and by the actual magnetic fields generated by the electron motion around the protons. Well, I hope I'm not overwhelming you. And you might say, well, how come sometimes a metal bar can be magnetized and sometimes it won't be? Well, when all of the electrons are doing random different things in a metal bar, then it's not magnetized. Because the magnetic spins, or the magnetism created by the electrons are all canceling each other out, because it's random. But if you align the spins of the electrons, and if you align their rotations, then you will have a magnetically charged bar. But anyway, I'm past the ten-minute mark, but hopefully that gives you a little bit of a working knowledge of what a magnet is. And in the next video, I will show what the effect is. Well, one, I'll explain how we think about a magnetic field. And then what the effect of a magnetic field is on an electron. Or not an electron, on a moving charge. See you in the next video." + }, + { + "Q": "Why is the tan of pi + 0.46 also a positive slope? If you start at the origin, isn't it then going downward since the arrow is going from right to left in a downward direction?", + "A": "It might help not to think of the terminal ray of pi+.46 as going from the origin outward. On the unit circle there are 3 variables, x, y, and the angle t. All the variation is along the circumference of the unit circle, which determines x, y, and t. The radius doesn t change. The slope of any line is delta y over delta x, which you can verify is y/x = tan(t) (In the 3rd quadrant, as x increases, so does y).", + "video_name": "C3HFAyigqoY", + "transcript": "Voiecover:One angle whose tangent is half is 0.46 radians. So we're saying that the tangent right over here is... So the tangent... So we're gonna write this down. So we're saying that the tangent of 0.46 radians is equal to half. And another way of thinking about the tangent of an angle is that's the slope of that angle's terminal ray. So it's the slope of this ray right over here. Yeah that makes sense that that slope is about half. Now what other angles have a tangent of 1 half? So let's look at these choices. So this is our original angle, 0.46 radians, plus pi over 2. If you think in degrees, pi is 180. pi over 2 is 90 degrees. So this one... Actually let me do in a color you're more likely to see. This one is gonna look like this. Where this is an angle of pi over 2. And just eyeballing it, you immediately see that the slope of this ray is very different than the slope of this ray right over here. In fact they look like they are. They are perpendicular because they have an angle of pi over 2 between them. But they're definitely not going to have the same tangent. They don't have the same slope. Let's think about pi minus 0.46. So that's essentially pi is going along the positive x axis. You go all the way around. Or half way around to your pi radians. But then we're gonna subtract 0.46. So it's gonna look something like this. It's gonna look something like that where this is 0.46 that we have subtracted. Another way to think about it, if we take our original terminal ray and we flip it over the y axis, we get to this terminal ray right over here. And you could immediately see that the slope of the terminal ray is not the same as the slope of this one, of our first one, of our original, in fact they look like the negatives of each other. So we can rule that one out as well. 0.46 plus pi or pi plus 0.46. So that's going to take us... If you add pi to this you're essentially going half way around the unit circle and you're getting to a point that is... Or you're forming a ray that is collinear with the original ray. So that's that angle right over here. So pi plus 0.46 is this entire angle right over there. And when you just look at this ray, you see its collinear is going to have the exact same slope as the terminal ray for the 0.46 radion. So just that tells you that the tangent is going to be the same. So I could check that there. And in previous videos when we explore the symmetries of the tangent function, we in fact saw that. That if you took an angle and you add pi to it, you're going to have the same tangent. And if you wanna dig a little bit deeper, I encourage you to look at that video on the symmetries of unit circle symmetries for the tangent function. So let's look at these other choices. 2 pi minus 0.46. So 2 pi... If this is 0 degrees, 2 pi gets you back to the positive x axis and then you're going to subtract 0.46. So that's going to be this angle right over here. And that looks like it has the negative slope of this original ray right up here. So these aren't going to have the same tangent. Now this one, you're taking 0.46 and you're adding 2 pi. So you're taking 0.46 and then you're adding 2 pi which essentially is just going around the unit circle once and you get to the exact same point. So you add 2 pi to any angle measure, you're going to not only have the same tangent value, you're gonna have the same sine value, cosine value because you're essentially going back around to the exact same angle when you add 2 pi. So this is definitely also going to be true." + }, + { + "Q": "If they used stem cells to cure diseases, wouldn't that stop one of natures ways to lower human population? I don't want to take sides on this one, but I think it would be unfriendly to the environment as a whole,", + "A": "There is no Nature s way that we can know of. That s a religious question. It is true though that overpopulation is a serious problem, and we haven t even cured that many diseases yet. The earth s biosphere is currently undergoing a mass extinction period almost certainly because of the rise of the human species.", + "video_name": "-yCIMk1x0Pk", + "transcript": "Where we left off after the meiosis videos is that we had two gametes. We had a sperm and an egg. Let me draw the sperm. So you had the sperm and then you had an egg. Maybe I'll do the egg in a different color. That's the egg, and we all know how this story goes. The sperm fertilizes the egg. And a whole cascade of events start occurring. The walls of the egg then become impervious to other sperm so that only one sperm can get in, but that's not the focus of this video. The focus of this video is how this fertilized egg develops once it has become a zygote. So after it's fertilized, you remember from the meiosis videos that each of these were haploid, or that they had-- oh, I added an extra i there-- that they had half the contingency of the DNA. As soon as the sperm fertilizes this egg, now, all of a sudden, you have a diploid zygote. Let me do that. So now let me pick a nice color. So now you're going to have a diploid zygote that's going to have a 2N complement of the DNA material or kind of the full complement of what a normal cell in our human body would have. So this is diploid, and it's a zygote, which is just a fancy way of saying the fertilized egg. And it's now ready to essentially turn into an organism. So immediately after fertilization, this zygote starts experiencing cleavage. It's experiencing mitosis, that's the mechanism, but it doesn't increase a lot in size. So this one right here will then turn into-- it'll just split up via mitosis into two like that. And, of course, these are each 2N, and then those are going to split into four like that. And each of these have the same exact genetic complement as that first zygote, and it keeps splitting. And this mass of cells, we can start calling it, this right here, this is referred to as the morula. And actually, it comes from the word for mulberry because it looks like a mulberry. So actually, let me just kind of simplify things a little bit because we don't have to start here. So we start with a zygote. This is a fertilized egg. It just starts duplicating via mitosis, and you end up with a ball of cells. It's often going to be a power of two, because these cells, at least in the initial stages are all duplicating all at once, and then you have this morula. Now, once the morula gets to about 16 cells or so-- and we're talking about four or five days. This isn't an exact process-- they started differentiating a little bit, where the outer cells-- and this kind of turns into a sphere. Let me make it a little bit more sphere like. So it starts differentiating between-- let me make some outer cells. This would be a cross-section of it. It's really going to look more like a sphere. That's the outer cells and then you have your inner cells on the inside. These outer cells are called the trophoblasts. Let me do it in a different color. Let me scroll over. I don't want to go there. And then the inner cells, and this is kind of the crux of what this video is all about-- let me scroll down a little bit. The inner cells-- pick a suitable color. The inner cells right there are called the embryoblast. And then what's going to happen is some fluid's going to start filling in some of this gap between the embryoblast and the trophoblast, so you're going to start having some fluid that comes in there, and so the morula will eventually look like this, where the trophoblast, or the outer membrane, is kind of this huge sphere of cells. And this is all happening as they keep replicating. Mitosis is the mechanism, so now my trophoblast is going to look like that, and then my embryoblast is going to look like this. Sometimes the embryoblast-- so this is the embryoblast. Sometimes it's also called the inner cell mass, so let me write that. And this is what's going to turn into the organism. And so, just so you know a couple of the labels that are involved here, if we're dealing with a mammalian organism, and we are mammals, we call this thing that the morula turned into is a zygote, then a morula, then the cells of the morula started to differentiate into the trophoblast, or kind of the outside cells, and then the embryoblast. And then you have this space that forms here, and this is just fluid, and it's called the blastocoel. A very non-intuitive spelling of the coel part of blastocoel. But once this is formed, this is called a blastocyst. That's the entire thing right here. Let me scroll down a little bit. This whole thing is called the blastocyst, and this is the case in humans. Now, it can be a very confusing topic, because a lot of times in a lot of books on biology, you'll say, hey, you go from the morula to the blastula or the blastosphere stage. Let me write those words down. So sometimes you'll say morula, and you go to blastula. Sometimes it's called the blastosphere. And I want to make it very clear that these are essentially the same stages in development. These are just for-- you know, in a lot of books, they'll start talking about frogs or tadpoles or things like that, and this applies to them. While we're talking about mammals, especially the ones that are closely related to us, the stage is the blastocyst stage, and the real differentiator is when people talk about just blastula and blastospheres. There isn't necessarily this differentiation between these outermost cells and these embryonic, or this embryoblast, or this inner cell mass here. But since the focus of this video is humans, and really that's where I wanted to start from, because that's what we are and that's what's interesting, we're going to focus on the blastocyst. Now, everything I've talked about in this video, it was really to get to this point, because what we have here, these little green cells that I drew right here in the blastocysts, this inner cell mass, this is what will turn into the organism. And you say, OK, Sal, if that's the organism, what's all of these purple cells out here? This trophoblast out there? That is going to turn into the placenta, and I'll do a future video where in a human, it'll turn into a placenta. So let me write that down. It'll turn into the placenta. And I'll do a whole future video about I guess how babies are born, and I actually learned a ton about that this past year because a baby was born in our house. But the placenta is really kind of what the embryo develops inside of, and it's the interface, especially in humans and in mammals, between the developing fetus and its mother, so it kind of is the exchange mechanism that separates their two systems, but allows the necessary functions to go on between them. But that's not the focus of this video. The focus of this video is the fact that these cells, which at this point are-- they've differentiated themselves away from the placenta cells, but they still haven't decided what they're going to become. Maybe this cell and its descendants eventually start becoming part of the nervous system, while these cells right here might become muscle tissue, while these cells right here might become the liver. These cells right here are called embryonic stem cells, and probably the first time in this video you're hearing a term that you might recognize. So if I were to just take one of these cells, and actually, just to introduce you to another term, you know, we have this zygote. As soon as it starts dividing, each of these cells are called a blastomere. And you're probably wondering, Sal, why does this word blast keep appearing in this kind of embryology video, these development videos? And that comes from the Greek for spore: blastos. So the organism is beginning to spore out or grow. I won't go into the word origins of it, but that's where it comes from and that's why everything has So these are blastomeres. So when I talk what embryonic stem cells, I'm talking about the individual blastomeres inside of this embryoblast or inside of this inner cell mass. These words are actually unusually fun to say. So each of these is an embryonic stem cell. Let me write this down in a vibrant color. So each of these right here are embryonic stem cells, and I wanted to get to this. And the reason why these are interesting, and I think you already know, is that there's a huge debate around these. One, these have the potential to turn into anything, that they have this plasticity. That's another word that you might hear. Let me write that down, too: plasticity. And the word essentially comes from, you know, like a plastic can turn into anything else. When we say that something has plasticity, we're talking about its potential to turn into a lot of different things. So the theory is, and there's already some trials that seem to substantiate this, especially in some lower organisms, that, look, if you have some damage at some point in your body-- let me draw a nerve cell. Let me say I have a-- I won't go into the actual mechanics of a nerve cell, but let's say that we have some damage at some point on a nerve cell right there, and because of that, someone is paralyzed or there's some nerve dysfunction. We're dealing with multiple sclerosis or who knows what. The idea is, look, we have these cell here that could turn into anything, and we're just really understanding how it knows what to turn into. It really has to look at its environment and say, hey, what are the guys around me doing, and maybe that's what helps dictate what it does. But the idea is you take these things that could turn to anything and you put them where the damage is, you layer them where the damage is, and then they can turn into the cell that they need to turn into. So in this case, they would turn into nerve cells. They would turn to nerve cells and repair the damage and maybe cure the paralysis for that individual. So it's a huge, exciting area of research, and you could even, in theory, grow new organs. If someone needs a kidney transplant or a heart transplant, maybe in the future, we could take a colony of these embryonic stem cells. Maybe we can put them in some type of other creature, or who knows what, and we can turn it into a replacement heart or a So there's a huge amount of excitement about what these can do. I mean, they could cure a lot of formerly uncurable diseases or provide hope for a lot of patients who might otherwise die. But obviously, there's a debate here. And the debate all revolves around the issue of if you were to go in here and try to extract one of these cells, you're going to kill this embryo. You're going to kill this developing embryo, and that developing embryo had the potential to become a human being. It's a potential that obviously has to be in the right environment, and it has to have a willing mother and all of the rest, but it does have the potential. And so for those, especially, I think, in the pro-life camp, who say, hey, anything that has a potential to be a human being, that is life and it should not be killed. So people on that side of the camp, they're against the destroying of this embryo. I'm not making this video to take either side to that argument, but it's a potential to turn to a human being. It's a potential, right? So obviously, there's a huge amount of debate, but now, now you know in this video what people are talking about when they say embryonic stem cells. And obviously, the next question is, hey, well, why don't they just call them stem cells as opposed to embryonic stem cells? And that's because in all of our bodies, you do have what are called somatic stem cells. Let me write that down. Somatic or adults stem cells. And we all have them. They're in our bone marrow to help produce red blood cells, other parts of our body, but the problem with somatic stem cells is they're not as plastic, which means that they can't form any type of cell in the human body. There's an area of research where people are actually maybe trying to make them more plastic, and if they are able to take these somatic stem cells and make them more plastic, it might maybe kill the need to have these embryonic stem cells, although maybe if they do this too good, maybe these will have the potential to turn into human beings as well, so that could become a debatable issue. But right now, this isn't an area of debate because, left to their own devices, a somatic stem cell or an adult stem cell won't turn into a human being, while an embryonic one, if it is implanted in a willing mother, then, of course, it will turn into a human being. And I want to make one side note here, because I don't want to take any sides on the debate of-- well, I mean, facts are facts. This does have the potential to turn into a human being, but it also has the potential to save millions of lives. Both of those statements are facts, and then you can decide on your own which side of that argument you'd like to or what side of that balance you would like to kind of put your own opinion. But there's one thing I want to talk about that in the public debate is never brought up. So you have this notion of when you-- to get an embryonic stem cell line, and when I say a stem cell line, I mean you take a couple of stem cells, or let's say you take one stem cell, and then you put it in a Petri dish, and then you allow it to just duplicate. So this one turns into two, those two turn to four. Then someone could take one of these and then put it in their own Petri dish. These are a stem cell line. They all came from one unique embryonic stem cell or what initially was a blastomere. So that's what they call a stem cell line. So the debate obviously is when you start an embryonic stem cell line, you are destroying an embryo. But I want to make the point here that embryos are being destroyed in other processes, and namely, in-vitro fertilization. And maybe this'll be my next video: fertilization. And this is just the notion that they take a set of eggs out of a mother. It's usually a couple that's having trouble having a child, and they take a bunch of eggs out of the mother. So let's say they take maybe 10 to 30 eggs out of the mother. They actually perform a surgery, take them out of the ovaries of the mother, and then they fertilize them with semen, either it might come from the father or a sperm donor, so then all of these becomes zygotes once they're fertilized with semen. So these all become zygotes, and then they allow them to develop, and they usually allow them to develop to the blastocyst stage. So eventually all of these turn into blastocysts. They have a blastocoel in the center, which is this area of fluid. They have, of course, the embryo, the inner cell mass in them, and what they do is they look at the ones that they deem are healthier or maybe the ones that are at least just not unhealthy, and they'll take a couple of these and they'll implant these into the mother, so all of this is occurring in a Petri dish. So maybe these four look good, so they're going to take these four, and they're going to implant these into a mother, and if all goes well, maybe one of these will turn into-- will give the couple a child. So this one will develop and maybe the other ones won't. But if you've seen John & Kate Plus 8, you know that many times they implant a lot of them in there, just to increase the probability that you get at least one child. But every now and then, they implant seven or eight, and then you end up with eight kids. And that's why in-vitro fertilization often results in kind of these multiple births, or reality television shows on cable. But what do they do with all of these other perfectly-- well, I won't say perfectly viable, but these are embryos. They may or may not be perfectly viable, but you have these embryos that have the potential, just like this one right here. These all have the potential to turn into a human being. But most fertility clinics, roughly half of them, they either throw these away, they destroy them, they allow them to die. A lot of these are frozen, but just the process of freezing them kills them and then bonding them kills them again, so most of these, the process of in-vitro fertilization, for every one child that has the potential to develop into a full-fledged human being, you're actually destroying tens of very viable embryos. So at least my take on it is if you're against-- and I generally don't want to take a side on this, but if you are against research that involves embryonic stem cells because of the destruction of embryos, on that same, I guess, philosophical ground, you should also be against in-vitro fertilization because both of these involve the destruction of zygotes. I think-- well, I won't talk more about this, because I really don't want to take sides, but I want to show that there is kind of an equivalence here that's completely lost in this debate on whether embryonic stem cells should be used because they have a destruction of embryos, because you're destroying just as many embryos in this-- well, I won't say just as many, but you are destroying embryos. There's hundreds of thousands of embryos that get destroyed and get frozen and obviously destroyed in that process as well through this in-vitro fertilization process. So anyway, now hopefully you have the tools to kind of engage in the debate around stem cells, and you see that it all comes from what we learned about meiosis. They produce these gametes. The male gamete fertilizes a female gamete. The zygote happens or gets created and starts splitting up the morula, and then it keeps splitting and it differentiates into the blastocyst, and then this is where the stem cells are. So you already know enough science to engage in kind of a very heated debate." + }, + { + "Q": "This question is not about the video but one I have had for awhile, What is the best way to write this: 3/4 or .75, I prefer decimals I do not know why but i do, Khan Academy normally asks for fractions, I have the right decimal for the fraction but it is considered wrong. What is best fraction or decimals or is it all opinion?", + "A": "Using fraction will help you in higher level math, and you will need to be able to use them well. They are just preparing you now for what s ahead.", + "video_name": "iI_2Piwn_og", + "transcript": "I now want to solve some inequalities that also have absolute values in them. And if there's any topic in algebra that probably confuses people the most, it's this. But if we kind of keep our head on straight about what absolute value really means, I think you will find that it's not that bad. So let's start with a nice, fairly simple warm-up problem. Let's start with the absolute value of x is less than 12. So remember what I told you about the meaning of absolute value. It means how far away you are from 0. So one way to say this is, what are all of the x's that are less than 12 away from 0? Let's draw a number line. So if we have 0 here, and we want all the numbers that are less than 12 away from 0, well, you could go all the way to positive 12, and you could go all the way to negative 12. Anything that's in between these two numbers is going to have an absolute value of less than 12. It's going to be less than 12 away from 0. So this, you could say, this could be all of the numbers where x is greater than negative 12. Those are definitely going to have an absolute value less than 12, as long as they're also-- and, x has to be less than 12. So if an x meets both of these constraints, its absolute value is definitely going to be less than 12. You know, you take the absolute value of negative 6, that's only 6 away from 0. The absolute value of negative 11, only 11 away from 0. So something that meets both of these constraints will satisfy the equation. And actually, we've solved it, because this is only a one-step equation there. But I think it lays a good foundation for the next few problems. And I could actually write it like this. In interval notation, it would be everything between negative 12 and positive 12, and not including those numbers. Or we could write it like this, x is less than 12, and is greater than negative 12. That's the solution set right there. Now let's do one that's a little bit more complicated, that allows us to think a little bit harder. So let's say we have the absolute value of 7x is greater than or equal to 21. So let's not even think about what's inside of the absolute value sign right now. In order for the absolute value of anything to be greater than or equal to 21, what does it mean? It means that whatever's inside of this absolute value sign, whatever that is inside of our absolute value sign, it must be 21 or more away from 0. Let's draw our number line. And you really should visualize a number line when you do this, and you'll never get confused then. You shouldn't be memorizing any rules. So let's draw 0 here. Let's do positive 21, and let's do a negative 21 here. So we want all of the numbers, so whatever this thing is, that are greater than or equal to 21. They're more than 21 away from 0. Their absolute value is more than 21. Well, all of these negative numbers that are less than negative 21, when you take their absolute value, when you get rid of the negative sign, or when you find their distance from 0, they're all going to be greater than 21. If you take the absolute value of negative 30, it's going to be greater than 21. Likewise, up here, anything greater than positive 21 will also have an absolute value greater than 21. So what we could say is 7x needs to be equal to one of these numbers, or 7x needs to be equal to one of these numbers out here. So we could write 7x needs to be one of these numbers. Well, what are these numbers? These are all of the numbers that are less than or equal to negative 21, or 7x-- let me do a different color here-- or 7x has to be one of these numbers. And that means that 7x has to be greater than or equal to positive 21. I really want you to kind of internalize what's going on here. If our absolute value is greater than or equal to 21, that means that what's inside the absolute value has to be either just straight up greater than the positive 21, or less than negative 21. Because if it's less than negative 21, when you take its absolute value, it's going to be more than 21 away from 0. Hopefully that make sense. We'll do several of these practice problems, so it really gets ingrained in your brain. But once you have this set up, and this just becomes a compound inequality, divide both sides of this equation by 7, you get x is less than or equal to negative 3. Or you divide both sides of this by 7, you get x is greater than or equal to 3. So I want to be very clear. This, what I drew here, was not the solution set. This is what 7x had to be equal to. I just wanted you to visualize what it means to have the absolute value be greater than 21, to be more than 21 away from 0. This is the solution set. x has to be greater than or equal to 3, or less than or equal to negative 3. So the actual solution set to this equation-- let me draw a number line-- let's say that's 0, that's 3, that is negative 3. x has to be either greater than or equal to 3. That's the equal sign. Or less than or equal to negative 3. Let's do a couple more of these. Because they are, I think, confusing, but if you really start to get the gist of what absolute value is saying, they become, I think, intuitive. So let's say that we have the absolute value-- let me get a good one. Let's say the absolute value of 5x plus 3 is less than 7. So that's telling us that whatever's inside of our absolute value sign has to be less than 7 away from 0. So the ways that we can be less than 7 away from 0-- let me draw a number line-- so the ways that you can be less than 7 away from 0, you could be less than 7, and greater than negative 7. You have to be in this range. So in order to satisfy this thing in this absolute value sign, it has to be-- so the thing in the absolute value sign, which is 5x plus 3-- it has to be greater than negative 7 and it has to be less than 7, in order for its absolute value to be less than 7. If this thing, this 5x plus 3, evaluates anywhere over here, its absolute value, its distance from 0, will be less than 7. And then we can just solve these. You subtract 3 from both sides. 5x is greater than negative 10. Divide both sides by 5. x is greater than negative 2. Now over here, subtract 3 from both sides. 5x is less than 4. Divide both sides by 5, you get x is less than 4/5. And then we can draw the solution set. We have to be greater than negative 2, not greater than or equal to, and less than 4/5. So this might look like a coordinate, but this is also interval notation, if we're saying all of the x's between negative 2 and 4/5. Or you could write it all of the x's that are greater than negative 2 and less than 4/5. These are the x's that satisfy this equation. And I really want you to internalize this visualization here. Now, you might already be seeing a bit of a rule here. And I don't want you to just memorize it, but I'll give it to you just in case you want it. If you have something like f of x, the absolute value of f of x is less than, let's say, some number a. So this was the situation. We have some f of x less than a. That means that the absolute value of f of x, or f of x has to be less than a away from 0. So that means that f of x has to be less than positive a or greater than negative a. That translates to that, which translates to f of x greater than negative a and f of x less than a. But it comes from the same logic. This has to evaluate to something that is less than a away from 0. Now, if we go to the other side, if you have something of the form f of x is greater than a. That means that this thing has to evaluate to something that is further than a away from 0. So that means that f of x is either just straight up greater than positive a, or f of x is less than negative a. If it's less than negative a, maybe it's negative a minus another 1, or negative 5 plus negative a. Then, when you take its absolute value, it'll become a plus 5. So its absolute value is going to be greater than a. So I just want to-- you could memorize this if you want, but I really want you to think about this is just saying, OK, this has to evaluate, be less than a away from 0, this has to be more than a away from 0. Let's do one more, because I know this can be a little bit confusing. And I encourage you to watch this video over and over and over again, if it helps. Let's say we have the absolute value of 2x-- let me do another one over here. Let's do a harder one. Let's say the absolute value of 2x over 7 plus 9 is greater than 5/7. So this thing has to evaluate to something that's more than 5/7 away from 0. So this thing, 2x over 7 plus 9, it could just be straight up greater than 5/7. Or it could be less than negative 5/7, because if it's less than negative 5/7, its absolute value is going to be greater than 5/7. Or 2x over 7 plus 9 will be less than negative 5/7. We're doing this case right here. And then we just solve both of these equations. See if we subtract-- let's just multiply everything by 7, just to get these denominators out of the way. So if you multiply both sides by 7, you get 2x plus 9 times 7 is 63, is greater than 5. Let's do it over here, too. You'll get 2x plus 63 is less than negative 5. Let's subtract 63 from both sides of this equation, and you get 2x-- let's see. 5 minus 63 is 58, 2x is greater than 58. If you subtract 63 from both sides of this equation, you get 2x is less than negative 68. Oh, I just realized I made a mistake here. You subtract 63 from both sides of this, 5 minus 63 is negative 58. I don't want to make a careless mistake there. And then divide both sides by 2. You get, in this case, x is greater than-- you don't have to swap the inequality, because we're dividing by a positive number-- negative 58 over 2 is negative 29, or, here, if you divide both sides by 2, or, x is less than negative 34. 68 divided by 2 is 34. And so, on the number line, the solution set to that equation will look like this. That's my number line. I have negative 29. I have negative 34. So the solution is, I can either be greater than 29, not greater than or equal to, so greater than 29, that is that right there, or I could be less than negative 34. So any of those are going to satisfy this absolute value inequality." + }, + { + "Q": "Doesn't he mean cross product at 2:00?", + "A": "Yes, he meant cross product.", + "video_name": "b7JTVLc_aMk", + "transcript": "What I want to do with this video is cover something called the triple product expansion-- or Lagrange's formula, sometimes. And it's really just a simplification of the cross product of three vectors, so if I take the cross product of a, and then b cross c. And what we're going to do is, we can express this really as sum and differences of dot products. Well, not just dot products-- dot products scaling different vectors. You're going to see what I mean. But it simplifies this expression a good bit, because cross products are hard to take. They're computationally intensive and, at least in my mind, they're confusing. Now this isn't something you have to know if you're going to be dealing with vectors, but it's useful to know. My motivation for actually doing this video is I saw some problems for the Indian Institute of Technology entrance exam that seems to expect that you know Lagrange's formula, or the triple product expansion. So let's see how we can simplify this. So to do that, let's start taking the cross product of b and c. And in all of these situations, I'm just going to assume-- let's say I have vector a. That's going to be a, the x component of vector a times the unit of vector i plus the y component of vector a times the unit vector j plus the z component of vector a times unit vector k. And I could do the same things for b and c. So if I say b sub y, I'm talking about what's scaling the j component in the b vector. So let's first take this cross product over here. And if you've seen me take cross products, you know that I like to do these little determinants. Let me just take it over here. So b cross c is going to be equal to the determinant. And I put an i, j, k up here. This is actually the definition of the cross product, so no proof necessary to show you why this is true. This is just one way to remember the dot product, if you remember how to take determinants of three-by-threes. And we'll put b's x term, b's y coefficient, and b's z component. And then you do the same thing for the c, cx, cy, cz. And then this is going to be equal to-- so first you have the i component. So it's going to be the i component times b. So you ignore this column and this row. So bycz minus bzcy. So I'm just ignoring all of this. And I'm looking at this two-by-two over here, minus bzcy. And then we want to subtract the j component. Remember, we alternate signs when we take our determinant. Subtract that. And then we take out that column and that row, so it's going to be bxcz-- this is a little monotonous, but hopefully, it'll have an interesting result-- bxcz minus bzcx. And then finally, plus the k component. OK, we're going to have bx times cy minus bycx. We just did the dot product, and now we want to take the-- oh, sorry, we just did the cross product. I don't want to get you confused. We just took the cross product of b and c. And now we need take the cross product of that with a, or the cross product of a with this thing right over here. Instead of rewriting the vector, let me just set up another matrix here. So let me write my i j k up here. And then let me write a's components. So we have a sub x, a sub y, a sub z. And then let's clean this up a little bit. We're just looking at-- no, I want to do that in black. Let's do this in black, so that we can kind of erase that. Now this is a minus j times that. So what I'm going to do is I'm going to get rid of the minus and the j, but I am going to rewrite this with the signs swapped. So if you swap the signs, it's actually bzcx minus bxcz. So let me delete everything else. So I just took the negative and I multiplied it by this. I hope I'm not making any careless mistakes here, so let me just check and make my brush size little bit bigger, so I can erase that a little more efficiently. And then we also want to get rid of that right over there. Now let me get my brush size back down to normal size. All right. So now let's just take this cross product. So once again, set it up as a determinant. And what I'm only going to focus on-- because it'll take the video, or it'll take me forever if I were to do the i, j, and k components-- I'm just going to focus on the i component, just on the x component of this cross product. And then we can see that we'll get the same result for the j and the k. And then we can see what, hopefully, this simplifies down to. So if we just focus on the i component here, this is going to be i times-- and we just look at this two-by-two matrix right over here. We ignore i's column, i's row. And we have ay times all of this. So let me just multiply it out. So it's ay times bxcy, minus ay times by, times bycx. And then we're going to want to subtract. We're going to have minus az times this. So let's just do that. So it's minus, or negative, azbzcx. And then we have a negative az times this, so it's plus azbxcz. And now what I'm going to do-- this is a little bit of a trick for this proof right here, just so that we get the results that I want. I'm just going to add and subtract the exact same thing. So I'm going to add an axbxcx. And then I'm going to subtract an axbxcx, minus axbxcx. So clearly, I have not changed this expression. I've just added and subtracted the same thing. And let's see what we can simplify. Remember, this is just the x component of our triple product. Just the x component. But to do this, let me factor out. I'm going to factor out a bx. So let me do this, let me get the bx. So if I were to factor it out-- I'm going to factor it out of this term that has a bx. I'm going to factor it out of this term. And then I'm going to factor it out of this term. So if I take the bx out, I'm going to have an aycy. Actually, let me write it a little bit differently. Let me factor it out of this one first. So then it's going to have an axcx. a sub x, c sub x. So I used this one up. And then I'll do this one now. Plus, if I factor the bx out, I get ay cy. I've used that one now. And now I have this one. I'm going to factor the bx out. So I'm left with a plus az, cz. So that's all of those. So I've factored that out. And now, from these right over here, let me factor out a negative cx. And so, if I do that-- let me go to this term right over here-- I'm going to have an axbx when I factor it out. So an axbx, cross that out. And then, over here, I'm going to have an ayby. Remember, I'm factoring out a negative cx, so I'm going to have a plus ay, sub by. And then, finally, I'm going to have a plus az, az bz. And what is this? Well, this right here, in green, this is the exact same thing as the dot products of a and c. This is the dot product of the vectors a and c. It's the dot product of this vector and that vector. So that's the dot of a and c times the x component of b minus-- I'll do this in the same-- minus-- once again, this is the dot product of a and b now, minus a dot b times the x component of c. And we can't forget, all of this was multiplied by the unit vector i. We're looking at the x component, or the i component of that whole triple product. So that's going to be all of this. All of this is times the unit vector i. Now, if we do this exact same thing-- and I'm not going to do it, because it's computationally intensive. But I think it won't be a huge leap of faith for you. This is for the x component. If I were to do the exact same thing for the y component, for the j component-- so it'll be plus-- if I do the same thing for the j component, we can really just pattern match. We have bx, cx, that's for the x component. We'll have by and c y for the j component. And then this is not component-specific, so it will be a dot c over here, and minus a dot b over here. You can verify any of these for yourself, if you don't believe me. But it's the exact same process we just did. And then, finally, for the z component, or the k component-- let me put parentheses over here-- same idea. You're going to have bz, cz. And then you're going to have a dot b over there. And then you're going to have a dot c over here. Now what does this become? How can we simplify this? Well, this right over here, we can expand this out. We can factor out an a dot c from all of these terms over here. And remember, this is going to be multiplied times i. Actually, let me not skip too many steps, just because I want you to believe what I'm doing. So if we expand the i here-- instead of rewriting it, let me just do it like this. It's a little bit messier, but let me just-- so I could write this i there and that i there. I'm kind of just distributing that x unit vector, or the i unit vector. And let me do the same thing for j. So I could put the j there. And I could put the j right over there. And then I could do the same thing for the k, put the k there, and then put the k there. And now what are these? Well, this part right over here is exactly the same thing as a dot c times-- and I'll write it out here-- bx times i plus by times j, plus bz times k. And then, from that, we're going to subtract all of this, a dot b. We're going to subtract a dot b times the exact same thing. And you're going to notice, this right here is the same thing as vector b. That is vector b. When you do it over here, you're going to get vector c. So I'll just write it over here. You're just going to get vector c. So just like that, we have a simplification for our triple product. I know it took us a long time to get here, but this is a simplification. It might not look like one, but computationally it is. It's easier to do. If I have-- I'll try to color-code it-- a cross b cross-- let me do it in all different colors-- c, we just saw that this is going to be equivalent to-- and one way to think about it is, it's going to be, you take the first vector times the dot product of-- the first vector in this second dot product, the one that we have our parentheses around, the one we would have to do first-- you take your first vector there. So it's vector b. And you multiply that times the dot product of the other two vectors, so a dot c. And from that, you subtract the second vector multiplied by the dot product of the other two vectors, of a dot b. And we're done. This is our triple product expansion. Now, once again, this isn't something that you really have to know. You could always, obviously, multiply it. You could actually do it by hand. You don't have to know this. But if you have really hairy vectors, or if this was some type of math competition, and sometimes it simplifies real fast when you reduce it to dot products, this is a useful thing to know, Lagrange's formula, or the triple product expansion." + }, + { + "Q": "What are new operators useful for?", + "A": "They are just definitions. Using these arbitrary operators helps us to be more flexible and comfortable in using algebra, and gives us a better understanding of numerical reasoning and computer science.", + "video_name": "ND-Bbp_q46s", + "transcript": "We're all used to the traditional operators like addition and subtraction and multiplication and division. And we've seen there's multiple ways to represent this. But what we're going to do in this video is a little fun. We're actually going to define our own operators. And what's neat about this is it kind of shows how broad mathematics can be. And on a more practical sense, it's actually something that you might see on some standardized tests. And the reason why they do that is so that you can appreciate that these aren't the only operators out there-- plus exponentiation and all those-- that in mathematics, you can define a whole new set of operators. So let's just do that. So let me just define x diamond y. And I'm going to define that as 5x minus y. So you could view this as defining it a function. But we're defining it using an operator. So if I have x diamond y, by definition, we have defined this operator. That means that's going to be equal to 5x minus y. So given that definition, what would 7 diamond 11 be? Well, you just go to the definition. 7 diamond 11. Instead of an x we have a 7. So it's going to be 5 times 7. So let me do it 5 times 7 minus, and instead of a y, we have an 11. So one way to think about it is, in our definition, every place you saw an x, you could replace with a 7, every place you saw a y, you replace with an 11. So you have minus 11 over here. Let me make the number-- So this is the 7. This 7 is this 7. And this 11 is this 11 right over here. And then we just evaluate that. So 5 times 7 is 35. So this is equal to 35 minus 11, which is equal to 24. So 7 diamond 11 is equal to 24. We can define other things. We can define something crazy like, let me define a-- well, I mentioned a star, let me use a star-- a star-- let me write it this way-- a star b. Let's say that that is the same thing as-- I don't know-- a over a plus b. And so same idea. What would 5 star 6 be? Well, you go back to the definition. By definition, every place where you see the a, you would now replace with a 5. Every time you saw the b, you would now replace with a 6. So this is going to be equal to 5 over 5 plus 6. a plus b. a is 5, b is 6 over 5 plus 6. So this would be 5/11. And then you can compound them. And we haven't defined any order of operations for these particular operations that we've just defined. So we're going to be careful to use parentheses when we put some of these together, but you can do something like, something interesting, like negative 1 diamond 0 star 5. And once again, we just focus on parentheses, because that's the only thing that's telling us what to start on first. Because we haven't figured out, we haven't defined whether diamond takes precedence over star, or star takes precedence over diamond the way that we have that saying that, hey, you do multiplication before you do addition. We haven't defined it for those operations, but that's what the parentheses helps us do. So we want to evaluate these parentheses first. 0 star 5 that is 0-- because you could view this 0 as the a and the 5 as the b--so it's going to be 0 over 0 plus 5, which is just going to be 0. So this over here, 0 or 5 just goes to 0. So this whole expression simplifies to negative 1 diamond-- this diamond right over here-- diamond 0. And now we go to the definition of the diamond operator. Well, that's five times the first number in our operator, or the first term that we're giving the operator. I guess you could think of it that way. So 5 times that. So it'll be 5 times negative 1, x is negative 1 minus y. Well, y here is the 0. Minus 0. So 5 times negative 1 is negative 5. And you will see-- and the idea here is just to make you feel comfortable defining new operators like this. And not being daunted if all of a sudden you see a diamond, and they're defining the diamond for you. And you're like, wait, I never saw a diamond. They're actually defining it for you, so you shouldn't say, I never saw a diamond. You should say, well, they've defined a diamond for me. This is how I use that operator. And sometimes you'll see even wackier things. You'll see things like this. Let me draw. So they'll define. I don't know if you would even consider this an operator. But you'll see something like this, that by definition, if someone writes a symbol like this, and they put-- a, b, c-- let me write it this way-- a, b, c, d. They'll say this is the same thing as ad minus b, all of that over c. And once again, this is just a definition. They have this weird symbolic way of representing these variables in all this. But they're defining how do you evaluate this crazy expression. And so, if someone were to give you, were to say, evaluate this diamond. Let me evaluate the diamond. So evaluate the diamond where, in my little sections of the diamond, I have a negative 1, a 5, a 3, and a 2. We would just use the definition of how to evaluate this diamond. And we'd say, OK, every time we see an a, that is going to be negative 1. So we have a negative 1 times d. Well, d is whatever is in the bottom right section of this diameter or this kite. So d is going to be 2. Let me write it this way. This is b. This is c. And this is d. So it's going to be negative 1 times 2 minus b-- well b is 5-- minus 5, all of that over c, which is 3. So this is going to be equal to negative 2 minus 5. So that is negative 7 over 3. And you could go crazy like this. And it might be a fun thing, actually, if you have some spare time. Define your own operators and see how creative you can get with those operators." + }, + { + "Q": "Why do we need parametric equations to draw lines in 3 dimensions? I'm not really familiar with the 3-dimensional coordinate plane, so this is new to me.", + "A": "Because we have 3 directions. With parametric equations you can describe all the coordinates of any point on the line. You just have to split the equation into x,y and z to get the coordinates of a point.", + "video_name": "hWhs2cIj7Cw", + "transcript": "Everything we've been doing in linear algebra so far, you might be thinking, it's kind of a more painful way of doing things that you already knew how to do. You've already dealt with vectors. I'm guessing that some of you all have already dealt with vectors in your calculus or your pre-calculus or your physics classes. But in this video I hope to show you something that you're going to do in linear algebra that you've never done before, and that it would have been very hard to do had you not been exposed to these videos. Well I'm going to start with, once again, a different way of doing something you already know how to do. So let me just define some vector here, instead of making them bold, I'll just draw it with the arrow on top. I'm going to define my vector to be-- I can do with the arrow on top or I can just make it super bold. I'm just going to define my vectors, it's going to be a vector in R2. Let's just say that my vector is the vector 2, 1. If I were to draw it in standard position, it looks like this. You go two to the right, up one, like that. That's my, right there, that is my vector v. Now, if I were to ask you, what are all of the possible vectors I can create? So let me define a set. Let me define a set, s, and it's equal to-- all of the vectors I can create, if I were to multiply v times some constant, so I multiply some constant, some scalar, times my vector v, and just to maybe be a little bit formal, I'll say such that c is a member of the real numbers. Now what would be a graphical representation of this set? Well, if we draw them all in standard position, c could be any real number. So if I were to multiply, c could be 2. If c is 2, let me do it this way. If I do 2 times our vector, I'm going to get the vector 4, 2. Let me draw that in standard position, 4, 2. It's right there. It's this vector right there. It's collinear with this first vector. It's along the same line, but just goes out further 2. Now I could've done another. I could have done 1.5 times our vector v. Let me do that in a different color. And maybe that would be, that would be what? That'd be 1.5 times 2, which is 3, 1.5. Where would that vector get me? I'd go one 1.5 and then I'd go 3, and then 1.5, I'd get right there. And I can multiply by anything. I can multiply 1.4999 times vector v, and get right over here. I could do minus 0.0001 times vector v. Let me write that down. I could do 0.001 times our vector v. And where were that put me? It would put me little super small vector right there. If I did minus 0.01, it would make a super small vector right there pointing in that direction. If I were to do minus 10, I would get a vector going in this direction that goes way like that. But you can imagine that if I were to plot all of the vectors in standard position, all of them that could be represented by any c in real numbers, I'll essentially get-- I'll end up drawing a bunch of vectors where their arrows are all lined up along this line right there, and all lined up in even in negative direction-- let me make sure I draw it properly-- along that line, like that. I think you get the idea. So it's a set of collinear vectors. So let me write that down. And if we view these vectors as position vectors, that this vector represents a point in space in R2-- this R2 is just our Cartesian coordinate plane right here in every direction-- if we view this vector as a position vector-- let me write that down-- if we view it as kind of a coordinate in R2, then this set, if we visually represent it as a bunch of position vectors, it'll be represented by this whole line over here. And I want to make that point clear because it's essentially a line, of slope 2. Right? Sorry, slope 1/2. Your rise is 1. Your rise is 1 for going over 2. But I don't want to go back to our Algebra 1 But I want to make this point that this line of slope 2 that goes through the origin, this is if we draw all of the vectors in the set as in their standard form, or if we draw them all as position vectors. If I didn't make that clarification, or that qualification, I could have drawn these vectors anywhere. Because this 4, 2 vector, I could have drawn over here. And then, to say that it's collinear probably wouldn't have made as much visual sense to you. But I think this collinearity of it makes more sense to you if you say, oh, let's draw them all in standard form. All of them start at the origin, and then their tails are at the origin, and their heads go essentially to the coordinate they represent. That's what I mean by their position vectors. They don't necessarily have to be position vectors, but for the visualization in this video, let's stick to that. Now I was only able to represent something that goes through the origin with this slope. So you can almost view that this vector kind of represented its slope. You almost want to view it as a slope vector, if you wanted to tie it in to what you learned in Algebra 1. What if we wanted to represent other lines that had that slope? What if we wanted to represent the the same line, or I guess a parallel line-- that goes through that point over there, the point 2 comma 4? Or if we're thinking in position vectors, we could say that point is represented by the vector, and we will call that x. It's represented by the vector x. And the vector x is equal to 2, 4. That point right there. What if I want to represent the line that's parallel to this that goes through that point 2, 4? So I want to represent this line right here. I'll draw it as parallel to this as I can. I think you get the idea, and it just keeps going like that in every direction. These two lines are parallel. How can I represent the set of all of these vectors, drawn in standard form, or all of the vectors, that if I were to draw them in standard form, would show this line? Well, you can think about it this way. If every one of the vectors that represented this line, if I start with any vector that was on this line, and I add my x vector to it, I'll show up at a corresponding point on this line that I want to be at. Right? Let's say I do negative 2 times my original, so minus 2 times my vector v, that equaled what? Minus 4, minus 2, so that's that vector there. But if I were to add x to it, if I were to add my x vector. So if I were to do minus 2 times my vector v, but I were to add x to it, so plus x. I'm adding this vector 2 comma 4 to it, so from here I'd go right 2 and up 4, so I'd go here. Or visually you could just say, heads to tails, so I would go right there. So I would end up at a corresponding point over there. So when I define my set, s, as the set of all points where I just multiply v times the scalar, I got this thing that went through the origin. But now let me define another set. Let me define a set l, maybe l for line, that's equal to the set of all of vectors where the vector x, I could do it bold or I'll just draw an arrow on it, plus some scalar-- I could use c, but let me use t, because I'm going to call this a parametrization of the line-- so plus some scalar, t times my vector v such that t could be any member of the real numbers. So what is this going to be? This is going to be this blue line. If I were to draw all of these vectors in standard position, I'm going to get my blue line. For example, if I do minus 2, this is minus 2, times my vector v, I get here. Then if I add x, I go there. So this vector right here that has its endpoint right there-- its endpoint sits on that line. I can do that with anything. If I take this vector, this is some scalar times my vector v, and I add x to it, I end up with this vector, whose endpoint, if I view it as a position vector, it's endpoint dictates some coordinate in the xy plane. So it will [UNINTELLIGIBLE] that point. So I can get to any of these vectors. This is a set of vectors right here, and all of these vectors are going to point-- they're essentially going to point to something-- when I draw them in standard form, if I draw them in standard form-- they're going to point to a point on that blue line. Now you might say, hey Sal, this was a really obtuse way of defining a line. I mean we do it in Algebra 1, where we just say, hey you know, y is equal to mx plus b. And we figure out the slope by figuring out the difference of two points, and then we do a little substitution. And this is stuff you learned in seventh or eighth grade. This was really straightforward. Why am I defining this obtuse set here and making you think in terms of sets and vectors and adding vectors? And the reason is, is because this is very general. This worked well in R2. So in R2, this was great. I mean, we just have to worry about x's and y's. But what about the situation, I mean notice, in your algebra class, your teacher never really told you much, at least in the ones I took, about how do you present lines in three dimensions? Maybe some classes go there, but they definitely didn't tell you how do you represent lines in four dimensions, or a hundred dimensions. And that's what this is going to do for us. Right here, I defined x and v as vectors in R2. They're two-dimensional vectors, but we can extend it to an arbitrary number of dimensions. So just to kind of hit the point home, let's do one more example in R2, where, it's kind of the classic algebra problem where you need to find the equation for the line. But here, we're going to call it the set definition for the line. Let's say we have two vectors. Let's say we have the vector a, which I'll define as-- let me just says it's 2, 1. So if I were draw it in standard form, it's 2, 1. That's my vector a, right there. And let's say I have vector b, let me define vector b. I'm going to define it as, I don't know, let me define it as 0, 3. So my vector b, 0-- I don't move to the right at all and I go up. So my vector b will look like that. Now I'm going to say that these are position vectors, that we draw them in standard form. When you draw them in standard form, their endpoints represent some position. So you can almost view these as coordinate points in R2. This is R2. All of these coordinate axes I draw are going be R2. Now what if I asked you, give me a parametrization of the line that goes through these two points. So essentially, I want the equation-- if you're thinking in Algebra 1 terms-- I want the equation for the line that goes through these two points. So the classic way, you would have figured out the slope and all of that, and then you would have substituted back in. But instead, what we can do is, we can say, hey look, this line that goes through both of those points-- you could almost say that both of those vectors lie on-- I guess that's a better-- Both of these vectors lie on this line. Now, what vector can be represented by that line? Or even better, what vector, if I take any arbitrary scalar-- can represent any other vector on that line? Now let me do it this way. What if I were to take-- so this is vector b here-- what if I were to take b minus a? We learned in, I think it was the previous video, that b minus a, you'll get this vector right here. You'll get the difference in the two vectors. This is the vector b minus the vector a. And you just think about it. What do I have to add to a to get to b? I have to add b minus a. So if I can get the vector b minus a-- right, we know how We just subtract the vectors, and then multiply it by any scalar, then we're going to get any point along that line. We have to be careful. So what happens if we take t, so some scalar, times our vector, times the vectors b minus a? What will we get then? So b minus a looks like that. But if we were to draw it in standard form-- remember, in standard form b minus a would look something like this. It would start at 0, it would be parallel to this, and then from 0 we would draw its endpoint. So if we just multiplied some scalar times b minus a, we would actually just get points or vectors that lie on this line. Vectors that lie on that line right there. Now, that's not what we set out to do. We wanted to figure out an equation, or parametrization, if you will, of this line, or this set. Let's call this set l. So we want to know what that set is equal to. So in order to get there, we have to start with this, which is this line here, and we have to shift it. And we could shift it either by shifting it straight up, we could add vector b to it. So we could take this line right here, and add vector b to it. And so any point on here would have its corresponding point there. So when you add vector b, it essentially shifts it up. That would work. So we could, say, we could add vector b to it. And now all of these points for any arbitrary-- t is a member of the real numbers, will lie on this green line. Or the other option we could have done is we could have added vector a. Vector a would have taken any arbitrary point here and shifted it that way. You would be adding vector a to it. But either way, you're going to get to the green line that we cared about, so you could have also defined it as the set of vector a plus this line, essentially, t times vector b minus a, where t is a member of the reals. So my definition of my line could be either of these things. The definition of my line could be this set, or it could be this set. And all of this seems all very abstract, but when you actually deal with the numbers, it actually becomes very simple. It becomes arguably simpler than what we did in Algebra 1. So this l, for these particular case of a and b, let's figure it out. My line is equal to-- let me just use the first example. It's vector b, so it's the vector 0, 3 plus t, times the vector b minus a. Well what's b minus a? 0 minus 2 is minus 2, 3, minus 1 is 2, for t is a member of the reals. Now, if this still seems kind of like a convoluted set definition for you, I could write it in terms that you might recognize better. If we want to plot points, if we call this the y-axis, and we call this the x-axis, and if we call this the x-coordinate, or maybe more properly that the x-coordinate and call this the y-coordinate, then we can set This actually is the x-slope. This is the x-coordinate, that's the y-coordinate. Or actually, even better, whatever-- actually, let me be very careful there. This is always going to end up becoming some vector, l1, l2. This is a set of vectors, and any member of this set is going to look something like this. So this could be li. So, this is the x-coordinate, and this is the y-coordinate. And just to get this in a form that you recognize, so we're saying that l is the set of this vector x plus t times this vector b minus a here. If we wanted to write it in kind of a parametric form, we can say, since this is what determines our x-coordinate, we would say that x is equal to 0 plus t times minus 2, or minus 2 times t. And then we can say that y, since this is what determines our y-coordinate, y is equal to 3 plus t times 2 plus 2t. So we could have rewritten that first equation as just x is equal to minus 2t, and y is equal to 2t plus 3. So if you watch the videos on parametric equations, this is just a traditional parametric definition of this line right there. Now, you might have still viewed this as, Sal, this was a waste of time, this was convoluted. You have to define these sets and all that. But now I'm going to show you something that you probably-- well, unless you have done this before, but I guess that's true of anything. But you probably haven't seen in your traditional algebra class. Let's say I have two points, and now I'm going to deal in three dimensions. So let's say I have one vector. I'll just call it point 1, because these are position vectors. We'll just call it position 1. This is in three dimensions. Just make up some numbers, negative 1, 2, 7. Let's say I have Point 2. Once again, this is in three dimensions, so you have to specify three coordinates. This could be the x, the y, and the z coordinate. Point 2, I don't know. Let's make it 0, 3, and 4. Now, what if I wanted to find the equation of the line that passes through these two points in R3? So this is in R3. Well, I just said that the equation of this line-- so I'll just call that, or the set of this line, let me just call this l. It's going to be equal to-- we could just pick one of these guys, it could be P1, the vector P1, these are all vectors, be careful here. The vector P1 plus some random parameter, t, this t could be time, like you learn when you first learn parametric equations, times the difference of the two vectors, times P1, and it doesn't matter what order you take it. So that's a nice thing too. P1 minus P2. It could be P2 minus P1-- because this can take on any positive or negative value-- where t is a member of the real numbers. So let's apply it to these numbers. Let's apply it right here. What is P1 minus P2? P1 minus P2 is equal to-- let me get some space here. P1 minus P2 is equal, minus 1 minus 0 is minus 1. 2 minus 3 is minus 1. 7 minus 4 is 3. So that thing is that vector. And so, our line can be described as a set of vectors, that if you were to plot it in standard position, it would be this set of position vectors. It would be P1, it would be-- let me do that in green-- it would be minus 1, 2, 7. I could've put P2 there, just as easily-- plus t times minus 1, minus 1, 3, where, or such that, t is a member of the real numbers. Now, this also might not be satisfying for you. You're like, gee, how do I plot this in three dimensions? Where's my x, y's, and z's? And if you want to care about x, y's, and z's, let's say that this is the z-axis. This is the x-axis, and let's say the y-axis. It kind of goes into our board like this, so the y-axis comes out like that. So what you can do, and actually I probably won't graph, so the determinate for the x-coordinate, just our convention, is going to be this term right here. So we can write that x-- let me write that down. So that term is going to determine our x-coordinate. So we can write that x is equal to minus 1-- be careful with the colors-- minus 1, plus minus 1 times t. That's our x-coordinate. Now, our y-coordinate is going to be determined by this part of our vector addition because these are the y-coordinates. So we can say the y-coordinate is equal to-- I'll just write it like this-- 2 plus minus 1 times t. And then finally, our z-coordinate is determined by that there, the t shows up because t times 3-- or I could just put this t into all of this. So that the z-coordinate is equal to 7 plus t times 3, or I could say plus 3t. And just like that, we have three parametric equations. And when we did it in R2, I did a parametric equation, but we learned in Algebra 1, you can just have a regular y in terms x. You don't have to have a parametric equation. But when you're dealing in R3, the only way to define a line is to have a parametric equation. If you have just an equation with x's, y's, and z's, if I just have x plus y plus z is equal to some number, this is not a line. And we'll talk more about this in R3. This is a plane. The only way to define a line or a curve in three dimensions, if I wanted to describe the path of a fly in three dimensions, it has to be a parametric equation. Or if I shoot a bullet in three dimensions and it goes in a straight line, it has to be a parametric equation. So these-- I guess you could call it-- these are the equations of a line in three dimensions. So hopefully you found that interesting. And I think this will be the first video where you have an appreciation that linear algebra can solve problems or address issues that you never saw before. And there's no reason why we have to just stop at three, three coordinates, right here. We could have done this with fifty dimensions. We could have defined a line in fifty dimensions-- or the set of vectors that define a line, that two points sit on, in fifty dimensions-- which is very hard to visualize, but we can actually deal with it mathematically." + }, + { + "Q": "At 4:00, Sal says \"should.\" Can some cancer cells be so mutated that they do not produce MHC I at all?", + "A": "Yes. Sometimes cancer cells are so mutated or too close to your actual cells that if your cyto cells attack them, they will also attack your healthy cells. That s why to be safe they don t attack them. They also don t recognize them so they don t kill them. That s why all people need help to fight off cancers since your body can t kill most cancers.", + "video_name": "YdBXHm3edL8", + "transcript": "Voiceover: Everything we've discussed so far involved recognizing and tagging or engulfing shady things that were found outside of cells. We've seen things like a B cell. A B cell has its membrane bound antibodies. Maybe one of these might recognize something shady out in the outside of the cell and of course this part over here as we know they all have a variable portion right over here. This is specific to this shady thing then this will be engulfed and then parts of it will be attached to an MHC two complex. Let me do that in a different color. An MHC two complex and then that will go to the surface, that will go to the membrane of the cell to present itself. That's an MHC two complex. It has little bit of the little piece of this shady thing out here we call this little shady piece, this is an antigen presenting cell here. We've seen as especially if this is a B cell then a helper T cell that it also has a variable portion that corresponds to this specific antigen. This would be helper T cell right over here. This is a helper, T helper cell. When it recognizes then it will start dividing into memory helper T cells and effector helper T cells. The effector helper T cells essentially ring the alarm bells and start kind of accelerating the B cell replication or I guess you could say that the B cell activation. The theory is that this is kind of a double handshake process. Once again this is what's occurring outside of cells. When we found stuff outside of cells, we engulf them and then we presented them on MHC two complexes. Now you're probably thinking well, I mean that's the outside of cells but there's MHC two, there's this helper T cells but we've also referred to cytotoxic T cells. What do those do? We've also, if there's MHC two, there's probably an MHC one complex. What does the MHC one complex do? We can recognize shady things that are happening outside of cells but don't shady things sometimes happen inside cells and how does our immune system respond to that? Actually as you can imagine all of those things will be answered in the rest of this video. Let's think about what happens when shady things start to happen inside the cell. For example it might not even be due to a virus or due to some type of bacteria could be the cell itself is gone awry. Let's say that this right over here is a cancer cell. It's had some mutations. It's starting to multiply like crazy. This is a cancer cell and a cancer cell because it had mutations are going to produce, it's going to produce some weird proteins. These cancer cells are going to produce some weird proteins. Every cell with a nucleus in your body and that's pretty much every cell except for red blood cells has MHC one complexes. The whole point of the MHC one complex is to bind two shady things that are produced inside of the cell, and then present them to the membrane. Even a malfunctioning cancer cell should be doing this. This MHC one complex is bound to this strange proteins that are produced by the mutations inside of the nucleus and then it can present them. You could imagine what the appropriate immune response should be. These cancer cells should be killed and actually let me label this properly. That was MHC two, you're presenting an antigen that was found, those initially found outside of the cells engulfing and taken out. MHC one, it's binding to shady things inside the cell and then presenting it out. This thing should be killed. Now as you can imagine, what's going to kill it? Well that's where the cytotoxic T cell comes into the picture. The cytotoxic T cell is going to have, that's a receptor right there. It will have a variable portion that's specific to this type of an antigen and so it will bind there. Once it does that, it says, oh boy, there's all this shady stuff here, this shady proteins that are being produce. This cell and all the other ones like it need to be killed. The cytotoxic T cell will begin to replicate once again like other types of immune cells is going to replicate into the memory cells just in case this type of thing shows up 10 years in the future and also the effector themselves. This is memory and also effector cells, effector cytotoxic T cells. As we always know the effector version is the thing that actual does something, it starts to actually affect things. What it is going to affect, it is going to start binding two things that are presenting the same antigen as part of their MR on top of their MHC one complex. This character right over here, so it's presenting that same antigen on this MHC one complex. Remember, the variable portions need to match up. Let's say that this is an effector cytotoxic T cell and actually let me draw in a little bit different. Let me draw it like this. This is effector cytotoxic T cell. Its receptor, its variable portion is the one that's compatible with this antigen that's being presented right over here. Let me just label this again. This is the MHC one complex. This is an effector cytotoxic T cell. We'll put the C there for cytotoxic and what it does is essentially kind of latches on to the cell that needs to die and it does it, not only have this receptor interfacing with the MHC one complex but actually has a whole series of proteins and I'm not drawing this to scale really. This would be much smaller or relative to the scale cell. It essentially latches on between the two and I'm not going to go in detail but essentially forms what you can call an immuno synapse which is kind of where the two things are interacting with each other. When it identifies this, it says, okay, I need to kill this thing or essentially I need to make this thing kill itself. It starts releasing all of these molecules. It can release molecules like preference so to release this preference which will essentially cause gaps or holes to form in the membrane of the cell that needs to die and it could release other things like granzymes that can go in and essentially cause this thing to kill itself. The whole point of this video is to appreciate I guess what we haven't talked about yet. We had already talked about what happens when you identify shady things outside of the cells and then how you can kind of bring them in and then present them and then use that to further activate the immune response. Now we're talking about identifying shady things inside the cells. Those get presented by MHC one complexes and then the cytotoxic T cells recognize them and then force the cell to kill themselves. This wouldn't just be cancer cells, this could also apply to a cell that has already been affected by a virus. For example a cell like this all ready, so that's its nucleus. It's already been infected by some virus so the virus has hijacked the cell's replication machinery in order to replicate itself. The proper immune response is hey look, I'm a virus making machine. I should kill myself. It will wreck some of the antigens that are being produced inside by the viruses. They're going to bind to MHC one complexes. Pieces of the virus are going to bind to MHC one complexes and then they're going to be presented on the surface. Let me do it this way, presented on the surface. This exact process can happen again." + }, + { + "Q": "using the squeeze theorem how can I evaluate: lim x-> infinity e^-8x cos x\n\ncos x will always be bound between -1 and 1 so do I set it up so do I set it up where\n\n-e^-8x (x - y)^3 + 3xy(x-y) = x^3 - y^3 => x^3 - y^3 = (x - y) [(x - y)^2 + 3xy] => x^3 - y^3 = (x - y)(x^2 + y^2 - 2xy + 3xy) => x^3 - y^3 = (x - y)(x^2 + xy + y^2 )", + "video_name": "rU222pVq520", + "transcript": "Let's try to find the limit as x approaches 1 of x to the third minus 1 over x squared minus 1. And at first when you just try to substitute x equals 1, you get 0/0 1 minus 1 over 1 minus 1. So that doesn't help us. So let's see if we can try to simplify this in some way. So you might immediately recognize-- so let's rewrite this expression right over here so it's x to the third minus 1 over x squared minus 1. This on the bottom immediately jumps out as a difference of squares. So we know on the bottom that this could be factored as x minus 1 times x plus 1. And so if somehow this thing on the top also has an x minus 1 as a factor, then that x minus 1 will cancel with this, and then we're not going to have an issue of dividing by 0. The reason why I care about the x minus 1 term is that this is what's making our denominator equal 0. When you say x equals 1, you have 1 minus 1 times 1 plus 1. So 0 times 2, it's this 0 that's making our denominator 0. So if we can have an x minus 1 up here, then we can cancel these out for any x not equal to 1. And then we might have a much simpler thing to find the limit of. So let's think about whether x to the third minus 1 is the product of x minus 1 and something else. And to do that we can do a little bit of algebraic long division. Some of you guys might already recognize a pattern here, but we'll try to do-- well, let's divide x minus 1 into it to see whether it divides evenly into x to the third minus 1. So x minus 1-- we just look at the highest degree term-- x goes into x to the third x squared times. Goes x squared times. Actually, let me do it this way so that way we can keep track of the place. So this would be x-- this would be the second degree place, first degree place, and this would be the constant. So x to the third minus 1. x goes into x to the third x squared times. x squared times x is x to the third. x squared times negative 1 is minus x squared. And now we're going to want to subtract this. So we are then left with x squared. x goes into x squared x times plus x. x times x is x squared. x times minus 1 is minus x. And once again we're going to subtract this. We'll swap the signs, negative and positive. And so these cancel out, and we're left with x. And then we bring down a minus 1. x minus 1 goes into x minus 1 exactly one time. 1 times x minus 1 is x minus 1. And then you subtract, and then you have no remainder. So this numerator right over here can be factored as x minus 1 times x squared plus x plus 1. And so we can say that this is the same exact thing. We can have these cancel out if we assume x does not equal 1. So that is equal to x squared plus x plus 1 over x plus 1, for x does not equal 1. And that's completely fine, because we're not evaluating x equals 1. We're evaluating as x approaches 1. So this is going to be the same thing as the limit as x approaches 1 of x squared plus x plus 1 over x plus 1. And now this is much easier to find. You could literally just say, well, what happens as we get right to x equals 1? Then you have 1 squared, which is 1 plus 1 plus 1, which is 3, over 1 plus 1, which is 2. So we get that equaling 3/2." + }, + { + "Q": "i have a question from my exam.\n\"In what way will the temperature of water at the bottom of a waterfall be different from the temperature at the top? Give a reason for your answer.\"\n\nIs water at the bottom hotter?\nI think that as water falls down- Potential.E. is converted to Kinetic E. and some of PE is also converted to heat.\nAm I right?", + "A": "That s correct, but you could also argue that evaporation takes place during the fall and the splashing at the bottom. The evaporation would have a cooling effect. Can t tell how it balances out.", + "video_name": "kw_4Loo1HR4", + "transcript": "Welcome back. At the end of the last video, I left you with a bit of a question. We had a situation where we had a 1 kilogram object. This is the 1 kilogram object, which I've drawn neater in this video. That is 1 kilogram. And we're on earth, and I need to mention that because gravity is different from planet to planet. But as I mentioned, I'm holding it. Let's say I'm holding it 10 meters above the ground. So this distance or this height is 10 meters. And we're assuming the acceleration of gravity, which we also write as just g, let's assume it's just 10 meters per second squared just for the simplicity of the math instead of the 9.8. So what we learned in the last video is that the potential energy in this situation, the potential energy, which equals m times g times h is equal to the mass is 1 kilogram times the acceleration of gravity, which is 10 meters per second squared. I'm not going to write the units down just to save space, although you should do this when you do it on your test. And then the height is 10 meters. And the units, if you work them all out, it's in newton meters or joules and so it's equal to 100 joules. That's the potential energy when I'm holding it up there. And I asked you, well when I let go, what happens? Well the block obviously will start falling. And not only falling, it will start accelerating to the ground at 10 meters per second squared roughly. And right before it hits the ground-- let me draw that in brown for ground-- right before the object hits the ground or actually right when it hits the ground, what will be the potential energy of the object? Well it has no height, right? Potential energy is mgh. The mass and the acceleration of gravity stay the same, but the height is 0. So they're all multiplied by each other. So down here, the potential energy is going to be equal to 0. And I told you in the last video that we have the law of conservation of energy. That energy is conserved. It cannot be created or destroyed. It can just be converted from one form to another. But I'm just showing you, this object had 100 joules of energy or, in this case, gravitational potential energy. And down here, it has no energy. Or at least it has no gravitational potential energy, and that's the key. That gravitational potential energy was converted into something else. And that something else it was converted into is kinetic energy. And in this case, since it has no potential energy, all of that previous potential energy, all of this 100 joules that it has up here is now going to be converted into kinetic energy. And we can use that information to figure out its velocity right before it hits the ground. So how do we do that? Well what's the formula for kinetic energy? And we solved it two videos ago, and hopefully it shouldn't be too much of a mystery to you. It's something good to memorize, but it's also good to know how we got it and go back two videos if you forgot. So first we know that all the potential energy was converted into kinetic energy. We had 100 joules of potential energy, so we're still going to have 100 joules, but now all of it's going to be kinetic energy. And kinetic energy is 1/2 mv squared. So we know that 1/2 mv squared, or the kinetic energy, is now going to equal 100 joules. What's the mass? The mass is 1. And we can solve for v now. 1/2 v squared equals 100 joules, and v squared is equal to 200. And then we get v is equal to square root of 200, which is something over 14. We can get the exact number. Let's see, 200 square root, 14.1 roughly. The velocity is going to be 14.1 meters per second squared downwards. Right before the object touches the ground. Right before it touches the ground. And you might say, well Sal that's nice and everything. We learned a little bit about energy. I could have solved that or hopefully you could have solved that problem just using your kinematics formula. So what's the whole point of introducing these concepts of energy? And I will now show you. So let's say they have the same 1 kilogram object up here and it's 10 meters in the air, but I'm going to change things a little bit. Let me see if I can competently erase all of this. Nope, that's not what I wanted to do. OK, there you go. I'm trying my best to erase this, all of this stuff. OK. So I have the same object. It's still 10 meters in the air and I'll write that in a second. And I'm just holding it there and I'm still going to drop it, but something interesting is going to happen. Instead of it going straight down, it's actually going to drop on this ramp of ice. The ice has lumps on it. And then this is the bottom. This is the ground down here. This is the ground. So what's going to happen this time? I'm still 10 meters in the air, so let me draw that. That's still 10 meters. I should switch colors just so not everything is ice. So that's still 10 meters, but instead of the object going straight down now, it's going to go down here and then start It's going to go sliding along this hill. And then at this point it's going to be going really fast in the horizontal direction. And right now we don't know how fast. And just using our kinematics formula, this would have been a really tough formula. This would have been difficult. I mean you could have attempted it and it actually would have taken calculus because the angle of the slope changes continuously. We don't even know the formula for the angle of the slope. You would have had to break it out into vectors. You would have to do all sorts of complicated things. This would have been a nearly impossible problem. But using energy, we can actually figure out what the velocity of this object is at this point. And we use the same idea. Here we have 100 joules of potential energy. We just figured that out. Down here, what's the height above the ground? Well the height is 0. So all the potential energy has disappeared. And just like in the previous situation, all of the potential energy is now converted into kinetic energy. And so what is that kinetic energy going to equal? It's going to be equal to the initial potential energy. So here the kinetic energy is equal to 100 joules. And that equals 1/2 mv squared, just like we just solved. And if you solve for v, the mass is 1 kilogram. So the velocity in the horizontal direction will be, if you solve for it, 14.1 meters per second. Instead of going straight down, now it's going to be going in the horizontal to the right. And the reason why I said it was ice is because I wanted this to be frictionless and I didn't want any energy lost to heat or anything like that. And you might say OK Sal, that's kind of interesting. And you kind of got the same number for the velocity than if I just dropped the object straight down. And that's interesting. But what else can this do for me? And this is where it's really cool. Not only can I figure out the velocity when all of the potential energy has disappeared, but I can figure out the velocity of any point-- and this is fascinating-- along this slide. So let's say when the box is sliding down here, so let's say the box is at this point. It changes colors too as it falls. So this is the 1 kilogram box, right? It falls and it slides down here. And let's say at this point it's height above the ground is 5 meters. So what's its potential energy here? So let's just write something. All of the energy is conserved, right? So the initial potential energy plus the initial kinetic energy is equal to the final potential energy plus the final kinetic energy. I'm just saying energy is conserved here. Up here, what's the initial total energy in the system? Well the potential energy is 100 and the kinetic energy is 0 because it's stationary. I haven't dropped it. I haven't let go of it yet. It's just stationary. So the initial energy is going to be equal to 100 joules. That's cause this is 0 and this is 100. So the initial energy is 100 joules. At this point right here, what's the potential energy? Well we're 5 meters up, so mass times Mass is 1, times gravity, 10 meters per second squared. Times height, times 5. So it's 50 joules. That's our potential energy at this point. And then we must have some kinetic energy with the velocity going roughly in that direction. Plus our kinetic energy at this point. And we know that no energy was destroyed. It's just converted. So we know the total energy still has to be 100 joules. So essentially what happened, and if we solve for this-- it's very easy, subtract 50 from both sides-- we know that the kinetic energy is now also going to be equal to 50 joules. Halfway down, essentially half of the potential energy got converted to kinetic energy. And we can use this information that the kinetic energy is 50 joules to figure out the velocity at this point. 1/2 mv squared is equal to 50. The mass is 1. Multiply both sides by 2. You get v squared is equal to 100. The velocity is 10 meters per second along this crazy, icy slide. And that is something that I would have challenged you to solve using traditional kinematics formulas, especially considering that we don't know really much about the surface of this slide. And even if we did, that would have been a million times harder than just using the law of conservation of energy and realizing that at this point, half the potential energy is now kinetic energy and it's going along the direction of the slide. I will see you in the next video." + }, + { + "Q": "What kind of change occurs when one scratches a steel surface. Is it chemical or physical? I understand that rust/tarnishing a steel surface is a chemical change, I would imagine this holds true for a scratch, correct?", + "A": "No, if you scratch something, it doesn t change chemically. The part that you scratch off is still steel. Anything something breaks physically, it is a physical change.", + "video_name": "pKvo0XWZtjo", + "transcript": "I think we're all reasonably familiar with the three states of matter in our everyday world. At very high temperatures you get a fourth. But the three ones that we normally deal with are, things could be a solid, a liquid, or it could be a gas. And we have this general notion, and I think water is the example that always comes to at least my mind. Is that solid happens when things are colder, relatively colder. And then as you warm up, you go into a liquid state. And as your warm up even more you go into a gaseous state. So you go from colder to hotter. And in the case of water, when you're a solid, you're ice. When you're a liquid, some people would call ice water, but let's call it liquid water. I think we know what that is. And then when it's in the gas state, you're essentially vapor or steam. So let's think a little bit about what, at least in the case of water, and the analogy will extend to other types of molecules. But what is it about water that makes it solid, and when it's colder, what allows it to be liquid. And I'll be frank, liquids are kind of fascinating because you can never nail them down, I guess is the best way to view them. Or a gas. So let's just draw a water molecule. So you have oxygen there. You have some bonds to hydrogen. And then you have two extra pairs of valence electrons in the oxygen. And a couple of videos ago, we said oxygen is a lot more electronegative than the hydrogen. It likes to hog the electrons. So even though this shows that they're sharing electrons here and here. At both sides of those lines, you can kind of view that hydrogen is contributing an electron and oxygen is contributing an electron on both sides of that line. But we know because of the electronegativity, or the relative electronegativity of oxygen, that it's hogging these electrons. And so the electrons spend a lot more time around the oxygen than they do around the hydrogen. And what that results is that on the oxygen side of the molecule, you end up with a partial negative charge. And we talked about that a little bit. And on the hydrogen side of the molecules, you end up with a slightly positive charge. Now, if these molecules have very little kinetic energy, they're not moving around a whole lot, then the positive sides of the hydrogens are very attracted to the negative sides of oxygen in other molecules. Let me draw some more molecules. When we talk about the whole state of the whole matter, we actually think about how the molecules are interacting with Not just how the atoms are interacting with each other within a molecule. I just drew one oxygen, let me copy and paste that. But I could do multiple oxygens. And let's say that that hydrogen is going to want to be near this oxygen. Because this has partial negative charge, this has a partial positive charge. And then I could do another one right there. And then maybe we'll have, and just to make the point clear, you have two hydrogens here, maybe an oxygen wants to hang out there. So maybe you have an oxygen that wants to be here because it's got its partial negative here. And it's connected to two hydrogens right there that have their partial positives. But you can kind of see a lattice structure. Let me draw these bonds, these polar bonds that start forming between the particles. These bonds, they're called polar bonds because the molecules themselves are polar. And you can see it forms this lattice structure. And if each of these molecules don't have a lot of kinetic energy. Or we could say the average kinetic energy of this matter is fairly low. And what do we know is average kinetic energy? Well, that's temperature. Then this lattice structure will be solid. These molecules will not move relative to each other. I could draw a gazillion more, but I think you get the point that we're forming this kind of fixed structure. And while we're in the solid state, as we add kinetic energy, as we add heat, what it does to molecules is, it just makes them vibrate around a little bit. If I was a cartoonist, they way you'd draw a vibration is to put quotation marks there. That's not very scientific. But they would vibrate around, they would buzz around a little bit. I'm drawing arrows to show that they are vibrating. It doesn't have to be just left-right it could be up-down. But as you add more and more heat in a solid, these molecules are going to keep their structure. So they're not going to move around relative to each other. But they will convert that heat, and heat is just a form of energy, into kinetic energy which is expressed as the vibration of these molecules. Now, if you make these molecules start to vibrate enough, and if you put enough kinetic energy into these molecules, what do you think is going to happen? Well this guy is vibrating pretty hard, and he's vibrating harder and harder as you add more and more heat. This guy is doing the same thing. At some point, these polar bonds that they have to each other are going to start not being strong enough to contain the vibrations. And once that happens, the molecules-- let me draw a couple more. Once that happens, the molecules are going to start moving past each other. So now all of a sudden, the molecule will start shifting. But they're still attracted. Maybe this side is moving here, that's moving there. You have other molecules moving around that way. But they're still attracted to each other. Even though we've gotten the kinetic energy to the point that the vibrations can kind of break the bonds between the polar sides of the molecules. Our vibration, or our kinetic energy for each molecule, still isn't strong enough to completely separate them. They're starting to slide past each other. And this is essentially what happens when you're in a liquid state. You have a lot of atoms that want be touching each other but they're sliding. They have enough kinetic energy to slide past each other and break that solid lattice structure here. And then if you add even more kinetic energy, even more heat, at this point it's a solution now. They're not even going to be able to stay together. They're not going to be able to stay near each other. If you add enough kinetic energy they're going to start looking like this. They're going to completely separate and then kind of bounce around independently. Especially independently if they're an ideal gas. But in general, in gases, they're no longer touching They might bump into each other. But they have so much kinetic energy on their own that they're all doing their own thing and they're not touching. I think that makes intuitive sense if you just think about what a gas is. For example, it's hard to see a gas. Why is it hard to see a gas? Because the molecules are much further apart. So they're not acting on the light in the way that a liquid or a solid would. And if we keep making that extended further, a solid-- well, I probably shouldn't use the example with ice. Because ice or water is one of the few situations where the solid is less dense than the liquid. That's why ice floats. And that's why icebergs don't just all fall to the bottom of the ocean. And ponds don't completely freeze solid. But you can imagine that, because a liquid is in most cases other than water, less dense. That's another reason why you can see through it a little Or it's not diffracting-- well I won't go into that too much, than maybe even a solid. But the gas is the most obvious. And it is true with water. The liquid form is definitely more dense than the gas form. In the gas form, the molecules are going to jump around, not touch each other. And because of that, more light can get through the substance. Now the question is, how do we measure the amount of heat that it takes to do this to water? And to explain that, I'll actually draw a phase change diagram. Which is a fancy way of describing something fairly straightforward. Let me say that this is the amount of heat I'm adding. And this is the temperature. We'll talk about the states of matter in a second. So heat is often denoted by q. Sometimes people will talk about change in heat. They'll use H, lowercase and uppercase H. They'll put a delta in front of the H. Delta just means change in. And sometimes you'll hear the word enthalpy. Let me write that. Because I used to say what is enthalpy? It sounds like empathy, but it's quite a different concept. At least, as far as my neural connections could make it. But enthalpy is closely related to heat. It's heat content. For our purposes, when you hear someone say change in enthalpy, you should really just be thinking of change in heat. I think this word was really just introduced to confuse chemistry students and introduce a non-intuitive word into their vocabulary. The best way to think about it is heat content. Change in enthalpy is really just change in heat. And just remember, all of these things, whether we're talking about heat, kinetic energy, potential energy, enthalpy. You'll hear them in different contexts, and you're like, I thought I should be using heat and they're talking about enthalpy. These are all forms of energy. And these are all measured in joules. And they might be measured in other ways, but the traditional way is in joules. And energy is the ability to do work. And what's the unit for work? Well, it's joules. Force times distance. But anyway, that's a side-note. But it's good to know this word enthalpy. Especially in a chemistry context, because it's used all the time and it can be very confusing and non-intuitive. Because you're like, I don't know what enthalpy is in my everyday life. Just think of it as heat contact, because that's really But anyway, on this axis, I have heat. So this is when I have very little heat and I'm increasing my heat. And this is temperature. Now let's say at low temperatures I'm here and as I add heat my temperature will go up. Temperature is average kinetic energy. Let's say I'm in the solid state here. And I'll do the solid state in purple. No I already was using purple. I'll use magenta. So as I add heat, my temperature will go up. Heat is a form of energy. And when I add it to these molecules, as I did in this example, what did it do? It made them vibrate more. Or it made them have higher kinetic energy, or higher average kinetic engery, and that's what temperature is a measure of; average kinetic energy. So as I add heat in the solid phase, my average kinetic energy will go up. And let me write this down. This is in the solid phase, or the solid state of matter. Now something very interesting happens. Let's say this is water. So what happens at zero degrees? Which is also 273.15 Kelvin. Let's say that's that line. What happens to a solid? Well, it turns into a liquid. Ice melts. Not all solids, we're talking in particular about water, about H2O. So this is ice in our example. All solids aren't ice. Although, you could think of a rock as solid magma. Because that's what it is. I could take that analogy a bunch of different ways. But the interesting thing that happens at zero degrees. Depending on what direction you're going, either the freezing point of water or the melting point of ice, something interesting happens. As I add more heat, the temperature does not to go up. As I add more heat, the temperature does not go up for a little period. Let me draw that. For a little period, the temperature stays constant. And then while the temperature is constant, it stays a solid. We're still a solid. And then, we finally turn into a liquid. Let's say right there. So we added a certain amount of heat and it just stayed a solid. But it got us to the point that the ice turned into a liquid. It was kind of melting the entire time. That's the best way to think about it. And then, once we keep adding more and more heat, then the liquid warms up too. Now, we get to, what temperature becomes interesting again for water? Well, obviously 100 degrees Celsius or 373 degrees Kelvin. I'll do it in Celsius because that's what we're familiar with. That's the temperature at which water will vaporize or which water will boil. But something happens. And they're really getting kinetically active. But just like when you went from solid to liquid, there's a certain amount of energy that you have to contribute to the system. And actually, it's a good amount at this point. Where the water is turning into vapor, but it's not getting any hotter. So we have to keep adding heat, but notice that the temperature didn't go up. We'll talk about it in a second what was happening then. And then finally, after that point, we're completely vaporized, or we're completely steam. Then we can start getting hot, the steam can then get hotter as we add more and more heat to the system. So the interesting question, I think it's intuitive, that as you add heat here, our temperature is going to go up. But the interesting thing is, what was going on here? We were adding heat. So over here we were turning our heat into kinetic energy. Temperature is average kinetic energy. But over here, what was our heat doing? Well, our heat was was not adding kinetic energy to the system. The temperature was not increasing. But the ice was going from ice to water. So what was happening at that state, is that the kinetic energy, the heat, was being used to essentially break these bonds. And essentially bring the molecules into a higher energy state. So you're saying, Sal, what does that mean, higher energy state? Well, if there wasn't all of this heat and all this kinetic energy, these molecules want to be very close to each other. For example, I want to be close to the surface of the earth. When you put me in a plane you have put me in a higher energy state. I have a lot more potential energy. I have the potential to fall towards the earth. Likewise, when you move these molecules apart, and you go from a solid to a liquid, they want to fall towards each other. But because they have so much kinetic energy, they never quite are able to do it. But their energy goes up. Their potential energy is higher because they want to fall towards each other. By falling towards each other, in theory, they could do some work. So what's happening here is, when we're contributing heat-- and this amount of heat we're contributing, it's called the heat of fusion. Because it's the same amount of heat regardless how much direction we go in. When we go from solid to liquid, you view it as the heat of melting. It's the head that you need to put in to melt the ice into liquid. When you're going in this direction, it's the heat you have to take out of the zero degree water to turn it into ice. So you're taking that potential energy and you're bringing the molecules closer and closer to each other. So the way to think about it is, right here this heat is being converted to kinetic energy. Then, when we're at this phase change from solid to liquid, that heat is being used to add potential energy into the system. To pull the molecules apart, to give them more potential energy. If you pull me apart from the earth, you're giving me potential energy. Because gravity wants to pull me back to the earth. And I could do work when I'm falling back to the earth. A waterfall does work. It can move a turbine. You could have a bunch of falling Sals move a turbine as well. And then, once you are fully a liquid, then you just become a warmer and warmer liquid. Now the heat is, once again, being used for kinetic energy. You're making the water molecules move past each other faster, and faster, and faster. To some point where they want to completely disassociate from each other. They want to not even slide past each other, just completely jump away from each other. And that's right here. This is the heat of vaporization. And the same idea is happening. Before we were sliding next to each other, now we're pulling apart altogether. So they could definitely fall closer together. And then once we've added this much heat, now we're just heating up the steam. We're just heating up the gaseous water. And it's just getting hotter and hotter and hotter. But the interesting thing there, and I mean at least the interesting thing to me when I first learned this, whenever I think of zero degrees water I'll say, oh it must be ice. But that's not necessarily the case. If you start with water and you make it colder and colder and colder to zero degrees, you're essentially taking heat out of the water. You can have zero degree water and it hasn't turned into ice yet. And likewise, you could have 100 degree water that hasn't turned into steam yeat. You have to add more energy. You can also have 100 degree steam. You can also have zero degree water. Anyway, hopefully that gives you a little bit of intuition of what the different states of matter are. And in the next problem, we'll talk about how much heat exactly it does take to move along this line. And maybe we can solve some problems on how much ice we might need to make our drink cool." + }, + { + "Q": "The simplest way I can see to solve this problem is to take the first valve: 15 subtract it by the second valve 15-9=6. Now you have the common difference, then you do this:\n15-6(100-1) then you compute that should give you get the answer.", + "A": "PEMDAS rules always apply. You can t do the subtraction 1st. 1) Do the parentheses 1st: 100-1 = 99 2) Muliply: 99 * (-6) = -594 3) Then subtract: 15 - 594 = - 579", + "video_name": "JtsyP0tnVRY", + "transcript": "We are asked, what is the value of the 100th term in this sequence? And the first term is 15, then 9, then 3, then negative 3. So let's write it like this, in a table. So if we have the term, just so we have things straight, and then we have the value. and then we have the value of the term. I'll do a nice little table here. So our first term we saw is 15. Our second term is 9. Our third term is 3. I'm just really copying this down, but I'm making sure we associate it with the right term. And then our fourth term is negative 3. And they want us to figure out what the 100th term of this sequence is going to be. So let's see what's happening here, if we can discern some type of pattern. So when we went from the first term to the second term, what happened? 15 to 9. Looks like we went down by 6. It's always good to think about just how much the numbers changed by. That's always the simplest type of pattern. So we went down by 6, we subtracted 6. Then to go from 9 to 3, well, we subtracted 6 again. And then to go from 3 to negative 3, well, we subtracted 6 again. So it looks like, every term, you subtract 6. So the second term is going to be 6 less than the first term. The third term is going to be 12 from the first term, or negative 6 subtracted twice. So in the third term, you subtract negative 6 twice. In the fourth term, you subtract negative 6 three times. So whatever term you're looking at, you subtract negative 6 one less than that many times. Let me write this down just so-- Notice when your first term, you have 15, and you don't subtract negative 6 at all. Or you could say you subtract negative 6 0 times. So you can say this is 15 minus negative 6 times-- or let me write it better this way --minus 0 times negative 6. That's what that first term is right there. What's the second term? This is 15. We just subtracted negative 6 once, or you could say, minus 1 times 6. Or you could say plus 1 times negative 6. Either way, we're subtracting the 6 once. Now what's happening here? This is 15 minus 2 times negative 6-- or, sorry --minus 2 times 6. We're subtracting a 6 twice. What's the fourth term? This is 15 minus-- We're subtracting the 6 three times from the 15, so minus 3 times 6. So, if you see the pattern here, when we have our fourth term, we have the term minus 1 right there. The fourth term, we have a 3. The third term, we have a 2. The second term, we have a 1. So if we had the nth term, if we just had the nth term here, what's this going to be? It's going to be 15 minus-- You see it's going to be n minus 1 right here. When n is 4, n minus 1 is 3. When n is 3, n minus 1 is 2. When n is 2, n minus 1 is 1. When n is 1, n minus 1 is 0. So we're going to have this term right here is n minus 1. So minus n minus 1 times 6. So if you want to figure out the 100th term of this sequence, I didn't even have to write it in this general term, you can just look at this pattern. It's going to be-- and I'll do it in pink --the 100th term in our sequence-- I'll continue our table down --is going to be what? It's going to be 15 minus 100 minus 1, which is 99, times 6. I just follow the pattern. 1, you had a 0 here. 2, you had a 1 here. 3, you had a 2 here. 100, you're going to have a 99 here. So let's just calculate what this is. What's 99 times 6? So 99 times 6-- Actually you can do this in your head. You could say that's going to be 6 less than 100 times 6, which is 600, and 6 less is 594. But if you didn't want to do it that way, you just do it the old-fashioned way. 6 times 9 is 54. Carry the 5. 9 times 6, or 6 times 9 is 54. 54 plus 5 is 594. So this right here is 594. And then to figure out what 15-- So we want to figure out what 15 minus 594 is. And this can sometimes be confusing, but the way I always process this in my head is, I say that this is the exact same thing as the negative of 594 minus 15. And if you don't believe me, distribute out this negative sign. Negative 1 times 594 is negative 594. Negative 1 times negative 15 is positive 15. So these two statements are equivalent. This is much easier for my brain to understand. So what's 594 minus 15? We can do this in our heads. 594 minus 14 would be 580, and then 580 minus 1 more would be 579. So that right there is 579, and then we have this negative sign sitting out there. So the 100th term in our sequence will be negative 579." + }, + { + "Q": "Can a Van Der Graaf Generator light a light bulb up if you put it on it's metallic top because isn't it in a way generating a steady current of negative charge which is what a light bulb requires.", + "A": "Hello Alex, Yes it could (in theory) but it would take a special light bulb. The bulb would need to operate at thousands of volts with a very small current. This is nearly impossible to construct with a traditional filament. There is another option, a fluorescent tube operates with high voltage and minimal current. Google de graaf fluorescent and you will find many examples. Regards, APD", + "video_name": "ZRLXDiiUv8Q", + "transcript": "- [Voiceover] All right, now we're gonna talk about the idea of an electric current. The story about currents starts with the idea of charge. We've learned that we have two kinds of charges, positive and negative charge. We'll just make up two little charges like that. And we know if they're the opposite sign, that there'll be a force of attraction between them. And if they have two like signs, here's two charges that are both positive, and these charges are gonna repel each other. So this is the basic electrostatics idea, and the same thing for two minus charges. They also repel. So like charges repel, and unlike charges attract. That's one idea. We have the idea of charge. And now we need a place to get some charge. One of the places we like to get charge from is copper, copper wires. A copper atom looks like this. Copper atom has a nucleus with some protons in it, and it also has electrons flying around the outside, electrons in orbits around the outside. So we'll draw the electrons like this. There'll be orbits around this nucleus. Pretty good circles. And there'll be electrons in these. Little minus signs. There's electrons stacked up in this. And even farther out, there's electrons. So there's kind of a interesting looking copper atom. Copper, the symbol for copper is Cu, and its atomic number is 29. That means there's 29 protons inside here, and there's 29 electrons outside. It turns out, just as a coincidence for copper, that the last orbital out here has just one electron in it, that guy right there. And that's the one that is the easiest to pull away from copper and have it go participate in conduction, in electric current. If I have a chunk of copper, every copper atom will have the opportunity to contribute one, this one lonely electron out here. If we look at another element, like for instance silver, silver has this same kind of electron configuration, where there's just one out here. And that's why silver and copper are such good, good conductors. Now we're gonna build, let's build a copper wire. Here's sort of a copper wire. It's just made of solid copper. It's all full of copper atoms. And I'm gonna put a voltage across this. There's our little battery. This is the minus sign, this is the plus side. And we'll hook up a battery to this. What's going on in here? Inside this copper is a whole bunch of electrons that are associated with atoms. It's a neutral piece of metal. There's the same number of protons as there is electrons. But these electrons are a little bit loose. So if I put a plus over there, that's this situation right here, where a plus is attracting a minus. So an electron is gonna sort of wander over this way and go like that. And that's gonna leave a net positive charge in this region. So these electrons are all gonna start moving in this direction. And down at the end, here, an electron is gonna come out of this battery, travel in here, and it's gonna go in there and make up the difference. So if I had a net positive charge here from the electrons leaving and going to the left, this battery would fill those in. And I'm gonna get a net movement of charge, of negative charge, around in this direction, like this. The question is, how do I measure that? How do I measure or give a number to that amount of stuff that's going on? So we wanna quantify that, we wanna assign a number to the amount of current happening here. What we do is, in our heads, we put a boundary across here. So just make that up in your head. And it cuts all the way through the copper. And what we know, we're gonna stand right here. We're gonna keep our eye right on this boundary down in here. As we watch, what we're gonna do is, we're gonna count the number of electrons that move by here, and we're gonna have a stop watch and we're gonna time that. So we're gonna get, basically, this is charge, it's negative charge, and it's moving to the side. What we're gonna do is, at one little spot right here, we're just gonna count the number that go by in one second. So we're gonna get charge per second. It's gonna be a negative charge moving by. That's what we call current. It's the same as water flowing by in a river. That's the same idea. Now I'm gonna set up a different situation that also produces a current. And this time, we're gonna do it with water, water and salt. Let's build a tube of salt, of salt water, like this. We're gonna pretend this is some tube that's all full of water. I'm also gonna put a battery here. Let's put another battery. And we'll stick the wire into there. We'll stick the wire into there. This is the plus side of the battery, and this is the minus side of the battery. Water is H20, and this does not conduct. There's no free electrons available here. But what I'm gonna do is, I'm gonna put some table salt in it. This is ordinary salt that you put on your food. It's made of sodium, that's the symbol for sodium, and chloride, Cl is chloride. Sodium chloride is table salt. If we sprinkle some table salt into water, what happens is, these dissolve and we get a net plus charge here and a net minus charge on the chlorine. So out here is floating around Na's with plus signs and Cl's nearby, really close nearly, with minuses. Let's keep it even. Now, when I dip my battery wires into this water, what's gonna happen is, this plus charge, this plus charge over here from the battery is gonna attract the minus Cl's. So the Cl's gonna move that way a little bit. And over here, the same thing is happening. There's a minus sign here. There's a minus from the battery. That's going to attract this, and it's also gonna repel Cl minuses. So what we get is a net motion of positive charge, plus q going this way, and we get minus q going this way. How do we measure that current? How do we measure that current? Well, we do it the same way as we did up there with copper. We put a boundary through here in our heads. We stand here and we watch the charges moving by. What we're gonna get is some sodiums, Na's, moving this way, and chlorines moving this way. Just like we showed here. Na moving this way. So there's gonna be plus charges moving through the boundary and minus charges moving through the boundary in the opposite direction. If I take the total sum of that. For example, if I see one Na go this way and one Cl go this way, that's equivalent to two charges moving through the boundary. Hope that makes sense. It's equivalent to two charges, one going this way and one going this way. Because they have opposite signs, they add together and make two charges. In this case, current is equal, again, to charge per second." + }, + { + "Q": "Are there still more printed copies of Mein Kampf in the world?", + "A": "Yes india has lots of copies", + "video_name": "5qWI2pEv0wg", + "transcript": "Narrator: By the end of 1923, Hitler sees it as his chance to seize power in Germany. He's getting this popularity, the Nazis are getting this follower-ship because the Weimar Republic is falling apart. You have the hyperinflation, the German people feel insulted by this French occupation of the Ruhr region. It isn't just regular people who are starting to support the Nazis, it's very notable people as well. This right over here is General Ludendorff, we've already talked about him, he's one of the believers in the 'stab-in-the-back' theory that Germany would have won World War I if it wasn't stabbed in the back by the November Criminals who had taken control of the government during the revolution in October and November. He becomes a supporter of Hitler as well. In 1922 you have Mussolini come to power, this inspires Hitler. So, as we get in to November, Hitler sees this as his chance and the way that he wants to take control, is he wants to abduct or kidnap the leaders of the Bavarian region, and there's three of them, in particular at this time. Then from there, try to take control of the nation as a whole. So, in November of 1923 you have a gathering of the three gentlemen who are essentially in charge of Bavaria, a gathering of them and several thousand officials in Bavaria at a local beer hall in Munich. (writing) at a beer hall. Hitler sees this as the opportunity to take control. This is where he launches his Beer Hall Putsch, and I know I'm mispronouncing it, but Putsch literally means coup d'\u00e9tat, to try to overthrow the government. So, Hitler and his Nazi's the go to that Beer Hall meeting of the government officials, they surround it with their paramilitary group, their storm troopers, Hitler enters into the hall, gets on stage, shoots into the air twice and says, look this is the revolution, it is beginning. He forces the three leaders of Bavaria at gunpoint to pledge allegiance to the Nazi party and to this Putsch and to Hitler, in particular. Then things start to go a little bit ... get a little bit ... start to dissolve. As Hitler tries to address some issues that are going on outside, the members who they were going to kidnap are allowed to leave, you have chaos in the area amongst the Nazi's and, frankly, amongst the government throughout that evening, into that morning, at which point Hitler and his followers, and Ludendorff is one of them, decide to march (writing) decide to march into central Munich. All of this is happening ... all of this is happening in Munich, which is in Bavaria. They decide to march, and it's during that march that they have a confrontation with the official government troops. It's unclear who fired the first shot, but you do have an exchange of fire and during that exchange of fire, I've seen estimates of about 14-16 Nazi's are shot. A few days later ... and a few policemen, or a few soldiers are shot as well, and then a few days later Hitler is arrested. (writing) Hitler, Hitler is arrested. He's tried in early 1924 and then he is sentenced to jail, so all of his ambitions were lead to nothing. In jail, he still continued to develop his philosophy. He actually continued to develop his following. He spent roughly the second two-thirds of 1924, in 1924, he spent it in jail. (writing) 1924 was spent primarily in jail, but while he was in jail he had dictated his autobiography and his, frankly, his belief system in Mein Kampf, which literally means 'My struggle.' It's actually banned in many countries, it's not banned in the U.S. It does make for interesting reading because you get a sense for, on one level, how bizarre Hitler's brain was and how disturbed Hitler's brain was, but on the other side, you can appreciate that he was a very, he was a strong communicator. Even before any of this people would talk about how transfixing his eyes were, how much attention people paid to him when he would give a speech. You can even see this in his writing, and you can do a web search on it and you can get the entire text of Mein Kampf. It's disturbing and fascinating at the same time, but this is a little passage. In this passage, it gives you an idea of Hitler's view of why Germany was having these failures and what he, in his bizarrely disturbed mind, thought what the solution was. \"If we pass all the causes of the German collapse \"in review, the ultimate and most decisive \"remains the failure to recognize the racial problem, \"and especially the Jewish menace.\" He's blaming all of Hitler's difficulty on a racial problem and in particular on Jews. \"The defeats on the battlefield in August 1918 \"would have been child's play to bear. \"They stood in no proportion to the victories \"of our people. \"It was not they that cause our downfall, \"no, it was brought about by that power which \"prepared these defeats by systematically, \"over many decades, robbing our people of \"the political and moral instincts and forces \"which alone make nations capable, \"and hence worthy of existence.\" If you read a lot of the other text, what he's talking about is this decades of, essentially, watering down their society, watering down their society with other people. If they didn't water it down, they say the defeats in the battlefield would've been child's play to bear. \"In heedlessly ignoring the question \"of the preservation of the racial foundations of our \"nation, the old Reich disregarded the sole right \"which gives life in this world.\" He views this racial, in his mind, racial impurity as the reason why Germany was facing all of this difficulty. As we'll see over the next few videos, this leads to one of the ugliest and bloodiest periods of human history." + }, + { + "Q": "Is this about video showing about the value of american coins or teaching word problems of money?", + "A": "It is definitely teaching you about the value of American coins. Remember: 1 dollar = 100 cents 1 quarter = 25 cents 1 dime = 10 cents 1 nickel = 5 cents 1 penny = 1 cent As for your second question, there are many annoyances with currency; for example, the value of currency fluctuates, and there is not one universal currency. That sure would make things a whole lot simpler!", + "video_name": "pJ8KwRztfF0", + "transcript": "- Let's get some practice counting money! So I have six coins right over here, and these are all United States coins, we're counting money in the United States for these examples, and what is this first coin? Well, this is called a quarter, or a quarter-dollar, so it represents 25 cents, and we could write it out as 25 cents, but I'll just keep that one like that. Now this one's another quarter, so it is also going to be 25 cents. Now this one looks different, but it's just the other side of these coins. This is what the other side looks like. So this is also going to be 25 cents. These are three quarters right over here. So how much money do these three quarters represent? Well, it's going to be 25 plus 25, which is 50, plus 25, which is going to be 75, so these three quarters are going to be 75 cents. And remember, 100 cents make a dollar, so this is still less than one dollar. But we're not done yet. We have this nickel, this is a nickel right over here, that represents five cents, and then we have another nickel here, it looks different, but it's just the other side, this is the head side, this is the tail side. So this is also another five cents, and so these two nickels, if you add them together, they are going to represent 10 cents, and then finally, you have a penny, and a penny, and it even says it right over here, is one cent, in fact, they all say it here, this is five cents, this is one cent. So this right over here's gonna be one cent. So what is 75 plus 10 plus 1? Well 75 plus 10 is 85, plus 1 is 86, so this is equal to 86 cents. And if it felt a little bit too fast to count up 25, 50, 75 in your head, you could also add them up. 25, 25, 25, plus 5, another 5, plus 1. You could add them up this way, and then what would you get? 5 plus 5 is 10, plus 5 is 15, plus 5 is 20, plus 5 is 25, plus 1 is 26, so that's two tens and one six, put the two tens up here. 2 plus 2 is 4, 4 plus 2 is 6, 6 plus 2 is 8. So you could get 86 cents, either way. Let's do one that has even more coins in it. So here we go, we have, so what is it, what's going on here? This right over here is a quarter, that's going to be 25 cents. 25 cents, and then, we have one that we didn't see in the previous example, we have two dimes. A dime represents 10 cents, so we have two dimes, where we can have those each represent 10 cents, then we have two nickels, we've already seen those each represent five cents, so 5 and 5, and then we have four pennies, one, two, three, four. Now we could put each penny separately, like that, or we could say, look, four pennies, each of them represent one cent, so that's going to be four cents. So let me do it that way. So, this one, two, three, four, that's going to be four cents. And then we could just add everything up, so 5 plus 0 plus 0 plus 5 is 10, plus 5 is 15, plus 4 is 19. 19 is one ten and nine ones, so I could put the one ten in the tens place, 1 plus 2 is 3, plus 1 is 4, plus 1 is 5. So it's five tens, five tens and nine ones, so 59 cents. This right over here is 59 cents. And notice, we have more coins, but it represents less value than the previous example. That's because we had a lot of coins that didn't represent a lot of values, like, we had these four pennies here, while the previous example, we had three quarters! Each of these quarters is equivalent to 25 pennies, so we're able to represent more money with fewer coins in the first example." + }, + { + "Q": "At 18:45 How does khan figure out that when derivative of -x_0-1/2x_0 is equal to 0 is actually minimum or maximum of the function?", + "A": "derivative = 0 ==> (implies) a maximum or minimum", + "video_name": "viaPc8zDcRI", + "transcript": "I just got sent this problem, and it's a pretty meaty problem. A lot harder than what you'd normally find in most textbooks. So I thought it would help us all to work it out. And it's one of those problems that when you first read it, your eyes kind of glaze over, but when you understand what they're talking about, it's reasonably interesting. So they say, the curve in the figure above is the parabola y is equal to x squared. So this curve right there is y is equal to x squared. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. So this is the first quadrant, right here. And they're saying that a normal line is something, when the first quadrant intersection with the parabola is normal to the parabola. So if I were to draw a tangent line right there, this line is normal to that tangent line. That's all that's saying. So this is a normal line, right there. Normal line. Fair enough. 5 normal lines are shown in the figure. 1, 2, 3, 4, 5. Good enough. And these all look perpendicular, or normal to the parabola in the first quadrant intersection, so that makes sense. For a while, the x-coordinate of the second quadrant intersection of the normal line of the parabola gets smaller, as the x-coordinate of the first quadrant intersection gets smaller. So let's see what happens as the x-quadrant of the first intersection gets smaller. So this is where I left off in that dense text. So if I start at this point right here, my x-coordinate right there would look something like this. Let me go down. my x-coordinate is right around there. And then as I move to a smaller x-coordinate to, say, this one right here, what happened to the normal line? Or even more important, what happened to the intersection of the normal line in the second quadrant? This is the second quadrant, right here. So when I had a larger x-value here, my normal line intersected here, in the second quadrant. Then when I brought my x-value in, when I lowered my x-value, my x-value here, because this is the next point, right here, my x-value at the intersection here, went-- actually, their wording is bad. They're saying that the second quadrant intersection gets smaller. But actually, it's not really getting smaller. It's getting less negative. I guess smaller could be just absolute value or magnitude, but it's just getting less negative. It's moving there, but it's actually becoming a larger number, right? It's becoming less negative, but a larger number. But if we think in absolute value, I guess it's getting smaller, right? As we went from that point to that point, as we moved the x in for the intersection of the first quadrant, the second quadrant intersection also moved in a bit, from that line to that line. Fair enough. But eventually, a normal line second quadrant intersection gets as small as it can get. So if we keep lowering our x-value in the first quadrant, so we keep on pulling in the first quadrant, as we get to this point. And then this point intersects the second quadrant, right there. And then, if you go even smaller x-values in the first quadrant then your normal line starts intersecting in the second quadrant, further and further negative numbers. So you can kind of view this as the highest value, or the smallest absolute value, at which the normal line can intersect in the second quadrant Let me make that clear. Up here, you were intersecting when you had a large x in the first quadrant, you had a large negative x in the second quadrant intersection. And then as you lowered your x-value, here, you had a smaller negative value. Up until you got to this point, right here, you got this, which you can view as the smallest negative value could get, and then when you pulled in your x even more, these normal lines started to push out again, out in the second quadrant. That's, I think, what they're talking about. The extreme normal line is shown as a thick line in the figure. This is the extreme normal line, right there. So this is the extreme one, that deep, bold one. Extreme normal line. After this point, when you pull in your x-values even more, the intersection in your second quadrant starts to push out some. And you can think of the extreme case, if you draw the normal line down here, your intersection with the second quadrant is going to be way out here someplace, although it seems like it's kind of asymptoting a little bit. Let's read the rest of the problem. Once the normal line passes the extreme normal line, the x-coordinates of their second quadrant intersections what the parabola start to increase. And they're really, when they say they start to increase, they're actually just becoming more negative. That wording is bad. I should change this to more, more negative. Or they're becoming larger negative numbers. Because once you get below this, then all of a sudden the x-intersections start to push out more in the second quadrant. Fair enough. The figures show 2 pairs of normal lines. Fair enough. The 2 normal lines of a pair have the same second quadrant intersection with the parabola, but 1 is above the extreme normal line, in the first quadrant, the other is below it. Right, fair enough. For example, this guy right here, this is when we had a large x-value. He intersects with the second quadrant there. Then if you lower and lower the x-value, if you lower it enough, you pass the extreme normal line, and then you get to this point, and then this point, he intersects, or actually, you go to this point. So if you pull in your x-value enough, you once again intersect at that same point in the second quadrant. So hopefully I'm making some sense to you, as I try to make some sense of this problem. Now what do they want to know? And I think I only have time for the first part of this. Maybe I'll do the second part in the another video. Find the equation of the extreme normal line. Well, that seems very daunting at first, but I think our toolkit of derivatives, and what we know about equations of a line, should be able to get us there. So what's the slope of the tangent line at any point on this curve? Well, we just take the derivative of y equals x squared, and y prime is just equal to 2x. This is the slope of the tangent at any point x. So if I want to know the slope of the tangent at x0, at some particular x, I would just say, well, let me just say, slope, it would be 2 x0. Or let me just say, f of x0 is equal to 2 x0. This is the slope at any particular x0 of the tangent line. Now, the normal line slope is perpendicular to this. So the perpendicular line, and I won't review it here, but the perpendicular line has a negative inverse slope. So the slope of normal line at x0 will be the negative inverse of this, because this is the slope of the tangent line x0. So it'll be equal to minus 1 over 2 x0. Fair enough. Now, what is the equation of the normal line at x0 let's say that this is my x0 in question. What is the equation of the normal line there? Well, we can just use the point-slope form of our equation. So this point right here will be on the normal line. And that's the point x0 squared. Because this the graph of y equals x0, x squared. So this normal line will also have this point. So we could say that the equation of the normal line, let me write it down, would be equal to, this is just a point-slope definition of a line. You say, y minus the y-point, which is just x0 squared, that's that right there, is equal to the slope of the normal line minus 1 over 2 x0 times x minus the x-point that we're at. Minus x, minus x0. This is the equation of the normal line. So let's see. And what we care about is when x0 is greater than 0, right? We care about the normal line when we're in the first quadrant, we're in all of these values right there. So that's my equation of the normal line. And let's solve it explicitly in terms of x. So y is a function of x. Well, if I add x0 squared to both sides, I get y is equal to, actually, let me multiply this guy out. I get minus 1/2 x0 times x, and then I have plus, plus, because I have a minus times a minus, plus 1/2. The x0 and the over the x0, they cancel out. And then I have to add this x0 to both sides. So all I did so far, this is just this part right there. That's this right there. And then I have to add this to both sides of the equation, so then I have plus x0 squared. So this is the equation of the normal line, in mx plus b form. This is its slope, this is the m, and then this is its y-intercept right here. That's kind of the b. Now, what do we care about? We care about where this thing intersects. We care about where it intersects the parabola. And the parabola, that's pretty straightforward, that's just y is equal to x squared. So to figure out where they intersect, we just have to set the 2 y's to be equal to each other. So they intersect, the x-values where they intersect, x squared, this y would have to be equal to that y. Or we could just substitute this in for that y. So you get x squared is equal to minus 1 over 2 x0 times x, plus 1/2 plus x0 squared. Fair enough. And let's put this in a quadratic equation, or try to solve this, so we can apply the quadratic equation. So let's put all of this stuff on the lefthand side. So you get x squared plus 1 over 2 x0 times x minus all of this, 1/2 plus x0 squared is equal to 0. All I did is, I took all of this stuff and I put it on the lefthand side of the equation. Now, this is just a standard quadratic equation, so we can figure out now where the x-values that satisfy this quadratic equation will tell us where our normal line and our parabola intersect. So let's just apply the quadratic equation here. So the potential x-values, where they intersect, x is equal to minus b, I'm just applying the quadratic equation. So minus b is minus 1 over 2 x0, plus or minus the square root of b squared. So that's that squared. So it's one over four x0 squared minus 4ac. So minus 4 times 1 times this minus thing. So I'm going to have a minus times a minus is a plus, so it's just 4 times this, because there was one there. So plus 4 times this, right here. 4 times this is just 2 plus 4 x0 squared. All I did is, this is 4ac right here. Well, minus 4ac. The minus and the minus canceled out, so There's a 1. So 4 times c is just 2 plus 4x squared. I just multiply this by two, and of course all of this should be over 2 times a. a is just 2 there. So let's see if I can simplify this. We're just figuring out where the normal line and the parabola intersect. Now, what do we get here. This looks like a little hairy beast here. Let me see if I can simplify this a little bit. So let us factor out-- let me rewrite this. I can just divide everything by 1/2, so this is minus 1 over 4 x0, I just divided this by 2, plus or minus 1/2, that's just this 1/2 right there, times the square root, let me see what I can simplify out of here. So if I factor out a 4 over x0 squared, then what does my expression become? This term right here will become an x to the fourth, x0 to the fourth, plus, now, what does this term become? This term becomes a 1/2 x0 squared. And just to verify this, multiply 4 times 1/2, you get 2, and then the x0 squares cancel out. So write this term times that, will equal 2, and then you have plus-- now we factored a four out of this and the x0 squared, so plus 1/16. Let me scroll over a little bit. And you can verify that this works out. If you were to multiply this out, you should get this business right here. I see the home stretch here, because this should actually factor out quite neatly. So what does this equal? So the intersection of our normal line and our parabola is equal to this. Minus 1 over 4 x0 plus or minus 1/2 times the square root of this business. And the square root, this thing right here is 4 over x0 squared. This is actually, lucky for us, a perfect square. And I won't go into details, because then the video will get too long, but I think you can recognize that this is x0 squared, plus 1/4. If you don't believe me, square this thing right here. You'll get this expression right there. And luckily enough, this is a perfect square, so we can actually take the square root of it. And so we get, the point at which they intersect, our normal line and our parabola, and this is quite a hairy problem. The points where they intersect is minus 1 over 4 x0, plus or minus 1/2 times the square root of this. The square root of this is the square root of this, which is just 2 over x0 times the square root of this, which is x0 squared plus 1/4. And if I were to rewrite all of this, I'd get minus 1 over 4 x0 plus, let's see, this 1/2 and this 2 cancel out, right? So these cancel out. So plus or minus, now I just have a one over x0 times x0 squared. So I have 1 over x0-- oh sorry, let me, we have to be very careful there-- x0 squared divided by x0 is just x0, let me do that in a yellow color so you know what I'm dealing with. This term multiplied by this term is just x0, and then you have a plus 1/4 x0. And this is all a parentheses here. So these are the two points at which the normal curve and our parabola intersect. Let me just be very clear. Those 2 points are, for if this is my x0 that we're dealing with, right there. It's this point and this point. And we have a plus or minus here, so this is going to be the plus version, and this is going to be the minus version. In fact, the plus version should simplify into x0. Let's see if that's the case. Let's see if the plus version actually simplifies to x0. So these are our two points. If I take the plus version, that should be our first quadrant intersection. So x is equal to minus 1/4 x0 plus x0 plus 1/4 x0. And, good enough, it does actually cancel out. That cancels out. So x0 is one of the points of intersection, which makes complete sense. Because that's how we even defined the problem. But, so this is the first quadrant intersection. So that's the first quadrant intersection. The second quadrant intersection will be where we take the minus sign right there. So x, I'll just call it in the second quadrant intersection, it'd be equal to minus 1/4 x0 minus this stuff over here, minus the stuff there. So minus x0 minus 1 over 4 mine x0. Now what do we have? So let's see. We have a minus 1 over 4 x0, minus 1 over 4 x0. So this is equal to minus x0, minus x0, minus 1 over 2 x0. So if I take minus 1/4 minus 1/4, I get minus 1/2. And so my second quadrant intersection, all this work I did got me this result. My second quadrant intersection, I hope I don't run out of space. My second quadrant intersection, of the normal line and the parabola, is minus x0 minus 1 over 2 x0. Now this by itself is a pretty neat result we just got, but we're unfortunately not done with the problem. Because the problem wants us to find that point, the maximum point of intersection. They call this the extreme normal line. The extreme normal line is when our second quadrant intersection essentially achieves a maximum point. I know they call it the smallest point, but it's the smallest negative value, so it's really a maximum point. So how do we figure out that maximum point? Well, we have our second quadrant intersection as a function of our first quadrant x. I could rewrite this as, my second quadrant intersection as a function of x0 is equal to minus x minus 1 over 2 x0. So this is going to reach a minimum or a maximum point when its derivative is equal to 0. This is a very unconventional notation, and that's probably the hardest thing about this problem. But let's take this derivative with respect to x0. So my second quadrant intersection, the derivative of that with respect to x0, is equal to, this is pretty straightforward. It's equal to minus 1, and then I have a minus 1/2 times, this is the same thing as x to the minus 1. So it's minus 1 times x0 to the minus 2, right? I could have rewritten this as minus 1/2 times x0 to the minus 1. So you just put its exponent out front and decrement it by 1. And so this is the derivative with respect to my first quadrant intersection. So let me simplify this. So x, my second quadrant intersection, the derivative of it with respect to my first quadrant intersection, is equal to minus 1, the minus 1/2 and the minus 1 become a positive when you multiply them, and so plus 1/2 over x0 squared. Now, this'll reach a maximum or minimum when it equals 0. So let's set that equal to 0, and then solve this problem right there. Well, we add one to both sides. We get 1 over 2 x0 squared is equal to 1, or you could just say that that means that 2 x0 squared must be equal to 1, if we just invert both sides of this equation. Or we could say that x0 squared is equal to 1/2, or if we take the square roots of both sides of that equation, we get x0 is equal to 1 over the square root of 2. So we're really, really, really close now. We've just figured out the x0 value that gives us our extreme normal line. This value right here. Let me do it in a nice deeper color. This value right here, that gives us the extreme normal line, that over there is x0 is equal to 1 over the square root of 2. Now, they want us to figure out the equation of the extreme normal line. Well, the equation of the extreme normal line we already figured out right here. It's this. The equation of the normal line is that thing, right there. So if we want the equation of the normal line at this extreme point, right here, the one that creates the extreme normal line, I just substitute 1 over the square root of 2 in for x0. So what do I get? I get, and this is the home stretch, and this is quite a beast of a problem. y minus x0 squared. x0 squared is 1/2, right? 1 over the square root of 2 squared is 1/2. Is equal to minus 1 over 2 x0. So let's be careful here. So minus 1/2 times 1 over x0. One over x0 is the square root of 2, right? All of that times x minus x0. So that's 1 over the square root of 2. x0 is one So let's simplify this a little bit. So the equation of our normal line, assuming I haven't made any careless mistakes, is equal to, so y minus 1/2 is equal to, let's see. If we multiply this minus square root of 2 over 2x, and then if I multiply these square root of 2 over this, it becomes one. And then I have a minus and a minus, so that I have a plus 1/2. I think that's right. Yeah, plus 1/2, this times this times that is equal to plus 1/2. And then, we're at the home stretch. So we just add 1/2 to both sides of this equation, and we get our extreme normal line equation, which is y is equal to minus square root of 2 over 2x. If you add 1/2 to both sides of this equation, you get plus 1. And there you go. That's the equation of that line there, assuming I haven't made any careless mistakes. But even if I have, I think you get the idea of hopefully how to do this problem, which is quite a beastly one." + }, + { + "Q": "so from 00:01 to 23:17 he talking about the common cold and flu (influenza)? :|", + "A": "He s talking about Viruses in general and not about a specific one. At the beginning he says that because he has a cold that he s going to talk about Viruses. In between those times he covers how a most viruses interact with living cells and also how a retrovirus might.", + "video_name": "0h5Jd7sgQWY", + "transcript": "Considering that I have a cold right now, I can't imagine a more appropriate topic to make a video on than a virus. And I didn't want to make it that thick. A virus, or viruses. And in my opinion, viruses are, on some level, the most fascinating thing in all of biology. Because they really blur the boundary between what is an inanimate object and what is life? I mean if we look at ourselves, or life as one of those things that you know it when you see it. If you see something that, it's born, it grows, it's constantly changing. Maybe it moves around. Maybe it doesn't. But it's metabolizing things around itself. It reproduces and then it dies. You say, hey, that's probably life. And in this, we throw most things that we see-- or we throw in, us. We throw in bacteria. We throw in plants. I mean, I could-- I'm kind of butchering the taxonomy system here, but we tend to know life when we see it. But all viruses are, they're just a bunch of genetic information inside of a protein. Inside of a protein capsule. So let me draw. And the genetic information can come in any form. So it can be an RNA, it could be DNA, it could be single-stranded RNA, double-stranded RNA. Sometimes for single stranded they'll write these two little S's in front of it. Let's say they are talking about double stranded DNA, they'll put a ds in front of it. But the general idea-- and viruses can come in all of these forms-- is that they have some genetic information, some chain of nucleic acids. Either as single or double stranded RNA or single or double stranded DNA. And it's just contained inside some type of protein structure, which is called the capsid. And kind of the classic drawing is kind of an icosahedron type looking thing. Let me see if I can do justice to it. It looks something like this. And not all viruses have to look exactly like this. There's thousands of types of viruses. And we're really just scratching the surface and understanding even what viruses are out there and all of the different ways that they can essentially replicate themselves. We'll talk more about that in the future. And I would suspect that pretty much any possible way of replication probably does somehow exist in the virus world. But they really are just these proteins, these protein capsids, are just made up of a bunch of little proteins put together. And inside they have some genetic material, which might be DNA or it might be RNA. So let me draw their genetic material. The protein is not necessarily transparent, but if it was, you would see some genetic material inside of there. So the question is, is this thing life? It seems pretty inanimate. It doesn't grow. It doesn't change. It doesn't metabolize things. This thing, left to its own devices, is just It's just going to sit there the way a book on a table just sits there. It won't change anything. But what happens is, the debate arises. I mean you might say, hey Sal, when you define it that way, just looks like a bunch of molecules put together. That isn't life. But it starts to seem like life all of a sudden when it comes in contact with the things that we normally consider life. So what viruses do, the classic example is, a virus will attach itself to a cell. So let me draw this thing a little bit smaller. So let's say that this is my virus. I'll draw it as a little hexagon. And what it does is, it'll attach itself to a cell. And it could be any type of cell. It could be a bacteria cell, it could be a plant cell, it could be a human cell. Let me draw the cell here. Cells are usually far larger than the virus. In the case of cells that have soft membranes, the virus figures out some way to enter it. Sometimes it can essentially fuse-- I don't want to complicate the issue-- but sometimes viruses have their own little membranes. And we'll talk about in a second where it gets their membranes. So a virus might have its own membrane like that. That's around its capsid. And then these membranes will fuse. And then the virus will be able to enter into the cell. Now, that's one method. And another method, and they're seldom all the same way. But let's say another method would be, the virus convinces-- just based on some protein receptors on it, or protein receptors on the cells-- and obviously this has to be kind of a Trojan horse type of thing. The cell doesn't want viruses. So the virus has to somehow convince the cell that it's a non-foreign particle. We could do hundreds of videos on how viruses work and it's a continuing field of research. But sometimes you might have a virus that just gets consumed by the cell. Maybe the cell just thinks it's something that it needs to consume. So the cell wraps around it like this. And these sides will eventually merge. And then the cell and the virus will go into it. This is called endocytosis. I'll just talk about that. It just brings it into its cytoplasm. It doesn't happen just to viruses. But this is one mechanism that can enter. And then in cases where the cell in question-- for example in the situation with bacteria-- if the cell has a very hard shell-- let me do it in a good color. So let's say that this is a bacteria right here. And it has a hard shell. The viruses don't even enter the cell. They just hang out outside of the cell like this. Not drawing to scale. And they actually inject their genetic material. So there's obviously a huge-- there's a wide variety of ways of how the viruses get into cells. But that's beside the point. The interesting thing is that they do get into the cell. And once they do get into the cell, they release their genetic material into the cell. So their genetic material will float around. If their genetic material is already in the form of RNA-- and I could imagine almost every possibility of different ways for viruses to work probably do exist in nature. We just haven't found them. But the ones that we've already found really do kind of do it in every possible way. So if they have RNA, this RNA can immediately start being used to essentially-- let's say this is the nucleus of the cell. That's the nucleus of the cell and it normally has the DNA in it like that. Maybe I'll do the DNA in a different color. But DNA gets transcribed into RNA, normally. So normally, the cell, this a normal working cell, the RNA exits the nucleus, it goes to the ribosomes, and then you have the RNA in conjunction with the tRNA and it produces these proteins. The RNA codes for different proteins. And I talk about that in a different video. So these proteins get formed and eventually, they can form the different structures in a cell. But what a virus does is it hijacks this process here. Hijacks this mechanism. This RNA will essentially go and do what the cell's own RNA would have done. And it starts coding for its own proteins. Obviously it's not going to code for the same things there. And actually some of the first proteins it codes for often start killing the DNA and the RNA that might otherwise compete with it. So it codes its own proteins. And then those proteins start making more viral shells. So those proteins just start constructing more and more viral shells. At the same time, this RNA is replicating. It's using the cell's own mechanisms. Left to its own devices it would just sit there. But once it enters into a cell it can use all of the nice machinery that a cell has around to replicate itself. And it's kind of amazing, just the biochemistry of it. That these RNA molecules then find themselves back in these capsids. And then once there's enough of these and the cell has essentially all of its resources have been depleted, the viruses, these individual new viruses that have replicated themselves using all of the cell's mechanisms, will find some way to exit the cell. The most-- I don't want to say, typical, because we haven't even discovered all the different types of viruses there are-- but one that's, I guess, talked about the most, is when there's enough of these, they'll release proteins or they'll construct proteins. Because they don't make their own. That essentially cause the cell to either kill itself or its membrane to dissolve. So the membrane dissolves. And essentially the cell lyses. Let me write that down. The cell lyses. And lyses just means that the cell's membrane just And then all of these guys can emerge for themselves. Now I talked about before that have some of these guys, that they have their own membrane. So how did they get there, these kind of bilipid membranes? Well some of them, what they do is, once they replicate inside of a cell, they exit maybe not even killing-- they don't have to lyse. Everything I talk about, these are specific ways that a virus might work. But viruses really kind of explore-- well different types of viruses do almost every different combination you could imagine of replicating and coding for proteins and escaping from cells. Some of them just bud. And when they bud, they essentially, you can kind of imagine that they push against the cell wall, or the membrane. I shouldn't say cell wall. The cell's outer membrane. And then when they push against it, they take some of the membrane with them. Until eventually the cell will-- when this goes up enough, this'll pop together and it'll take some of the membrane with it. And you could imagine why that would be useful thing to have with you. Because now that you have this membrane, you kind of look like this cell. So when you want to go infect another cell like this, you're not going to necessarily look like a foreign particle. So it's a very useful way to look like something that you're not. And if you don't think that this is creepy-crawly enough, that you're hijacking the DNA of an organism, viruses can actually change the DNA an organism. And actually one of the most common examples is HIV virus. Let me write that down. HIV, which is a type of retrovirus, which is fascinating. Because what they do is, so they have RNA in them. And when they enter into a cell, let's say that they got into the cell. So it's inside of the cell like this. They actually bring along with them a protein. And every time you say, where do they get this protein? All of this stuff came from a different cell. They use some other cell's amino acids and ribosomes and nucleic acids and everything to build themselves. So any proteins that they have in them came from another cell. But they bring with them, this protein reverse transcriptase. And the reverse transcriptase takes their RNA and codes it into DNA. So its RNA to DNA. Which when it was first discovered was, kind of, people always thought that you always went from DNA to RNA, but this kind of broke that paradigm. But it codes from RNA to DNA. And if that's not bad enough, it'll incorporate that DNA into the DNA of the host cell. So that DNA will incorporate itself into the DNA of the host cell. Let's say the yellow is the DNA of the host cell. And this is its nucleus. So it actually messes with the genetic makeup of what it's infecting. And when I made the videos on bacteria I said, hey for every one human cell we have twenty bacteria cells. And they live with us and they're useful and they're part of us and they're 10% of our dry mass and all of that. But bacteria are kind of along for the ride. They don't change who we are. But these retroviruses, they're actually changing our I mean, my genes, I take very personally. They define who I am. But these guys will actually go in and change my genetic makeup. And then once they're part of the DNA, then just the natural DNA to RNA to protein process will code their actual proteins. Or their-- what they need to-- so sometimes they'll lay dormant and do nothing. And sometimes-- let's say sometimes in some type of environmental trigger, they'll start coding for themselves again. And they'll start producing more. But they're producing it directly from the organism's cell's DNA. They become part of the organism. I mean I can't imagine a more intimate way to become part of an organism than to become part of its DNA. I can't imagine any other way to actually define an organism. And if this by itself is not eerie enough, and just so you know, this notion right here, when a virus becomes part of an organism's DNA, this is called a provirus. But if this isn't eerie enough, they estimate-- so if this infects a cell in my nose or in my arm, as this cell experiences mitosis, all of its offspring-- but its offspring are genetically identical-- are going to have this viral DNA. And that might be fine, but at least my children won't get it. You know, at least it won't become part of my species. But it doesn't have to just infect somatic cells, it could infect a germ cell. So it could go into a germ cell. And the germ cells, we've learned already, these are the ones that produce gametes. For men, that's sperm and for women it's eggs. But you could imagine, once you've infected a germ cell, once you become part of a germ cell's DNA, then I'm passing on that viral DNA to my son or my daughter. And they are going to pass it on to their children. And just that idea by itself is, at least to my mind. vaguely creepy. And people estimate that 5-8%-- and this kind of really blurs, it makes you think about what we as humans really are-- but the estimate is 5-8% of the human genome-- so when I talked about bacteria I just talked about things that were along for the ride. But the current estimate, and I looked up this a lot. I found 8% someplace, 5% someplace. I mean people are doing it based on just looking at the DNA and how similar it is to DNA in other organisms. But the estimate is 5-8% of the human genome is from viruses, is from ancient retroviruses that incorporated themselves into the human germ line. So into the human DNA. So these are called endogenous retroviruses. Which is mind blowing to me, because it's not just saying these things are along for the ride or that they might help us or hurt us. It's saying that we are-- 5-8% of our DNA actually comes from viruses. And this is another thing that speaks to just genetic variation. Because viruses do something-- I mean this is called horizontal transfer of DNA. And you could imagine, as a virus goes from one species to the next, as it goes from Species A to B, if it mutates to be able to infiltrate these cells, it might take some-- it'll take the DNA that it already has, that makes it, it with it. But sometimes, when it starts coding for some of these other guys, so let's say that this is a provirus right here. Where the blue part is the original virus. The yellow is the organism's historic DNA. Sometimes when it codes, it takes up little sections of the other organism's DNA. So maybe most of it was the viral DNA, but it might have, when it transcribed and translated itself, it might have taken a little bit-- or at least when it translated or replicated itself-- it might take a little bit of the organism's previous DNA. So it's actually cutting parts of DNA from one organism and bringing it to another organism. Taking it from one member of a species to another member of But it can definitely go cross-species. So you have this idea all of a sudden that DNA can jump between species. It really kind of-- I don't know, for me it makes me appreciate how interconnected-- as a species, we kind of imagine that we're by ourselves and can only reproduce with each other and have genetic variation within But viruses introduce this notion of horizontal transfer via transduction. Horizontal transduction is just the idea of, look when I replicate this virus, I might take a little bit of the organism that I'm freeloading off of, I might take a little bit of their DNA with me. And infect that DNA into the next organism. So you actually have this DNA, this jumping, from organism to organism. So it kind of unifies all DNA-based life. Which is all the life that we know on the planet. And if all of this isn't creepy enough-- and actually maybe I'll save the creepiest part for the end. But there's a whole-- we could talk all about the different classes of viruses. But just so you're familiar with some of the terminology, when a virus attacks bacteria, which they often do. And we study these the most because this might be a good alternative to antibiotics. Because viruses that attack bacteria might-- sometimes the bacteria is far worse for the virus-- but these are called bacteriaphages. And I've already talked to you about how they have their DNA. But since bacteria have hard walls, they will just inject the DNA inside of the bacteria. And when you talk about DNA, this idea of a provirus. So when a virus lyses it like this, this is called the lytic cycle. This is just some terminology that's good to know if you're going to take a biology exam about this stuff. And when the virus incorporates it into the DNA and lays dormant, incorporates into the DNA of the host organism and lays dormant for awhile, this is called the lysogenic cycle. And normally, a provirus is essentially experiencing a lysogenic cycle in eurkaryotes, in organisms that have a nuclear membrane. Normally when people talk about the lysogenic cycle, they're talking about viral DNA laying dormant in the DNA of bacteria. Or bacteriophage DNA laying dormant in the DNA of bacteria. But just to kind of give you an idea of what this, quote unquote, looks like, right here. I got these two pictures from Wikipedia. One is from the CDC. These little green dots you see right here all over the surface, this big thing you see here, this is a white blood cell. Part of the human immune system. This is a white blood cell. And what you see emerging from the surface, essentially budding from the surface of this white blood cell-- and this gives you a sense of scale too-- these are HIV-1 viruses. And so you're familiar with the terminology, the HIV is a virus that infects white blood cells. AIDS is the syndrome you get once your immune system is weakened to the point. And then many people suffer infections that people with a strong immune system normally won't suffer from. But this is creepy. These things went inside this huge cell, they used the cell's own mechanism to reproduce its own DNA or its own RNA and these protein capsids. And then they bud from the cell and take a little bit of the membrane with it. And they can even leave some of their DNA behind in this cell's own DNA. So they really change what the cell is all about. This is another creepy picture. These are bacteriaphages. And these show you what I said before. This is a bacteria right here. This is its cell wall. And it's hard. So it's hard to just emerge into it. Or you can't just merge, fuse membranes with it. So they hang out on the outside of this bacteria. And they are essentially injecting their genetic material into the bacteria itself. And you could imagine, just looking at the size of these things. I mean, this is a cell. And it looks like a whole planet or something. Or this is a bacteria and these things are so much smaller. Roughly 1/100 of a bacteria. And these are much less than 1/100 of this cell we're talking about. And they're extremely hard to filter for. To kind of keep out. Because they are such, such small particles. If you think that these are exotic things that exist for things like HIV or Ebola , which they do cause, or SARS, you're right. But they're also common things. I mean, I said at the beginning of this video that I have a cold. And I have a cold because some viruses have infected the tissue in my nasal passage. And they're causing me to have a runny nose and whatnot. And viruses also cause the chicken pox. They cause the herpes simplex virus. Causes cold sores. So they're with us all around. I can almost guarantee you have some virus with you as you speak. They're all around you. But it's a very philosophically puzzling question. Because I started with, at the beginning, are these life? And at first when I just showed it to you, look they are just this protein with some nucleic acid molecule in it. And it's not doing anything. And that doesn't look like life to me. It's not moving around. It doesn't have a metabolism. It's not eating. It's not reproducing. But then all of a sudden, when you think about what it's doing to cells and how it uses cells to kind of reproduce. It kind of like-- in business terms it's asset light. It doesn't need all of the machinery because it can use other people's machinery to replicate itself. You almost kind of want to view it as a smarter form of life. Because it doesn't go through all of the trouble of what every other form of life has. It makes you question what life is, or even what we are. Are we these things that contain DNA or are we just transport mechanisms for the DNA? And these are kind of the more important things. And these viral infections are just battles between different forms of DNA and RNA and whatnot. Anyway I don't want to get too philosophical on you. But hopefully this gives you a good idea of what viruses are and why they really are, in my mind, the most fascinating pseudo organism in all of biology." + }, + { + "Q": "I still don't get how to divide notes how dose that work?", + "A": "Dividing notes is basically just dividing a note into smaller subsections of notes to get shorter duration notes. (different rhythms) (Depends on the time signature and such.) Just keep practicing, watching and researching more! It ll help! ;)", + "video_name": "esbqpgAD-EM", + "transcript": "- [Voiceover] At times, especially in popular dance forms the meter will remain constant. All ballroom dancing fits into this category. A march would also fit into this category, remaining constant, usually in two-four. It is also the case with most music from the 18th and 19th centuries. In the latter part of the 1800's and into the 1900's, composers started to feel free to change meters during a movement or work, sometimes quite often. The actual meters remain as we have discussed. If we look at the last movement of the Sam Jones \"Cello Concerto\", we can see some simple changes of meter, from two-four to three-four, back to two-four, then three-four and four-four. (\"Cello Concerto\" by Sam Jones) Another simple example is in Phillip Glass' \"Harmonium Mountain\". At this excerpt, he mixes the meters two-four, three-four, and four-four. (\"Harmonium Mountain\" by Phillip Glass) If we look at David Stock's work called \"Blast\" written in 2010, we find a more complicated section of meter changers, using five-eight, seven-eight, three-four, and four-four. (\"Blast\" by David Stock) If we look at a four-four measure, we have learned that the measure can easily be divided by using various note lengths. Half notes, quarter notes, sixteenth notes, thirty-second notes and so forth. But what if a composer would like to divide one of those quarter notes into three equal parts to create more rhythmic interest? We accomplish this by adding a three above or below a group of three eighth notes. The three signifies that three notes are performed during the time of one quarter note. Let's again look at Phillip Glass' \"Harmonium Mountain\". In this passage we see the violins playing the groups of three called triplets, and the violas and cellos are playing quarter notes. Then the violas join the violins playing triplets, the cellos play the eighth notes and the double basses play the quarter notes. (\"Harmonium Mountain\" by Phillip Glass) This method of changing duple notes to triple notes can work in any duple meter. The composer can also divide the triple beat in different ways. For example, instead of three eighth notes in a beat, we could see an eighth note and a quarter note, or a quarter note and an eighth note. We still need the number three above or below the notes. If we look at Ravel's \"Daphnis et Chloe\" the meter is five-four, but Ravel adds the three for each part to create triplets. You will see that he doesn't continue to write the three during the continuation of the excerpt, assuming that the performer understands the pattern. (\"Daphnis et Chloe\" by Ravel) The triplet is the most common variation within a meter, but there could also be, for example, five notes within a quarter, again with a five above or below, or six, or quite frankly any number that is not common to the meter. In a triple meter like six-eight, one could do the same. Six-eight can be one dotted half note or two dotted quarter notes, or six eighths, or twelve sixteenths. We could also have a rhythm of quarter, eighth, or eighth, quarter. Or any combination that adds up to six eighth notes. If a composer wanted four notes during a dotted quarter note, the number four would go above or below the group of notes. As you can see, the notation of rhythm can become very complicated. We will discuss this in later lessons." + }, + { + "Q": "what does circumscribed mean?", + "A": "2. GEOMETRY draw (a figure) around another, touching it at points but not cutting it", + "video_name": "KjQ1KN5GgoE", + "transcript": "Line AC is tangent to circle O at point C. So this is line AC, tangent to circle O at point C. What is the length of segment AC? What is this distance right over here, between point A and point C? And I encourage you now to pause this video and try this out on your own. So I'm assuming you've given a go at it. So the key thing to realize here, since AC is tangent to the circle at point C, that means it's going to be perpendicular to the radius between the center of the circle and point C. So this right over here is a right angle. And the reason why that is useful is now we know that triangle AOC is a right triangle. So if we know two of its sides, we could use the Pythagorean theorem to figure out the third. Now, we clearly know OC. Now OA, we don't know the entire side. They only give us that AB is equal to 2. But the thing that might jump out in your mind is OB is a radius. It's going to be the same length as any radius. So this is going to be 3 as well. It's the distance between the center of the circle and a point on the circle, just like the distance between O and C. So this is going to be 3 as well. And so now we are able to figure out that the hypotenuse of this triangle has length 5. And so we need to figure out what the length of segment AC is. So let's just call that, I don't know. I'll call that x. And so we know that x squared plus 3 squared-- I'm just applying the Pythagorean theorem here-- is going to be equal to the length of the hypotenuse squared, is going to be equal to 5 squared. And I know this is the hypotenuse. It's the side opposite the 90-degree angle. It's the longest side of the right triangle. So x squared plus 9 is equal to 25. Subtract 9 from both sides, and you get x squared is equal to 16. And so it should jump out at you that x is going to be equal to 4. So x is equal to 4. x is the same thing as the length of segment AC, so the length of segment AC is 4." + }, + { + "Q": "at 5:07 i got confused.", + "A": "There, 15 goes into 75 ----> 5 times r 1. To review it, when he was writing 5*5 = 25 in the division table, after writing 5, he takes 2 to add with (1*5). But, instead of writing 2 there, he faultily wrote it as 7. Then, after finding the error, he rectified it.", + "video_name": "gHTH6PKfpMc", + "transcript": "Welcome to the presentation on level 4 division. So what makes level 4 division harder than level 3 division is instead of having a one-digit number being divided into a multi-digit number, we're now going to have a two-addition number divided into a multi-digit number. So let's get started with some practice problems. So let's start with what I would say is a relatively straightforward example. The level 4 problems you'll see are actually a little harder than this. But let's say I had 25 goes into 6,250. So the best way to think about this is you say, OK, I have 25. Does 25 go into 6? Well, no. Clearly 6 is smaller than 25, so 25 does not go into 6. So then ask yourself, well, then if 25 doesn't go into 6, does 25 go into 62? Well, sure. 62 is larger than 25, so 25 will go into 62? Well, let's think about it. 25 times 1 is 25. 25 times 2 is 50. So it goes into 62 at least two times. And 25 times 3 is 75. So that's too much. So 25 goes into 62 two times. And there's really no mechanical way to go about figuring this out. You have to kind of think about, OK, how many times do I think 25 will go into 62? And sometimes you get it wrong. Sometimes you'll put a number here. Say if I didn't know, I would've put a 3 up here and then I would've said 3 times 25 and I would've gotten a 75 here. And then that would have been too large of a number, so I would have gone back and changed it to a 2. Likewise, if I had done a 1 and I had done 1 tmes 25, when I subtracted it out, the difference I would've gotten would be larger than 25. And then I would know that, OK, 1 is too small. I have to increase it to 2. I hope I didn't confuse you too much. I just want you to know that you shouldn't get nervous if you're like, boy, every time I go through the step it's kind of like- I kind of have to guess what the numbers is as opposed to kind of a method. And that's true; everyone has to do that. So anyway, so 25 goes into 62 two times. Now let's multiply 2 times 25. Well, 2 times 5 is 10. And then 2 times 2 plus 1 is 5. And we know that 25 times 2 is 50 anyway. Then we subtract. 2 minus 0 is 2. 6 minus 5 is 1. And now we bring down the 5. So the rest of the mechanics are pretty much just like a level 3 division problem. Now we ask ourselves, how many times does 25 go into 125? Well, the way I think about it is 25-- it goes into 100 about four times, so it will go into 125 one more time. It goes into it five times. If you weren't sure you could try 4 and then you would see that you would have too much left over. Or if you tried 6 you would see that you would actually get 6 times 25 is a number larger than 125. So you can't use 6. So if we say 25 goes into 125 five times then we just multiply 5 times 5 is 25. 5 times 2 is 10 plus 2, 125. So it goes in exact. So 125 minus 125 is clearly 0. Then we bring down this 0. And 25 goes into 0 zero times. 0 times 25 is 0. Remainder is 0. So we see that 25 goes into 6,250 exactly 250 times. Let's do another problem. Let's say I had-- let me pick an interesting number. Let's say I had 15 and I want to know how many times it goes into 2,265. Well, we just do the same thing we did before. We say OK, does 15 go into 2? No. So does 15 go into 22? Sure. 15 goes into 22 one time. Notice we wrote the 1 above the 22. If it go had gone into 2 we would've written the 1 here. But 15 goes into 22 one time. 1 times 15 is 15. 22 minus 15-- we could do the whole carrying thing-- 1, 12. 12 minus 5 is 7. 1 minus 1 is 0. 22 minus 15 is 7. Bring down the 6. OK, now how many times does 15 go into 76? Once again, there isn't a real easy mechanical way to do it. You can kind of eyeball it and estimate. Well, 15 times 2 is 30. 15 times 4 is 60. 15 times 5 is 75. That's pretty close, so let's say 15 goes into 76 five times. So 5 times 5 once again, I already figured it out in my head, but I'll just do it again. 5 times 1 is 5. Plus 7. Oh, sorry. 5 times 5 is 25. 5 times 1 is 5. Plus 2 is 7. Now we just subtract. 76 minus 75 is clearly 1. Bring down that 5. Well, 15 goes into 15 exactly one time. 1 times 15 is 15. Subtract it and we get a remainder of 0. So 15 goes into 2,265 exactly 151 times. So just think about what we're doing here and why it's a little bit harder than when you have a one-digit number here. Is that you have to kind of think about, well, how many times does this two-digit number go into this larger number? And since you don't know two-digit multiplication tables-- very few people do-- you have to do a little bit of guesswork. Sometimes you can look at this first digit and look at the first digit here and make an estimate. But sometimes it's trial and error. You'll try and when you multiply it out you might get it wrong on the first try. Let's do another problem. And actually, I'm going to pick numbers at random, so it might not have an easy remainder. But I think you'll get the point. I won't teach you decimals now, so I'll just leave the remainder if there is one. Let's say I had 67 going into 5,978. So I just picked these numbers randomly out of my head, so I'll show you that I also sometimes have to do a little bit of guesswork to figure out how many times one of these two-digit numbers go into a larger number. So 67 goes into 5 zero times. 67 goes into 59 zero times. 67 goes into 597-- so let's see. 67 is almost 70 and 597 is almost 600. So if it was 70 goes into-- 70 times 9 to 630. Because 7 times 9 is 63. So I'm going to just eyball approximate. I'm going to say that it goes into it eight times. I might be wrong. And you can always check, but well, we're going to actually check in this step essentially. 8 times 7-- well that's 56. And then 8 times 6 is 48. Plus 2 is 53. 7 minus 6 is 1. 9 minus 9 is 6. 5 minus 5 is 0. 61. So good. I got it right because if I got a number here that was larger than-- 67 or larger, than that means that this number up here wasn't large enough. But here, I got a number that's positive because 536 is less than 597. And it's less than 67, so I did that step right. So now we bring down this 8. Now this one might be a little bit trickier this time. Once again, we have almost 70 and here we have almost 630. So maybe it will go into it 9 times. Well, let's give it a try and see if it does. 9 times 7 is 63. 9 times 6 is 54. Plus 6 is 60. Good. So it did actually go into it nine times because 603 is less than 618. 8 minus 3 is 5. 1 minus 0 is 1. And 6 minus 6 is 0. We have a remainder of 15, which is smaller than 67. So I'm not going to teach you decimals right now, so we can just leave that remainder. So what we could say is that 67 goes into 5,978 89 times. And when it goes into it 89 times, you're left with a remainder of 15. hopefully you're ready now to try some level 4 division problems. Have fun." + }, + { + "Q": "I have a question: if the quantity supplied of an item in the market decreases, according to the law of supply, the price would decrease too. But wouldn't the fact that it is less abundant, mean that the price actually goes up, as there is less of it.", + "A": "Well, it depends on the demand for the product. If it is a private good, often highly demanded, a reduction in supply will result in higher demand, which increases the inelasticity of the good, meaning the firm can confidently raise their prices. So yea, it comes down to the product and it s demand in home and overseas markets.", + "video_name": "3xCzhdVtdMI", + "transcript": "We've talked a lot about demand. So now let's talk about supply, and we'll use grapes as this example. We'll pretend to be grape farmers of some sort. So I will start by introducing you-- and maybe I'll do it in purple in honor of the grapes-- to the law of supply, which like the law of demand, makes a lot of intuitive sense. If we hold all else equal-- in the next few videos, we'll talk about what happens when we change some of those things that we're going to hold equal right now-- but if you hold all else equal and the only thing that you're doing is you're changing price, then the law of supply says that if the price goes up-- I'll just say p for price-- if the price goes up, then the supply-- now, let me be careful-- the quantity supplied goes up. And then you can imagine, if the price goes down, the quantity supplied goes down. And you might already notice that I was careful to say quantity supplied. And it's just like we saw with demand. When we talk about demand going up or down, we're talking about the entire price-quantity relationship shifting. When we talk about a particular quantity demanded, we say quantity demanded. We don't just say demand. This is the exact same thing for supply. When we're talking about a particular quantity, we'll be careful to say quantity. If we talk about supply increasing, we're talking about the entire relationship shifting either up or down. So let's just make sure that this makes intuitive sense for us. And I think it probably does. Let's think about ourselves as grape farmers. And I'll make a little supply schedule right over here. So Grape Supply Schedule, which is really just a table showing the relationship between, all else equal, the price and the quantity supplied. So let's label some scenarios over here, just like we did with the demand schedule. Scenarios. And then let's put our Price over here. This will be in price per pound, the per pound price of grapes. And then this is the quantity produced over the time period. And whenever we do any of these supply or demand schedules, we're talking over a particular time period. It could be per day, it could be per month, it could be per year. But that's the only way to make some sense of, OK, what is the quantity per day going to be produced if that's the price? So if we didn't say per day, we don't know what we're really talking about. Quantity Supplied. And so let's just say Scenario A, if the price per pound of grapes is $0.50-- if it's $0.50 per pound-- actually, let me just do round numbers, but you get the idea. If the price per pound is $1, let's just say for us, we consider that to be a relatively low price. And so we'll only kind of do the easiest land, our most fertile land, where it's easy to produce grapes. And maybe the fertile-- and sheep land. So no one else wants to use that land for other things. It's only good for growing grapes. And so we will provide-- so this is price per pound. And in that situation, we can produce 1,000 pounds in this year. And I've never been a grape farmer, so I actually don't know if that's a reasonable amount or not, but I'll just go with it, 1,000 pounds. Now, let's take Scenario B. Let's say the price goes up to $2. Well now, not only would we produce what we were producing before, but we might now want to maybe buy some more land, land that might have had other uses, land that's maybe not as productive for grapes. But we would, because now we can get more for grapes. And so maybe now we are willing to produce 2,000 pounds. And we can keep going. The same dynamics keep happening. So let's say the price-- if the price were $3 per pound, now we do want to produce more. Maybe we're even willing to work a little harder or plant things closer to each other, or maybe I'll get even more land involved than I would have otherwise used for other crops. And so then I'm going to produce 2,500 pounds. And I'll do one more scenario. Let's say Scenario D, the price goes to $4 a pound. Same dynamic, I will stop planting other crops, use them now for grapes, because grape prices are so high. And so I will produce 2,750 pounds. And so we can draw a supply curve just like we have drawn demand curves. And it's the same exact convention, which I'm not a fan of, putting price on the vertical axis. Because as you see, we tend to talk about price as the independent variable. We don't always talk about it that way. And in most of math and science, you put the independent variable on the horizontal axis. But the convention in economics is to put it on the vertical axis. So price on the vertical axis. So then this is really Price per pound. And then in the horizontal axis, Quantity Produced, or-- let me just write it. Quantity Produced, I'll say in the next year. We're assuming all of this is for the next year, so next year. And it's in thousands of pounds, so I'll put it in thousands of pounds. And so let's see, we go all the way from 1,000 to close to 3,000. So let's say this is 1,000, that's 1 for 1,000, that's 2,000, and that is 3,000. And then the price goes all the way up to 4. So it's 1, 2, 3, and then 4. So we can just plot these points. These are specific points on the supply curve. So at $1, we would supply 1,000 pounds, at $1, 1,000 pounds. That's Scenario A. At $2, we would supply 2,000 pounds, $2, we'd supply 2,000 pounds. That's scenario B. At $3, we'd supply 2,500 pounds, $3-- oh, Now, when we look up-- See, now notice, I get my axes confused. This is Price. This isn't, when we talk about it this way, that we're viewing the thing that's changing. Although, you don't always have to do it that way. So at one $1, 1,000 pounds. $1, 1,000 pounds. $2, 2,000 pounds. $2, 2,000 pounds. $3-- this isn't $3, this is $3. $3, 2,500 pounds. So right about there. That's about 2,500. But I want to do it in that blue color, so we don't get confused. So $3, 2,500 pounds. That's about right. So this is Scenario C. And then Scenario D, at $4-- actually, let me be a little bit clearer with that, because we're getting close. So this is 2,500 pounds, gets us right over here. This is Scenario C. And then Scenario D at $4, 2,750. So 2,750 is like right over there. So that is $4. That is Scenario D. And if we connect them, they should all be on our supply curve. So they will all be-- it will look something like that. And there's some minimum price we would need to supply some grapes at all. We wouldn't give them away for free. So maybe that's something-- that minimum price is over here, that just even gets started producing grapes. So this right over here is what our supply curve would look like. Now remember, the only thing we're varying here is the price. So if the price were to change, all else equal, we would move along this curve here. Now, in the next few videos, I'll talk about all those other things we've been holding equal and what they would do at any given price point to this curve or, in general, what they would do to the curve." + }, + { + "Q": "If Hitler hated communists so much, why didn't he alliance with France, Great Britain, and the United States to to attack the USSR?", + "A": "First off, France, Great Britain and the United States wanted peace in Europe, they would NEVER support his future Operation Barbarossa which would launch Europe into war. Second, Hitler wanted to conquer all of Europe, he didn t want to be allies with them. He was fueled more by his desire to expand than his hatred of Communism. Hope this helped :) Happy holidays!!", + "video_name": "X3bqQI7-sCg", + "transcript": "- [Voiceover] As we've already seen in the last few videos, with the war officially starting in September of 1939, the Axis powers get momentum through the end of 1939, all the way into 1940. That was the last video that we covered and that takes us to 1941 and what we're gonna see in 1941 which is the focus of this video is that the Axis powers only seem to gain more momentum. Because of all of that momentum they perhaps gets a little bit overconfident and stretch themselves or begin to stretch themselves too thin. So let's think about what happens in 1941. So, if we talk about early 1941 or the Spring of 1941, in March, Bulgaria decides to join the Axis powers. You can imagine there's a lot of pressuring applied to them and they kind of see where the momentum is. Let's be on that side. Bulgaria joins the Axis and then in North Africa you might remember that in 1940, the Allies, in particular, the British, were able to defeat the Italians and push them back into Libya but now in March of 1941, the Italians get reinforcements, Italian reinforcements and also German reinforcements under the command of Rommel the Desert Fox, famous desert commander and they are able to push the British back to the Egyptian border and they also take siege of the town of Tobruk. Now, you might have noticed something that I just drew. The supply lines in the North Africa campaign are very, very, very long and that's part of the reason why there's one side. One side has supply lines and as they start to make progress and as the Allies make progress and push into Libya, their supply lines got really long and so the other side has an easier timely supply. Then as the Axis pushes the Allies back into Egypt, then their supply lines get really long and the other side...it makes it easier for them to resupply and so North Africa is kind of defined by this constant back and forth. But, by early 1941, it looks like the Axis is on the offensive, able to push the British back into Egypt lay siege to the town of Tobruk. So, let me write this down as North Africa. So, I'll just say North Africa over here or I'll could say Rommel in North Africa pushing the British back. And then we can start talking about what happens in the Balkans and this is still in Spring as we go into April of 1941 and just as a little bit of background here, and frankly I should have covered it a couple of videos ago. As far back as 1939, actually before World War II officially started, in Spring of 1939, Italy actually occupies Albania so this actually should have already been red. This is in 1939 that this happens and then at the end of 1940, Italy uses Albania as a base of operation to try to invade Greece but they are pushed back. Actually one of the reasons why the British we able to be pushed back in North Africa is after they were successful against the Italians, most of the bulk of the British forces we sent to Greece to help defend Greece at the end of 1940. So, in 1939, Albania gets taken over by Italy and at the end of 1940...October 1940, Greece is invaded by Italy but they are then pushed back but to help the Greeks, the Allies send many of the forces that were in North Africa after they were successful against the Italians in Libya. Now, as we go into April of 1941... that was all background, remember Albania before the war started in April 1939, October 1940 was Italy's kind of first push into Greece and it was unsuccessful. Then the Greeks get support from the Allies in North Africa and now as we go into 1941, the Germans start supporting and really take charge in Balkans and in Greece and so with the help of the Germans the Axis is able to take over Yugoslavia and Greece and start aerial bombardment of Crete. So, once again, we're not even halfway through the year in 1941 and we see a huge swath of Europe is under the control of the Axis powers. And now we go into the summer of 1941. This is actually a pivotal move, what's about to happen. Now you can imagine that the Axis powers, in particular, Hitler, are feeling pretty confident. We are only about that far into the war. So we're not even two years into the war yet and it looks like the Axis is going to win. Now you might remember that they have a pact with the Soviet Union. Hey, we're gonna split a lot of Eastern Europe into our spheres of influence so to speak, but now Hitler's like, well, I think I'm ready to attack and when you attack the Soviet Union really matters. You do not want to attack the Soviet Union in the winter...or Russia in the winter. Russia's obviously at the heart of the Soviet Union. That something that Napoleon learned. Many military commanders have learnt. You do not want to be fighting in Russia over the winter, so summer of 1941, Hitler figures, hey, this is the Axis chance. And so, in June, he decides to attack the Soviet Union. So, this is a very, very, very bold move because now they're fighting the British. Remember, the British are kind of not a joke to be battling out here in Western Europe and now they're going to be taking on the Soviet Union in the east, a major, major world power. But at first, like always, it seems like it's going well for the Germans. By September, they're able to push up all the way to Leningrad. So, this is September of 1941 and lay lay siege and begin laying siege to that town. This is kind of a long bloody siege that happens there. So, we're right now, right about there. And most historians would tell you that this was one of the mistakes of Adolph Hitler because now he is stretched very, very, very thin. He has to fight two world powers, Soviet Union and Great Britain and the United States hasn't entered into the war yet and that's what we're about to get into because if we go into Asia it was still in 1941 what happens in July. So, little bit after Hitler decides to start invading the Soviet Union going back on the pact, the non- aggression pact. In July, you could imagine the US, they were never pleased with what's been happening, what the Empire of Japan has been doing in the Pacific, what they've been doing in China, in Manchuria or even in terms of the war in China, the second Sino Japanese War. They weren't happy of the Japanese taking over French Indochina. There's a big world power here, the Empire of Japan. There's a big world power here, the United States, that has a lot of possessions in the Pacific and so, the United States in July of 1941...So remember, this is still all 1941, this is the same year...decides to freeze the assets of Japan and probably the most important part of that was an oil embargo of Japan. This is a big, big deal. Japan is fighting a major conflict with the Chinese. It's kind of flexing it's imperial muscles but it does not have many natural resources in and of itself and in fact, that's one of the reasons why it's trying to colonize other places to get more control of natural resources. And now if it's fighting a war it doesn't have it's own oil resources and now there's an oil embargo of Japan and the United States at the time was major oil producer and even today, it's major oil producer. This was a big deal to the Japanese because some estimates say they only had about two years of reserves and they were fighting a war where they might have to touch their reserves even more. So, you could imagine the Japanese, they want to have their imperial ambitions. They probably want, especially now with this oil embargo, they probably want to take over more natural resources and they probably want to knock out the US or at least keep the US on its heels so the US can't stop Japan from doing what it wants to do. So, all it wants in December 1941, that's over the course of December 7th and 8th, and it gets a little confusing because a lot of this happens across the International Date Line. But over the course of December 7th and 8th, Japan goes on the offensive in a major way in the Pacific. Over the course of several hours, at most, a day, Japan is able to attack Malaya, which is a British possession. It's able to attack Pearl Harbor, where the US Pacific fleet is in hope to knock out the US Pacific fleet so the US will have trouble stopping Japan from doing whatever Japan wants to do. In the US, we focus a lot on Pearl Harbor but this was just one of the attacks in this whole kind of several hours of attacks where Japan went on the offensive. So, we have Malaya, we have Pearl Harbor, we have Singapore, we have Guam, we have (which was the US military base), Wake Island. was a US possession ever since the Spanish American war. You have Hong Kong, which is a British possession and then shortly after that as you get further into December, so this is kind of when you have Japan offensive. Then as you go on into later Decemeber, the kind of real prize for Japan was what we would now call Indonesia but the Dutch East Indies. On this map it says Netherlands East Indies. You have to remember the Netherlands had been overrun. They're the low countries they were already overrun by German forces so the Japanese say hey, look there are a lot of resources here, natural resources, especially oil. Let's go for this and so by the end of 1941, they're also going for the Dutch East Indies and for Burma so you could imagine it's a very aggressive, very, very bold move on Japan but they kind of had imperial ambitions. They were afraid of they access to natural resources so they went for it but obviously one of the major consequence of this is the United States was not happy about this and they were already sympathetic to the Allies. They didn't like what was going on in Europe either. They didn't like what was going on in China and so that causes the United States to enter into World War II on the side of the Allies and then the Axis powers to declare war on the United States, which was a big deal." + }, + { + "Q": "Did Sal make this website. We'll anyway, Salman is awesome!", + "A": "Yes he did", + "video_name": "jxA8MffVmPs", + "transcript": "Find the place value of 3 in 4,356. Now, whenever I think about place value, and the more you do practice problems on this it'll become a little bit of second nature, but whenever I see a problem like this, I like to expand out what 4,356 really is, so let me rewrite So if I were to write it-- and I'll write it in different colors. So 4,356 is equal to-- and just think about how I just said it. It is equal to 4,000 plus 300 plus 50 plus 6. And you could come up with that just based on how we said it: four thousand, three hundred, and fifty-six. Now another way to think about this is this is just like saying this is 4 thousands plus-- or you could even think of \"and\"-- so plus 3 hundreds plus 50, you could think of it as 5 tens plus 6. And instead of 6, we could say plus 6 ones. And so if we go back to the original number 4,356, this is the same thing as 4-- I'll write it down. Let me see how well I can-- I'll write it up like this. This is the same thing is 4 thousands, 3 hundreds, 5 tens and then 6 ones. So when they ask what is the place value of 3 into 4,356, we're concerned with this 3 right here, and it's place value. It's in the hundreds place. If there was a 4 here, that would mean we're dealing with 4 hundreds. If there's a 5, 5 hundreds. It's the third from the right. This is the ones place. That's 6 ones, 5 tens, 3 hundreds. So the answer here is it is in the hundreds place." + }, + { + "Q": "Hi\nWhen I have substituted the value of vi in the eq s= ut +1/2at^2, I have got the answer correct with 2.5 sec, but when I am doing it with t= 5sec I am getting zero as answer and with your equation taken t=5s I get72.5 m kindly clarify", + "A": "The equation he uses only works for the first half of the trajectory. To work for the second half, you have to find the average velocity for the ball going down, now. The average velocity will be the same value, but now in the negative direction (Vavg = -12.25 m/s). Using his equation, you have to use t = 2.5 s ( cause it s the time for the second half), and the displacement will be s = -30.625 m. Since the ball was at the height of 30.625 m, the final position would be 0 m.", + "video_name": "IYS4Bd9F3LA", + "transcript": "Let's say you and I are playing a game or I'm trying to figure out how high a ball is being thrown in the air How fast would we throwing that ball in the air? And what we do is one of us has a ball and the other one has a stop watch over here So this is my best attempt to make it more like a cat than a stop watch but I think you get the idea And what we do is one of us throw the ball the other one times how long the ball is in the air And what we do is gonna use that time in the air to figure out how fast the ball was thrown straight up and how long it was in the air or how high it got And there is going to be one assumption I make here frankly that's an assumption we are gonna make in all of these projectile motion type problem is that air resistance is negligible And for something like it, this is a baseball or something like that That's a pretty good approximation So when can I get the exact answer, I encourage you experiment it on your own or even to see what air resistance does to your calculations We gonna assume for this projectile motion in future one at least in the basic Physics playlist We gonna assume air resistance is negligible And what that does for us is we can assume that the time up That the time for the ball to go up to its peak height is the same thing as the time that takes it to go down If you look at this previous video, we've plot it displacement verse time You see after 2 seconds the ball went from being on the ground or I guess the thrower's hand all the way to its peak height And then the next 2 seconds it took the same amount of time to go back down to the ground which makes sense whatever the initial velocity is, it take half the time to go to zero and it takes the same amount of time to now be accelerated into downward direction back to that same magnitude of velocity but now in the downward direction So let's play around with some numbers here Just so you get a little bit more of concrete sense So let's say I throw a ball in the air And you measure using the stop watch and the ball is in the air for 5s So how do we figure out how fast I threw the ball? Well the first thing we could do is we could say look at the total time in the air was 5 seconds that mean the time, let me write it, that means the change in time to go up during the first half, I guess the ball time in the air is going to be 2.5 seconds and which tells us that over this 2.5 seconds we went from our initial velocity, whatever it was We went from our initial velocity to our final velocity which is a velocity of 0 m/s in the 2.5 seconds And this is a graph for that example, This is the graph for the previous one, The previous example we knew the initial velocity but in whatever the time is you are going from you initial velocity to be stationery at the top, right with the ball being stationery and then start getting increasing velocity in the downward direction So it takes 2.5 seconds to go from some initial velocity to 0 seconds So we do know what the acceleration of the gravity is We know that the acceleration We know the acceleration of gravity here, we are assuming it's constant or slightly not constant but we are gonna assume it's constant We are just dealing close to the surface of the earth is negative 9.8 m/s*s, so let's think about it This change in velocity, are change in velocity Their change in velocity is the final velocity minus the initial velocity which is the same thing as zero minus the initial velocity which is the negative of the initial velocity That's another way to think about change in velocity We just shown the definition of acceleration change in velocity is equal to acceleration, is equal to acceleration negative 9.8 m/s*s times time or times change in time, our change in time, we are just talking about the first half of the ball's time in the air So the change in time is 2.5 s, times 2.5 s So what is our change in velocity which is also the same thing as negative of our initial velocity Get the calculator out, let me get my calculator, bring it on to the screen, so it is negative 9.8 m/s times 2.5 s Times 2.5 s, it gives us negative 24.5, so this gives us I will write it in new color This gives us negative 24.5 m/s, this seconds cancels out With one of these seconds in the denominator we only have one of the denominator out m/s, and this is the same thing as the negative, as the negative initial velocity Negative initial velocity that's the same thing as change in velocity So you multiply both side by a negative, you will get our initial velocity So that simply we are able to figure out what our velocity is So literally you take the time, the total time in the air divide by two And multiply that by acceleration of gravity and if you take I guess you can take the absolute value of that or take the positive version of that And that gives you your initial velocity So your initial velocity here is literally 24.5 m/s Since it's a positive quantity it is upwards in this example So that's my initial velocity, so we already figure out part of this game The initial velocity that threw upward That's also going to be, we gonna have the same magnitude of velocity The balls about to hit the ground although is gonna be in the other direction So what is the distance or let me make it clear what is the displacement of the ball from its lowest point right when it leaves your hand all the way to the peak, all the way to the peak? We just have to remember, all of these come from very straight forward ideas Change in velocity is equal to acceleration times change in time And the other simple idea is that displacement is equal to average velocity, average velocity times change in time Now what is our average velocity? Our average velocity is your initial velocity plus your final velocity Divided by 2, or we assume that acceleration is constant So literally just the arithmetic mean of your initial and final velocity So what is that? That's gonna be 24.5 m/s plus our final velocity In this situation we are just going over to the first 2.5 s So our final velocity is once again 0 m/s We are just talking about when we get to his point over here So our final velocity is just 0 m/s And we just gonna divide that by 2 This will give us the average velocity And we wanna multiply that by 2.5 s, times 2.5 s So we get this part right over here 24.5 divided by 2 When you go with the 0, it is still 24.5 It gives us 12.25 times 2.5, and remember this right over here is in seconds let me write the units down So this is 12.25 m/s times 2.5 seconds And just to remind ourselves We are calculating the displacement over the first 2.5 seconds So this gives us, I get the calculator out once again We have 12.25 times 2.5 seconds gives us 30.625 So this gives us, this gives us, the displacement is 30.625 m The seconds cancel out This is actually a ton, you know, roughly give or take about 90 feet throw into the air, this looks like a nine stories building And I frankly do not have the arm for that But if someone is able to throw the ball for 5 seconds in the air They have thrown 30 meters in the air Hopefully you will find that entertaining In my next video I'll generalize this maybe we can get a little bit of formula so maybe you can generalize it So regardless of the measurement of time you can get the displacement in the air Or even better, try to derive it yourself And you will see how, at least how I tackle it in the next video" + }, + { + "Q": "Is it possible in a real life situation for an atom like chlorine to give 7 valence electrons to Sodium to create a new element ClNa", + "A": "No, that s not possible! But, even if it were, it would not be a new element because what you describe is two atoms and the term element refers to a single atom with a single nucleus.", + "video_name": "Rr7LhdSKMxY", + "transcript": "Voiceover: What I want to talk about in this video are the notions of Electronegativity, electro, negati, negativity, and a closely, and a closely related idea of Electron Affinity, electron affinity. And they're so closely related that in general, if something has a high electronegativity, they have a high electron affinity, but what does this mean? Well, electron affinity is how much does that atom attract electrons, how much does it like electrons? Does it want, does it maybe want more electrons? Electronegativity is a little bit more specific. It's when that atom is part of a covalent bond, when it is sharing electrons with another atom, how likely is it or how badly does it want to hog the electrons in that covalent bond? Now what do I mean by hogging electrons? So let me make, let me write this down. So how badly wants to hog, and this is an informal definition clearly, hog electrons, keep the electrons, to spend more of their time closer to them then to the other party in the covalent bond. And this is how, how much they like electrons, or how much affinity they have towards electrons. So how much they want electrons. And you can see that these are very, these are very related notions. This is within the context of a covalent bond, how much electron affinity is there? Well this, you can think of it as a slightly broader notion, but these two trends go absolutely in line with each other. And to think about, to just think about electronegativity makes it a little bit more tangible. Let's think about one of the most famous sets of covalent bonds, and that's what you see in a water molecule. Water, as you probably know, is H two O, you have an oxygen atom, and you have two hydrogens. Each of the hydrogen's have one valence electron, and the oxygen has, we see here, at it's outermost shell, it has one, two, three, four, five, six valence electrons. One, two, three, four, five, six valence electrons. And so you can imagine, hydrogen would be happy if it was able to somehow pretend like it had another electron then it would have an electron configuration a stable, first shell that only requires two electrons, the rest of them require eight, hydrogen would feel, hey I'm stable like helium if it could get another electron. And oxygen would feel, hey I'm stable like neon if I could get two more electrons. And so what happens is they share each other's electrons. This, this electron can be shared in conjunction with this electron for this hydrogen. So that hydrogen can kind of feel like it's using both and it gets more stable, it stabilizes the outer shell, or it stabilizes the hydrogen. And likewise, that electron could be, can be shared with the hydrogen, and the hydrogen can kind of feel more like helium. And then this oxygen can feel like it's a quid pro quo, it's getting something in exchange for something else. It's getting the electron, an electron, it's sharing an electron from each of these hydrogens, and so it can feel like it's, that it stabilizes it, similar to a, similar to a neon. But when you have these covalent bonds, only in the case where they are equally electronegative would you have a case where maybe they're sharing, and even there what happens in the rest of the molecule might matter, but when you have something like this, where you have oxygen and hydrogen, they don't have the same electronegativity. Oxygen likes to hog electrons more than hydrogen does. And so these electrons are not gonna spend an even amount of time. Here I did it kind of just drawing these, you know, these valence electrons as these dots. But as we know, the electrons are in this kind of blur around, around the, around the actual nuclei, around the atoms that make up the atoms. And so, in this type of a covalent bond, the electrons, the two electrons that this bond represents, are going to spend more time around the oxygen then they are going to spend around the hydrogen. And these, these two electrons are gonna spend more time around the oxygen, then are going to spend around the hydrogen. And we know that because oxygen is more electronegative, and we'll talk about the trends in a second. This is a really important idea in chemistry, and especially later on as you study organic chemistry. Because, because we know that oxygen is more electronegative, and the electrons spend more time around oxygen then around hydrogen, it creates a partial negative charge on this side, and partial positive charges on this side right over here, which is why water has many of the properties that it does, and we go into much more in depth in that in other videos. And also when you study organic chemistry, a lot of the likely reactions that are going to happen can be predicted, or a lot of the likely molecules that form can be predicted based on elecronegativity. And especially when you start going into oxidation numbers and things like that, electronegativity will tell you a lot. So now that we know what electronegativity is, let's think a little bit about what is, as we go through, as we start, as we go through, as we go through a period, as say as we start in group one, and we go to group, and as we go all the way all the way to, let's say the halogens, all the way up to the yellow column right over here, what do you think is going to be the trend for electronegativity? And once again, one way to think about it is to think about the extremes. Think about sodium, and think about chlorine, and I encourage you to pause the video and think about that. Assuming you've had a go at it, and it's in some ways the same idea, or it's a similar idea as ionization energy. Something like sodium has only one electron in it's outer most shell. It'd be hard for it to complete that shell, and so to get to a stable state it's much easier for it to give away that one electron that it has, so it can get to a stable configuration like neon. So this one really wants to give away an electron. And we saw in the video on ionization energy, that's why this has a low ionization energy, it doesn't take much energy, in a gaseous state, to remove an electron from sodium. But chlorine is the opposite. It's only one away from completing it's shell. The last thing it wants to do is give away electron, it wants an electron really, really, really, really badly so it can get to a configuration of argon, so it can complete it's third shell. So the logic here is that sodium wouldn't mind giving away an electron, while chlorine really would love an electron. So chlorine is more likely to hog electrons, while sodium is very unlikely to hog electrons. So this trend right here, when you go from the left to the right, your electronegativity, let me write this, your getting more electronegative. More electro, electronegative, as you, as you go to the right. Now what do you think the trend is going to be as you go down, as you go down in a group? What do you think the trend is going to be as you go down? Well I'll give you a hint. Think about, think about atomic radii, and given that, what do you think the trend is? Are we gonna get more or less electronegative as we move down? So once again I'm assuming you've given a go at it, so as we know, from the video on atomic radii, our atom is getting larger, and larger, and larger, as we add more and more and more shells. And so cesium has one electron in it's outer most shell, in the sixth shell, while, say, lithium has one electron. Everything here, all the group one elements, have one electron in it's outer most shell, but that fifty fifth electron, that one electron in the outer most shell in cesium, is a lot further away then the outer most electron in lithium or in hydrogen. And so because of that, it's, well one, there's more interference between that electron and the nucleus from all the other electrons in between them, and also it's just further away, so it's easier to kind of grab it off. So cesium is very likely to give up, it's very likely to give up electrons. It's much more likely to give up electrons than hydrogen. So, as you go down a given group, you're becoming less, less electronegative, electronegative. So what, what are, based on this, what are going to be the most electronegative of all the atoms? Well they're going to be the ones that are in the top and the right of the periodic table, they're going to be these right over here. These are going to be the most electronegative, Sometimes we don't think as much about the noble gases because they aren't, they aren't really that reactive, they don't even form covalent bond, because they're just happy. While these characters up here, they sometimes will form covalent bonds, and when they do, they really like to hog those electrons. Now what are the least electronegative, sometimes called very electropositive? Well these things down here in the bottom left. These, over here, they have only, you know in the case of cesium, they have one electron to give away that would take them to a stable state like, like xenon, or in the case of these group two elements they might have to give away two, but it's much easier to give away two then to gain a whole bunch of them. And they're big, they're big atoms. So those outer most electrons are getting less attracted to the positive nucleus. So the trend in the periodic table as you go from the bottom left, to the top right, you're getting more, more electro, electronegative." + }, + { + "Q": "So what do you do if you're given a hypotenuse that, when squared, has no combination of natural numbers squared to equal itself? How do you find the length of the two missing legs?", + "A": "In a 45-45-90, you have an isosceles triangle, which means the two legs are equal. Taking the Pythagorean formula a^2 + b^2 = c^2, we will now be able to simplify it to a^2 + a^2 = c^2 or 2\u00e2\u0080\u00a2a^2 = c^2. Given the hypotenuse, isolate equation for a: a^2 = c^2/2 a = sqrt(c^2/2)", + "video_name": "McINBOFCGH8", + "transcript": "In the last video, we showed that the ratios of the sides of a 30-60-90 triangle are-- if we assume the longest side is x, if the hypotenuse is x. Then the shortest side is x/2 and the side in between, the side that's opposite the 60 degree side, is square root of 3x/2. Or another way to think about it is if the shortest side is 1-- Now I'll do the shortest side, then the medium size, then the longest side. So if the side opposite the 30 degree side is 1, then the side opposite the 60 degree side is square root of 3 times that. So it's going to be square root of 3. And then the hypotenuse is going to be twice that. In the last video, we started with x and we said that the 30 degree side is x/2. But if the 30 degree side is 1, then this is going to be twice that. So it's going to be 2. This right here is the side opposite the 30 degree side, opposite the 60 degree side, and then the hypotenuse opposite the 90 degree side. And so, in general, if you see any triangle that has those ratios, you say hey, that's a 30-60-90 triangle. Or if you see a triangle that you know is a 30-60-90 triangle, you could say, hey, I know how to figure out one of the sides based on this ratio right over here. Just an example, if you see a triangle that looks like this, where the sides are 2, 2 square root of 3, and 4. Once again, the ratio of 2 to 2 square root of 3 is 1 to square root of 3. The ratio of 2 to 4 is the same thing as 1 to 2. This right here must be a 30-60-90 triangle. What I want to introduce you to in this video is another important type of triangle that shows up a lot in geometry and a lot in trigonometry. And this is a 45-45-90 triangle. Or another way to think about is if I have a right triangle that is also isosceles. You obviously can't have a right triangle that is equilateral, because an equilateral triangle has all of their angles have to be 60 degrees. But you can have a right angle, you can have a right triangle, that is isosceles. And isosceles-- let me write this-- this is a right isosceles triangle. And if it's isosceles, that means two of the sides are equal. So these are the two sides that are equal. And then if the two sides are equal, we have proved to ourselves that the base angles are equal. And if we called the measure of these base angles x, then we know that x plus x plus 90 have to be equal to 180. Or if we subtract 90 from both sides, you get x plus x is equal to 90 or 2x is equal to 90. Or if you divide both sides by 2, you get x is equal to 45 degrees. So a right isosceles triangle can also be called-- and this is the more typical name for it-- it can also be called a 45-45-90 triangle. And what I want to do this video is come up with the ratios for the sides of a 45-45-90 triangle, just like we did for a 30-60-90 triangle. And this one's actually more straightforward. Because in a 45-45-90 triangle, if we call one of the legs x, the other leg is also going to be x. And then we can use the Pythagorean Theorem to figure out the length of the hypotenuse. So the length of the hypotenuse, let's call that c. So we get x squared plus x squared. That's the square of length of both of the legs. So when we sum those up, that's going to have to be equal to c squared. This is just straight out of the Pythagorean theorem. So we get 2x squared is equal to c squared. We can take the principal root of both sides of that. I wanted to just change it to yellow. Last, take the principal root of both sides of that. The left-hand side you get, principal root of 2 is just square root of 2, and then the principal root of x squared is just going to be x. So you're going to have x times the square root of 2 is equal to c. So if you have a right isosceles triangle, whatever the two legs are, they're going to have the same length. That's why it's isosceles. The hypotenuse is going to be square root of 2 times that. So c is equal to x times the square root of 2. So for example, if you have a triangle that looks like this. Let me draw it a slightly different way. It's good to have to orient ourselves in different ways every time. So if we see a triangle that's 90 degrees, 45 and 45 like this, and you really just have to know two of these angles to know what the other one is going to be, and if I tell you that this side right over here is 3-- I actually don't even have to tell you that this other side's going to be 3. This is an isosceles triangle, so those two legs are going to be the same. And you won't even have to apply the Pythagorean theorem if you know this-- and this is a good one to know-- that the hypotenuse here, the side opposite the 90 degree side, is just going to be square root of 2 times the length of either of the legs. So it's going to be 3 times the square root of 2. So the ratio of the size of the hypotenuse in a 45-45-90 triangle or a right isosceles triangle, the ratio of the sides are one of the legs can be 1. Then the other leg is going to have the same measure, the same length, and then the hypotenuse is going to be square root of 2 times either of those. 1 to 1, 2 square root of 2. So this is 45-45-90. That's the ratios. And just as a review, if you have a 30-60-90, the ratios were 1 to square root of 3 to 2. And now we'll apply this in a bunch of problems." + }, + { + "Q": "you can use remainders", + "A": "Yeah, what s the question?", + "video_name": "xXIG8ouHcsc", + "transcript": "Let's divide 7,182 by 42. And what's different here is we're now dividing by a two-digit number, not a one-digit number, but the same idea holds. So we say, hey, how many times does 42 go into 7? Well, it doesn't really go into 7 at all, so let's add one more place value. How many times does 42 go into 71? Well, it goes into 71 one time. Just a reminder, whoever's doing the process where you say, hey, 42 goes into 71 one time. But what we're really saying, 42 goes into 7,100 100 times because we're putting this one in the hundreds place. But let's put that on the side for a little bit and focus on the process. So 1 times 42 is 42, and now we subtract. Now, you might be able to do 71 minus 42 in your head, knowing, hey, 72 minus 42 would be 30. So 71 minus 42 would be 29, but we could also do it by regrouping. To regroup, you want to subtract a 2 from a 1. You can't really do that in any traditional way. So let's take a 10 from the 70, so that it becomes a 60, and give that 10 to the ones place, and then that becomes an 11. And so 11 minus 2 is 9, and 6 minus 4 is 2. So you get 29. And we can bring down the next place value. Bring down an 8. And now, this is where the art happens when we're dividing by a multi-digit number right over here. We have to estimate how many times does 42 go into 298. And sometimes it might involve a little bit of trial and error. So you really just kind of have to eyeball it. If you make a mistake, try again. The way you know you make a mistake is, if say it goes into it 9 times, and you do 9 times 42 and you get a number larger than 298, then you overestimated. If you say it goes into it three times, you do 3 times 42, you get some number here. When you subtract, you get something larger than 42, then you also made a mistake, and you have to adjust upwards. Well, let's see if we can eyeball it. So this is roughly 40. This is roughly 300. 40 goes into 300 the same times as 4 goes into the 30, so it's going to be roughly 7. Let's see if that's right. 7 times 2 is 14. 7 times 1 is 28, plus 1 is 29. So I got pretty close. My remainder here-- notice 294 is less than 298. So I'm cool there. And my remainder is less than 42, so I'm cool as well. So now let's add another place value. Let's bring this 2 down. And here we're just asking ourselves, how many times does 42 go into 42? Well, 42 goes into 42 exactly one time. 1 times 42 is 42, and we have no remainder. So this one luckily divided exactly. 42 goes into 7,182 exactly 171 times." + }, + { + "Q": "What is vector xstar? What is its significance, and magnitude? I mean aside from what is explained in the video... I noticed that if you draw it, it is just there...", + "A": "xstar is the vector we need to multiply A by to find our solution, representing slope and y-intercept. Doing so multiplies the slope (x*0) by the x value of our points, and adds it (add in dot product is equivalent to add in mx + b in this case) to the y intercept (X* 1) times 1 to equal our y points.", + "video_name": "QkepM8Vv3kw", + "transcript": "So I've got four Cartesian coordinates here. This first one is minus 1, 0. I tried to draw them ahead of time. So minus 1, 0 is this point right there. Doing this in these new colors. The next point is a 0, 1, which is that point right there. Then the next point is 1, 2, which is that point right up there. And then the last point is 2, 1, which is that point there. Now my goal in this video is to find some line, y equals mx plus v, that goes through these points. Now the first thing I'd say is, hey Sal, there is not going to be any line that goes through these points, and you can see that immediately. You could find a line that maybe goes through these points, but it's not going to go through this point over here. If you try to make a line to goes through these two points, it's not going to go through those points there. So you're not going to be able to find a solution that goes through those points. Let's set up the equation that we know we can't find the solution to and maybe we can use our least squares approximation to find a line that almost goes through all these points. Or it's at least the best approximation for a line that goes through those points. So this first one, I can express my line, y Let me just express it as f of x is equal to mx plus b, or y is equal to f of x. We can write it that way. So our first point right there -- let me do it in that color, that orange -- that tells us that f of minus 1, which is equal to m times -- let me just write this way -- minus 1 times m, it's minus m plus b, that that is going to be equal to 0. That's what that first equation tells us. The second equation tells us that f of 0, which is equal to 0 times m, which is just 0 plus b is equal to 1. f of 0 is 1. This is f of x. The next one -- let me do it in this yellow color -- tells us that f of 1, which is equal to 1 times m, or just m, plus b, is going to be equal to 2. And then this last one down here tells us that f of 2, which is of course 2 times m plus b, that that is going to be equal to 1. These are the constraints. If we assume that our line can go through all of these points, then all of these things must be true. Now you could immediately, if you wish, try to solve this equation, but you'll find that you won't find a solution. We want to find some m's and b's that satisfy all of these equations. Or another way of writing this -- We want to write it as a matrix vector or a matrix equation . We could write it like this. Minus 1, 1, 0, 1, 1, 1, 2, 1, times the vector mv has got to be equal to the vector 0, 1, 2, 1. These two systems, this system and this system right here, are equivalent statements, right? Minus 1 times m plus 1 times b has got to be equal to that 0. 0 times m plus 1 times b has got to be equal to that 1 That's equivalent to that statement right here. And this isn't going to have a solution. The solution would have to go through all of those points. So let's at least try to find a least squares solution. So if we call this a, if we call that x, and let's call this b, there is no solution to ax is equal to b. Now maybe we can find a least -- Well, we can definitely find a least squares solution. So let's find our least squares solution such that a transpose a times our least squares solution is equal to a transpose times b. Our least squares solution is the one that satisfies this equation. We proved it two videos ago. So let's figure out what a transpose a is and what a transpose b is, and then we can solve. So a transpose will look like this. b minus 1, 1, 0, 1, 1, 1, and then 2, 1. This first column becomes this first row; this second column becomes this second row. So we're going to take the product of a transpose and then a-- a is that thing right there --minus 1, 0, 1, 2, and we just get a bunch of 1's. So what does this equal to? We have a 2 by 4 times a 4 by 2. So we're going to have a 2 by 2 matrix. So this is going to be -- Let's do it this way. Well, we're going to have minus 1 times minus 1, which is 1, plus 0 times 0, which is 0 -- so we're at 1 right now --plus 1 times 1. So that's 1 plus the other 1 up there, so that's 2, plus 2 times 2. 2 times 2 is 4, so we get 6. That's that row, dotted with that column, was equal to 6. Now let's take this row dotted with this column. So it's going to be negative 1 times 1, plus 0 times 1, so all of these guys times 1 plus each other. So minus 1 plus 0 plus 1 -- that's all 0's --plus 2. So it's going to get a 2. I just dotted that guy with that guy. Now I need to take the dot of this guy with this column. So it's just going to be 1 times minus 1 plus 1 times 0 plus 1 times 1 plus 1 times 2. Well, these are all 1 times everything, so it's minus 1 plus 0 plus 1, which is 0 plus 2. It's going to be 2. And then finally -- Well. I mean, I think you see some symmetry here. We're going to have to take the dot of this guy and this guy over here. That's 1 times 1, which is 1, plus 1 times 1, which is 2, plus 1 times 1. So we're going to have 1 plus itself four times. So we're going to get that it's equal to 4. So this is a transpose a. And let's figure out what a transpose b looks like. Scroll down a little bit. So a transpose is this matrix again-- let me switch colors --minus 1, 0, 1, 2. We get all of our 1's just like that. And then the matrix b is 0, 1, 2, 1. We have a 2 by 4 times a 4 by 1, so we're just going to get a 2 by 1 matrix. So this is going to be equal to a 2 by 1 matrix. We have here, let's see, minus 1 times 0 is 0, plus 0 times 1 is still 0. Plus 1 times 2, which is 2, plus 2 times 1, which is 4, right? This is 2 plus 2, so it's going to be 4 right there. And then we have 1 times 0, plus 1 times 2, plus-- So 1 times all of these guys added up. So 0 plus 1 is 1, 1 plus 2 is 3, 3 plus 1 is 4. So this right here is a transpose b. So just like that, we know that the least squares solution will be the solution to this system. 6, 2, 2, 4, times our least squares solution, is going to be equal to 4, 4. Or we could write it this way. We could write it 6, 2, 2, 4, times our least squares solution, which I'll write-- Remember, the first entry was m . I'll write it as m star. That's our least square m, and this is our least square b, is equal to 4, 4. And I can do this as an augmented matrix or I could just write this as a system of two unknowns, which is actually probably easier. So let's do it that way. So this, if I were to write it as a system of equations, is 6 times m star plus 2 times b star, is equal to 4. And then I get 2 times m star plus 4 times b star is equal to this 4. So let me solve for my m stars and my b stars. So let's multiply this second equation, actually let's multiply that top equation by 2. This is just straight Algebra 1. So times 2, what do we get? We get 12m star plus 4b star is equal to 8. We just multiplied that top guy by 2. Now let's multiply this magenta 1 by negative 1. So this becomes a minus, this becomes a minus, that becomes a minus, and now we can add these two equations. So we get minus 2 plus 12m star, that's 10m star. And then the minus 4b and the 4b cancel out, is equal to 4, or m star is equal to 4 over 10, which is equal to 2/5. Now we can just go and back-substitute into this. We can say 6 times m star-- This is just straight Algebra 1. So 6 times our m star, so 6 times 2 over 5, plus 2 times our b star is equal to 4. Enough white, let me use yellow. So we get 12 over 5 plus 2b star is equal to 4, or we could say 2b star-- let me scroll down a little bit --2b star is equal to 4. Which is the same thing as 20 over 5, minus 12 over 5, which is equal to-- I'm just subtracting the 12 over 5 from both sides --which is equal to 8 over 5. And you divide both sides of the equation by 2, you get b star is equal to 4/5. And just like that, we got our m star and our b star. Our least squares solution is equal to 2/5 and 4/5. So m is equal to 2/5 and b is equal to 4/5. And remember, the whole point of this was to find an equation of the line. y is equal to mx plus b. Now we can't find a line that went through all of those points up there, but this is going to be our least squares solution. This is the one that minimizes the distance between a times our vector and b. No vector, when you multiply times that matrix a-- that's not a, that's transpose a --no other solution is going to give us a closer solution to b than when we put our newly-found x star into this equation. This is going to give us our best solution. It's going to minimize the distance to b. So let's write it out. y is equal to mx plus b. So y is equal to 2/5 x plus 2/5. Let's graph that out. y is equal to 2/5 x plus 2/5. So its y-intercept is 2/5, which is about there . This is at 1. 2/5 is right about there. And then its slope is 2/5. Let's think of it this way: for every 2 and 1/2 you go to the right, you're going to go up 1. So if you go 1, 2 and 1/2, we're going to go up 1. We're going to go up 1 like that. So our line-- and obviously this isn't precise --but our line is going to look something like this. I want to do my best shot at drawing it because this is the fun part. It's going to look something like that. And that right there is my least squares estimate for a line that goes through all of those points. And you're not going to find a line that minimizes the error in a better way, at least when you measure the error as the distance between this vector and the vector a times our least squares estimate. Anyway, thought you would find that neat." + }, + { + "Q": "Magnets need energy to apply force in the form of attracting and repelling.\nFrom where does they get this energy?\nDoes it violates the law of conservation of energy?", + "A": "Your first statement is false, so the first question is moot and the answer to the second one is no . Consider an analogy: If you drop a book, do you say the earth needs energy to apply force to the book to attract it ? Does it violate the law of conservation of energy when the book gets attracted to the earth? No, right? Why not? Now see if you can apply the same thinking to your magnet question.", + "video_name": "8Y4JSp5U82I", + "transcript": "We've learned a little bit about gravity. We've learned a little bit about electrostatic. So, time to learn about a new fundamental force of the universe. And this one is probably second most familiar to us, And that's magnetism. Where does the word come from? Well, I think several civilizations-- I'm no historian-- found these lodestones, these objects that would attract other objects like it, other magnets. Or would even attract metallic objects like iron. Ferrous objects. And they're called lodestones. That's, I guess, the Western term for it. And the reason why they're called magnets is because they're named after lodestones that were found near the Greek province of Magnesia. And I actually think the people who lived there were called Magnetes. But anyway, you could Wikipedia that and learn more about it than I know. But anyway let's focus on what magnetism is. And I think most of us have at least a working knowledge of what it is; we've all played with magnets and we've dealt with compasses. But I'll tell you this right now, what it really is, is pretty deep. And I think it's fairly-- I don't think anyone has-- we can mathematically understand it and manipulate it and see how it relates to electricity. We actually will show you the electrostatic force and the magnetic force are actually the same thing, just viewed from different frames of reference. I know that all of that sounds very complicated and all of that. But in our classical Newtonian world we treat them as two different forces. But what I'm saying is although we're kind of used to a magnet just like we're used to gravity, just like gravity is also fairly mysterious when you really think about what it is, so is magnetism. So with that said, let's at least try to get some working knowledge of how we can deal with magnetism. So we're all familiar with a magnet. I didn't want it to be yellow. I could make the boundary yellow. No, I didn't want it to be like that either. So if this is a magnet, we know that a magnet always has two poles. It has a north pole and a south pole. And these were just labeled by convention. Because when people first discovered these lodestones, or they took a lodestone and they magnetized a needle with that lodestone, and then that needle they put on a cork in a bucket of water, and that needle would point to the Earth's north pole. They said, oh, well the side of the needle that is pointing to the Earth's north, let's call that the north pole. And the point of the needle that's pointing to the south pole-- sorry, the point of the needle that's pointing to the Earth's geographic south, we'll call that the south pole. Or another way to put it, if we have a magnet, the direction of the magnet or the side of the magnet that orients itself-- if it's allowed to orient freely without friction-- towards our geographic north, we call that the north pole. And the other side is the south pole. And this is actually a little bit-- obviously we call the top of the Earth the north pole. You know, this is the north pole. And we call this the south pole. And there's another notion of magnetic north. And that's where-- I guess, you could kind of say-- that is where a compass, the north point of a compass, will point to. And actually, magnetic north moves around because we have all of this moving fluid inside of the earth. And a bunch of other interactions. It's a very complex interaction. But magnetic north is actually roughly in northern Canada. So magnetic north might be here. So that might be magnetic north. And magnetic south, I don't know exactly where that is. But it can kind of move around a little bit. It's not in the same place. So it's a little bit off the axis of the geographic north pole and the south pole. And this is another slightly confusing thing. Magnetic north is the geographic location, where the north pole of a magnet will point to. But that would actually be the south pole, if you viewed the Earth as a magnet. So if the Earth was a big magnet, you would actually view that as a south pole of the magnet. And the geographic south pole is the north pole of the magnet. You could read more about that on Wikipedia, I know it's a little bit confusing. But in general, when most people refer to magnetic north, or the north pole, they're talking about the geographic north area. And the south pole is the geographic south area. But the reason why I make this distinction is because we know when we deal with magnets, just like electricity, or electrostatics-- but I'll show a key difference very shortly-- is that opposite poles attract. So if this side of my magnet is attracted to Earth's north pole then Earth's north pole-- or Earth's magnetic north-- actually must be the south pole of that magnet. And vice versa. The south pole of my magnet here is going to be attracted to Earth's magnetic south. Which is actually the north pole of the magnet we call Earth. Anyway, I'll take Earth out of the equation because it gets a little bit confusing. And we'll just stick to bars because that tends to be a little bit more consistent. Let me erase this. There you go. I'll erase my Magnesia. I wonder if the element magnesium was first discovered in Magnesia, as well. Probably. And I actually looked up Milk of Magnesia, which is a laxative. And it was not discovered in Magnesia, but it has magnesium in it. So I guess its roots could be in Magnesia if magnesium was discovered in Magnesia. Anyway, enough about Magnesia. Back to the magnets. So if this is a magnet, and let me draw another magnet. Actually, let me erase all of this. All right. So let me draw two more magnets. We know from experimentation when we were all kids, this is the north pole, this is the south pole. That the north pole is going to be attracted to the south pole of another magnet. And that if I were to flip this magnet around, it would actually repel north-- two north facing magnets would repel each other. And so we have this notion, just like we had in electrostatics, that a magnet generates a field. It generates these vectors around it, that if you put something in that field that can be affected by it, it'll be some net force acting on it. So actually, before I go into magnetic field, I actually want to make one huge distinction between magnetism and electrostatics. Magnetism always comes in the form of a dipole. It means that we have two poles. A north and a south. In electrostatics, you do have two charges. You have a positive charge and a negative charge. So you do have two charges. But they could be by themselves. You could just have a proton. You don't have to have an electron there right next to it. You could just have a proton and it would create a positive electrostatic field. And our field lines are what a positive point charge would do. And it would be repelled. So you don't always have to have a negative charge there. Similarly you could just have an electron. And you don't have to have a proton there. So you could have monopoles. These are called monopoles, when you just have one charge when you're talking about electrostatics. But with magnetism you always have a dipole. If I were to take this magnet, this one right here, and if I were to cut it in half, somehow miraculously each of those halves of that magnet will turn into two more magnets. Where this will be the south, this'll be the north, this'll be the south, this will be the north. And actually, theoretically, I've read-- my own abilities don't go this far-- there could be such a thing as a magnetic monopole, although it has not been observed yet in nature. So everything we've seen in nature has been a dipole. So you could just keep cutting this up, all the way down to if it's just one electron left. And it actually turns out that even one electron is still a magnetic dipole. It still is generating, it still has a north pole and a south pole. And actually it turns out, all magnets, the magnetic field is actually generated by the electrons within it. By the spin of electrons and that-- you know, when we talk about electron spin we imagine some little ball of charge spinning. But electrons are-- you know, it's hard to-- they do have mass. But it starts to get fuzzy whether they are energy or mass. And then how does a ball of energy spin? Et cetera, et cetera. So it gets very almost metaphysical. So I don't want to go too far into it. And frankly, I don't think you really can get an intuition. It is almost-- it is a realm that we don't normally operate in. But even these large magnets you deal with, the magnetic field is generated by the electron spins inside of it and by the actual magnetic fields generated by the electron motion around the protons. Well, I hope I'm not overwhelming you. And you might say, well, how come sometimes a metal bar can be magnetized and sometimes it won't be? Well, when all of the electrons are doing random different things in a metal bar, then it's not magnetized. Because the magnetic spins, or the magnetism created by the electrons are all canceling each other out, because it's random. But if you align the spins of the electrons, and if you align their rotations, then you will have a magnetically charged bar. But anyway, I'm past the ten-minute mark, but hopefully that gives you a little bit of a working knowledge of what a magnet is. And in the next video, I will show what the effect is. Well, one, I'll explain how we think about a magnetic field. And then what the effect of a magnetic field is on an electron. Or not an electron, on a moving charge. See you in the next video." + }, + { + "Q": "4:48 Hey, does anyone know why Sal puts the = sign like a smiley face? =D", + "A": "It s not an equal sign, but rather an arrow. He just draws it in a way that it doesn t look quite connected. This is actually a common way to note progression of steps in mathematics.", + "video_name": "X2jVap1YgwI", + "transcript": "Let's do some more percentage problems. Let's say that I start this year in my stock portfolio with $95.00. And I say that my portfolio grows by, let's say, 15%. How much do I have now? I think you might be able to figure this out on your own, but of course we'll do some example problems, just in case it's a little confusing. So I'm starting with $95.00, and I'll get rid of the dollar sign. We know we're working with dollars. 95 dollars, right? And I'm going to earn, or I'm going to grow just because I was an excellent stock investor, that 95 dollars is going to grow by 15%. So to that 95 dollars, I'm going to add another 15% of 95. So we know we write 15% as a decimal, as 0.15, so 95 plus 0.15 of 95, so this is times 95-- that dot is just a times sign. It's not a decimal, it's a times, it's a little higher than a decimal-- So 95 plus 0.15 times 95 is what we have now, right? Because we started with 95 dollars, and then we made another 15% times what we started with. Hopefully that make sense. Another way to say it, the 95 dollars has grown by 15%. So let's just work this out. This is the same thing as 95 plus-- what's 0.15 times 95? Let's see. So let me do this, hopefully I'll have enough space here. 95 times 0.15-- I don't want to run out of space. Actually, let me do it up here, I think I'm about to run out of space-- 95 times 0.15. 5 times 5 is 25, 9 times 5 is 45 plus 2 is 47, 1 times 95 is 95, bring down the 5, 12, carry the 1, 15. And how many decimals do we have? 1, 2. 15.25. Actually, is that right? I think I made a mistake here. See 5 times 5 is 25. 5 times 9 is 45, plus 2 is 47. And we bring the 0 here, it's 95, 1 times 5, 1 times 9, then we add 5 plus 0 is 5, 7 plus 5 is 12-- oh. I made a mistake. It's 14.25, not 15.25. So I'll ask you an interesting question? How did I know that 15.25 was a mistake? Well, I did a reality check. I said, well, I know in my head that 15% of 100 is 15, so if 15% of 100 is 15, how can 15% of 95 be more than 15? I think that might have made sense. The bottom line is 95 is less than 100. So 15% of 95 had to be less than 15, so I knew my answer of 15.25 was wrong. And so it turns out that I actually made an addition error, and the answer is 14.25. So the answer is going to be 95 plus 15% of 95, which is the same thing as 95 plus 14.25, well, that equals what? 109.25. Notice how easy I made this for you to read, especially this 2 here. 109.25. So if I start off with $95.00 and my portfolio grows-- or the amount of money I have-- grows by 15%, I'll end up with $109.25. Let's do another problem. Let's say I start off with some amount of money, and after a year, let's says my portfolio grows 25%, and after growing 25%, I now have $100. How much did I originally have? Notice I'm not saying that the $100 is growing by 25%. I'm saying that I start with some amount of money, it grows by 25%, and I end up with $100 after it grew by 25%. To solve this one, we might have to break out a little bit of algebra. So let x equal what I start with. So just like the last problem, I start with x and it grows by 25%, so x plus 25% of x is equal to 100, and we know this 25% of x we can just rewrite as x plus 0.25 of x is equal to 100, and now actually we have a level-- actually this might be level 3 system, level 3 linear equation-- but the bottom line, we can just add the coefficients on the x. x is the same thing as 1x, right? So 1x plus 0.25x, well that's just the same thing as 1 plus 0.25, plus x-- we're just doing the distributive property in reverse-- equals 100. And what's 1 plus 0.25? That's easy, it's 1.25. So we say 1.25x is equal to 100. Not too hard. And after you do a lot of these problems, you're going to intuitively say, oh, if some number grows by 25%, and it becomes 100, that means that 1.25 times that number is equal to 100. And if this doesn't make sense, sit and think about it a little bit, maybe rewatch the video, and hopefully it'll, over time, start to make a lot of sense to you. This type of math is very very useful. I actually work at a hedge fund, and I'm doing this type of math in my head day and night. So 1.25 times x is equal to 100, so x would equal 100 divided by 1.25. I just realized you probably don't know what a hedge fund is. I invest in stocks for a living. Anyway, back to the math. So x is equal to 100 divided by 1.25. So let me make some space here, just because I used up too much space. Let me get rid of my little let x statement. Actually I think we know what x is and we know how we got to there. If you forgot how we got there, you can I guess rewatch the video. Let's see. Let me make the pen thin again, and go back to the orange color, OK. X equals 100 divided by 1.25, so we say 1.25 goes into 100.00-- I'm going to add a couple of 0's, I don't know how many I'm going to need, probably added too many-- if I move this decimal over two to the right, I need to move this one over two to the right. And I say how many times does 100 go into 100-- how many times does 125 go into 100? None. How many times does it go into 1000? It goes into it eight times. I happen to know that in my head, but you could do trial and error and think about it. 8 times-- if you want to think about it, 8 times 100 is 800, and then 8 times 25 is 200, so it becomes 1000. You could work out if you like, but I think I'm running out of time, so I'm going to do this fast. 8 times 125 is 1000. Remember this thing isn't here. 1000, so 1000 minus 1000 is 0, so you can bring down the 0. 125 goes into 0 zero times, and we just keep getting 0's. This is just a decimal division problem. So it turns out that if your portfolio grew by 25% and you ended up with $100.00 you started with $80.00. And that makes sense, because 25% is roughly 1/4, right? So if I started with $80.00 and I grow by 1/4, that means I grew by $20, because 25% of 80 is 20. So if I start with 80 and I grow by 20, that gets me to 100. Makes sense. So remember, all you have to say is, well, some number times 1.25-- because I'm growing it by 25%-- is equal to 100. Don't worry, if you're still confused, I'm going to add at least one more presentation on a couple of more examples like this." + }, + { + "Q": "Why couldnt they just settle it peacefully instead of aggrivating each other to the point of war?", + "A": "After decades of trying to compromise and to settle their differences, the South saw no alternative than to leave the Union. The remaining states in the US did not recognize the secession of the southern states and the formation of the Confederacy. Once the attack was made on Fort Sumter, there was no turning back.", + "video_name": "L87VpmRLAPg", + "transcript": "- [Voiceover] All right Kim, so we left off in I guess, early-mid 1861, you have Lincoln gets inagurated, Fort Sumter which is kind of the first real conflict of the war, if not the first major battle. Lincoln forms his volunteer army, and then the rest of the southern states secede, four more states secede. - [Kim] Right. - [Voiceover] And then what was the first major conflict? - [Kim] So the first major conflict comes after a number of months. There are a couple of little skirmishes here and there, but nothing super large until about 60,000 troops meet outside of Manassas, Virginia, at a place called Bull Run. An interesting fact, I think, is that Union armies and Confederate armies actually named battles different things, if you've ever been confused about this. The Union armies tended to name battles after bodies of water, whereas the Confederate armies tended to name them by nearby towns. So if you've ever heard the Battle of Manassas and the Battle of Bull Run, they're the same thing, it's just the Union officers are talking about this creek, Bull Run, whereas the Confederates are talking about the town nearby. - [Voiceover] I see, and the 60,000 troops between the two of them. - [Kim] Right. So they meet, and this is very close to Washington, D.C., so much so that people go out and they bring picnics to watch this battle. - [Voiceover] They think it's going to be entertaining. - [Kim] Yeah, they think it's going to be like a football game. And it is not like a football game. It is a gigantic battle, 800 people die that day, which doesn't sound like a lot to us, but it was the most deadly battle ever in American history up until that point. So it's a Confederate victory, which is very surprising to the Union, because they think that they have such superior forces that this is really going to be a very short war. And this is a quick rebellion, in 90 days we're going to be able to, you know, suppress this rebellion and that'll be it. But Bull Run is really the first sign that this is going to be a major war. It's not going to be quick and it is going to be very deadly. - [Voiceover] This was July of... - [Both] 1861. - [Voiceover] Okay, so now it's clear to both sides, especially, I guess you could say the North, that this is not going to be a short war. So they need to prepare. How are they approaching this? - [Kim] Well, so both sides have some advantages and disadvantages. For the South, they have some of the same advantages that the United States would have had during the war for independence. They have home court advantage, we could say, which is that they know the territory very well and also there's a real incentive for people to protect their homes, right. You're gonna care more about a war that's happening on your property than a war that's gonna happen very far away. The other advantage that they have is just really, really terrific military leadership. So they have Robert E. Lee, who is widely considered the greatest general of his era. He's truly a military genius. He, in fact, was offered a commission in the Union army but when Virginia seceded, he went with Virginia. He preferred his home state. So he is a terrific general. The Union is gonna really struggle to come up with the kind of military leadership that the South has. - [Voiceover] Who is in charge of the Union or the You said, the United States Army. - [Kim] The United States Army. The first general that Lincoln puts in charge is George B. McClellan. This is problematic for a lot of reasons. One is that George McClellan is a Democrat, so he doesn't agree politically with Lincoln. I think he would have preferred peace, in fact in 1864 he runs against Lincoln for President on a platform of letting the South go, basically. And so Lincoln is struggling to match the South when it comes to military leadership, but he does have other advantages. For one thing, there are four times as many free people in the North as there are in the South. - [Voiceover] And you made the point, free people. - [Kim] Right. - [Voiceover] Because the South, as you mentioned, it has a majority of the population was not free. - [Kim] I wouldn't say a majority of the population, in many states, - [Voiceover] In Deep South. - [Kim] In the Deep South states, right. But so there are only about 9,000,000 people living in the South, and of those 9,000,000 people 3,500,000 to 4,000,000 of them are enslaved. So they're not going to be fighting to continue the institution of slavery. By contrast, the North has 22,000,000 people and it also has a terrific industrial base. One of the major cultural differences between the North and South that leads to the Civil War is that the South is primarily agrarian, and the North becomes very industrial. But industry is really helpful in a war. They've got miles and miles of railroad tracks which means that they can move supplies very quickly, and they also have hundreds and hundreds of factories that make it easy for them to make munitions. - [Voiceover] This is the middle of the Industrial Revolution - [Kim] Right. - [Voiceover] and an industrial base matters a lot. And so what's, given the North's advantages and the South's advantages, what's their strategies, how do they try to play to their strengths? - [Kim] Right, so the South, they are basically trying to outlast the North. They know that they have this territory, and if the North wants them to come back into the Union, they're going to have to conquer this territory. And even though it's hard to kind of tell, the territory of the South is actually larger than Western Europe. - [Voiceover] Wow. - [Kim] So in a way, the North has a bigger job to conquer the South than the Allies did in World War II, to conquer Europe. So they know that the North is gonna have to fight a war to conquer them, whereas the South just needs to win the war of waiting. - [Voiceover] Of attrition. - [Kim] Yeah, they're hoping that the North will get tired of fighting. - [Voiceover] Fighting in another person's land, you're not defending your own land. - [Kim] Right, and they know that there are plenty of whites in the North who don't care about slavery. It's not in their.. - [Voiceover] They're indifferent, what do they care. - [Kim] Yeah, what do they care, in fact some people are afraid that if the slaves are freed in the South, they're all gonna come up North and they're going to compete for labor with poor white people. So there are plenty of whites in the North who have no interest in the slaves in the South being free, even if that's not an early war aim of the North. So the South is hoping that maybe they can win a couple of really big battles that show this isn't gonna be a big war. - [Voiceover] Or it'd be so painful for the North to try to conquer the South, so to speak. - [Kim] And they're also trying to show that they're serious, to an international audience, particularly England, because the South is producing 3/4 of the world's supply of cotton at this point, and England is an industrial nation which is built in many cases around textile manufacturing. So they're hoping that if they show that they are serious about their own nationhood that they're going to win this war against the North that England will intercede on their behalf to protect their supply of cotton. - [Voiceover] So this would be an appeal to England on purely economic grounds. - [Kim] Right. Because England, they didn't have slavery. - [Kim] No. - [Voiceover] But purely economically, at least, appeal to them. - [Kim] So on the other hand, the North's strategy is what they call \"The Anaconda Plan\". And the idea of the Anaconda Plan is that they are going to squeeze the South, economically. What they want to do, - [Voiceover] Like an anaconda. - [Kim] Like an anaconda, right. So they want to blockade the Atlantic ocean because they don't want the South to be able to sell their cotton to get money, and they also don't want the South to be able to buy the kinds of things that they're going to need to make a war happen. They also want to control the Mississippi River cuz that's the real main artery of commerce in the West. Anyone who is gonna be shipping their grain or their cotton is gonna be shipping it down the Mississippi to the port of New Orleans. So the Union hopes that if they can basically surround the South, and make sure nothing gets in or out, then eventually they're just gonna starve to death. - [Voiceover] This also goes through the industrial bays, it can also produce more ships and etc. - [Kim] Right, and it takes them a while to do that, in fact at the start of the war, the Union only has 90 ships. I've heard it compared to \"Five leaky boats\". Right, we're not a naval power at this point and so it's gonna take them a while to build up the kind of naval power they need to do that, cuz this is 3500 miles of coastline that they're gonna need to patrol. - [Voiceover] I'm just looking at this map, not getting too much into details, it looks like a lot of the battles are concentrated right in this Virginia/Maryland area, and then there's more, it's a little bit more sparse but you have a few that are in the Deep South and along this Mississippi corridor. - [Kim] There are two major theaters of the war. We'd say the Eastern Theater, and this is that 100-mile corridor between Washington and Richmond, where a huge amount of the fighting takes place. It's important to remember that the capital of the Confederacy and the capital of the United States are only 100 miles apart. - [Voiceover] This capital is, you can't see it on this map but it's someplace in the middle of Virginia, and then D.C., literally, as you mentioned you said it was 100 miles apart? - [Kim] Yep. - [Voiceover] Fascinating." + }, + { + "Q": "So , if A goes 5km and then goes back, is tht mean the displacement is 0?", + "A": "yes because the forward movement is positive and the backward movement is negative and the same in length so adding them up gives you zero", + "video_name": "oRKxmXwLvUU", + "transcript": "Now that we know a little bit about vectors and scalars, let's try to apply what we know about them for some pretty common problems you'd, one, see in a physics class, but they're also common problems you'd see in everyday life, because you're trying to figure out how far you've gone, or how fast you're going, or how long it might take you to get some place. So first I have, if Shantanu was able to travel 5 kilometers north in 1 hour in his car, what was his average velocity? So one, let's just review a little bit about what we know about vectors and scalars. So they're giving us that he was able to travel 5 kilometers to the north. So they gave us a magnitude, that's the 5 kilometers. That's the size of how far he moved. And they also give a direction. So he moved a distance of 5 kilometers. Distance is the scalar. But if you give the direction too, you get the displacement. So this right here is a vector quantity. He was displaced 5 kilometers to the north. And he did it in 1 hour in his car. What was his average velocity? So velocity, and there's many ways that you might see it defined, but velocity, once again, is a vector quantity. And the way that we differentiate between vector and scalar quantities is we put little arrows on top of vector quantities. Normally they are bolded, if you can have a typeface, and they have an arrow on top of them. But this tells you that not only do I care about the value of this thing, or I care about the size of this thing, I also care about its direction. The arrow isn't necessarily its direction, it just tells you that it is a vector quantity. So the velocity of something is its change in position, including the direction of its change in position. So you could say its displacement, and the letter for displacement is S. And that is a vector quantity, so that is displacement. And you might be wondering, why don't they use D for displacement? That seems like a much more natural first letter. And my best sense of that is, once you start doing calculus, you start using D for something very different. You use it for the derivative operator, and that's so that the D's don't get confused. And that's why we use S for displacement. If someone has a better explanation of that, feel free to comment on this video, and then I'll add another video explaining that better explanation. So velocity is your displacement over time. If I wanted to write an analogous thing for the scalar quantities, I could write that speed, and I'll write out the word so we don't get confused with displacement. Or maybe I'll write \"rate.\" Rate is another way that sometimes people write speed. So this is the vector version, if you care about direction. If you don't care about direction, you would have your rate. So this is rate, or speed, is equal to the distance that you travel over some time. So these two, you could call them formulas, or you could call them definitions, although I would think that they're pretty intuitive for you. How fast something is going, you say, how far did it go over some period of time. These are essentially saying the same thing. This is when you care about direction, so you're dealing with vector quantities. This is where you're not so conscientious about direction. And so you use distance, which is scalar, and you use rate or speed, which is scalar. Here you use displacement, and you use velocity. Now with that out of the way, let's figure out what his average velocity was. And this key word, average, is interesting. Because it's possible that his velocity was changing over that whole time period. But for the sake of simplicity, we're going to assume that it was kind of a constant velocity. What we are calculating is going to be his average velocity. But don't worry about it, you can just assume that it wasn't changing over that time period. So his velocity is, his displacement was 5 kilometers to the north-- I'll write just a big capital. Well, let me just write it out, 5 kilometers north-- over the amount of time it took him. And let me make it clear. This is change in time. This is also a change in time. Sometimes you'll just see a t written there. Sometimes you'll see someone actually put this little triangle, the character delta, in front of it, which explicitly means \"change in.\" It looks like a very fancy mathematics when you see that, but a triangle in front of something literally means \"change in.\" So this is change in time. So he goes 5 kilometers north, and it took him 1 hour. So the change in time was 1 hour. So let me write that over here. So over 1 hour. So this is equal to, if you just look at the numerical part of it, it is 5/1-- let me just write it out, 5/1-- kilometers, and you can treat the units the same way you would treat the quantities in a fraction. 5/1 kilometers per hour, and then to the north. Or you could say this is the same thing as 5 kilometers per hour north. So this is 5 kilometers per hour to the north. So that's his average velocity, 5 kilometers per hour. And you have to be careful, you have to say \"to the north\" if you want velocity. If someone just said \"5 kilometers per hour,\" they're giving you a speed, or rate, or a scalar quantity. You have to give the direction for it to be a vector quantity. You could do the same thing if someone just said, what was his average speed over that time? You could have said, well, his average speed, or his rate, would be the distance he travels. The distance, we don't care about the direction now, is 5 kilometers, and he does it in 1 hour. His change in time is 1 hour. So this is the same thing as 5 kilometers per hour. So once again, we're only giving the magnitude here. This is a scalar quantity. If you want the vector, you have to do the north as well. Now, you might be saying, hey, in the previous video, we talked about things in terms of meters per second. Here, I give you kilometers, or \"kil-om-eters,\" depending on how you want to pronounce it, kilometers per hour. What if someone wanted it in meters per second, or what if I just wanted to understand how many meters he travels in a second? And there, it just becomes a unit conversion problem. And I figure it doesn't hurt to work on that right now. So if we wanted to do this to meters per second, how would we do it? Well, the first step is to think about how many meters we are traveling in an hour. So let's take that 5 kilometers per hour, and we want to convert it to meters. So I put meters in the numerator, and I put kilometers in the denominator. And the reason why I do that is because the kilometers are going to cancel out with the kilometers. And how many meters are there per kilometer? Well, there's 1,000 meters for every 1 kilometer. And I set this up right here so that the kilometers cancel out. So these two characters cancel out. And if you multiply, you get 5,000. So you have 5 times 1,000. So let me write this-- I'll do it in the same color-- 5 times 1,000. So I just multiplied the numbers. When you multiply something, you can switch around the order. Multiplication is commutative-- I always have trouble pronouncing that-- and associative. And then in the units, in the numerator, you have meters, and in the denominator, you have hours. Meters per hour. And so this is equal to 5,000 meters per hour. And you might say, hey, Sal, I know that 5 kilometers is the same thing as 5,000 meters. I could do that in my head. And you probably could. But this canceling out dimensions, or what's often called dimensional analysis, can get useful once you start doing really, really complicated things with less intuitive units than something But you should always do an intuitive gut check right here. You know that if you do 5 kilometers in an hour, that's a ton of meters. So you should get a larger number if you're talking about meters per hour. And now when we want to go to seconds, let's do an intuitive gut check. If something is traveling a certain amount in an hour, it should travel a much smaller amount in a second, or 1/3,600 of an hour, because that's how many seconds there are in an hour. So that's your gut check. We should get a smaller number than this when we want to say meters per second. But let's actually do it with the dimensional analysis. So we want to cancel out the hours, and we want to be left with seconds in the denominator. So the best way to cancel this hours in the denominator is by having hours in the numerator. So you have hours per second. So how many hours are there per second? Or another way to think about it, 1 hour, think about the larger unit, 1 hour is how many seconds? Well, you have 60 seconds per minute times 60 minutes per hour. The minutes cancel out. 60 times 60 is 3,600 seconds per hour. So you could say this is 3,600 seconds for every 1 hour, or if you flip them, you would get 1/3,600 hour per second, or hours per second, depending on how you want to do it. So 1 hour is the same thing as 3,600 seconds. And so now this hour cancels out with that hour, and then you multiply, or appropriately divide, the numbers right here. And you get this is equal to 5,000 over 3,600 meters per-- all you have left in the denominator here is second. Meters per second. And if we divide both the numerator and the denominator-- I could do this by hand, but just because this video's already getting a little bit long, let me get my trusty calculator out. I get my trusty calculator out just for the sake of time. 5,000 divided by 3,600, which would be really the same thing as 50 divided by 36, that is 1.3-- I'll just round it over here-- 1.39. So this is equal to 1.39 meters per second. So Shantanu was traveling quite slow in his car. Well, we knew that just by looking at this. 5 kilometers per hour, that's pretty much just letting the car roll pretty slowly." + }, + { + "Q": "think what can go into that # so break it up so we have 2x256 then 2x 128 then 2x64 then 2x32 then 2x16 then 2x8 then 2x4 then 2x2. he could just divide by 4 or 8 or 16 or 32 but he just adds more work but all those things make up 512", + "A": "2 comments in response to your observation. 1. It is easier to divide by 2 in your head. 2. The goal is to get to prime factors so any method that ends up with 2^9 will be useful. You could factor like this. 512 = 16*32 = 4*4 * 8*4 = 2^2*2^2 * 2^3*2^2 = 2^9 It s less steps but the steps took me longer.", + "video_name": "DKh16Th8x6o", + "transcript": "We are asked to find the cube root of negative 512. Or another way to think about it is if I have some number, and it is equal to the cube root of negative 512, this just means that if I take that number and I raise it to the third power, then I get negative 512. And if it doesn't jump out at you immediately what this is the cube of, or what we have to raise to the third power to get negative 512, the best thing to do is to just do a prime factorization of it. And before we do a prime factorization of it to see which of these factors show up at least three times, let's at least think about the negative part a little bit. So negative 512, that's the same thing-- so let me rewrite the expression-- this is the same thing as the cube root of negative 1 times 512, which is the same thing as the cube root of negative 1 times the cube root of 512. And this one's pretty straightforward to answer. What number, when I raise it to the third power, do I get negative 1? Well, I get negative 1. This right here is negative 1. Negative 1 to the third power is equal to negative 1 times negative 1 times negative 1, which is equal to negative 1. So the cube root of negative 1 is negative 1. So it becomes negative 1 times this business right here, times the cube root of 512. And let's think what this might be. So let's do the prime factorization. So 512 is 2 times 256. 256 is 2 times 128. 128 is 2 times 64. We already see a 2 three times. 64 is 2 times 32. 32 is 2 times 16. We're getting a lot of twos here. 16 is 2 times 8. 8 is 2 times 4. And 4 is 2 times 2. So we got a lot of twos. So essentially, if you multiply 2 one, two, three, four, five, six, seven, eight, nine times, you're going to get 512, or 2 to the ninth power is 512. And that by itself should give you a clue of what the cube root is. But another way to think about it is, can we find-- there's definitely three twos here. But can we find three groups of twos, or we could also find-- let me look at it this way. We can find three groups of two twos over here. So that's 2 times 2 is 4. 2 times 2 is 4. So definitely 4 multiplied by itself three times is divisible into this. But even better, it looks like we can get three groups of three twos. So one group, two groups, and three groups. So each of these groups, 2 times 2 times 2, that's 8. That is 8. This is 2 times 2 times 2. That's 8. And this is also 2 times 2 times 2. So that's 8. So we could write 512 as being equal to 8 times 8 times 8. And so we can rewrite this expression right over here as the cube root of 8 times 8 times 8. So this is equal to negative 1, or I could just put a negative sign here, negative 1 times the cube root of 8 times 8 times 8. So we're asking our question. What number can we multiply by itself three times, or to the third power, to get 512, which is the same thing as 8 times 8 times 8? Well, clearly this is 8. So the answer, this part right over here, is just going to simplify to 8. And so our answer to this, the cube root of negative 512, is negative 8. And we are done. And you could verify this. Multiply negative 8 times itself three times. Negative 8 times negative 8 times negative 8. Negative 8 times negative 8 is positive 64. You multiply that times negative 8, you get negative 512." + }, + { + "Q": "At 2:25 Sal says that anatomically modern humans have been on earth for about 200,000 years. Then what kinds of characteristics of human anatomy count as the characteristics of modern humans?", + "A": "Mainly it means a smaller jaw, bigger brain, walking upright almost all of the time, Ability to use hands more like modern day hands instead of really weird feet.", + "video_name": "13E90TAtZ30", + "transcript": "LeBron: If the history of the earth was a basketball game, at what point in the game will the humans show up? Voiceover: Let's first think about how long a basketball game is in the NBA. We have 4 quarters that each lasts 12 minutes, so we're talking about 48 minutes. 48 minutes of regulation play. I'm not considering half time and the time outs and the commercial breaks and potential overtime. I am just taking about regulation play. We could think about how the numbers might change if you think about the total duration of the game, including time outs and commercial breaks and everything that might get you closer to 2 or 2 1/2 hours. But 48 minutes we can actually convert to seconds because we know that there are 60 seconds per minute, times 60 seconds, 60 seconds per minute and this is pretty straight forward multiplication. We can just say 48 times 60 gives us, so we got this 0 here and then 6 times 48, 6 times 8 is 48 and then 6 times 4 is 24 plus 4 is 28. So there's 2,880 seconds during regulation play. Now let's think about the actual age of earth. We estimate that the earth, and actually the entire solar system, which was all formed roughly at the same time, is 4.54 billion years old. So let me draw that here. so I am going to draw it as the same length. So 4.54 billion years old and just to give a sense of how large of a number that is, a billion is a 1,000 million. So we could also write this as 4,540 million years. or we could write it as 4,540,000 thousand years, or millennia, or we could just write out the number as 4,540,000,000 years which seems kind of old but let's think about how long anatomically, modern humans have been roaming the surface of the earth and here we estimate that anatomically modern humans have been on the surface of the earth for about 200,000 years, which seems like a reasonable amount of time but we'll see it's a very small fraction when you compare it to 4.54 billion years. So let's say that that's that little there and I am actually overdoing it when I'm drawing the diagram. So that right over there is the amount of time humans have been 200,000 years and I am actually drawing this way too big. But what we want to figure out is what is that equivalent length in seconds on a basketball game or another way to think about it, 200,000 years is to 4.54 billion years as the number of seconds. Let's call this thing right over here x as x is to the total number of seconds in a game. So 2,880 seconds, 2,880 seconds just like that. And one way that we can do this, to solve for x and this is kind of a more basic algebra, but just as a reminder if we want to solve for x here, the easiest way is to multiply by 2,880 and that will cancel with this right over here. but we can't just do it to the right hand side, we also have to do it to the left hand side and so if you multiply both sides by 2,880, so multiply both sides by 2,880, you get that x, the number of the equivalent number of seconds in a basketball game. If the history of the earth was a basketball game, when the humans would show up is equal to the fraction of earth's history that humans have been around. That's this part right over here, times the number of seconds in a game. So let's think about what we get there. We are going to have the fraction of earth's history, so 200,000 divided by 4.45 billion. There's a couple of ways I could write it. I could write it 4.54e9, which literally means 4.54 times 10 to the 9th or 4.54 times 1 followed with 9 zeros or 4.54 times a billion, which is exactly 4.54 billion. So I could write it like that or I could just write it out. I could write 4, 5 so 4 billion, we'll be careful, 4, 540,000,000 and now I am doing the thousands 2, 3 and now I am just doing the 1, 2, 3. So this is 4 billion and then I should have 9 places after that 1,2,3,4,5,6,7,8,9. So this expression right here is a fraction of earth's history that anatomically modern humans have been around and then I am going to multiply that, I am going to multiply that, times I am going to multiply that times the number of seconds of regulation play, 2,880 and now drum roll, we get .12, we could round up .13 seconds. So we get x is equal to, or I could say maybe approximately equal to 0.13 seconds and so, just to imagine if the history of earth were regulation play of an NBA game and let's say this game is kind of, at the end of this game there is a buzzer beater shot that wins the game, the humans don't show up in the game until the ball has left that final shot taker's hands and it has just about to enter the basket that's when the first humans, 200,000 years ago will show up. A little over a tenth of a second before the end of the game and since we already have our brains in this mindset, I will throw out another interesting question. Okay, humans are you know, just a flash that they've actually shown up. Just as the ball is about to go into the basket, we have a little over a tenth of a second left in the game. Think about when the dinosaurs went extinct. I will give you a hint here. The dinosaurs went extinct. So we believe an asteroid hit the earth. So, this kind of a meteorite. This is a current theory, meteorite hit the earth, mass extinction event 65.5 million years ago. So if we think of that 65.5 million years ago relative to the history of the earth and if the history of the earth were just regulation play of a basketball game, when did that happen? Well, once again let's get our calculator out. What fraction of the earth's history ago was this? 65.5 million, I could write this as 65.5 times 10 to the 6th, which is that and then I'll just divide that by 4.54 billion years. So times 10 to the 9th and so this is the fraction of earth's history that has happened since the extinction of the dinosaurs. So a little more, about a percent and a half of earth's history has happened since the extinction of the dinosaurs 65.5 million years ago and if we are talking about a basketball game, let's just multiply that times the number of seconds in a basketball game, 2,880 and we get about 41 seconds. So with less than a minute left in the 4th quarter is when the meteorite hits the earth" + }, + { + "Q": "32 plus 32 equals 64", + "A": "Yes because you solve addition problems on top of each other from right to left: 2+2=4, there s your last number. 3+3=6, there s your first. They collide to make 64 so that is true, 32+32=64", + "video_name": "nFsQA2Zvy1o", + "transcript": "Most of us are familiar with the equal sign from our earliest days of arithmetic. You might see something like 1 plus 1 is equal to 2. Now, a lot of people might think when they see something like this that somehow equal means give me the answer. 1 plus 1 is the problem. Equal means give me the answer and 1 plus 1 is 2. That's not what equal actually means. Equal is actually just trying to compare two quantities. When I write 1 plus 1 equals 2, that literally means that what I have on the left hand side of the equal sign is the exact same quantity as what I have on the right hand side of the equal sign. I could have just as easily have written 2 is equal to 1 plus 1. These two things are equal. I could have written 2 is equal to 2. This is a completely true statement. These two things are equal. I could have written 1 plus 1 is equal to 1 plus 1. I could have written 1 plus 1 minus 1 is equal to 3 minus 2. These are both equal quantities. What I have here on the left hand side, this is 1 plus 1 minus 1 is 1 and this right over here is 1. These are both equal quantities. Now I will introduce you to other ways of comparing numbers. The equal sign is when I have the exact same quantity on both sides. Now we'll think about what we can do when we have different quantities on both sides. So let's say I have the number 3 and I have the number 1 and I want to compare them. So clearly 3 and 1 are not equal. In fact, I could make that statement with a not equal sign. So I could say 3 does not equal 1. But let's say I want to figure out which one is a larger and which one is smaller. So if I want to have some symbol where I can compare them, where I can tell, where I can state which of these is larger. And the symbol for doing that is the greater than symbol. This literally would be read as 3 is greater than 1. 3 is a larger quantity. And if you have trouble remembering what this means-- greater than-- the larger quantity is on the opening. I guess if you could view this as some type of an arrow, or some type of symbol, but this is the bigger side. Here, you have this little teeny, tiny point and here you have the big side, so the larger quantity is on the big side. This would literally be read as 3 is greater than-- so let me write that down-- greater than, 3 is greater than 1. And once again, it just doesn't have to be numbers like this. I could write an expression. I could write 1 plus 1 plus 1 is greater than, let's say, well, just one 1 right over there. This is making a comparison. But what if we had things the other way around. What if I wanted to make a comparison between 5 and, let's say, 19. So now the greater than symbol wouldn't apply. It's not true that 5 is greater than 19. I could say that 5 is not equal to 19. So I could still make this statement. But what if I wanted to make a statement about which one is larger and which one is smaller? Well, as in plain English, I would want to say 5 is less than 19. So I would want to say-- let me write that down-- I want to write 5 is less than 19. That's what I want to say. And so we just have to think of a mathematical notation for writing \"is less than.\" Well, if this is greater than, it makes complete sense that let's just swap it around. Let's make, once again, the point point towards the smaller quantity and the big side of the symbol point to the larger quantity. So here 5 is a smaller quantity so I'll make the point point there. And 19 is a larger quantity, so I'll make it open like this. And so this would be read as 5 is less than 19. 5 is a smaller quantity than 19. I could also write this as 1 plus 1 is less than 1 plus 1 plus 1. It's just saying that this statement, this quantity, 1 plus 1 is less than 1 plus 1 plus 1." + }, + { + "Q": "Does anybody have an idea of the approximate cost of the parts necessary for this project, assuming one doesn't just have any of them lying around but needs to buy everything new?", + "A": "Depends where you live, robot part are about twice as expensive in Australia then they are in America. You re probably looking at $40. (he might tell you in another video).", + "video_name": "VnfpSf6YxuU", + "transcript": "This is all our parts all laid out for the Bit-zee bot, the things that you'll need to make one. Now you can make yours out of a broad variety of things, and we highly recommend that you do that. The only thing that you really have to have is the Arduino. Everything else you can switch out for other things. You can use different types of batteries. You can use different motors, et cetera. I'm going to go through what I've got here and where the products came from, or where the parts came from, I should say. And then we're going to start to put a Bit-zee together on this board so you can see how it's all wired up. But if you don't happen to have two hair dryers that you can take apart, you can either go and buy two electric motors and get some wheels for them. Or there's a variety of things you can do to solve that problem. So again, these are two motors from our hair dryer. You can see the hair dryer blower fan there. And underneath that is a sheet of Lexan. It's a stiff plastic that's really resilient. And that sheet of Lexan, it's easy to machine. You can drill holes in it and do stuff like that. So it's going to be used for mounting some of our devices. And you can get that at a hardware store for a few dollars. And this is a universal remote, and it can be gotten at Target for around $8. And we're going to use that to control our Bit-zee bot. And then we've got some electrical tape and different 22-gauge wire. And then we have some solder here. We'll use that to make our solder connections. Just like if you saw the video for the motor controller, it was used to solder that together. And this is a motor controller, which will allow us to control the speed and direction of our motors. And this is our Arduino. It's our microprocessor that we can plug into-- I should say a microcontroller that we can plug into our computer and download code to it to get the motor controller and other things to function the way we want them to. So this is a breadboard, and it's used for prototyping. And we're going to show you how to wire it up and how to connect different electronic components to that. And this is our digital recording module, and it's for basically recording sounds and playing them back. And we're going to use the Arduino to trigger that so that when the little bot drives around it can make some sounds. Of course, these are just double-A batteries, and they're going to go in this battery holder. The double-As are 1.5 volts. But when we connect them in series together, they're going to be 12 volts. So that'll be great for powering our motors, because they want to run on a higher voltage than 1.5. And so we have our different transistors here that we're going to use to do some switching in our circuits. And we've got some three-color LED and some screws and nuts and then a bunch of resistors. And these are 330 ohm, 10K ohm, 220 ohm. We'll go into the details on that kind of stuff later. And then I have some of the board of our alarm clock radio, so we're going to use some components off of that board. And we've got our coffee maker here. Or I should say our coffee carafe; it's just the holder for the coffee. And we're going to use some of the components on this for the Bit-zee. And then we have some-- over here you can see some zip ties. And so we also have an-- we're going to need a infrared sensor for the Bit-zee. And that infrared sensor will be used for sensing from the remote control. And so it looks like this, and we'll have one of those as well." + }, + { + "Q": "How do you convert a fractional percent into a percent", + "A": "convert the fraction to decimal and multiply with 100", + "video_name": "FaDtge_vkbg", + "transcript": "Let's give ourselves a little bit of practice with percentages. So let's ask ourselves, what percent of-- I don't know, let's say what percent of 16 is 4? And I encourage you to pause this video and to try it out yourself. So when you're saying what percent of 16 is 4, percent is another way of saying, what fraction of 16 is 4? And we just need to write it as a percent, as per 100. So if you said what fraction of 16 is 4, you would say, well, look, this is the same thing as 4/16, which is the same thing as 1/4. But this is saying what fraction 4 is of 16. You'd say, well, 4 is 1/4 of 16. But that still doesn't answer our question. What percent? So in order to write this as a percent, we literally have to write it as something over 100. Percent literally means \"per cent.\" The word \"cent\" you know from cents and century. It relates to the number 100. So it's per 100. So you could say, well, this is going to be equal to question mark over 100, the part of 100. And there's a bunch of ways that you could think about this. You could say, well, look, if in the denominator to go from 4 to 100, I have to multiply by 25. In the numerator to go from-- I need to also multiply by 25 in order to have an equivalent fraction. So I'm also going to multiply by 25. So 1/4 is the same thing as 25/100. And another way of saying 25/100 is this is 25 per 100, or 25%. So this is equal to 25%. Now, there's a couple of other ways you could have thought about it. You could have said well, 4/16, this is literally 4 divided by 16. Well, let me just do the division and convert to a decimal, which is very easy to convert to a percentage. So let's try to actually do this division right over here. So we're going to literally divide 4 by 16. Now, 16 goes into 4 zero times. 0 times 16 is 0. You subtract, and you get a 4. And we're not satisfied just having this remainder. We want to keep adding zeroes to get a decimal answer right over here. So let's put a decimal right over here. We're going into the tenths place. And let's throw some zeroes right over here. The decimal makes sure we keep track of the fact that we are now in the tenths, and in the hundredths, and in the thousandths place if we have to go that far. But let's bring another 0 down. 16 goes into 40 two times. 2 times 16 is 32. If you subtract, you get 8. And you could bring down another 0. And we have 16 goes into 80. Let's see, 16 goes into 80 five times. 5 times 16 is 80. You subtract, you have no remainder, and you're done. 4/16 is the same thing as 0.25. Now, 0.25 is the same thing as twenty-five hundredths. Or, this is the same thing as 25/100, which is the same thing as 25%." + }, + { + "Q": "You said that the diastole occurs between the dub and lub but isnt the dub sound or the 2nd heart sound supposed to be diastole?", + "A": "The dub and lub sounds are actually the sounds created by the valves closing; the first heart sound (S1) are the atrioventricular (AV) valves closing, and the second heart sounds (S2) are the semilunar valves closing. So diastole (the heart filling with blood) occurs when the AC valves are open and the semilunar are closed, and vice versa for systole.", + "video_name": "-4kGMI-qQ3I", + "transcript": "If you take a good long listen to your heart, you'll actually notice that it makes sounds. And those sounds are usually described as lub dub, lub dub, lub dub. And if you actually try to figure out what that would spell out like, usually it's something L- U- B, D- U- B. And it just repeats over and over and over. And to sort of figure out where those sounds come from, what I did is I took that diagram of the heart that we've been using and actually exaggerated the valves, made them really, really clear to see in this picture. And we'll use those valves to kind of talk through where those sounds are coming from. So let's start by labeling our heart. So we've got at the top, blood is coming into the right atrium and going to the right ventricle. It goes off to the lungs, comes back into the left atrium and then the left ventricle. So these are the chambers of our heart. Now, keep your eye on the valves. And we'll actually talk about them as the blood moves through. So let's start with blood going from the right atrium this way into the right ventricle. Now, at the same moment that blood is actually going from the right atrium to the right ventricle, blood is actually also going from the left atrium to the left ventricle. Now, you might think, well, how's that possible? How can blood be in two places at one time? But now remember that blood is constantly moving through the heart. So in a previous cycle, you actually had some blood that was coming back from the lungs, and that's what's dumping into the left ventricle. And in a new cycle, you have a bit of blood that's going from the right atrium to the right ventricle. So you have simultaneously two chambers that are full of blood-- the right and left ventricle. Now, to get the blood into those ventricles, the valves had to open. And specifically, let's label all the valves now. So here you have our tricuspid valve, and I'm going to label that as just a T. And then up here, you have the pulmonary valve, and this'll be just a P. And on the other side, you've got the mitral valve, which separates the left atrium from the left ventricle. And you've got the aortic valve. So these are the four valves of the heart. And as the blood is now in the ventricles, you can see that the tricuspid and the mitral valve are open. So far, so good. Now, I've actually drawn the pulmonary valve as being open. But is that really the case? And the answer is no, because what happens is that as blood is moving down from the right atrium to the right ventricle, let's say that-- and I'm going to draw it in black. Black arrows represent the bad or the wrong direction of flow. So let's say some blood is actually trying to go that way, which is not the way it should be going. What happens is that these two valves, they, based on their shape, are actually not-- they're going to jam up. They're going to basically just jam up like this, and they're not going to let the blood pass through. So this is what happens as that valve closes down. And the same thing happens on this side. Let's imagine you have some backwards flow of blood by accident, meaning that it's going in the wrong direction. Well, then these valves are going to close down. So the white arrows represent the correct flow of blood, and the black arrows represent the incorrect flow of blood. So these valves shut down like that. So now you can see how the valves, the aortic and pulmonary valve, are actually closed when the mitral and tricuspid valve are open. So what happens after this? So now our ventricles are full of blood, right? They're full of blood. And let's say they squeeze down, and they jettison all the blood into those arteries. Well, now you're going to have-- this is actually going to close down. Let's say this arrow flips around. These arrows become white, because the direction of flow is going to be in the direction we want it. It's going to go this way and this way And to allow that, of course, I need to show you that these open up. And they allow the blood to go the way that we want it to go, so now blood is going to flow through those two valves. But similar to before, you could have some backflow here. You could have backflow here. And you can have backflow here. So you can imagine now, let's say you have a little bit of backflow that wants to go this way, which is the wrong direction. Right? Well, then these valves are going to close up. They're going to say, no, you can't go that way. They're going to close right up, and they're going to not allow blood to go that way. So this is going to happen on both sides, both ventricles. And the valves shut. And so basically the backflow of blood is not allowed, because the valves keep shutting. And when the valves snap shut-- so for example, right now the tricuspid valve and the mitral valve snapped shut. Well, that makes a noise. So when T and M snap shut, that makes a noise that we call lub. That's that first noise, that first heart sound. In fact, sometimes people don't even call it lub dub. They say, well, it's the first heart sound. And to make that even shorter, sometimes people call that S1. So if you hear S1, you know they're talking about that same exact thing. And this dub is called the second heart sound. And, no surprise, just as before, if that's S1, this is S2. So you'll hear S1 when the tricuspid and mitral valve snap shut. So far, so good. But you also know that if that's what's making noise, you can kind of guess-- and it's a very smart guess-- that at the same time, the pulmonic valve and the aortic valve just opened. So if the other valves snap shut, these just opened. You can kind of assume that, although the noise you're hearing is actually from here. So what's happening with dub? Well, the opposite. And what I mean by that is-- let me now show you what happens a moment later. Well, after the ventricles are done squeezing, then we get to a point where you might have a little bit of flow that way and that way, just as I drew before. And these valves snap shut as well. So now these snap shut. And as these snap shut-- because they don't want to allow backflow, right? They're going to snap shut like that. They make noise. And so when you have dub, you actually have noise coming from the pulmonic and aortic valve snapping shut. And that must mean that then the other two valves just opened up-- the tricuspid and mitral just opened. You can assume that, right? And I didn't draw that in the picture. Let me update my picture now to show that. So now these two have opened up, and blood is coming into the ventricles again. So it's actually a nice little rhythm that you get going. And every time these valves go open and shut, you hear noise. So you can kind of figure out what's happening based on-- and these actually-- let me erase that. And now you have white arrows going this way. And we've returned to where we started from. So you basically have a full cycle, and between these two-- so let's say from lub to dub, because there's a little bit of space there. If you were to follow it over time, over time, this is what it might look like if this is a little timeline. You might hear lub here, or the first heart sound. I'll just call it S1. And you might hear S2 here, the second heart sound. And then you'll hear S1 again over here and S2. And what's happening between the two-- so between these two, this time lag here-- is that blood is actually squeezing out, because the pulmonary and aortic valves just opened. It is squeezing out and going out to the whole body. So this is when blood is going to the body, and sometimes we call that systole. And between dub and the next lub-- so in this area right here-- well, at that point, blood is kind of refilling from the atriums into the ventricles, and we call that diastole. So now you can actually listen to your heart. And you can actually figure out, well, if you're listening to the sound between lub and dub or the space in time between lub and dub, that's when you're having systole. And if you're listening to or waiting for the sound to start up again-- so you just heard dub, and you're waiting for lub again-- then that space in time is diastole." + }, + { + "Q": "can someone explain how 5 /5\u00e2\u0088\u009a61/61 = \u00e2\u0088\u009a61? I don't understand how the bottom 61 cancels out. Thanks : )", + "A": "What is 61 divided by square root of 61?", + "video_name": "l5VbdqRjTXc", + "transcript": "We're asked to solve the right triangle shown below. Give the links to the nearest tenth. So when they say solve the right triangle, we can assume that they're saying, hey figure out the lengths of all the sides. So whatever a is equal to, whatever b is equal to. And also what are all the angles of the right triangle? They've given two of them. We might have to figure out this third right over here. So there's multiple ways to tackle this, but we'll just try to tackle side XW first, try to figure out what a is. And I'll give you a hint. You can use a calculator, and using a calculator, you can use your trigonometric functions that we've looked at a good bit now. So I'll give you a few seconds to think about how to figure out what a is. Well, what do we know? We know this angle y right over here. We know the side adjacent to angle y. And length a, this is the side that's the length of the side that is opposite to angle Y. So what trigonometric ratio deals with the opposite and the adjacent? So if we're looking at angle Y, relative to angle Y, this is the opposite. And this right over here is the adjacent. Well if we don't remember, we can go back to SohCahToa. Sine deals with opposite and hypotenuse. Cosine deals with adjacent and hypotenuse. Tangent deals with opposite over adjacent. So we can say that the tangent of 65 degrees, of that angle of 65 degrees, is equal to the opposite, the length of the opposite side, which we know has length a over the length of the adjacent side, which they gave us in the diagram, which has length five. And you might say, how do I figure out a? Well we can use our calculator to evaluate what the tangent of 65 degrees are. And then we can solve for a. And actually if we just want to get the expression explicitly solving for a, we could just multiply both sides of this equation times 5. So let's do that. 5 times, times 5. These cancel out, and we are left with, if we flip the equal around, we're left with a is equal to 5 times the tangent of 65 degrees. So now we can get our calculator out and figure out what this is to the nearest tenth. That's my handy TI-85 out and I have 5 times the tangent-- I didn't need to press that second right over there, just a regular tangent-- of 65 degrees. And I will get, if I round to the nearest tenth like they ask me to, I get 10.7. So a is approximately equal to 10.7. I say approximately because I rounded it down. This is not the exact number. But a is equal to 10.7. So we now know that this has length 10.7, approximately. There are several ways that we can try to tackle b. And I'll let you pick the way you want to. But then I'll just do it the way I would like to. So my next question to you is, what is the length of the side YW? Or what is the value of b? Well there are several ways to do it. This is the hypotenuse. So we could use trigonometric functions that deal with adjacent over hypotenuse or opposite over hypotenuse. Or we could just use the Pythagorean theorem. We know two sides of a right triangle. We can come up with the third side. I will go with using trigonometric ratios since that's what we've been working on a good bit. So this length of b, that's the length of the hypotenuse. So this side WY is the hypotenuse. And so what trigonometric ratios-- or we can decide what we want to use. We could use opposite and hypotenuse. We could use adjacent and hypotenuse. Since we know that XY is exactly 5 and we don't have to deal with this approximation, let's use that side. So what trigonometric ratios deal with adjacent and hypotenuse? Well we see from SohCahToa cosine deals with adjacent over hypotenuse. So we could say that the cosine of 65 degrees is equal to the length of the adjacent side, which is 5 over the length of the hypotenuse, which has a length of b. And then we can try to solve for b. You multiply both sides times b, you're left with b times cosine of 65 degrees is equal to 5. And then to solve for b, you could divide both sides by cosine of 65 degrees. This is just a number here. So we're just dividing-- we have to figure it out what our calculator, but this is just going to evaluate to some number. So we can divide both sides by that, by cosine of 65 degrees. And we're left with b is equal to 5 over the cosine of 65 degrees. So let us now use our calculator to figure out the length of b. Length of b is 5 divided by cosine of 65 degrees. And I get, if I round to the nearest tenth, 11.8. So b is approximately equal to, rounded to the nearest tenth, 11.8. So b is equal to 11.8. And then we're almost done solving this right triangle. And you could have figured this out using the Pythagorean theorem as well, saying that 5 squared plus 10.7 squared should be equal to b squared. And hopefully you would get the exact same answer. And the last thing we have to figure out is the measure of angle W right over here. So I'll give you a few seconds to think about what the measure of angle W is. Well here we just have to remember that the sum of the angles of a triangle add up to 180 degrees. So angle w plus 65 degrees, that's this angle right up here, plus the right angle, this is a right triangle, they're going to add up to 180 degrees. So all we need to do is-- well we can simplify the left-hand side right over here. 65 plus 90 is 155. So angle W plus 155 degrees is equal to 180 degrees. And then we get angle W-- if we subtract 155 from both sides-- angle W is equal to 25 degrees. And we are done solving the right triangle shown below." + }, + { + "Q": "what would happen if point a is underneath the x-axis?", + "A": "You would follow the same procedure.", + "video_name": "0rlNHYHhrWs", + "transcript": "Find the point B on segment AC, such that the ratio of AB to BC is 3 to 1. And I encourage you to pause this video and try this on your own. So let's think about what they're asking. So if that's point C-- I'm just going to redraw this line segment just to conceptualize what they're asking for. And that's point A. They're asking us to find some point B that the distance between C and B, so that's this distance right over here. So if this distance is x, then the distance between B and A is going to be 3 times that. So this will be 3x. That the ratio of AB to BC is 3 to 1. So that would be the ratio-- let me write this down. It would be AB-- that looks like an HB-- it would be AB to BC is going to be equal to 3x to x, which is the same thing as 3 to 1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted to say, oh, well, you could use the distance formula to find the distance, which by itself isn't completely uncomplicated. And then this will be 1/4 of the way. Because if you think about it, this entire distance is going to be 4x. Let me draw that a little bit neater. This entire distance, if you have an x plus a 3x, is going to be 4x. So you'd say, well, this is 1 out of the 4 x's along the way. This is going to be 1/4 of the distance between the two points. Let me write that down. This is 1/4 of the way between C and B, going from C to A. B is going to be 1/4 of the way. So maybe you try to find the distance. And you say, well, what are all the points that are 1/4 of the way? But it has to be 1/4 of that distance away. But then it has to be on that line. But that makes it complicated, because this line is at an incline. It's not just horizontal. It's not just vertical. What we can do, however, is break this problem down into the vertical change between A and C, and the horizontal change between A and C. So for example, the horizontal change between A and C, A is at 9 right over here, and C is at negative 7. So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. And you see that here. 9 plus 7, this total distance is 16. That's the horizontal distance change going from A to C, or going from C to A. And the vertical change, and you could even just count that, that's going to be 4. C is at 1. A is at 5. Going from 1 to 5, you've changed vertically 4. So what we can say, going from C to B in each direction, in the vertical direction and the horizontal direction, we're going to go 1/4 of the way. So if we go 1/4 in the vertical direction, we're going to end up at y is equal to 2. So I'm just going, starting at C, 1/4 of the way. 1/4 of 4 is 1. So I've just moved up 1. So our y is going to be equal to 2. And if we go 1/4 in the horizontal direction, 1/4 of 16 So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that point right over there. We end up at this point. This is the point negative 3 comma 2. And if you were really careful with your drawing, you could have actually just drawn-- well, actually you don't have to be that careful, since this is graph paper. You actually could have just said, hey, we're going to go 1/4 this way. Where does that intersect the line? Hey, it intersects the line right over there. Or you could have said, we're going to go 1/4 this way. Where does that intersect the line? And that would have let you figure it out either way. So this point right over here is B. It is 1/4 of the way between C and A. Or another way of thinking about the distance between C and B, which we haven't even figured out. We could do that using the distance formula or the Pythagorean theorem, which it really is. This distance, the distance CB, is 1/3 the distance BA. The ratio of AB to BC is 3 to 1." + }, + { + "Q": "what if the numerator is larger than the denominator how would you solve it?", + "A": "Same thing just it would be 1.", + "video_name": "NM8qTo361ic", + "transcript": "Let's see if we can express 16/21 as a decimal. Or we could call this 16 twenty-firsts. This is also 16 divided by 21. So we can literally just divide 21 into 16. And because 21 is larger than 16, we're going to get something less than 1. So let's just literally divide 21 into 16. And we're going to have something less than 1. So let's add some decimal places here. We're going to round to the nearest thousandths in case our digits keep going on, and on, and on. And let's start dividing. 21 goes into 1 zero times. 21 goes into 16 zero times. 21 goes into 160-- well, 20 would go into 160 eight times. So let's try 7. Let's see if 7 is the right thing. So 7 times 1 is 7. 7 times 2 is 14. And then when we subtract it, we should get a remainder less than 21. If we pick the largest number here where, if I multiply it by 21, I get close to 160 without going over. And so if we subtract, we do get 13. So that worked. 13 is less than 21. And you could just subtract it. I did it in my head right there. But you could regroup. You could say this is a 10. And then this would be a 5. 10 minus 7 is 3. 5 minus 4 is 1. 1 minus 1 is 0. Now let's bring down a 0. 21 goes into 130. Would 6 work? It looks like 6 would work. 6 times 21 is 126. So that looks like it works. So let's put a 6 there. 6 times 1 is 6. 6 times 2 is 120. There's a little bit of an art to this. All right, now let's subtract. And once again, we can regroup. This would be a 10. We've taken 10 from essentially this 30. So now this becomes a 2. 10 minus 6 is 4. 2 minus 2 is 0. 1 minus 1 is 0. Now let's bring down another 0. 21 goes into 40, well, almost two times, but not quite, so only one time. 1 times 21 is 21. And now let's subtract. This is a 10. This becomes a 3. 10 minus 1 is 9. 3 minus 2 is 1. And we're going have to get this digit. Because we want to round to the nearest thousandth. So if this is 5 or over, we're going round up. If this is less than 5, we're going to round down. So let's bring another 0 down here. And 21 goes into 190. Let's see, I think 9 will work. 9 times 1 is 9. 9 times 2 is 18. When you subtract, 190 minus 189 is 1. And we could keep going on, and on, and on. But we already have enough digits to round to the nearest thousandth. This digit right over here is greater than or equal to 5. So we will round up in the thousandths place. So if we round to the nearest thousandths, we can say that this is 0.76. And then this is where we're going around up-- 762." + }, + { + "Q": "is there a formula to con vert the remainder of a problem into a fraction in an easy way?", + "A": "take the remainder and put it over the divisor in a fraction then simplify into lowest terms. if there is a remainder or 2 and a divisor of 10 the fraction is 2/10 with both numbers reduced to 1/5", + "video_name": "NcADzGz3bSI", + "transcript": "It never hurts to get a lot of practice, so in this video I'm just going to do a bunch more of essentially, what we call long division problems. And so if you have 4 goes into 2,292. And I don't know exactly why they call it long division, and we saw this in the last video a little bit. I didn't call it long division then, but I think the reason why is it takes you a long time or it takes a long piece of your paper. As you go along, you kind of have this thing, this long tail that develops on the problem. So all of those are, at least, reasons in my head why it's called long division. But we saw in the last video there's a way to tackle any division problem while just knowing your multiplication tables up to maybe 10 times 10 or 12 times 12. But just as a bit of review, this is the same thing as 2,292 divided by 4. And it's actually the same thing, and you probably haven't seen this notation before, as 2,292 divided by 4. This, this, and this are all equivalent statements on some level. And you could say, hey Sal, that looks like a fraction in case you have seen fractions already. And that is exactly what it is. It is a fraction. But anyway, I'll just focus on this format and in future videos we'll think about other ways to represent division. So let's do this problem. So 4 goes into 2 how many times? It goes into 2 no times, so let's move on to-- let me just switch colors. So let's move on to the 22. 4 goes into 22 how many times? 4 times 5 is equal to 20. 4 times 6 is equal to 24. So 6 is too much. So 4 goes into 22 five times. 5 times 4 is 20. There's going to be a little bit of a leftover. And then we subtract 22 minus 20. Well that's just 2. And then you bring down this 9. And you saw in the last video exactly what this means. When you wrote this 5 up here-- notice we wrote So this is really a 500. But in this video I'm just going to focus more on the process, and you can think more about what it actually means in terms of where I'm writing the numbers. But I think the process is going to be crystal clear hopefully, by the end of this video. So we brought down the 9. 4 goes into 29 how many times? It goes into at least six times. What's 4 times 7? 4 times 7 is 28. So it goes into it at least seven times. What's 4 times 8? 4 times 8 is 32, so it can't go into it eight times so it's going to go into it seven. 4 goes into 29 nine seven times. 7 times 4 is 28. 29 minus 28 to get our remainder for this step in the problem is 1. And now we're going to bring down this 2. We're going to bring it down and you get a 12. 4 goes into 12? That's easy. 4 times 3 is 12. 4 goes into 12 three times. 3 times 4 is 12. 12 minus 12 is 0. We have no remainder. So 4 goes into 2,292 exactly 573 times. So this 2,292 divided by 4 we can say is equal to 573. Or we could say that this thing right here is equal to 573. Let's do a couple of more. Let's do a few more problems. So I'll do that red color. Let's say we had 7 going into 6,475. Maybe it's called long division because you write it nice and long up here and you have this line. I don't know. There's multiple reasons why it could be called long division. So you say 7 goes into 6 zero times. So we need to keep moving forward. So then we go to 64. 7 goes into 64 how many times? Let's see. 7 times 7 is? Well, that's way too small. Let me think about it a little bit. Well 7 times 9 is 63. That's pretty close. And then 6 times 10 is going to be too big. 7 times 10 is 70. So that's too big. So 7 goes into 64 nine times. 9 times 7 is 63. 64 minus 63 to get our remainder of this stage 1. Bring down the 7. 7 goes into 17 how many times? Well, 7 times 2 is 14. And then 7 times 3 is 21. So 3 is too big. So 7 goes into 17 two times. 2 times 7 is 14. 17 minus 14 is 3. And now we bring down the 5. And 7 goes into 35? That's in our 7 multiplication tables, five times. 5 times 7 is 35. And there you go. So the remainder is zero. So all the examples I did so far had no remainders. Let's do one that maybe might have a remainder. And to ensure it has a remainder I'll just make up the problem. It's much easier to make problems that have remainders than the ones that don't have remainders. So let's say I want to divide 3 into-- I'm going to divide it into, let's say 1,735,092. This will be a nice, beastly problem. So if we can do this we can handle everything. So it's 1,735,092. That's what we're dividing 3 into. And actually, I'm not sure if this will have a remainder. In the future video I'll show you how to figure out whether something is divisible by 3. Actually, we can do it right now. We can just add up all these digits. 1 plus 7 is 8. 8 plus 3 is 11. 11 5 five is 16. 16 plus 9 is 25. 25 plus 2 is 27. So actually, this number is divisible by 3. So if you add up all of the digits, you get 27. And then you can add up those digits-- 2 plus 7 is 9. So that is divisible by 9. That's a trick that only works for 3. So this number actually is divisible by 3. So let me change it a little bit, so it's not divisible by 3. Let me make this into a 1. Now this number will not be divisible by 3. I definitely want a number where I'll end up with a remainder. Just so you see what it looks like. So let's do this one. 3 goes into 1 zero times. You could write a 0 here and multiply that out, but that just makes it a little bit messy in my head. So we just move one to the right. 3 goes into 17 how many times? Well, 3 times 5 is equal to 15. And 3 times 6 is equal to 18 and that's too big. So 3 goes into 17 right here five times. 5 times 3 is 15. And we subtract. 17 minus 15 is 2. And now we bring down this 3. 3 goes into 23 how many times? Well, 3 times 7 is equal to 21. And 3 times 8 is too big. That's equal to 24. So 3 goes into 23 seven times. 7 times 3 is 21. Then we subtract. 23 minus 21 is 2. Now we bring down the next number. We bring down the 5. I think you can appreciate why it's called long division now. We bring down this 5. 3 goes into 25 how many times? Well, 3 times 8 gets you pretty close and 3 times 9 is too big. So it goes into it eight times. 8 times 3 is 24. I'm going to run out of space. You subtract, you get 1. 25 minus 24 is 1. Now we can bring down this 0. And you get 3 goes into 10 how many times? It goes into it three times. 3 times 3 is 9. That's about as close to 10 as we can get. 3 times 3 is 9. 10 minus 9, I'm going to have to scroll up and down here a little bit. 10 minus 9 is 1, and then we can bring down the next number. I'm running out of colors. I can bring down that 9. 3 goes into 19 how many times? Well, 6 is about as close as we can get. That gets us to 18. 3 goes into 19 six times. 6 times 3-- let me scroll down. 6 times 3 is 18. 19 minus 18-- we subtract it up here too. 19 minus 18 is 1 and then we're almost done. I can revert back to the pink. We bring down this 1 right there. 3 goes into 11 how many times? Well, that's three times because 3 times 4 is too big. 3 times 4 is 12, so that's too big. So it goes into it three times. So 3 goes into 11 three times. 3 times 3 is 9. And then we subtract and we get a 2. And there's nothing left to bring down. When we look up here there's nothing left to bring down, so we're done. So we're left with the remainder of 2 after doing this entire problem. So the answer, 3 goes into 1,735,091-- it goes into it 578,363 remainder 2. And that remainder 2 was what we got all the way down there. So hopefully you now appreciate and you can tackle pretty much any division problem. And you also, through this exercise, can appreciate why it's called long division." + }, + { + "Q": "Did you notice that the numbers that have an odd number of factors all are perfect square numbers! I wonder if it has to do with the extra one factor...He explains that later in the video", + "A": "You mean the loss of one factor, since you can t count the square root of the number twice.", + "video_name": "WNhxkpmVQYw", + "transcript": "Let's say we have 100 light bulbs. Well, let me actually just draw them. So I have one light bulb there. I have another light bulb. And I have 100 of them. 100 light bulbs. And what I'm going to do is-- Well actually, before I even start turning these light bulbs on and off, let me let you know that they are all off. So they start off. Now, the next thing I'm going to do is I'm going to number the hundred light bulbs. I'm going to number them one through 100. So the first light bulb is light bulb one. The second light bulb is light bulb two. All the way to light bulb 100. And what I'm going to do is, first I'm going to go and I'm going to switch essentially every light bulb. So if they all start off, I'm going to turn them all on. So let me do it. So on my first pass-- let's call this pass one-- so in pass one, I'm going to turn all of these on. On, on, on. They're all going to be turned on. On. And then in pass two, what I'm going to do is I'm going to switch only every other light bulb. So, for example I'll say, OK, I won't switch light bulb one. I'll only switch light bulb two. So light bulb one will stay on. Light bulb two will be off. Light bulb three will be on. Light bulb four will be off. And so essentially, every light bulb-- if you look at their numbers-- that is a multiple of two, will be switched. So 100 will be switched, so that'll also be off. Then I'm going to come-- and ignore this right here-- and I'm going to switch every third light bulb. So what's going to happen? This one's going to-- let me switch colors arbitrarily-- this one's going to stay on. This one's going to stay off. And I'm going to switch the third light bulb. So this one was on. Now this one will be off. The fourth light bulb will stay off, because I'm not touching it. The fifth light bulb would have been on and it'll stay on. Now the sixth light bulb, in this case we switched it off, and now it'll be on again. But I think you get the point. Every third light bulb, or if we look at the numbers of the light bulb, every numbered light bulb would that is a multiple of three is going to be switched. And if it was a multiple of three and two, it would have been switched on the first time and then off the second time. But I think you're getting the point. But what I'm going to do is, I'm going to do 100 passes. So the first pass, I switch every light bulb. They all started off, so they're all going to be turned on. The second pass, I switch every other light bulb or every second light bulb. The third pass, I do every third one, or that's a multiple of three. And my question to you is, after 100 passes, how many light bulbs are still on? Or how many are on, period? And that is the brain teaser? How do you figure out, of the hundred, which ones are going to be on? You should be able to do this in your head. You don't have to make an Excel spreadsheet and actually do all the on and off switches. So the first question is, how many of these are going to be on after I do 100 passes? And just to make it clear, what's the 100th path going to be? Well, I'm only going to switch every 100th light bulb. So whatever this light bulb was already doing, I'm just going to switch it. If it was off, it'll come on. If it was on, it'll become off. So the first question is, how many of these are going to be on after 100 passes? And then the bonus question is, which of these are going to be on? And so that's the question. Pause it if you don't want the answer. And then try to solve it. I think it shouldn't take you too much time. But now I'm going to give you the answer. Or maybe I'll start with a couple of hints. So, when do we know that a light bulb is being switched? So if I'm on the second pass, I will-- I don't want to say turn on. I will switch every light bulb that's divisible by two. And then if I'm on the third pass, I'll switch every light bulb that's divisible by three. So on every pass, what am I doing? If I'm on pass n-- this is a hint if you want it-- what do I know? I know all the light bulbs that are numbered where n is a factor of that light bulb, that will get switched. So we know that switched if n is a factor of the light bulb number. And that's just a fancy way of saying, look, if I'm on pass 17, I'm going to switch all the multiples of 17. Or I could say, I know that I'm going to switch light bulb 51, because 17 is a factor of 51. So that tells you that we're always going to be switching one of these light bulbs on or off when one of its factors is our pass. So for example, if we're looking at light bulb eight. This is light bulb eight. So when will it be switched? So on pass one, we're definitely going So pass one, it's going to be switched on. Pass two, it'll be switched off. I know that because two is divisible into eight. Pass three, nothing's going to happen. On pass three, nothing's going to happen because this isn't a multiple of three. Pass four, what'll happen? It will be switched. It'll be switched back on. And then pass eight is the next time we'll touch this light bulb, and it'll be switched back. So every time one of its factors go by, we're going to switch this thing. And as you can see, in order for it to be on at the end, you have to have an odd number of factors. So that's an interesting thing. So in order for a light bulb to be on, it has to have odd number of factors. Now that's an interesting question. What numbers have an odd number of factors. And this is something that I think they should teach you in grade school, and they never do. But it's a really interesting kind of number theory. It's a simple one, but it's interesting to think about. So what numbers are true? Let's do all the factors for some of the starting numbers. So all the factors of one. Well one, the only factor is just one. So one works? One has an odd number of factors. So that means that one will remain on. Because you're only going to turn it on in the first pass. Makes sense. Two. What are all the factors of two. Well you have one and two. So two has an even number. You're going to switch it on the first time, then off the second time. Then you're never going to touch it again. So this is going to stay off. Your factors are one and three. Four. Your factors are one, two and four. Interesting. Here we have three factors. We have an odd number of factors. So four is going to stay on. We're going to turn it on in our first pass, we're going to turn it off on our second pass. And we're going to turn it on again in our fourth pass. Let's keep going. So five. The factors of five are one and five. The factors are one, two, three and six. It's an even number, so they're going to be off when we're done with it. Seven. it's one and seven. We just did that. It's one, two, four and eight. Still going to be off. Nine. Let's see. The factors are one, three and nine. Interesting. Once again we have an odd number of factors. So the light bulb number nine is also going to be on when everything's done. Let's keep going. I don't know, I actually did this at our mental boot camp with some of the kids. And they immediately said, the distance between one and four is three. The distance between four and nine is five. And maybe the distance between nine and the next number is going to be seven. It increases by odd numbers. What's nine plus seven? 16. What are the factors of 16? They're one, two, four, eight and 16. Interesting. From nine to 16, you incremented by seven. From four to nine, you incremented by five. So it seems like we have a pattern. But can you see something even more interesting about the numbers one, four, nine and 16. And you could try all the numbers between nine and 16, and you'll see that they have an even number of factors. But what's interesting about all of these numbers? Why do they have an odd number of factors? In all of these other cases, every factor is paired with another number. One times two is two. One time six is six. Two times three is six. There's always a pair. Except for these numbers. There's no pair. Why isn't there? One times four is four. But two times two is also four. So we only write the two once. Three times three is nine. Four times four is 16. So all the lights that are going to be on are actually perfect squares. That's why they have an odd number of factors. So what's our question? So this problem of what light bulbs are going to be on, boils down to how many perfect squares are there between one and 100? And you could just list them out. And you could say, oh well the perfect squares are one, four, nine, 16. And you could try to think of them all. Or you could say, well how many numbers can I square and get a number less than or equal to 100? Well 100 is equal to 10 squared. So you can only square the numbers between one and 10 to get a perfect square. So there's only 10 perfect squares between one and 100. Hopefully you didn't lose that, but if that confuses you, just list them all out. Given that 100 is the largest number, it's the whole largest perfect square there. And that's 10 squared. The only other perfect squares in our range we're talking about are one squared, two squared, three squared, all And we could do that. Four squared is 16. Five squared is 25. 36, 49, 64, 81, and 100. So the light bulbs with these numbers on it are the ones that will stay lit when they're all done. Anyway, hope you enjoyed that." + }, + { + "Q": "At 5:17 why do you have to times everything with -2?", + "A": "Because by doing so, we can get 200m in one equation and -200m in the other, which allows us to solve using elimination. We can add the two equations and be left with only one variable, w.", + "video_name": "VuJEidLhY1E", + "transcript": "Everyone in the kingdom is very impressed with your ability to help with the party planning, everyone except for this gentleman right over here. This is Arbegla, and he is the king's top adviser and also chief party planner, And he seems somewhat threatened by your ability to solve these otherwise unsolvable problems or, at least, from his point of view because he keeps over-ordering or under-ordering things like cupcakes, and so he asks... he says, \"King, that cupcake problem was easy.\" \"Ask them about the potato chip issue...\" \"because we can never get the potato chips right.\" And so the king says,\"Arbegla, that's a good idea. We need to get the potato chips right.\" So he comes to you and says \"How do we figure out, on average, how many potato chips we need to order?\" And to do that, we have to figure out, how much, on average does each man and each woman eat? You say, \"Well, what about the children?\" He says, the king says,\"In our kingdom, we forbid the potato chips for children.\" You say, \"Oh well, that's...that's always good. So tell me what happened at the previous parties.\" And so, the king says ,\" You might remember, at the last party, in fact the last two parties we had 500 adults, at the last party, 200 of them were men and 300 of them were women, and in total they ate 1200 bags of potato chips.\" You say, \"Well, what about the party before that?\" He says,\" That we had a bigger sque towards women. We only had a 100 men, and 400 women. And THAT time, we actually had fewer bags consumed; 1100 bags of potato chips.\" So you say, \"King and Arbegla, this seems like a fairly straightfoward thing, let me define some variables to represent our unknowns. So you go ahead, and you say, \"Lets let m=the number of bags eaten by each man. You could think of it on average, or maybe everyone, all the men in that kingdom are completely identical, or its its the average bags eaten by each man.\" \"And lets let w= the number of bags eaten by each woman. And so with that, these definitions of our variables, lets think about how we can represent this first piece of information. \" \" In green. Well, let's think about the total number of bags that the men ate. You had 200 men, [Let me scroll over a little bit] and they each ate m bags per man. \" \"So the man at this first party collectively ate 200 times m bags. If m is 10 bags per man, then this would be 2000. If m was 5 bags per man, then this would be 5000. We don't know what m is, but 200 times m is the total eaten by the man.\" \"How do we figure out, on average, how many potato chips we need to order?\"" + }, + { + "Q": "Is there a differents if you say something like 7.2 does it matter if you put the point or do you have to only put the R because when i was taught division with remainders we put the point.", + "A": "If you put a point your answer could be confusing: it looks like the decimal number 7.2. For example: 5 / 2 = 2, with remainder 1 or: 5 / 2 = 2.5 Writing 2.1 in this case would be very confusing. Note that there is no universally accepted way to write remainders - nor for decimal numbers (e.g. many European countries use a decimal comma instead of a point). Explicitly using the word remainder makes it immediately clear.", + "video_name": "8Ft5iHhauJ0", + "transcript": "Let's now see if we can divide into larger numbers. And just as a starting point, in order to divide into larger numbers, you at least need to know your multiplication tables from the 1-multiplication tables all the way to, at least, the 10-multiplication. So all the way up to 10 times 10, which you know is 100. And then, starting at 1 times 1 and going up to 2 times 3, all the way up to 10 times 10. And, at least when I was in school, we learned through 12 times 12. But 10 times 10 will probably do the trick. And that's really just the starting point. Because to do multiplication problems like this, for example, or division problems like this. Let's say I'm taking 25 and I want to divide it by 5. So I could draw 25 objects and then divide them into groups of 5 or divide them into 5 groups and see how many elements are in each group. But the quick way to do is just to think about, well 5 times what is 25, right? 5 times question mark is equal to 25. And if you know your multiplication tables, especially your 5-multiplication tables, you know that 5 times 5 is equal to 25. So something like this, you'll immediately just be able to say, because of your knowledge of multiplication, that 5 goes into 25 five times. And you'd write the 5 right there. Not over the 2, because you still want to be careful of the place notation. You want to write the 5 in the ones place. It goes into it 5 ones times, or exactly five times. And the same thing. If I said 7 goes into 49. That's how many times? Well you say, that's like saying 7 times what-- you could even, instead of a question mark, you could put a blank there --7 times what is equal to 49? And if you know your multiplication tables, you know that 7 times 7 is equal to 49. All the examples I've done so far is a number multiplied by itself. Let me do another example. Let me do 9 goes into 54 how many times? Once again, you need to know your multiplication tables to do this. 9 times what is equal to 54? And sometimes, even if you don't have it memorized, you could say 9 times 5 is 45. And 9 times 6 would be 9 more than that, so that would be 54. So 9 goes into 54 six times. So just as a starting point, you need to have your multiplication tables from 1 times 1 all the way up the 10 times 10 memorized. In order to be able to do at least some of these more basic problems relatively quickly. Now, with that out of the way, let's try to do some problems that's might not fit completely cleanly into your multiplication tables. So let's say I want to divide-- I am looking to divide 3 into 43. And, once again, this is larger than 3 times 10 or 3 times 12. Well, let me do another problem. Let me do 3 into 23. And, if you know your 3-times tables, you realize that there's 3 times nothing is exactly 23. 3 times 1 is 3. 3 times 2 is 6. Let me just write them all out. 3 times 3 is 9, 12, 15, 18, 21, 24, right? There's no 23 in the multiples of 3. So how do you do this division problem? Well what you do is you think of what is the largest multiple of 3 that does go into 23? And that's 21. And 3 goes into 21 how many times? Well you know that 3 times 7 is equal to 21. So you say, well 3 will go into 23 seven times. But it doesn't go into it cleanly because 7 times 3 is 21. So there's a remainder left over. So if you take 23 minus 21, you have a remainder of 2. So you could write that 23 divided by 3 is equal to 7 remainder-- maybe I'll just, well, write the whole word out --remainder 2. So it doesn't have to go in completely cleanly. And, in the future, we'll learn about decimals and fractions. But for now, you just say, well it goes in cleanly 7 times, but that only gets us to 21. But then there's 2 left over. So you can even work with the division problems where it's not exactly a multiple of the number that you're dividing into the larger number. But let's do some practice with even larger numbers. And I think you'll see a pattern here. So let's do 4 going into-- I'm going to pick a pretty large number here --344. And, immediately when you see that you might say, hey Sal, I know up to 4 times 10 or 4 times 12. 4 times 12 is 48. This is a much larger number. This is way out of bounds of what I know in my 4-multiplication tables. And what I'm going to show you right now is a way of doing this just knowing your 4-multiplication tables. So what you do is you say 4 goes into this 3 how many times? And you're actually saying 4 goes into this 3 how many hundred times? So this is-- Because this is 300, right? This is 344. But 4 goes into 3 no hundred times, or 4 goes into-- I guess the best way to think of it --4 goes 3 0 times. So you can just move on. 4 goes into 34. So now we're going to focus on the 34. So 4 goes into 34 how many times? And here we can use our 4-multiplication tables. 4-- Let's see, 4 times 8 is equal to 32. 4 times 9 is equal to 36. So 4 goes into 34-- 30-- 9 is too many times, right? 36 is larger than 34. So 4 goes into 34 eight times. There's going to be a little bit left over. 4 goes in the 34 eights times. So let's figure out what's left over. And really we're saying 4 goes into 340 how many ten times? We're actually saying 4 goes into 340 eighty times. Because notice we wrote this 8 in the tens place. But just for our ability to do this problem quickly, you just say 4 goes into 34 eight times, but make sure you write the 8 in the tens place right there. 8 times 4. 8 times 4 is 32. And then we figure out the remainder. 34 minus 32. Well, 4 minus 2 is 2. And then these 3's cancel out. So you're just left with a 2. But notice we're in the tens column, right? This whole column right here, that's the tens column. So really what we said is 4 goes into 340 eighty times. 80 times 4 is 320, right? Because I wrote the 3 in the hundreds column. And then there is-- and I don't want to make this look like a-- I don't want to make this look like a-- Let me clean this up a little bit. I didn't want to make that line there look like a-- when I was dividing the columns --to look like a 1. But then there's a remainder of 2, but I wrote the 2 in the tens place. So it's actually a remainder of 20. But let me bring down this 4. Because I didn't want to just divide into 340. I divided into 344. So you bring down the 4. Let me switch colors. And then-- So another way to think about it. We just said that 4 goes into 344 eighty times, right? We wrote the 8 in the tens place. And then 8 times 4 is 320. The remainder is now 24. So how many times does 4 go into 24? Well we know that. 4 times 6 is equal to 24. So 4 goes into 24 six times. And we put that in the ones place. 6 times 4 is 24. And then we subtract. 24 minus 24. That's-- We subtract at that stage, either case. And we get 0. So there's no remainder. So 4 goes into 344 exactly eighty-six times. So if your took 344 objects and divided them into groups of 4, you would get 86 groups. Or if you divided them into groups of 86, you would get 4 groups. Let's do a couple more problems. I think you're getting the hang of it. Let me do 7-- I'll do a simple one. 7 goes into 91. So once again, well, this is beyond 7 times 12, which is 84, which you know from our multiplication tables. So we use the same system we did in the last problem. 7 goes into 9 how many times? 7 goes into 9 one time. 1 times 7 is 7. And you have 9 minus 7 is 2. And then you bring down the 1. And remember, this might seem like magic, but what we really said was 7 goes into 90 ten times-- 10 because we wrote the 1 in the tens place --10 times 7 is 70, right? You can almost put a 0 there if you like. And 90 with the remainder-- And 91 minus 70 is 21. So 7 goes into 91 ten times remainder 21. And then you say 7 goes into 21-- Well you know that. 7 times 3 is 21. So 7 goes into 21 three times. 3 times 7 is 21. You subtract these from each other. Remainder 0. So 91 divided by 7 is equal to 13. And I won't take that little break to explain the places and all of that. I think you understand that. I want, at least, you to get the process down really really well in this video. So let's do 7-- I keep using the number 7. Let me do a different number. Let me do 8 goes into 608 how many times? So I go 8 goes into 6 how many times? It goes into it 0 time. So let me keep moving. 8 goes into 60 how many times? Let me write down the 8. Let me draw a line here so we don't get confused. Let me scroll down a little bit. I need some space above the number. So 8 goes into 60 how many times? We know that 8 times 7 is equal to 56. And that 8 times 8 is equal to 64. So 8 goes into-- 64 is too big. So it's not this one. So 8 goes into 60 seven times. And there's going to be a little bit left over. So 8 goes into 60 seven times. Since we're doing the whole 60, we put the 7 above the ones place in the 60, which is the tens place in the whole thing. 7 times 8, we know, is 56. 60 minus 56. That's 4. Or if we wanted, we can borrow. That be a 10. That would be a 5. 10 minus 6 is 4. Then you bring down this 8. 8 goes into 48 how many times? Well what's 8 times 6? Well, 8 times 6 is exactly 48. So 8 times-- 8 goes into 48 six times. 6 times 8 is 48. And you subtract. We subtracted up here as well. 48 minus 48 is 0. So, once again, we get a remainder of 0. So hopefully, that gives you the hang of how to do these larger division problems. And all we really need to know to be able to do these, to tackle these, is our multiplication tables up to maybe 10 times 10 or 12 times 12." + }, + { + "Q": "what is hemoglobin?", + "A": "Hemoglobin is the iron-containing oxygen-transport metalloprotein in the red blood cells. And is responsible for aiding in the transportation of gases to and from the body. :)", + "video_name": "QhiVnFvshZg", + "transcript": "Where I left off in the last video, we talked about how the hemoglobin in red blood cells is what sops up all of the oxygen so that it increases the diffusion gradient-- or it increases the incentive, we could say, for the oxygen to go across the membrane. We know that the oxygen molecules don't know that there's less oxygen here, but if you watch the video on diffusion you know how that process happens. If there's less concentration here than there, the oxygen will diffuse across the membrane and there's less inside the plasma because the hemoglobin is sucking it all up like a sponge. Now, one interesting question is, why does the hemoglobin even have to reside within the red blood cells? Why aren't hemoglobin proteins just freely floating in the blood plasma? That seems more efficient. You don't have to have things crossing through, in and out of, these red blood cell membranes. You wouldn't have to make red blood cells. What's the use of having these containers of hemoglobin? It's actually a very interesting idea. If you had all of the hemoglobin sitting in your blood plasma, it would actually hurt the flow of the blood. The blood would become more viscous or more thick. I don't want to say like syrup, but it would become thicker than blood is right now-- and by packaging the hemoglobin inside these containers, inside the red blood cells, what it allows the blood to do is flow a lot better. Imagine if you wanted to put syrup in water. If you just put syrup straight into water, what's going to happen? The water's going to become a little syrupy, a little bit more viscous and not flow as well. So what's the solution if you wanted to transport syrup in water? Well, you could put the syrup inside little containers or inside little beads and then let the beads flow in the water and then the water wouldn't be all gooey-- and that's exactly what's happening inside of our blood. Instead of having the hemoglobin sit in the plasma and make it gooey, it sits inside these beads that we call red blood cells that allows the flow to still be non-viscous. So I've been all zoomed in here on the alveolus and these capillaries, these pulmonary capillaries-- let's zoom out a little bit-- or zoom out a lot-- just to understand, how is the blood flowing? And get a better understanding of pulmonary arteries and veins relative to the other arteries and veins that are in the body. So here-- I copied this from Wikipedia, this diagram of the human circulatory system-- and here in the back you can see the lungs. Let me do it in a nice dark color. So we have our lungs here. You can see the heart is sitting right in the middle. And what we learned in the last few videos is that we have our little alveoli and our lungs. Remember, we get to them from our bronchioles, which are branching off of the bronchi, which branch off of the trachea, which connects to our larynx, which connects to our pharynx, which connects to our mouth and nose. But anyway, we have our little alveoli right there and then we have the capillaries. So when we go away from the heart-- and we're going to delve a little bit into the heart in this video as well-- so when blood travels away from the heart, it's de-oxygenated. It's this blue color. So this right here is blood. This right here is blood traveling away from the heart. It's going behind these two tubes right there. So this is the blood going away from the heart. So this blue that I've been highlighting just now, these are the pulmonary arteries and then they keep splitting into arterials and all of that and eventually we're in capillaries-- super, super small tubes. They run right past the alveoli and then they become oxygenated and now we're going back to the heart. So we're talking about pulmonary veins. So we go back to the heart. So these capillaries-- in the capillaries we get oxygen. Now we're going to go back to the heart. Hope you can see what I'm doing. And we're going to enter the heart on this side. You actually can't even see where we're entering the heart. We're going to enter the heart right over here-- and I'm going to go into more detail on that. Now we have oxygenated blood. And then that gets pumped out to the rest of the body. Now this is the interesting thing. When we're talking about pulmonary arteries and veins-- remember, the pulmonary artery was blue. As we go away from the heart, we have de-oxygenated blood, but it's still an artery. Then as we go towards the heart from the lungs, we have a vein, but it's oxygenated. So that's this little loop here that we start and I'm going to keep going over the circulation pattern because the heart can get a little confusing, especially because of its three-dimensional nature. But what we have is, the heart pumps de-oxygenated blood from the right ventricle. You're saying, hey, why is it the right ventricle? That looks like the left side of the drawing, but it's this dude's right-hand side, right? This is this guy's right hand. And this is this dude's left hand. He's looking at us, right? We don't care about our right or left. We care about this guy's right and left. And he's looking at us. He's got some eyeballs and he's looking at us. So this is his right ventricle. Actually, let me just start off with the whole cycle. So we have de-oxygenated blood coming from the rest of the body, right? The name for this big pipe is called the inferior vena cava-- inferior because it's coming up below. Actually, you have blood coming up from the arms and the head up here. They're both meeting right here, in the right atrium. Let me label that. I'm going to do a big diagram of the heart in a second. And why are they de-oxygenated? Because this is blood returning from our legs if we're running, or returning from our brain, that had to use respiration-- or maybe we're working out and it's returning from our biceps, but it's de-oxygenated blood. It shows up right here in the right atrium. It's on our left, but this guy's right-hand side. From the right atrium, it gets pumped into the right ventricle. It actually passively flows into the right ventricle. The ventricles do all the pumping, then the ventricle contracts and pumps this blood right here-- and you don't see it, but it's going behind this part right here. It goes from here through this pipe. So you don't see it. I'm going to do a detailed diagram in a second-- into the pulmonary artery. We're going away from the heart. This was a vein, right? This is a vein going to the heart. This is a vein, inferior vena cava vein. This is superior vena cava. They're de-oxygenated. Then I'm pumping this de-oxygenated blood away from the heart to the lungs. Now this de-oxygenated blood, this is in an artery, right? This is in the pulmonary artery. It gets oxygenated and now it's a pulmonary vein. And once it's oxygenated, it shows up here in the left-- let me do a better color than that-- it shows up right here in the left atrium. Atrium, you can imagine-- it's kind of a room with a skylight or that's open to the outside and in both of these cases, things are entering from above-- not sunlight, but blood is entering from above. On the right atrium, the blood is entering from above. And in the left atrium, the blood is entering-- and remember, the left atrium is on the right-hand side from our point of view-- on the left atrium, the blood is entering from above from the lungs, from the pulmonary veins. Veins go to the heart. Then it goes into-- and I'll go into more detail-- into the left ventricle and then the left ventricle pumps that oxygenated blood to the rest of the body via the non-pulmonary arteries. So everything pumps out. Let me make it a nice dark, non-blue color. So it pumps it out through there. You don't see it right here, the way it's drawn. It's a little bit of a strange drawing. It's hard to visualize, but I'll show it in more detail and then it goes to the rest of the body. Let me show you that detail right now. So we said, we have de-oxygenated blood. Let's label it right here. This is the superior vena cava. This is a vein from the upper part of our body from our arms and heads. This is the inferior vena vaca. This is veins from our abdomen and from our legs and the rest of our body. So it it first enters the right atrium. Remember, we call the right atrium because this is someone's heart facing us, even though this is on the left-hand side. It enters through here. It's de-oxygenated blood. It's coming from veins. the body used the oxygen. Then it shows up in the right ventricle, right? These are valves in our heart. And it passively, once the right ventricle pumps and then releases, it has a vacuum and it pulls more blood from the It pumps again and then it pushes it through here. Now this blood right here-- remember, this one still is de-oxygenated blood. De-oxygenated blood goes to the lungs to become oxygenated. So this right here is the pulmonary-- I'm using the word pulmonary because it's going to or from the lungs. It's dealing with the lungs. And it's going away from the heart. It's the pulmonary artery and it is de-oxygenated. Then it goes to the heart, rubs up against some alveoli and then gets oxygenated and then it comes right back. Now this right here, we're going to the heart. So that's a vein. It's in the loop with the lungs so it's a pulmonary vein and it rubbed up against the alveoli and got the oxygen diffused into it so it is oxygenated. And then it flows into your left atrium. Now, the left atrium, once again, from our point of view, is on the right-hand side, but from the dude looking at it, it's his left-hand side. So it goes into the left atrium. Now in the left ventricle, after it's done pumping, it expands and that oxygenated blood flows into the left ventricle. Then the left ventricle-- the ventricles are what do all the pumping-- it squeezes and then it pumps the blood into the aorta. This is an artery. Why is it an artery? Because we're going away from the heart. Is it a pulmonary artery? No, we're not dealing with the lungs anymore. We dealt with the lungs when we went from the right ventricle, went to the lungs in a loop, back to the left atrium. Now we're in the left ventricle. We pump into the aorta. Now this is to go to the rest of the body. This is an artery, a non-pulmonary artery-- and it is oxygenated. So when we're dealing with non-pulmonary arteries, we're oxygenated, but a pulmonary artery has no oxygen. It's going away from the heart to get the oxygen. Pulmonary vein comes from the lungs to the heart with oxygen, but the rest of the veins go to the heart without oxygen because they want to go into that loop on the pulmonary loop right there. So I'll leave you there. Hopefully that gives-- actually, let's go back to that first diagram. I think you have a sense of how the heart is dealing, but let's go look at the rest of the body and just get a sense of things. You can look this up on Wikipedia if you like. All of these different branching points have different names to them, but you can see right here you have kind of a branching off, a little bit below the heart. This is actually the celiac trunk. Celiac, if I remember correctly, kind of refers to an abdomen. So this blood that-- your hepatic artery. Hepatic deals with the liver. Your hepatic artery branches off of this to get blood flow to the liver. It also gives blood flow to your stomach so it's very important in digestion and all that. And then let's say this is the hepatic trunk. Your liver is sitting like that. Hepatic trunk-- it delivers oxygen to the liver. The liver is doing respiration. It takes up the oxygen and then it gives up carbon dioxide. So it becomes de-oxygenated and then it flows back in and to the inferior vena cava, into the vein. I want to make it clear-- it's a loop. It's a big loop. The blood doesn't just flow out someplace and then come back someplace else. This is just one big loop. And if you want to know at any given point in time, depending on your size, there's about five liters of blood. And I looked it up-- it takes the average red blood cell to go from one point in the circulatory system and go through the whole system and come back, 20 seconds. That's an average because you can imagine there might be some red blood cells that get stuck someplace and take a little bit more time and some go through the completely perfect route. Actually, the 20 seconds might be closer to the perfect route. I've never timed it myself. But it's an interesting thing to look at and to think about what's connected to what. You have these these arteries up here that they first branch off the arteries up here from the aorta into the head and the neck and the arm arteries and then later they go down and they flow blood to the rest of the body. So anyway, this is a pretty interesting idea. In the next video, what I want to do is talk about, how does the hemoglobin know when to dump the oxygen? Or even better, where to dump the oxygen-- because maybe I'm running so I need a lot of oxygen in the capillaries around my thigh muscles. I don't need them necessarily in my hands. How does the body optimize where the oxygen is actually It's actually fascinating." + }, + { + "Q": "is tension always occurring when on a string or can it occur otherwise?", + "A": "any material that can withstand being stretched", + "video_name": "_UrfHFEBIpU", + "transcript": "I will now introduce you to the concept of tension. So tension is really just the force that exists either within or applied by a string or wire. It's usually lifting something or pulling on something. So let's say I had a weight. Let's say I have a weight here. And let's say it's 100 Newtons. And it's suspended from this wire, which is right here. Let's say it's attached to the ceiling right there. Well we already know that the force-- if we're on this planet that this weight is being pull down by gravity. So we already know that there's a downward force on this weight, which is a force of gravity. And that equals 100 Newtons. But we also know that this weight isn't accelerating, It also has no velocity. But the important thing is it's not accelerating. But given that, we know that the net force on it must be 0 by Newton's laws. So what is the counteracting force? You didn't have to know about tension to say well, the string's pulling on it. The string is what's keeping the weight from falling. So the force that the string or this wire applies on this weight you can view as the force of tension. Another way to think about it is that's also the force that's within the wire. And that is going to exactly offset the force of gravity on And that's what keeps this point right here stationery and keeps it from accelerating. That's pretty straightforward. Tension, it's just the force of a string. And just so you can conceptualize it, on a guitar, the more you pull on some of those higher-- what was it? The really thin strings that sound higher pitched. The more you pull on it, the higher the tension. It actually creates a higher pitched note. So you've dealt with tension a lot. I think actually when they sell wires or strings they'll probably tell you the tension that that wire or string can support, which is important if you're going to build a bridge or a swing or something. So tension is something that should be hopefully, a little bit intuitive to you. So let's, with that fairly simple example done, let's create a slightly more complicated example. So let's take the same weight. Instead of making the ceiling here, let's add two more strings. Let's add this green string. Green string there. And it's attached to the ceiling up here. That's the ceiling now. And let's see. This is the wall. And let's say there's another string right here attached to the wall. So my question to you is, what is the tension in these two strings So let's call this T1 and T2. Well like the first problem, this point right here, this red point, is stationary. It's not accelerating in either the left/right directions and it's not accelerating in the up/down directions. So we know that the net forces in both the x and y dimensions must be 0. My second question to you is, what is going to be the offset? Because we know already that at this point right here, there's going to be a downward force, which is the force of gravity again. The weight of this whole thing. We can assume that the wires have no weight for simplicity. So we know that there's going to be a downward force here, this is the force of gravity, right? The whole weight of this entire object of weight plus wire is pulling down. So what is going to be the upward force here? Well let's look at each of the wires. This second wire, T2, or we could call it w2, I guess. The second wire is just pulling to the left. It has no y components. It's not lifting up at all. So it's just pulling to the left. So all of the upward lifting, all of that's going to occur from this first wire, from T1. So we know that the y component of T1, so let's call-- so if we say that this vector here. Let me do it in a different color. Because I know when I draw these diagrams it starts to get confusing. Let me actually use the line tool. So I have this. Let me make a thicker line. So we have this vector here, which is T1. And we would need to figure out what that is. And then we have the other vector, which is its y component, and I'll draw that like here. This is its y component. We could call this T1 sub y. And then of course, it has an x component too, and I'll do that in-- let's see. I'll do that in red. Once again, this is just breaking up a force into its component vectors like we've-- a vector force into its x and y components like we've been doing in the last several problems. And these are just trigonometry problems, right? We could actually now, visually see that this is T sub 1 x and this is T sub 1 sub y. Oh, and I forgot to give you an important property of this problem that you needed to know before solving it. Is that the angle that the first wire forms with the ceiling, this is 30 degrees. So if that is 30 degrees, we also know that this is a parallel line to this. So if this is 30 degrees, this is also going to be 30 degrees. So this angle right here is also going to be 30 degrees. And that's from our-- you know, we know about parallel lines and alternate interior angles. We could have done it the other way. We could have said that if this angle is 30 degrees, this angle is 60 degrees. This is a right angle, so this is also 30. But that's just review of geometry that you already know. But anyway, we know that this angle is 30 degrees, so what's its y component? Well the y component, let's see. What involves the hypotenuse and the opposite side? Let me write soh cah toa at the top because this is really just trigonometry. soh cah toa in blood red. So what involves the opposite and the hypotenuse? So opposite over hypotenuse. So that we know the sine-- let me switch to the sine of 30 degrees is equal to T1 sub y over the tension in the string going in this direction. So if we solve for T1 sub y we get T1 sine of 30 degrees is equal to T1 sub y. And what did we just say before we kind of dived into the math? We said all of the lifting on this point is being done by the y component of T1. Because T2 is not doing any lifting up or down, it's only pulling to the left. So the entire component that's keeping this object up, keeping it from falling is the y component of this tension vector. So that has to equal the force of gravity pulling down. This has to equal the force of gravity. That has to equal this or this point. So that's 100 Newtons. And I really want to hit this point home because it might be a little confusing to you. We just said, this point is stationery. It's not moving up or down. It's not accelerating up or down. And so we know that there's a downward force of 100 Newtons, so there must be an upward force that's being provided by these two wires. This wire is providing no upward force. So all of the upward force must be the y component or the upward component of this force vector on the first wire. So given that, we can now solve for the tension in this first wire because we have T1-- what's sine of 30? Sine of 30 degrees, in case you haven't memorized it, sine of 30 degrees is 1/2. So T1 times 1/2 is equal to 100 Newtons. Divide both sides by 1/2 and you get T1 is equal to 200 Newtons. So now we've got to figure out what the tension in this second wire is. And we also, there's another clue here. This point isn't moving left or right, it's stationary. So we know that whatever the tension in this wire must be, it must be being offset by a tension or some other force in the opposite direction. And that force in the opposite direction is the x component of the first wire's tension. So it's this. So T2 is equal to the x component of the first wire's tension. And what's the x component? Well, it's going to be the tension in the first wire, 200 Newtons times the cosine of 30 degrees. It's adjacent over hypotenuse. And that's square root of 3 over 2. So it's 200 times the square root of 3 over 2, which equals 100 square root of 3. So the tension in this wire is 100 square root of 3, which completely offsets to the left and the x component of this wire is 100 square root of 3 Newtons to the right. Hopefully I didn't confuse you. See you in the next video." + }, + { + "Q": "In the equation:\n2KNO3-->2KNO2+O2\nwould the reduction agent be just oxygen or the entire conpound 2KNO3?", + "A": "It would be the compound as the compound itself is getting oxidised, not just the oxygen. Hope this helps!", + "video_name": "TOdHMORp4is", + "transcript": "Let's see how to identify the oxidizing and reducing agents in a redox reaction. So here, we're forming sodium chloride from sodium metal and chlorine gas. And so before you assign oxidizing and reducing agents, you need to assign oxidation states. And so let's start with sodium. And so the sodium atoms are atoms in their elemental form and therefore have an oxidation state equal to 0. For chlorine, each chlorine atom is also an atom in its elemental form, and therefore, each chlorine atom has an oxidation state equal to 0. We go over here to the right, and the sodium cation. A plus 1 charge on sodium, and for monatomic ions, the oxidation state is equal to the charge on the ion. And since the charge on the ion is plus 1, that's also the oxidation state. So plus 1. We're going to circle the oxidation state to distinguish it from everything else we have on the board here. And for chloride anion, a negative 1 charge. Therefore, the oxidation state is equal to negative 1. And so let's think about what happened in this redox reaction. Sodium went from an oxidation state of 0 to an oxidation state of plus 1. That's an increase in the oxidation state. 0 to plus 1 is an increase in oxidation state, so therefore, sodium, by definition, is being oxidized. So sodium is being oxidized in this reaction. We look at chlorine. Chlorine is going from an oxidation state of 0 to an oxidation state of negative 1. That's a decrease in the oxidation state, and therefore, chlorine is being reduced. So each chlorine atom is being reduced here. Now, before we assign oxidizing and reducing agents, let's just go ahead and talk about this one more time, except showing all of the valence electrons. So let's also assign some oxidation states using this way because there are two ways to assign oxidation states. So let's assign an oxidation state to sodium over here. So if you have your electrons represented as dots, you can assign an oxidation state by thinking about how many valence electrons the atom normally has and subtracting from that how many electrons you have in your picture here. So for sodium, being in group one, one valence electron normally, and that's exactly what we have in our picture. Each sodium has a valence electron right here. So 1 minus 1 gives us an oxidation state equal to 0, which is what we saw up here, as well. So sodium has an oxidation state equal to 0. Notice that I have two sodium atoms drawn here, and that's just what the two reflects in the balanced equation up here. Let's assign an oxidation state to each chlorine atom in the chlorine molecule. And so we have a bond between the two chlorine atoms, and we know that bond consists of two electrons. Now, when you're assigning oxidation states and dot structures, you want to give those electrons to the more electronegative elements. In this case, it's the same element, so there's no difference. And so we give one electron to one atom and the other electron to the other atom, like that. And so assigning an oxidation state, you would say chlorine normally has seven valence electrons, and in our picture here, this chlorine atom has seven electrons around it. So 7 minus 7 gives us an oxidation state equal to 0. And of course, that's what we saw up here as well, when we were just using the memorized rules. And so it's the same for this chlorine atom over here, an oxidation state equal to 0. So sometimes it just helps to see the electrons. We'll go over here for our products. We had two sodium chlorides, so here are two sodium chlorides. And let's see what happened with our electrons. So the electron in magenta, this electron over here in magenta on this sodium, added onto one of these chlorines here. And then this electron on this sodium added onto the other chlorine, like that, and so sodium lost its valence electron. Each sodium atom lost its valence electron, forming a cation. And when we calculate the oxidation state, we do the same thing. Sodium normally has one valence electron, but it lost that valence electron. So 1 minus 0 is equal to plus 1 for the oxidation state, which is also what we saw up here. And then when we do it for chlorine, chlorine normally has seven valence electrons, but it gained the one in magenta. So now it has eight around it. So 7 minus 8 gives us an oxidation state equal to negative 1. And so maybe now it makes more sense as to why these oxidation states are equal to the charge on the polyatomic monatomic ion here. And so now that we've figured out what exactly is happening to the electrons in magenta, let's write some half reactions and then finally talk about what's the oxidizing agent and what's the reducing agent. So let's break down the reaction a little bit more in a different way. So you can see we have two sodium atoms over here. So we're going to write two sodiums. And when we think about what's happening, those two sodium atoms are turning into two sodium ions over here on the right. And so we have two sodium ions on the right. Now, those sodium atoms turned into the ions by losing electrons, so each sodium atom lost one electron. So we have a total of two electrons that are lost. I'm going to put it in magenta here. So those 2 electrons are lost, and this is the oxidation half reaction. You know it's the oxidation half reaction, because you're losing electrons here. So remember, LEO the lion. So Loss of Electrons is Oxidation. So this is the oxidation half reaction. We're going to write the reduction half reaction. The chlorine molecule gained those two electrons in magenta. So those two electrons in magenta we're going to put over here this time. The chlorine molecule gained them, and that turned the chlorine atoms into chloride anions. And so we have two chloride anions over here. And so those are, of course, over here on the right, our two chloride anions. And so here we have those two electrons being added to the reactant side. That's a gain of electrons, so this is our reduction half reaction, because LEO the lion goes GER. Gain of Electrons is Reduction. And so if we add those two half reactions together, we should get back the original redox reaction, because those two electrons are going to cancel out. It's actually the same electrons. These two electrons in magenta that are lost by sodium are the same electrons that are gained by chlorine, and so when we add all of our reactants that are left, we get 2 sodiums and Cl2, so we get 2 sodiums plus chlorine gas. And then for our products, we would make 2 NaCl, so we get 2 NaCl for our products, which is, of course, our original balanced redox reaction. So finally, we're able to identify our oxidizing and reducing agents. I think it was necessary to go through all of that, because thinking about those electrons and the definitions are really the key to not being confused by these terms here. And so sodium is undergoing oxidation, and by sodium undergoing oxidation, it's supplying the two electrons for the reduction of chlorine. Therefore, you could say that sodium is the agent for the reduction of chlorine, or the reducing agent. So let's go ahead and write that here. So sodium, even though it is being oxidized, is the reducing agent. It is allowing chlorine to be reduced by supplying these two electrons. And chlorine, by undergoing reduction, is taking the electrons from the 2 sodium atoms. That allows sodium to be oxidized, so chlorine is the agent for the oxidation of sodium, or the oxidizing agent. Let me go ahead and write that in red here. Chlorine is the oxidizing agent. And so this is what students find confusing sometimes, because sodium is itself being oxidized, but it is actually the reducing agent. And chlorine itself is being reduced, but it is actually the oxidizing agent. But when you think about it by thinking about what happened with those electrons, those are the exact same electrons. The electrons that are lost by sodium are the same electrons gained by chlorine, and that allows sodium to be the reducing agent for chlorine, and that is allowing chlorine at the same time to oxidize sodium. And so assign your oxidation states, and then think about these definitions, and then you can assign oxidizing and reducing agents." + }, + { + "Q": "if we have a switch parallel to a resistor in a circuit, if the switch is closed will current pass through the resistor parallel to it? if yes, why? if no, why ?", + "A": "Hello Ram, If you have an ideal switch with zero resistance than all current will flow through the switch. If you have a real switch the current will flow through ALL parallel branches. Most of the current will flow through the switch. Here we assume that the switch has a low resistance relative to the resistor. Regards, APD", + "video_name": "3NcIK0s3IwU", + "transcript": "Let's see if we can apply what we've learned to a particularly hairy problem that I have constructed. So let me see how I can construct this. So let's say in parallel, I have this resistor up here. And I try to make it so the numbers work out reasonably neat. That is 4 ohms. Then I have another resistor right here. That is 8 ohms. Then I have another resistor right here. That is 16 ohms. And then, I have another resistor here, that's ohms. Actually, I'm now making it up on the fly. I think the numbers might work out OK. 16 ohms. And let's say that now here in series, I have a resistor that is 1 ohm, and then in parallel to this whole thing-- now you can see how hairy it's getting-- I have a resistor that is 3 ohms. And let's say I have a resistor here. Let's just make it simple: 1 ohm. And just to make the numbers reasonably easy-- I am doing this on the fly now-- that's the positive terminal, negative terminal. Let's say that the voltage difference is 20 volts. So what I want us to do is, figure out what is the current flowing through the wire at that point? Obviously, that's going to be different than the current at that point, that point, that point, that point, all of these different points, but it's going to be the same as the current flowing at this point. So what is I? So the easiest way to do this is try to figure out the equivalent resistance. Because once we know the equivalent resistance of this big hairball, then we can just use Ohm's law and be done. So first of all, let's just start at, I could argue, the simplest part. Let's see if we could figure out the equivalent resistance of these four resistors in parallel. Well, we know that that resistance is going to be equal to 1/4 plus 1/8 plus 1/16 plus 1/16. So that resistance-- and now it's just adding fractions-- over 16. 1/4 is 4/16 plus 2/16 plus 1 plus 1, so 1/R is equal to 4 plus 2 is equal to 8/16-- the numbers are working out-- is equal to 1/2, so that equivalent resistance is 2. So that, quickly, we just said, well, all of these resistors combined is equal to 2 ohms. So let me erase that and simplify our drawing. Simplify it. So that whole thing could now be simplified as 2 ohms. I lost some wire here. I want to make sure that circuit can still flow. So that easily, I turned that big, hairy mess into something that is a lot less hairy. Well, what is the equivalent resistance of this resistor and this resistor? Well, they're in series, and series resistors, they just add up together, right? So the combined resistance of this 2-ohm resistor and this 1-ohm resistor is just a 3-ohm resistor. So let's erase and simplify. So then we get that combined resistor, right? We had the 2-ohm that we had simplified and then we had a 1-ohm. So we had a 2-ohm and a 1-ohm in series, so those simplify to 3 ohms. Well, now this is getting really simple. So what do these two resistors simplify to? Well, 1 over their combined resistance is equal to 1/3 plus 1/3. 2/3. 1/R is equal to 2/3, so R is equal to 3/2, or we could say 1.5, right? So let's erase that and simplify our drawing. So this whole mess, the 3-ohm resistor in parallel with the other 3-ohm resistor is equal to one resistor with a 1.5 resistance. And actually, this is actually a good point to give you a little intuition, right? Because even though these are 3-ohm resistors, we have two of them, so you're kind of increasing the pipe that the electrons can go in by a factor of two, right? So it's actually decreasing the resistance. It's giving more avenues for the electrons to go through. Actually, they're going to be going in that direction. And that's why the combined resistance of both of these in parallel is actually half of either one of these I encourage you to think about that some more to give you some intuition of what's actually going on with the electrons, although I'll do a whole video on resistivity. OK so we said those two resistors combined-- I want to delete all of that. Those two resistors combined equal to a 1.5-ohm resistor. That's 1.5 ohms. And now all we're left with is two resistors in parallel, so the whole circuit becomes this, which is the very basic one. This is a resistor: 1.5 ohms, 1 ohm in series. Did I say parallel just now? No, they're in series. 1.5 plus 1, that's 2.5 ohms. The voltage is 20 volts across them. So what is the current? Ohm's law. V is equal to IR. Voltage is 20 is equal to current times our equivalent resistance times 2.5 ohms. Or another way to write 2.5 five is 5/2, right? So 20 is equal to I times 5/2. Or I is equal to 2/5 times 20, and what is that? 2/5 is equal to I is equal to 8. 8 amperes. That was not so bad, I don't think. Although when you saw it initially, it probably looked extremely intimidating. Anyway, if you understood that, you can actually solve fairly complicated circuit problems. I will see you in future videos." + }, + { + "Q": "Is Papal States Pronounced Pay-pal or Pa-pal", + "A": "Probably the closest is Pay-Pull with the emphasis on the first syllable.", + "video_name": "ALJGz4r_VF0", + "transcript": "In the video on the Fourth Coalition, I forgot to add to one super important consequence of the Treaties of Tilsit. And especially the Treaty of Tilsit with Prussia. I already talked about that it was all about carving up Prussia, and humiliating Prussia. And really removing it from the status of one the preeminent powers. And all I talked about was the loss of the territories of Prussia west of the Elbe. And that's about that area right there. But just as important as that, the Polish holdings of Prussia. So all of this area right over here, this also was removed from Prussia and became a French satellite state. It became the Duchy of Warsaw. So I just really want to emphasize. The Treaty of Tilsit, I only emphasized kind of what happened on the western side of Prussia, but the eastern side of Prussia also got carved up. And Prussia essentially lost half of its size. So it's very dramatic humiliation for Prussia at the end of the Treaty of Tilsit. Or the Treaties of Tilsit. Now with that out of the way, we talked about in the last video, that at the end of the Fourth Coalition, Napoleon was kind of near the peak of his power. He'd kind of done everything right. He had this kind of steady upward momentum, or France had a steady upward momentum in its power. But what we're going to see in this video, at least the beginnings of the downfall of Napoleon. And it's not going to be obvious when you look at the territory. Because from a territorial point of view, you're going to see in this video that he's actually gaining territory. But he is going to start doing some of the actions that end up undermining him. So we talked about in the last video, we talked about this whole notion of the Continental System, where Napoleon was obsessed with people on the continent of Europe boycotting England, not trading with England. And he figured this is the only way that he could really undermine England's dominance on the ocean. Or eventually maybe even undermine England generally. So as we said, in the Treaties of Tilsit he got Russia to participate in the Continental System. So he wanted everyone to buy into it. And one party that, at this point, we're talking about-- we're in 1807 now-- one party that wasn't all that keen in participating in the Continental System was Portugal. That's Portugal right there. So Napoleon goes and chats-- well they didn't chat directly-- but he gets the agreement of the King of Spain. This is Charles IV, and he's going to look like a bit of a fool and this video. And Napoleon says, hey Charles, let's go in there, let's go into Portugal, that little upstart country that doesn't want to participate in the Continental System. You and me, we'll invade together. We'll bring them into kind of our realm of influence. And we can both kind of pillage the lands and get the wealth of Portugal. Charles IV, he's all up for this. So a combined French and Spanish force invade Portugal. So in 1807, this is the end of 1807, it's actually in October. In October, you have a combined French and Spanish invasion of Portugal. And they are able to take Portugal, but we're going to see that it's reasonably temporary. Now I just mentioned that this guy is going to look like the fool of this video. And the reason is, because with the excuse of reinforcements, obviously to get to Portugal, you have to go through Spain. So with the excuse of sending in reinforcements, Napoleon in 1808-- and now we're talking about early 1808, in particular in March. So with the excuse of sending in reinforcements to support the Portugal campaign, and Spain is like your my ally, sure, send those hundreds of thousands of troops right through our territory. We're not going to worry about it. And with that excuse, Napoleon was able to send 100,000 troops and occupy Madrid. So this is one of those lessons of never get too greedy. This guy got greedy, wanted to help Napoleon. Or I guess the other lesson is be careful who your friends are. This guy wanted to invade Portugal, but the side effect of it is that Madrid gets occupied. And that actually he gets dethrowned. And so you have this situation here, the French are now in control of Spain. In May of 1808-- and this is really going to be the first little spark that is kind of the downfall of Napoleon. In May 2, 1808, a popular uprising starts in Madrid. Dos de Mayo. So a popular uprising in Madrid. And at the same time, a little bit after that-- So you can imagine, this is a hugely tumultuous time. You have this occupation of Portugal with the excuse of reinforcements in March. The French troops occupy Madrid. Then in May-- so a couple of months later-- a popular uprising starts in Madrid. This leads to popular uprisings throughout Spain. But at the very same time as this-- this is a little bit after the uprising in May-- Napoleon says, oh, this is just a little uprising, I'm still in control of Spain. He appoints his other brother-- remember, there's this whole business he's putting his brothers in charge of different parts of the Empire. He puts his brother Joseph, he appoints his brother Joseph-- or you could kind of say-- he inserts his brother Joseph as the King of Spain. So this is all in kind of early, mid-1808. Spain is in all of this turmoil. A new king has been appointed, who is Napoleon's brother. The old king is no longer in charge. You have this ongoing battle in Portugal. They don't have a firm hold on Portugal just yet. And in the rest of 1808, the uprising that occurs throughout Spain is actually pretty successful in enforcing the French troops to retreat. And a major, I guess, aspect of this uprising is it's one of the first real national uprisings in history. It's people saying we are Spanish, we do not like being controlled by the French. We do not like how they have treated our royalty. We as a nation are going to rise up. And the other interesting aspect of this whole uprising that starts in Madrid with Dos de Mayo, but then it starts continuing throughout the whole nation, is the idea of guerrilla warfare. Not gorilla warfare. And this comes from the Spanish for little war. Not from the large ape. And what it implies, you probably heard the word on the news before, is kind of a non-conventional style of fighting, where small little groups kind of engage their enemy in very nontraditional styles. So it becomes a very painful-- at least for it Napoleon's forces-- it became very difficult fighting these non-conventional battles all over Spain. So they were able to force the French to retreat. Napoleon says, gee, you know what? If you want a job well done, you've got to do it yourself. So Napoleon comes in at the end of the year, and then he retakes Madrid. So December of 1808, Napoleon back in Madrid. Now, you might say all is fine and well. Now Napoleon is back here. He has firm control of Spain. But not everything is good. Because as you could imagine, there's all these other characters here that keep forming coalitions for and against Napoleon. Even when they say that they're allied, you know that in the back of their minds they can't wait until they can declare the next war on Napoleon. So in 1809-- let me write this down-- Austria declares war. And since Great Britain was in-- at this point in time-- perpetual war with France, this becomes the Fifth Coalition. But this one is fairly short-lived. Napoleon says gee, I got these guys on my eastern front. Austria is re-declaring war on me. So he leaves Spain to go lead that fight. And he leaves 300,000 of his best troops in Spain to hold Spain. And frankly, this is the most important side effect of the Fifth Coalition, is that it makes Napoleon go to fight Austria, to lead that effort, as opposed to worrying about Spain. And essentially by doing that-- and I don't know if it's necessarily the fact that Napoleon wasn't there. But it could be because Napoleon wasn't there-- is that Spain just becomes a major thorn in Napoleon's side. This guerrilla warfare just continues on and on and on. And it just goes back and forth. And the French will win a battle and they'll win another battle. But they still don't have control. And these guerrillas will kind of peck at them and continue the uprising. And this really just drains the French army. And really just gets at them little bit by little bit, really over the remainder of Napoleon's reign. So all the way until 1814. We haven't gone over that yet. But this occurs all the way to 1814. So I said at the beginning the video, this is one of the starting points of Napoleon's downfall. That's just because he was just stuck in Spain from 1808 on, just continuing to have to send troops and supplies and reinforcements and wealth to support what they called the Peninsular Campaign. And it just drains him. It drains his resources, it drains his energy. And it really hurts his ability to fight wars with all of the other people who he needs to fight wars with. This is one of the major downfalls. The other one, which we'll probably talk about in the next video or video after that, is his invasion of Russia. Which he does in 1812. One could debate which one drains France's resources more. But the invasion of Russia really decimates Napoleon's forces. And really makes him susceptible to really conquest by England and all of the other allies. And we're going to see that in a couple of videos. So you have this been Peninsular Campaign continuing to drain Napoleon. It all started because he wanted to enforce the Continental System on Portugal. And he got a little bit greedy. And he also wanted to conquer Spain. And just to highlight why it's called the Peninsular Campaign. This right here, a little bit of geography, this is called the Iberian Peninsula right there that I'm circling. So you could call it the Iberian Peninsular Campaign, because it's everything that's going on in this Peninsula in Spain and Portugal. Now if we back up a little bit back to 1808, where we had this uprising in Spain, and they were able to push the French back. At the same time, you also had a popular uprising in Portugal roughly in the fall, late summer or fall of 1808. The British got excited. They saw it as their chance to push Napoleon out of Portugal. So you have this gentleman right here, Sir Arthur Wellesley. He's a future Duke of Wellington. And he's eventually going to be responsible for pushing Napoleon out of Spain entirely. Or at least out of Madrid. Him and along with the British and along with the Portuguese are able to push the French out in August of 1808. So let me put this in my not so neatly drawn timeline here. So in December, Napoleon is back, so right before that in August, out of Portugal. And this is another motivation for Napoleon to say, gee, you know what? Things aren't going well on the Iberian Peninsula, I have to take charge of things myself. Now, at the very same time as all of this is happening, and this is really just kind of out of interest. Well it's more than out of interest, because actually it has huge global repercussions. You might say, OK, well you have this Iberian Peninsula. Spain is going back and forth between the French and the guerillas. And Portugal has this whole situation where their king was dethrowned, but then the British help and take it back. But you could imagine, these nations are in just a super state of flux. Now you could also imagine, the King of Spain wasn't just the King of Spain, he was King of the Spanish Empire. And the Spanish Empire, the main land mass of the Spanish Empire was in the Americas. So this right here. That was the Spanish Empire at the time. This was a 400 year old Spanish Empire. Starting with Columbus sailing the ocean blue in 1492. You had this huge Spanish Empire. And one of the really important side effects of Napoleon invading Spain and having this long protracted engagement in Spain, is it catalyzed the ability of these colonies at the time to start looking for their And we're going to do whole videos on that in the future. But this really is one of the things that allowed them to get independence. Obviously, if the empire is in flux, these guys can say hey, gee, why do we have to listen to that nation anymore? We don't even know who's in charge there. At the same time, same thing in Portugal. Brazilian independence didn't come until a little bit later after this period, but Napoleon's invasion is what really sparked the beginning of a lot of turmoil in Portugal. And that eventually is one of the causes that leads to the eventual independence of Brazil. That doesn't happen for another 10 or 15 years. But you could imagine, this is where a lot of that action can be traced back. Now, another interesting point that occurred around this time. And actually, I didn't tell you what happened on the Fifth Coalition. I said Austria declared war. Obviously Britain was already at war. So it was the Fifth Coalition. Napoleon had to leave, that maybe made Spain a little bit harder to hold for France. And that's why it kind of bled France slowly. But Napoleon was able to take care of Austria. And then he was able to take a little bit more land from them. Actually Galicia, this area of Austria was given to the Duchy of Warsaw, which was a French satellite state. And then Austria once again, had to say oh Napoleon, we're your friend, we're going to do whatever you ask us to. So you can imagine at this time landwise, the empire of Napoleon seemed pretty dramatic. You could include Spain here. Although he had to spend a lot of resources to keep Spain. And then now we had Austria, at least it was in the fold. You know Prussia was not really happy about it. But this whole area here, the western half of Poland was under French control. Germany-- the Confederacy of the Rhine-- which is now Germany. And then a good bit of Italy, the Kingdom of Italy was also a French satellite state. But Napoleon, of course, he wanted everyone to participate in the Continental System. That's the only way to really strangle England. And the Papal States were not participating in the Continental System. So he sent some people over to kind of try to convince them to. And when they didn't, they occupied the Papal States. So French troops occupied the Papal States. And then once again, this was still back in 1808. This is actually early 1808, it's just on a different front. So in February, up here, in 1808. Actually, that's before they even occupied Madrid. So in 1808, February, French troops occupy Papal States. They essentially give them over to the Kingdom of Italy, which at that time was a French satellite state. So it's almost like annexing it to France. And then once the Fifth Coalition was done with, Napoleon felt so good about himself, that he formally annexed the Papal States. Now we're in 1809. In 1809, he formally-- The Papal States are actually annexed into the French Empire. Now you can imagine that the Pope wasn't that happy about this. This is the Pope at the time, this is Pius VII. He wasn't so happy about it. So he excommunicates Napoleon. And I'll do a whole video on excommunication. But it's really about as bad as something you could do to someone within the powers of the Catholic Church. And by implication, you're no longer part of the church, and you'll probably go to hell now, at least if the Pope has anything to do with it. So Napoleon wasn't happy about this. He sent some people, some officers, once again to talk to the Pope about it. To say hey, gee, why do you want to excommunicate Napoleon? Why don't you just play nice? Why don't you just agree to whatever Napoleon says? The Pope doesn't agree, and so he gets abducted. This is why it's interesting. Napoleon, he's not afraid to take some serious action. So he gets abducted in 1809 by French officers. And it's not clear, it's not obvious that Napoleon told But once he was abducted, and they actually started shuttling him all around France depending on who needed to talk to him. Or if they were afraid that the British might try to free him from one port, they would send them some place else. But it wasn't clear that Napoleon ordered this. But he never ordered his release. So in some ways, you got to say that it was sanctioned by Napoleon. So all of this mess starts, you know, Napoleon is messing with the Pope. He has this ongoing bleeding going on in Spain. And that ends actually in 1812 where Sir Arthur Wellesley finally retakes Madrid. But during this whole period, you can imagine it's really draining into the resources of the French Empire." + }, + { + "Q": "What does R^2 represent? Can we just use c or d? what's the significance of using R^2 as obtaining the members a and b?", + "A": "\u00e2\u0084\u009d\u00c2\u00b2 represents the set of all 2 dimensional vectors composed of real numbers. When Sal writes a\u00e2\u0083\u0097, b\u00e2\u0083\u0097 \u00e2\u0088\u0088 \u00e2\u0084\u009d\u00c2\u00b2 all he s saying is that the vectors a\u00e2\u0083\u0097 and b\u00e2\u0083\u0097 are 2 dimensional vectors composed of real numbers. I don t know what you mean by Can we just use c or d . If you mean as labels for the vectors, then yes, you can name the vectors with whatever symbol you want, Sal used a\u00e2\u0083\u0097 and b\u00e2\u0083\u0097 just to use the first letters of the alphabet, but there is no special significance in that.", + "video_name": "8QihetGj3pg", + "transcript": "So I have two 2-dimensional vectors right over here, vector a and vector b. And what I want to think about is how can we define or what would be a reasonable way to define the sum of vector a plus vector b? Well, one thing that might jump at your mind is, look, well, each of these are two dimensional. They both have two components. Why don't we just add the corresponding components? So for the sum, why don't we make the first component of the sum just a sum of the first two components of these two vectors. So why don't we just make it 6 plus negative 4? Well, 6 plus negative 4 is equal to 2. And why don't we just make the second component the sum of the two second components? So negative 2 plus 4 is also equal to 2. So we start with two 2-dimensional vectors. You add them together, you get another two 2-dimensional vectors. If you think about it in terms of real coordinates bases, both of these are members of R2-- I'll write this down here just so we get used to the notation. So vector a and vector b are both members of R2, which is just another way of saying that these are both two tuples. They are both two-dimensional vectors right over here. Now, this might make sense just looking at how we represented it, but how does this actually make visual or conceptual sense? And to do that, let's actually plot these vectors. Let's try to represent these vectors in some way. Let's try to visualize them. So vector a, we could visualize, this tells us how far this vector moves in each of these directions-- horizontal direction and vertical direction. So if we put the, I guess you could say the tail of the vector at the origin-- remember, we don't have to put the tail at the origin, but that might make it a little bit easier for us to draw it. We'll go 6 in the horizontal direction. 1, 2, 3, 4, 5, 6. And then negative 2 in the vertical. So negative 2. So vector a could look like this. Vector a looks like that. And once again, the important thing is the magnitude and the direction. The magnitude is represented by the length of this vector. And the direction is the direction that it is pointed in. And also just to emphasize, I could have drawn vector a like that or I could have put it over here. These are all equivalent vectors. These are all equal to vector a. All I really care about is the magnitude and the direction. So with that in mind, let's also draw vector b. Vector b in the horizontal direction goes negative 4-- 1, 2, 3, 4, and in the vertical direction goes 4-- 1, 2, 3, 4. So its tail if we start at the origin, if its tail is at the origin, its head would be at negative 4, 4. So let me draw that just like that. So that right over here is vector b. And once again, vector b we could draw it like that or we could draw it-- let me copy and let me paste it-- so this would also be another way to draw vector b. Once again, what I really care about is its magnitude and its direction. All of these green vectors have the same magnitude. They all have the same length and they all have the same direction. So how does the way that I drew vector a and b gel with what its sum is? So let me draw its sum like this. Let me draw its sum in this blue color. So the sum based on this definition we just used, the vector addition would be 2, 2. So 2, 2. So it would look something like this. So how does this make sense that the sum, that this purple vector plus this green vector is somehow going to be equal to this blue vector? I encourage you to pause the video and think about if that even makes sense. Well, one way to think about it is this first purple vector, it shifts us this much. It takes us from this point to that point. And so if we were to add it, let's start at this point and put the green vector's tail right there and see where it ends up putting us. So the green vector, we already have a version. So once again, we start the origin. Vector a takes us there. Now, let's start over there with the green vector and see where green vector takes us. And this makes sense. Vector a plus vector b. Put the tail of vector b at the head of vector a. So if you were to start at the origin, vector a takes you there then if you add on what vector b takes you, it takes you right over there. So relative to the origin, how much did you-- I guess you could say-- shift? And once again, vectors don't only apply to things like displacement. It can apply to velocity. It can apply to actual acceleration. It can apply to a whole series of things, but when you visualize it this way, you see that it does make complete sense. This blue vector, the sum of the two, is what results where you start with vector a. At that point right over there, vector a takes you there, then you take vector b's tail, start over there and it takes you to the tip of the sum. Now, one question you might be having is well, vector a plus vector b is this, but what is vector b plus vector a? Does this still work? Well, based on the definition we had where you add the corresponding components, you're still going to get the same sum vector. So it should come out the same. So this will just be negative 4 plus 6 is 2. 4 plus negative 2 is 2. But does that make visual sense? So if we start with vector b. So let's say you start right over here. Vector b takes you right over there. And then if you were to go there and you were to start with vector a-- so let's do that. So actually, let me make this a little bit-- actually, let me start with a new vector b. So let's say that that's our vector b right over there. And then-- actually, let me give this a place where I'll have some space to work with. So let's say that's my vector b right over there. And then let me get a copy of the vector a. That's a good one. So copy and let me paste it. So I could put vector a's tail at the tip of vector b, and then it'll take me right over there. So if I start right over here, vector b takes me there. And now I'm adding to that vector a, which starting here will take me there. And so from my original starting position, I have gone this far. Now, what is this vector? Well, this is exactly the vector 2, 2. Or another way of thinking about it, this vector shifts you 2 in the horizontal direction and 2 in the vertical direction. So either way, you're going to get the same result, and that should, hopefully, make visual or conceptual sense as well." + }, + { + "Q": "How many times cn u actually borrow?", + "A": "you can borrow (total number of digits - 1) times if needed. Its not necessary to borrow every time though.", + "video_name": "OJ-wajo6oa4", + "transcript": "We've got 9,601 minus 8,023. And immediately when we try to start subtracting in our ones place, we have a problem. This 3 is larger than this 1. And we also have that problem in the tens place. This 2 is larger than this 0. So we're going to have to do some type of borrowing or regrouping. And so the way I like to think about it-- I like to go to the first place value that has something to give. Obviously, the tens place is in no position to give anything to the ones place. It needs things itself. And so we're going to go to the hundreds place. And the hundreds place has an abundance of value that it can regroup into the tens and ones place. This 6 right over here represents 600. So why don't we take 100 from that 600-- so then this will become 500-- and then give that 100 over to the tens place. Now, if we give 100 to the tens place, how would I represent that in the tens place? Well, I have zero 10's. And now I'm going to give 100. 100 is the same thing as 10 10's. It's going to be 0 plus 100. 100 in the tens place is just 10. So let me write it this way. So this right over here is now going to be rewritten as 10. Now, you might be saying, wait, wait, wait. What's going on here, Sal? You took 100 from the hundreds place. That's why it became 500. Now, why did this become 10 and not 100? Now remember, this is 10 10's. So this is still representing 100. You have not changed the value of this top number. Before, the value was 9,000 plus 600 plus 1. Now it's 9,000 plus 500 plus 100-- 10 10's is 100-- plus 1. I have not changed the value here. Now, we're still not done yet. We don't want to just subtract because we still have the problem with the ones place. The ones place still doesn't have enough value. Now, the good thing is we've given some value to the tens So why don't we take 10 from the tens place? So if you have 10 10's, and you take one 10 away, you're going to be left with nine 10's, or 90. And then we can take that 10 and give it to the ones place So let's do that. You take that 10 we just took from there, and you give it to the ones place. You now have 11 here. And now we are ready to subtract. 11 minus 3 is 8. 9 minus 2-- and this is really 90 minus 20-- is 70. But in the tens place, we represent that as a 7. 500 minus zero hundred is 500, represented as a 5 in the hundreds place. 9,000 minus 8,000 is 1,000. And we're done. And just to make things really clear, I'm going to redo this problem now but with things expanded out. So this first number is 9,000 plus 600 plus zero 10's plus 1. And this number right here, we're subtracting 8,000. We're subtracting zero 100's. We're subtracting two 10's, which is 20. Subtracting 20. And subtracting three 1's. So I have just rewritten this exact same statement. But the regrouping and the borrowing is going to become a little bit clearer now. So the same exact thing-- we said, hey, we can't subtract the 3 from the 1 or the 20 from the 0. But we have a lot of value right over here in the 600. So why don't we take 100 from that? So this becomes 500. And we give that 100 to the tens place. So this becomes 100. Notice, the value has not changed. This is 9,000 plus 500 plus 100 plus 1. That's the same thing as 9,000 plus 600 plus 1. We've just put the value in different places. And here we have explicitly written 100. But when we represent it in the tens place, 10 10's is the same thing as 100. Now, we aren't done regrouping just yet. We want to give some value to the ones place. So we can take 10 from the tens place-- and this becomes a 90-- and give that 10 to the ones place. 10 plus 1 is 11. So notice, I did the exact borrowing, the exact regrouping, that I did here. I just represented it a little bit different. This 500 was represented by a 5 in the hundreds place. This 90 was represented by a 9 in the tens place. But either way, we're ready to subtract now. 11 minus 3 is 8. 90 minus 20 is 70. Write a plus there. 500 minus 0 is 500. And then 9,000 minus 8,000 is 1,000. And we got the same result because 1,000 plus 500 plus 70 plus 8 is 1,578." + }, + { + "Q": "I'm solving stuff like -1/4x+3=-3/2x-2 and i want to know how to solve it but i can't seem to find any thing that will help me.", + "A": "Its a linear equation, i would do something like this: -1/4x + 3 = -3/2x -2 i d take -3/2x by adding 3/2x to both sides of the equation to get -1.75x + 3 = -2 the i d take away 3 to get -1.75x = -5 then divide both sides to get x =2.8571428571428571428571428571429", + "video_name": "5a6zpfl50go", + "transcript": "Let's say I have the equation y is equal to x plus 3. And I want to graph all of the sets, all of the coordinates x comma y that satisfy this equation right there. And we've done this many times before. So we draw our axis, our axes. That's my y-axis. This is my x-axis. And this is already in mx plus b form, or slope-intercept form. The y-intercept here is y is equal to 3, and the slope here is 1. So this line is going to look like this. We intersect at 0 comma 3-- 1, 2, 3. At 0 comma 3. And we have a slope of 1, so every 1 we go to the right, we go up 1. So the line will look something like that. It's a good enough approximation. So the line will look like this. And remember, when I'm drawing a line, every point on this line is a solution to this equation. Or it represents a pair of x and y that satisfy this equation. So maybe when you take x is equal to 5, you go to the line, and you're going to see, gee, when x is equal to 5 on that line, y is equal to 8 is a solution. And it's going to sit on the line. So this represents the solution set to this equation, all of the coordinates that satisfy y is equal to x plus 3. Now let's say we have another equation. Let's say we have an equation y is equal to negative x plus 3. And we want to graph all of the x and y pairs that satisfy this equation. Well, we can do the same thing. This has a y-intercept also at 3, right there. But its slope is negative 1. So it's going to look something like this. Every time you move to the right 1, you're going to move down 1. Or if you move to the right a bunch, you're going to move down that same bunch. So that's what this equation will look like. Every point on this line represents a x and y pair that will satisfy this equation. Now, what if I were to ask you, is there an x and y pair that satisfies both of these equations? Is there a point or coordinate that satisfies both equations? Well, think about it. Everything that satisfies this first equation is on this green line right here, and everything that satisfies this purple equation is on the purple line right there. So what satisfies both? Well, if there's a point that's on both lines, or essentially, a point of intersection of the lines. So in this situation, this point is on both lines. And that's actually the y-intercept. So the point 0, 3 is on both of these lines. So that coordinate pair, or that x, y pair, must satisfy both equations. When x is 0 here, 0 plus 3 is equal to 3. When x is 0 here, 0 plus 3 is equal to 3. It satisfies both of these equations. So what we just did, in a graphical way, is solve a system of equations. Let me write that down. And all that means is we have several equations. Each of them constrain our x's and y's. So in this case, the first one is y is equal to x plus 3, and then the second one is y is equal to negative x plus 3. This constrained it to a line in the xy plane, this constrained our solution set to another line in the xy plane. And if we want to know the x's and y's that satisfy both of these, it's going to be the intersection of those lines. So one way to solve these systems of equations is to graph both lines, both equations, and then look at their intersection. And that will be the solution to both of these equations. In the next few videos, we're going to see other ways to solve it, that are maybe more mathematical and less graphical. But I really want you to understand the graphical nature of solving systems of equations. Let's do another one. Let's say we have y is equal to 3x minus 6. That's one of our equations. And let's say the other equation is y is equal to negative x plus 6. And just like the last video, let's graph both of these. I'll try to do it as precisely as I can. There you go. Let me draw some. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And then 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I should have just copied and pasted some graph paper here, but I think this'll do the job. So let's graph this purple equation here. Y-intercept is negative 6, so we have-- let me do another 1, 2, 3, 4, 5, 6. So that's y is equal to negative 6. And then the slope is 3. So every time you move 1, you go up 3. You moved to the right 1, your run is 1, your rise is 1, 2, 3. That's 3, right? 1, 2, 3. So the equation, the line will look like this. And it looks like I intersect at the point 2 comma 0, which is right. 3 times 2 is 6, minus 6 is 0. So our line will look something like that right there. That's that line there. What about this line? Our y-intercept is plus 6. 1, 2, 3, 4, 5, 6. And our slope is negative 1. So every time we go 1 to the right, we go down 1. And so this will intersect at-- well, when y is equal to 0, x is equal to 6. 1, 2, 3, 4, 5, 6. So right over there. So this line will look like that. The graph, I want to get it as exact as possible. And so we're going to ask ourselves the same question. What is an x, y pair that satisfies both of these equations? Well, you look at it here, it's going to be this point. This point lies on both lines. And let's see if we can figure out what that point is. Just eyeballing the graph here, it looks like we're at 1, 2, 3 comma 1, 2, 3. It looks like this is the same point right there, that this is the point 3 comma 3. I'm doing it just on inspecting my hand-drawn graphs, so maybe it's not the exact-- let's check this answer. Let's see if x is equal to 3, y equals 3 definitely satisfies both these equations. So if we check it into the first equation, you get 3 is equal to 3 times 3, minus 6. This is 9 minus 6, which is indeed 3. So 3 comma 3 satisfies the top equation. And let's see if it satisfies the bottom equation. You get 3 is equal to negative 3 plus 6, and negative 3 plus 6 is indeed 3. So even with our hand-drawn graph, we were able to inspect it and see that, yes, we were able to come up with the point 3 comma 3, and that does satisfy both of these equations. So we were able to solve this system of equations. When we say system of equations, we just mean many equations that have many unknowns. They don't have to be, but they tend to have more than And you use each equation as a constraint on your variables, and you try to find the intersection of the equations to find a solution to all of them. In the next few videos, we'll see more algebraic ways of solving these than drawing their two graphs and trying to find their intersection points." + }, + { + "Q": "when is it appropriate to use the prepositions at and in", + "A": "Use at to indicate where you are, like heidi mentioned - but never where are you at? the at in that usage is redundant. Use in to indicate position such as I am in the house or The hamster is in its cage or Your books are in the desk .", + "video_name": "O-6q-siuMik", + "transcript": "- [Voiceover] Hi, everyone. My name is David, and I'm here to introduce you to Grammar on Khan Academy. I'm so glad you could join me. So, let's start by asking the question, \"What is grammar?\" What is this thing, why is it worthwhile to study it, why would you wanna put up with listening to me? Well, first of all, grammar is a set of conventions and rules that govern language. So what's the difference between a convention and a rule? Well, a rule is kind of the bare minimum of what it takes to make your language understandable by other people, right? So in order to make a car work, for example, in order to make it move forward as intended, the wheels have to go on the bottom instead of the roof. That's a rule. The idea that all cars should be painted teal, for example, is a convention. Now is that true that all cars should be teal? No, not necessarily, but that leads me to my second point, that grammar is context dependent. The kind of grammar that you use throughout your day changes. It depends on who you're talking to, what you're trying to say, and how you're trying to say it. And so we use multiple kinds of grammar throughout our days and throughout our lives. Another thing you need to know is that you already know so much grammar. Just from living and existing in the world and talking to other people. You know how to put a sentence together. If you can understand me, then you know so much about grammar. You know more than I can teach you. What these videos are for is to give names to the things you already know. To give you a greater command of them. And I want to say, too, that these videos are only about a very specific kind of grammar. It's called Standard American English. But I want you to know that there are many Englishes. And you know what? They're all great. They are all wonderful and vibrant and important and special. And what I do not want for you to take away from these videos is that I'm trying to teach you what is right and what is wrong. If the kind of English you speak doesn't sound like the kind of English I speak, that is okay, you know? You are great. What I want to do is give you the tools to harness language. To harness English and use it any way you want. I mean, I'm saying I don't care what color your car is. It could be pink, it could be green, it could be purple, it could be paisley, you know. I'm just trying to make sure your wheels are on straight. You are a grammarian. You have made a study of grammar throughout your entire speaking and reading life. And I firmly believe that you can learn anything. Welcome to Grammar on Khan Academy. David out." + }, + { + "Q": "if on angel is 90 degrees and the other is 80.54 I really then the other angel should be 9.46 degrees and according to the picture its not", + "A": "Do not let pictures fool you. Unless they are indicated as drawn to scale, then you must assume that they are sketches for calculation purposes, The upper angle is obviously very small, but it does not have to be exactly 9.46 since we have no indication that the drawing is to scale. That is why architects get paid the big bucks, they almost always have to draw and model to scale.", + "video_name": "aHzd-u35LuA", + "transcript": "A tiny but horrible alien is standing at the top of the Eiffel Tower-- so this is where the tiny but horrible alien is-- which is 324 meters tall-- and they label that, the height of the Eiffel Tower-- and threatening to destroy the city of Paris. A Men In Black-- or a Men In Black agent. I was about to say maybe it should be a man in black. A Men In Black agent is standing at ground level, 54 meters across the Eiffel square. So 54 meters from, I guess you could say the center of the base of the Eiffel Tower, aiming his laser gun at the alien. So this is him aiming the laser gun. At what angle should the agent shoot his laser gun? Round your answer, if necessary, to two decimal places. So if we construct a right triangle here, and we can. So the height of this right triangle is 324 meters. This width right over here is 54 meters. It is a right triangle. What they're really asking us is what is this angle right over here. And they've given us two pieces of information. They gave us the side that is opposite the angle. And they've given us the side that is adjacent to the angle. So what trig function deals with opposite and adjacent? And to remind ourselves, we can write, like I always like to do, soh, cah, toa. And these are really by definition. So you just have to know this, and soh cah toa helps us. Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. We can write that the tangent of theta is equal to the length of the opposite side-- 324 meters-- over the length of the adjacent side-- over 54 meters. Now you might say, well, OK, that's fine. What angle, when I take its tangent, gives me 324/54? Well, for this, it will probably be useful to use a calculator. And the way that we'd use a calculator is we would use the Inverse Tan Function. So we could rewrite this as we're going to take the inverse tangent-- and sometimes it's written as tangent with this negative 1 superscript. So the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54. And just to be clear, what is this inverse tangent? This just literally says, this will return what is the angle that, when I take the tangent of it, gives me 324/54. This says, what is the angle that, when I take the tangent of it, gives me tangent of theta? So this right over here, this just simplifies to theta. Theta is the angle that when you get the tangent of it gets you tangent of theta. And so we get theta is equal to inverse tangent of 324/54. Once again, this inverse tangent thing you might find confusing. But all this is saying is, over here, we're saying tangent of some angle is 324/54. This is just saying my angle is whatever angle I need so that when I take the tangent of it, I get 324/54. It's how we will solve for theta. So let's get our calculator out. And let's say that we want our answer in degrees. Well, I'm just going to assume that they want our answers in degrees. So let me make sure my calculator is actually in degree mode. So I'll go to the 2nd mode right over here. And actually it's in radian mode right now. So let me make sure I'm in degree mode to get my answer in degrees. Now let me exit out of here. And let me just type in the inverse tangent-- so it's in this yellow color right here-- inverse tangent of 324 divided by 54 is going to be-- and they told us to round to two decimal places-- 80.54 degrees. So theta is equal to 80.54 degrees. That's the angle at which you should shoot the gun to help defeat this horrible alien." + }, + { + "Q": "what is the meaning of sub x and sub y ...\n\n\nplease tell me ..", + "A": "When you are dealing with a vector like velocity you often break it up into the amounts along each axis. So if you have the velocity vector V the amount along the X axis is usually V sub x and along the Y axes is V sub y. It is just a way of indicating that you are dealing with a portion of the vector.", + "video_name": "FaF3v-ezbSk", + "transcript": "Good afternoon. We've done a lot of work with vectors. In a lot of the problems, when we launch something into--- In the projectile motion problems, or when you were doing the incline plane problems. I always gave you a vector, like I would draw a vector like this. I would say something has a velocity of 10 meters per second. It's at a 30 degree angle. And then I would break it up into the x and y components. So if I called this vector v, I would use a notation, v sub x, and the v sub x would have been this vector right here. v sub x would've been this vector down here. The x component of the vector. And then v sub y would have been the y component of the vector, and it would have been this vector. So this was v sub x, this was v sub y. And hopefully by now, it's second nature of how we would figure these things out. v sub x would be 10 times cosine of this angle. 10 cosine of 30 degrees, which I think is square root of 3/2, but we're not worried about that right now. And v sub y would be 10 times the sine of that angle. This hopefully should be second nature to you. If it's not, you can just go through SOH-CAH-TOA and say, well, the sine of 30 degrees is the opposite of the And you would get back to this. But we've reviewed all of that, and you should review the initial vector videos. But what I want you to do now, because this is useful for simple projectile motion problems-- But once we start dealing with more complicated vectors-- and maybe we're dealing with multi-dimensional of vectors, three-dimensional vectors, or we start doing linear algebra, where we do end dimensional factors --we need a coherent way, an analytical way, instead of having to always draw a picture of representing vectors. So what we do is, we use something I call, and I think everyone calls it, unit vector notation. So what does that mean? So we define these unit vectors. Let me draw some axes. And it's important to keep in mind, this might seem a little confusing at first, but this is no different than what we've been doing in our physics problem so far. Let me draw the axes right there. Let's say that this is 1, this is 0, this is 2. 0, 1, 2. I don't know if must been writing an Arabic or something, going backwards. This is 0, 1, 2, that's not 20. And then let's say this is 1, this is 2, in the y direction. I'm going to define what I call the unit vectors in two dimensions. So I'm going to first define a vector. I'll call this vector i. And this is the vector. It just goes straight in the x direction, has no y component, and it has the magnitude of 1. And so this is i. We denote the unit vector by putting this little cap on top of it. There's multiple notations. Sometimes in the book, you'll see this i without the cap, and it's just boldface. There's some other notations. But if you see i, and not in the imaginary number sense, you should realize that that's the unit vector. It has magnitude 1 and it's completely in the x direction. And I'm going to define another vector, and that one is called j. And that is the same thing but in the y direction. That is the vector j. You put a little cap over it. So why did I do this? Well, if I'm dealing with two dimensions. And as later we'll see in three dimensions, so there will actually be a third dimension and we'll call that k, but don't worry about that right now. But if we're dealing in two dimensions, we can define any vector in terms of some sum of these two vectors. So how does that work? Well, this vector here, let's call it v. This vector, v, is the sum of its x component plus its y component. When you add vectors, you can put them head to tail like this. And that's the sum. So hopefully knowing what we already know, we knew that the vector, v, is equal to its x component plus its y component. When you add vectors, you essentially just put them head to tails. And then the resulting sum is where you end up. It would be if you added this vector, and then you put this tail to this head. So you end up there. So that's the vector. So can we define v sub x as some multiple of i, of this unit vector? Well, sure. v sub x completely goes in the x direction. But it doesn't have a magnitude of 1. It has a magnitude of 10 cosine 30 degrees. So its magnitude is ten. Let me draw the unit vector up here. This is the unit vector i. It's going to look something like this and this. So v sub x is in the exact same direction, and it's just a scaled version of this unit vector. And what multiple is it of that unit vector? Well, the unit vector has a magnitude of 1. This has a magnitude of 10 cosine of 30 degrees. I think that's like, 5 square roots of 3, or something like that. So we can write v sub x-- I keep switching colors to keep things interesting. We can write v sub x is equal to 10 cosine of 30 degrees times-- that's the degrees --times the unit vector i-- let me stay in that color, so you don't confused --times the unit vector i. Does that make sense? Well, the unit vector i goes in the exact same direction. But the x component of this vector is just a lot longer. It's 10 cosine 30 degrees long. And that's equal to-- cosine of 30 degrees is square root of 3/2 --so that's 5 square roots of 3 i. Similary, we can write the y component of this vector as some multiple of j. So we could say v sub y, the y component-- Well, what is sine of 30 degrees? Sine of 30 degrees is 1/2. 1/2 times 10, so this is 5. So the y component goes completely in the y direction. So it's just going to be a multiple of this vector j, of the unit vector j. And what multiple is it? Well, it has length 5, while the unit vector has just length 1. So it's just 5 times the unit vector j. So how can we write vector v? Well, we know the vector v is the sum of its x component and its y component. And we also know, so this is a whole vector v. What's its x component? Its x component can be written as a multiple of the x unit vector. That's that right there. So you can write it as 5 square roots of 3 i plus its y component. So what's its y component? Well, its y component is just a multiple of the y unit vector, which is called j, with the little funny hat on top. And that's just this. It's 5 times j. So what we've done now, by defining these unit vectors-- And I can switch this color just so you remember this is i. This unit vector is this. Using unit vectors in two dimensions, and we can eventually do them in multiple dimensions, we can analytically express any two dimensional vector. Instead of having to always draw it like we did before, and having to break out its components and always do it visually. We can stay in analytical mode and non graphical mode. And what makes this very useful is that if I can write a vector in this format, I can add them and subtract them without having to resort to visual means. And what do I mean by that? So if I had to find some vector a, is equal to, I don't know, 2i plus 3j. And I have some other vector b. This little arrow just means it's a vector. Sometimes you'll see it as a whole arrow. As, I don't know, 10i plus 2j. If I were to say what's the sum of these two vectors a plus b? Before we had this unit vector notation, we would have to draw them, and put them heads to tails. And you had to do it visually, and it would take But once you have it broken up into the x and y components, you can just separately add the x and y components. So vector a plus vector b, that's just 2 plus 10 times i plus 3 plus 2 times j. And that's equal to 12i plus 5j. And something you might want to do, maybe I'll do it in the future video, is actually draw out these two vectors and add them visually. And you'll see that you get this exact answer. And as we go into further videos, or future videos, you'll see how this is super useful once we start doing more complicated physics problems, or once we start doing physics with calculus. Anyway, I'm about to run out of time on the ten minutes. So I'll see you in the next video." + }, + { + "Q": "Why didn't any of the native American tribes help Spain?", + "A": "One, Spain wasn t really involved in this particular war. Two, at first they did but would really keep helping the people that is conquering and killing you?", + "video_name": "UAhbwYBAoe0", + "transcript": "- [Instructor] When we're talking about major wars in colonial North America, we tend to think about the American Revolution, not its earlier iteration, the Seven Years' War, and I think that's a shame because the Seven Years' War was incredibly influential not only on the American Revolution, but on the complexion of the world. Thanks to the Seven Years' War, Canada became a British country, not a French country. The Acadians moved down to Louisiana and became known as the Cajuns, and most importantly, England became the world's preeminent empire. So if you've been following along this far, you may have noticed two things. One, that the people who named this war seem to be very bad at math because 1754 to 1763 is nine years, not seven, and that this war seems to have two names, both the Seven Years' War and the French and Indian War, which is a name you perhaps have heard before. Well, lemme tackle those two oddities in reverse order. So not only does the Seven Years' War have two names, it has a whole number of names. It's called the Seven Years' War, the French and Indian War, the War of the Conquest, the Pomeranian War, the Third Silesian War, the Third Carnatic War. This is a war with a whole bunch of names, and the reason that it has a whole bunch of names is that it was fought in a whole bunch of places. The Seven Years' War was really the first global war, and we're talking 150 years before World War One. Aspects of the Seven Years' War, as you can kinda see from this map, were fought in Europe, in South America, the coast of Africa, in India, the Philippines, and of course, in North America. The many different names come from the many different fronts of this war, and I would say that French and Indian War is actually the name for the North American front of this war, or theater of this war. So there are two reasons why I think Seven Years' War is a better name than French and Indian War. One is that Seven Years' War gets at the idea that it was not just happening in North America. It was happening all over the world, so it shows that it was a global war, but I also think Seven Years' War is a better name than French and Indian War because I think French and Indian War is kind of confusing because you would think that it means that the principal parties in this war were the English versus the French and the Indians, when in fact it was the English and their Indian allies versus the French and their Indian allies. Native Americans fought on both sides of this conflict, so rather than the English and Indian versus French and Indian War, let's go with the shorter Seven Years' War, which brings us back to our awkward date range. So the reason that it's called the Seven Years' War is because the English didn't actually declare war on the French until 1756. So even though fighting started a little bit earlier in North America, the true range of dates, at least in legal terms, is from 1756 to 1763, or seven years. It's a complicated name for a complicated war, but really what it came down to was England and France duking it out over who was going to be the supreme imperial power in the world, and they were concerned about who was going to have the most territory in the world, therefore, their concern over who was going to control North America and their competing claims here, and also access to trade. So who was going to be able to trade with North Americans? Who was going to be able to trade with the lucrative Indian subcontinent, and who would be the leading power in Europe? So let's dial in a little closer on the North American theater of this war, which will have the most effect on the future United States. Alright, so here is map of territorial claims by European powers in North America before the Seven Years' War. Now you can see that there are some places where they overlap, which is really gonna be the heart of the problem in this conflict. So England, shown here in red, I'm gonna outline it a bit, was, as you know from your early American history, here along the eastern seaboard of what's today the United States, and also up into Canada. France claimed this interior region of Canada and today of the territorial United States, and Spain was in the mix here. Remember Spain has still been a fairly influential colonial power in Florida and in contemporary Mexico, and also down here in Cuba and South America. Alright, so we've got three major European powers in the mix here in North America, England, France, and Spain, but what this map doesn't show is the American Indian powers, who are also in this area. So most of this region really west of the Appalachian mountains, is Indian country, and the majority of inhabitants were Native Americans, and they really held the majority of power in this region as well. So major Native American groups that are in play in this conflict are Iroquois Confederacy, and also Cherokees, Hurons, Algonquians, Abenakis, and Mi'kmaqs, and that's just a small sampling. So you can see that there are a number of important Native American tribes who are specifically in this area of Canada, which is disputed, and also moving in the greater Appalachian region. So what does each of these groups want? Well, England definitely wants territory. They want to make sure that they're English settlers along the eastern seaboard, whom we'll soon be calling Americans, have room to expand. The French wanna make sure that they still have access to trade with Native Americans because their main concern is fur, which is a very valuable commodity in Europe, and Spain wants to make sure that they have access to their sugar islands and also their precious metals in the Caribbean and in South America. Now it's worth noting, 'cause I think this is really interesting to students of American history, that all of this territory, all of North America, was way less valuable than all of this territory because we're not talking about just value in land. We're talking about value in commodities, and what the Caribbean had was sugar, and sugar is the most valuable crop in this time period. So a tiny island down here in the Bahamas is probably worth more to a European power than the entire interior of North America, and what do these Native American groups want? Well, some of them want help with revenge on each other. Many other smaller Native American groups have been displaced by the Iroquois, who are here in upstate New York, kind of Quebec region. So the Iroquois is actually expanding and really defending their claim as the largest Native American empire, but the other thing that they want is to make sure that their territory is no longer encroached upon by English settlers in particular. Now one mistake I see early students of U.S. history making is thinking that all Native Americans kind of shared a cultural and political bond, right? That they saw themselves as one larger people who had to unite against the encroachment of Europeans, and that was definitely not the case. Native Americans had been living in this territory for thousands of years, and they had enemies and beef with other groups that went back way longer than the arrival of Europeans in North America. So when nations like England and France arrived with their weapons and their trade goods, the American Indians didn't look at each other and say, \"Oh wait, now we're all one race. \"We need to join together against \"the encroachment of whites.\" They saw England and France and Spain as possible avenues to getting one up on their older enemies. So when an English trader sold a gun to, say, a Huron, he was way more likely to go after, say, the Iroquois with that gun than he was to go after a French trader. So another reason why the Seven Years' War is a better name for the French and Indian War than French and Indian War is because these Native American groups did not ally all with France. In fact, the Iroquois and Cherokee ended up allied with England, and most of the other Native American groups ended up allied with France, but they were fighting each other in addition to fighting England. Alright, so the stage is set for this conflict with all of these competing groups in this unclear territory, and how this turns into a war, we'll get to in the next video." + }, + { + "Q": "Where's the manger animals? This is an encouraging subject for Leonardo...to use his anatomy studies..:)", + "A": "Wrong gospel. The Magi are in Matthew, where there is neither stable nor animals. Those participants are found in Luke, where there are no Magi, no star, and not even any camels. Leonardo was doing everyone a favor by depicting merely ONE of the stories, and not conflating or confusing the two.", + "video_name": "QxNqWZPzsGw", + "transcript": "SPEAKER 1: We're standing in front of an unfinished painting by Leonardo da Vinci. It's a big painting. And interestingly, it's almost a perfect square. SPEAKER 2: It's really unfinished. It's not just that it has parts that are unfinished, but it's really just the underpainting. SPEAKER 1: This is The Adoration of the Magi, a moment in the Christian story when Christ has been born and three kings from the east, guided by the star of Bethlehem, come to Mary and offer Christ three gifts, frankincense, myrrh, and gold. What's revealed to us here is Leonardo's working method-- not only his brilliant drawing, but the way in which he constructs figures. Remember that Leonardo is first and foremost not a painter. He's really a scientist. He's really an engineer. He's somebody who looks and understands nature. SPEAKER 2: We have a sense of Leonardo's deep understanding of human anatomy. Even when he's painting clothed figures, he's really understanding the skeletal structure. He's understanding the musculature of the body. SPEAKER 1: If you look at the group of figures to the right, about mid-level, you see one figure that almost looks like it's a skull. And it's as if Leonardo is literally constructing the bones before he'll put flesh on them, before he'll put clothes on them, before he'll add color. That's a group of figures that's often referred to as the philosophers. But maybe we should discuss the central group first. SPEAKER 2: So we have Mary and the Christ child front and center, forming a pyramid shape together with the Magi in front. And that's a shape that we see very often in paintings of the High Renaissance that provide a stable form. And you see that right here in the foreground with Mary and Christ. SPEAKER 1: That's especially important in this painting, which is so chaotic, where there's so much going on. On the upper right, for instance, there's actually a battle. You have two horses rearing up. On the upper left, you have the fragments of what look like some sort of classical architecture. You can see these wonderful steps in perfect linear perspective. Leonardo actually did some brilliant drawings in preparation for this painting. But let's look a little more closely at what you just said, and see if we can define those lines a little more exactly. If you start with the Virgin Mary and you look at her face, she's glancing across the top of her son's head, down his arm. He picks up, actually, her glance and brings our eye down until it's met by one of the Magi who's offering a gift. We can actually run that line down past his toes to the corner of the painting. Or we can actually pick up from Mary again and go the other way. If we go down the bridge of her nose, across her shoulder, picked up by the kneeling figure in the foreground at the left. What's interesting, as you said, this is not simply a triangle. But this is a pyramid that actually comes forward as it moves down. SPEAKER 2: --and exists in space. I'm struck by the way that Leonardo is paying attention to all of these human reactions to what's going on. And we've got lots of faces half hidden in the darkness and a lot of gestures. And it really reminds me, in a way, of Leonardo's Last Supper, where you have Christ in the middle forming a kind of pyramid shape with his outstretched arms. And all the chaos, and the reactions of the apostles around him, but this real sense of stability in the center. SPEAKER 1: That's such a characteristic of the High Renaissance-- this notion of balance, of a kind of perfection, of a sense of the eternal. But then, of course, how do we as humans, react? There's another element here, which is important and very characteristic of Leonardo. And even though this is just the underpainting, we can make it out. And that's this technique of sfumato, which in Italian is smoke. And it tends to create a kind of visual glue that creates a kind of harmony between forms within the paint, brings things together, and keeps paintings from having that sense of the isolated, so much a characteristic of the early Renaissance. SPEAKER 2: Right, so instead of figures being defined by lines, the figures are enveloped in atmospheric [? haziness ?] or softness, that kind of smokiness. And they almost seem to emerge out of the darkness into light and fade back into the darkness again. And so Leonardo's unifying the figures in yet another way. Not only in the pyramid composition and through their glances and gestures, but also into that smoky atmosphere. SPEAKER 1: We see this beautiful chiaroscuro, this beautiful smoke, this beautiful line, this beautiful composition, this complex sense of emotion. I'd love to know what this painting would have looked like had it been finished. SPEAKER 2: Me too." + }, + { + "Q": "So you need to have the same y change the same way each time? Example:\nx=1|2|3|\ny=10|20|30\nthat would be a direct variation right?", + "A": "Yes as x increases y should also increase at a constant interval", + "video_name": "rSadG6EtJmY", + "transcript": "We're told that the total cost of filling up your car with gas varies directly with the number of gallons of gasoline you are purchasing. So this first statement tells us that if x is equal to the number of gallons purchased, and y is equal to the cost of filling up the car, this first statement tells us that y varies directly with the number of gallons, with x. So that means that y is equal to some constant, we'll just call that k, times x. This is what it means to vary directly. If x goes up, y will go up. We don't know what the rate is. k tells us the rate. If x goes down, y will be down. Now, they give us more information, and this will help us figure out what k is. If a gallon of gas costs $2.25, how many gallons could you purchase for $18? So if x is equal to 1-- this statement up here, a gallon of gas-- that tells us if we get 1 gallon, if x is equal to 1, then y is $2.25, right? y is what it costs. They tell us 1 gallon costs $2.25, so you could write it right here, $2.25 is equal to k times x, times 1. Well, I didn't even have to write the times 1 there. It's essentially telling us exactly what the rate is, what k is. We don't even have to write that 1 there. k is equal to 2.25. That's what this told us right there. So the equation, how y varies with x, is y is equal to 2.25x, where x is the number of gallons we purchase. y is the cost of that purchase, so it's $2.25 a gallon. And then they ask us, how many gallons could you purchase for $18? So $18 is going to be our total cost. It is y cost of filling the car. So 18 is going to be equal to 2.25x. Now if we want to solve for x, we can divide both sides by 2.25, so let's do that. You divide 18 by 2.25, divide 2.25x by 2.25, and what do we get? Let me scroll down a little bit. The right-hand side, the 2.25's cancel out, you get x. And then what is 18 divided by 2.25? So let me write this down. So first of all, I just like to think of it as a fraction. 2.25 is the same thing-- let me write over here-- 2.25 is equal to 2 and 1/4, which is the same thing as 9 over 4. So 18 divided by 2.25 is equal to 18 divided by 9 over 4, which is equal to 18 times 4 over 9, or 18 over 1 times 4 over 9. And let's see, 18 divided by 9 is 2, 9 divided by 9 is 1. That simplifies pretty nicely into 8. So 18 divided by 2.25 is 8, so we can buy 8 gallons for $18." + }, + { + "Q": "What if it says negative 2/3? (Like it's not -2/3 or 2/-3, the negative sign is in the middle)", + "A": "-2/3 or 2/-3, they both are negative 2/3.", + "video_name": "pi3WWQ0q6Lc", + "transcript": "Let's do a few examples multiplying fractions. So let's multiply negative 7 times 3/49. So you might say, I don't see a fraction here. This looks like an integer. But you just to remind yourself that the negative 7 can be rewritten as negative 7/1 times 3/49. Now we can multiply the numerators. So the numerator is going to be negative 7 times 3. And the denominator is going to be 1 times 49. 1 times 49. And this is going to be equal to-- 7 times 3 is 21. And one of their signs is negative, so a negative times a positive is going to be a negative. So this is going to be negative 21. You could view this as negative 7 plus negative 7 plus negative 7. And that's going to be over 49. And this is the correct value, but we can simplify it more because 21 and 49 both share 7 as a factor. That's their greatest common factor. So let's divide both the numerator and the denominator by 7. Divide the numerator and the denominator by 7. And so this gets us negative 3 in the numerator. And in the denominator, we have 7. So we could view it as negative 3 over 7. Or, you could even do it as negative 3/7. Let's do another one. Let's take 5/9 times-- I'll switch colors more in this one. That one's a little monotonous going all red there. 5/9 times 3/15. So this is going to be equal to-- we multiply the numerators. So it's going to be 5 times 3. 5 times 3 in the numerator. And the denominator is going to be 9 times 15. 9 times 15. We could multiply them out, but just leaving it like this you see that there is already common factors in the numerator and the denominator. Both the numerator and the denominator, they're both divisible by 5 and they're both divisible by 3, which essentially tells us that they're divisible by 15. So we can divide the numerator and denominator by 15. So divide the numerator by 15, which is just like dividing by 5 and then dividing by 3. So we'll just divide by 15. Divide by 15. And this is going to be equal to-- well, 5 times 3 is 15. Divided by 15 you get 1 in the numerator. And in the denominator, 9 times 15 divided by 15. Well, that's just going to be 9. So it's equal to 1/9. Let's do another one. What would negative 5/9 times negative 3/15 be? Well, we've already figured out what positive 5/9 times positive 3/15 would be. So now we just have to care about the sign. If we were just multiplying the two positives, it would be 1/9. But now we have to think about the fact that we're multiplying by a negative times a negative. Now, we remember when you multiply a negative times a negative, it's a positive. The only way that you get a negative is if one of those two numbers that you're taking the product of is negative, not two. If both are positive, it's positive. If both are negative, it's positive. Let's do one more example. Let's take 5-- I'm using the number 5 a lot. So let's do 3/2, just to show that this would work with improper fractions. 3/2 times negative 7/10. I'm arbitrarily picking colors. And so our numerator is going to be 3 times negative 7. 3 times negative 7. And our denominator is going to be 2 times 10. 2 times 10. So this is going to be the numerator. Positive times a negative is a negative. 3 times negative 7 is negative 21. Negative 21. And the denominator, 2 times 10. Well, that is just 20. So this is negative 21/20. And you really can't simplify this any further." + }, + { + "Q": "how does Sal explain it so thoroughly?", + "A": "Because he has had much practice with mathmathics", + "video_name": "XkRD9lv_y44", + "transcript": "- [Voiceover] Let's give ourselves some practice substituting positive and negative values for variables. So we're told to evaluate X, we're told to evaluate X minus negative Y, where X is equal to negative two and Y is equal to five. So everywhere we see an X, we can replace with a negative two. Everywhere we see a Y, we replace with a five. So this is a Y right over here, and then of course, and then of course this is an X. Let's do that. So instead of that X, let's write negative two. So we have negative two minus and then we have in the parentheses a negative Y. Y is equal to, Y is equal to five. Now what is this going to evaluate to? Well, this is the same thing as negative two, now subtracting a negative five, so all of this business here, subtracting a negative five, that's the same thing as adding a five. So it's going to be negative two plus five, which is equal to three. This is going to be equal to three. And there's several ways to think about this. my brain thinks okay, I'm starting at negative two. If I add two I get to zero and then I would have to add another, and then I have to add another three so that gets me to three. You can even view this as, you can even view this, my brain kind of says negative two, well let's see, I have to add, i have to add two to get back to zero and then I have to add another three if I want to add a total of five. So this is going to be zero plus three gets me to three. Another way to think about it, negative two plus five is the same thing as five minus two. Five minus two, which is of course equal to three. Or you could of course draw it on a number line. And you would say if you start at negative two and you take five steps to the right, you get to positive three. Let's do another one of these. This one's a little bit more, a little bit more complex. So let's see. We're told to evaluate three minus negative six plus negative H plus negative four, where H is equal to negative seven. Well there's two ways you could do it. I could just take the negative seven and replace it, the H with that negative seven, or I could actually try to simplify this expression first and then do the substitution for H. Let's actually do that. My brain feels like doing that. So this expression, I have the three, but instead of subtracting a negative six, instead of subtracting a negative six, that's gonna be the same thing as just adding six. So three plus six. And adding a negative H, adding a negative H, adding a negative H, that's the same thing as just subtracting H. So three plus six minus H, and then adding a negative four, adding a negative four, that's the same thing as subtracting four. That's the same thing as subtracting four. And now of course we can do this in any, you know, addition and subtraction we have we can change the order in which we do it. So let's do that just to kind of simplify all of this. So, actually first of all, I can figure out what three plus six is. Three plus six is equal to nine. And then I have this minus four here. So I could say nine minus four, actually I want to be careful not to skip any steps. So three plus six, three plus six, in a color that you can see, so three plus six is nine. So that's nine minus H minus four, minus four. Now I could change the order in which I do this addition or subtraction, so this is going to be the equivalent of nine minus four. Nine minus four minus H, minus H. And I just did that so I can simplify and figure out what nine minus four is. Nine minus four, of course, let me do this in blue, navy blue. Nine minus four is five. So this whole thing simplified to five minus H before I even did the substitution. And now I can substitute H with negative seven. So this is going to be equal to, this is going to be equal to, when I do the substitution, I'll write it up here, it's going to be five minus, I'll do the minus in that magenta color, minus and now where I see an H, I'm gonna replace it with negative seven. Five minus negative seven. You want to be very careful there, you might be tempted to say, oh I have a negative here, negative here, let me just replace H with a seven. Remember, H is negative seven, so you're subtracting H. You're gonna subtract negative seven. So this is five minus negative seven, which is the same thing, which is the same thing as five plus seven. Five plus seven, which we all know is equal to 12. And we're done. Let's do a few more of these. You can't really get enough practice here, this is some important foundational skills for the rest of your mathematical lives. (laughing) Alright. So consider, I don't make you too stressed about it. Consider the following number line. Alright, so we've got a number line here and let's see, they didn't mark off all the numbers here. This is negative four, E is at this point, this is then we go to two. So it looks like we're counting by twos here, that this is negative two, this is zero, yep. Negative four, if you increase by two, negative four, negative two, zero, two, four, this would be a six. This would be a negative six. They intentionally left those numbers off, so we had to figure out that hey, look, between negative four and two, to go from negative four to two, you have to increase by six and we only have one, two, three hashmarks. So each of those hashmarks must be increasing by two. Well anyway, now we know, now we know what all the points in the number line are. Evaluate E minus F. Well we know, we know that E is equal to negative two. And we know, we know that F is equal to four. So this is going to be the same thing, E is equal to negative two minus, minus positive four. Minus positive four. Let me do it in that same blue color. Minus four. Well negative two minus four, we have the number line in front of us, this is just going to be negative six. If you start at negative two, you subtract, if you subtract two you get to negative four, you subtract another two, you get to negative six. So we are done. Let's do one more of these. Let's evaluate T plus negative U. Where once again T and U are on this number line and it looks like each of these hashmarks we're incrementing by three as we go up. Zero, three, six. And if we go down, zero, negative three, negative six. So it's clear that U is equal to three. U is equal to three and it is also clear that T is equal to negative six. T is equal to negative six. So T plus negative U is going to be, so it's going to be negative six plus, negative six plus negative U, U is three. U is three. You want to be very careful, you might say oh well U is positive so I'm just gonna put a positive number here, but remember, it's negative U. Wherever you see the U, replace it with the three. So it's negative three. So this is gonna be six plus negative three, or sorry, negative six plus negative three, which we can rewrite as negative six minus three. All of this business can be rewritten as negative six minus three. Negative six minus three. If you're at negative six and you go three to the left, you go three more negative, you're going to end up at negative nine. And we're all done." + }, + { + "Q": "How do you find a slope when you only have an x coordinate?", + "A": "You can t find a slope with only an X coordinate. You need an X and Y to find a slope", + "video_name": "xR9r38mZjK4", + "transcript": "what i want to do with this video is think about the relationship between variables and then think about what the graph of that relationship should look like. So let's say these two axis, the horizontal axis over here I plot the price of a product and lets say this vertical axis over here i plot the demand for the product, and i'm only plotting the first quadrant here because i'm assuming that the price can only be positive and i'm assuming that the demand can only be positive, that people aren't going to pay someone to take the product away from them. So let's think about what would happen for the price and demand for most normal products. So, if the price is low, you would expect that a lot of poeple are willing to buy that thing they're like \"Oh it's a good price, i would like to buy it.\" So, if the price is low, then the demand would be high so maybe it would be somplace over here, all the way that you would have really high demand if the price was zero. so if the price was low the demand would be high. Now what happens is the price -- so right here the price is low, demand is high, if the price were to go up a little bit then maybe the demand goes down a little bit, right? price went up a little bit demand went down a little bit. if the price went up a little bit more then maybe demand goes down a little bit more. as the price went up a bunch then demand would go down a bunch and so the line that represents how the demand relates to price might look something like this, and i'm just going to assume it's it a line. It might not be a line, it might be a curve. It might look something like that. Or it might look something like that. But in general is someone were to ask you, if you saw this magenta curve that as price increases what happens to demand. You just say \"Well look price increases, as price increases what happens to demand?\" Well demand is decreasing. Now let's think about a different scenario. Let's talk about the demand for real estate. For actual property, and lets say that on this axis that we plot the population. The population in the area, and this right over here this is demand for land. So when the population is very low, you can imagine, if the population is zero there is no one there that would want to buy land. So if the population is very low the demand is going to be very low. And as population increases, demand should increase. If the population increases, more people are going to want to buy land. And if the population goes up a bunch then a lot of people are going to want to buy land. So you'll see a line that looks something like this. And once again I drew a line, it doesn't have to be a line it could be a curve of some kind. It could be a curve that looks something like that, or a curve that looks something like that. We don't know but the general idea is that if someone showed you a graph that looked like this. And as population increases what happens to demand. We'll you'd say \"Look, this is population increasing, what happens to demand?\" Demand is going up. Where as price increased the demand went down. Here as population is increasing demand went up. And you can just make that more general with variables. We're talking about specific cases here. But if I were to plot something like this, if you were to see a graph that looked like that and this is the variable x and this is the variable y. And someone were to ask you what happens to y as x increases. Well you take any x that's the y that we have for that x. And as you increase x, as you move in the positive horizontal directions. As you increase x what is happening to y? Y is going up. So for this example, as x increases y is increasing. If we had a graph that looked like this. Let's call that the a axis and this is the b axis and maybe our graph looked like this. What happens as A increases? If you pick an A right over here. We're at that A and that B. As A increases what's happening to B? Well as A increases our B is lower. As A increases here B is decreasing. So, just wanting to give you a general idea, when X and Y increased together the line goes from the bottom left to the top right, we would call this an upward sloping line. We would call this a positive slope. Everytime X is increased Y also increases is upwards sloping. When our independant variable increases and our dependent variable decreases. When the independent variable is increasing, then you say it has a downward slope, when you go from the top left to the bottom right." + }, + { + "Q": "0:47 Why is it weak?\nIs it weak because both oxygens have the same electronegitivity?", + "A": "Not because of the electronegativity but because of how alcohols are more stable than cyclic peroxides and can easily form from a reaction with water.", + "video_name": "KfTosrMs5W0", + "transcript": "If you start with an alkene and add to that alkene a percarboxylic acid, you will get epoxide. So this is an epoxide right here, which is where you have oxygen in a three-membered ring with those two carbons there. You can open up this ring using either acid or base catalyzed, and we're going to talk about an acid catalyzed reaction in this video. And what ends up happening is you get two OH groups that add on anti, so anti to each other across from your double bond. So the net result is you end up oxidizing your alkene. So you could assign some oxidation numbers on an actual problem and find out that this is an oxidation reaction. All right. Let's look at the mechanism to form our epoxide. So we start with our percarboxylic acid here, which looks a lot like a carboxylic acid except it has an extra oxygen. And the bond between these two oxygen atoms is weak, so this bond is going to break in the mechanism. The other important thing to note about the structure of our percarboxylic acid is the particular confirmation that it's in. So this hydrogen ends up being very close to this oxygen because there's a source of attraction between those atoms. There's some intramolecular hydrogen bonding that keeps it in this conformation. When the percarboxylic acid approaches the alkene, when it gets close enough in this confirmation, the mechanism will begin. This is a concerted eight electron mechanism, which means that eight electrons are going to move at the same time. So the electrons in this bond between oxygen and hydrogen are going to move down here to form a bond with this carbon. The electrons in this pi bond here are going to move out and grab this oxygen. That's going to break this weak oxygen-oxygen bond, and those electrons move into here. And then finally, the electrons in this pi bond are going to move to here to form an actual bond between that oxygen and that hydrogen. So let's see if we can draw the results of this concerted eight electron mechanism. So, of course, at the bottom here we're going to form our epoxide. So we draw in our carbons, and then we can put in our oxygen here. And then we show the bond between those like that. And then up at the top here, here's my carbonyl carbon. So now there's only one bond between that carbon and this oxygen. There is a new bond that formed between that oxygen and that hydrogen, and there is an R group over here. And then there used to be only one bond to this oxygen, but another lone pair of electrons moved in to form a carbonyl here. So this is our other product, which you can see is a carboxylic acid. Let's color code these electrons so we can follow them a little bit better. So let's make these electrons in here, those electrons are going to form the bond on the left side between the carbon and the oxygen like that. Let's follow these electrons next. So now let's look at these electrons in here, the electrons in this pi bond. Those are the ones that are going to form this side of our epoxide ring like that. And let's make our oxygen-oxygen bond blue here. So the electrons in this bond, those are the ones that moved in here to form our carbonyl like that. And then let's go ahead and make these green right here, the electrons in this bond right here. These are the ones that moved out here to form the bond between our oxygen and our hydrogen. So our end result is to form a carboxylic acid and our epoxide. Let's look at a reaction, an actual reaction for the formation of epoxide, and then we'll talk about how to form a diol from that. So if we start with cyclohexene-- let's go ahead and draw cyclohexene in here. Let's do another one. That one wasn't very good. So we draw our cyclohexene ring like that. And to cyclohexene, we're going to add peroxyacetic acid. So what does peroxyacetic acid look like? Well, it's based on acetic acid. But it has one extra oxygen in there, so it looks like that. So that's our peroxyacetic acid. So we add cyclohexene to peroxyacetic acid, we're going to form an epoxide. So we're going to form a three-membered ring, including oxygen. I'm going to say the oxygen adds to the top face of our ring. It doesn't really matter for this example, but we'll go ahead and put in our epoxide using wedges here. And that must mean going away from us, those are hydrogens in space. So that's the epoxide that would form using the mechanism that we put above there. Let's go ahead and open this up epoxide using acid. So just to refresh everyone's memory, go back up here. Now we're going to look at this second part where we add H3O plus to form our diol. So let's take a look at that now. So we're going to add H3O plus to this epoxide. And I'm going to redraw our epoxide to give us a better view point here. So I'm going to put my oxygen right here, and then that's bonded to our two carbons like this. And then we see if we can draw the rest of the ring. And so in the back here, here is the rest of my cyclohexane ring like that. And we'll go ahead and put in our lone pairs of electrons. So this is the same exact drawing above here. Now I have my H3O plus in here like this, so my hydronium ion is present with a lone pair of electrons, giving us a plus one formal charge like that. So the oxygen on our epoxide is going to act as a base. It's going to take a proton. So this lone pair of electrons is going to take this proton right here, which would kick these electrons in here off onto my oxygen. So let's draw the result of that acid-base reaction. So I'm going to make a protonated epoxide. So let's go ahead and draw our oxygen here, and it's connected to those carbons down here. So I'll go ahead and draw the rest of my ring in the back here like that. And then one lone pair of electrons didn't do anything, so it's still there. One lone pair of electrons is the one that formed the bond on that proton, so this is my structure now. And this would give this oxygen a plus one formal charge, so it's positively charged now. So this is the same structure that we saw in the earlier videos, like with our cyclic halonium ion. And just like the cyclic halonium ion in those earlier videos-- check out the halohydrin video-- you're going to get a partial carbocation character with these carbons down here. So the resonance hybrid is going to give these carbons some partial positive character. So when water comes along as a nucleophile, the lone pair of electrons on water are going to be attracted to those carbons. So opposite charges attract. These two blue carbons are partially positive. The negative electrons are attracted to the partially positive carbon, and you're going to get nucleophilic attacks. So let's say this lone pair of electrons attacks right here. Well, that would kick the electrons in this bond off onto your oxygen. So let's go ahead and draw the result of an attack on the carbon on the left. So let's get some more room here. So what would happen in that instance? Well, let's go ahead and draw our cyclohexane ring back here. So here is our cyclohexane ring. The oxygen attacked the carbon on the left. So there is the oxygen that did the nucleophilic attack, so it has two hydrogens on it. It has one lone pair of electrons now, and it formed a plus one formal charge. Our epoxide opened. The electrons kicked off onto the top oxygen and that means that the top oxygen moves over here like that. So that would be your structure. Well, this lone pair electrons could have attacked this carbon, right? This carbon could have been the partially positive one in the resonance hybrid, which would kick these electrons off onto this oxygen. So let's go ahead and draw the result of that nucleophilic attack. So I'll go ahead and put in my cyclohexane ring like that. This time our oxygen is going to bond with a carbon on the left, two hydrogens attach to it, a lone pair of electrons, a positive one formal charge. And then this time the oxygen on top is going to kick off onto this oxygen over here in the left like that. So you're going to get an OH over there. So in the next step of the mechanism-- we're almost done, we've almost formed our diol. We're going to have water come along. And this time, instead of water acting as a nucleophile, water is going to act as a base. It's going to take a proton. So let's look at the product on the left, here. So this lone pair of electrons would take one of these protons, kick these electrons off onto your oxygen like that. So let's go ahead and draw the result of that acid-base reaction. So let's draw our cyclohexane ring like that, and now we have an OH down here. So this is now an OH, and this was an OH. So we've achieved our product. We've added 2 OHs anti to each other. Same thing can happen over here. You could grab these. You could grab this proton, kick these electrons off onto your oxygen like that. And so on the right, after we draw our cyclohexane ring, we're going to have an OH right here. And then we're going to have an OH over here like that. So we have two products. And if you look at them, they are mirror images of each other. If I were to put a mirror right here, you would see that they would be reflected in a mirror. And they are nonsuperimposable. So nonsuperimposable mirror images, enantiomers. So you're going to get two products for this reaction. So just to summarize this reaction, let's do it one more time. Let's start with the cyclohexene. And in the first step of our mechanism, the first of our reaction here, we added peroxyacetic acid. So we added CH3CO3H. And in the second step of our mechanism, in the second step of our reactions here, we added H3O plus. And so that opened up the epoxide that formed to form our diol, and we get two products. So this product over here on the left, let's go ahead and redraw this product over here on the left in a way that's a little bit more familiar. So once again, I put my eye right here. I stare down at this top carbon. If this is the top of my head, this OH is coming out at me. So I would draw that product. I would draw my cyclohexane ring, and at that top carbon I would show the OH coming out at me. And then, of course, at this carbon down here, the OH would be going away from me. So I go ahead and draw my OH as a dash here, down here. And then I do the same thing with this one right here. So if I stare down at that molecule, once again, if I stare down here and look this way, this time at this top carbon-- if my head's right here, top of my head, this OH would be down. So I go ahead and put a dash right here and put my OH. And then, over here, at this carbon it would be going up. So it might be easier to see that these are enantiomers when you look at them drawn like this. A different absolute configuration at both carbons. So this is how to form an epoxide and one way to make a diol. In the next video we will see another way to make a diol, although it will add in a different way to give you a slightly different product." + }, + { + "Q": "who's idea was the declaration of indepence to be made?", + "A": "It was Benjamin Franklin s idea to create a document that declared that the colonies wanted independence and freedom from Great Britain. Thomas Jefferson created the document with the advice of Benjamin Franklin and John Adams.", + "video_name": "er1L9BB6UoE", + "transcript": "Male: I'm here with Walter Isaacson and what are we about to talk about? Walter: We're talking about the Declaration of Independence, which happens, as it says up top, on July 4, 1776, but what we have to remember is that for more than a year, since April 1775 there had been a lot of fighting going on. There was a revolution happening, but up until this point the fighting was mainly against what they considered to be the acts of Parliament and Parliament's ministers and taxing the colonies. Finally, with this document they decide they're going to become ... The American colonies are going to become free and independent, a separate country, which means rebelling against the King himself, George the III. Male: And this is George the III in all of his regality. (laughing) Walter: Right. You know up until they met in Congress with the Continental Congress gathering themselves together they pretended at least to respect George the III, and they were blaming everything on the British Parliament, but it was a pretty difficult thing to decide you were going to overthrow the King himself. Male: And just to be clear, this Continental Congress it's easy now for us in hindsight. It seems like a very official thing, but this was really a rebel congress. It wasn't sanctioned in any way by kind of the formal government, by the government of Great Britain. Walter: Right, and as you see it says the 13 United States of America. This is the first time they really start using the phrase United States of America. They weren't really a country yet. They were 13 different colonies, and not all of them wanted to come to this Continental Congress. Getting them all together was quite difficult. They do so partly to help George Washington's troops get funded because they've started the skirmishes up in Lexington and Concord and Massachusetts, but by 1776 George Washington has a real army and they have to fund him, and eventually they'd figure out well this congress ought to decide are we really having a revolution? Are we trying to break away from the King himself? And the answer here is yes. Male: Wow. And so this is the beginning of the Declaration of Independence and these three fellows on the right-hand side look very familiar. Walter: They're on the committee that the Congress appoints to draft the Declaration. Actually there were five people on the committee, but these of course are the three most important. Thomas Jefferson only 33 years old, by far the youngest person on the committee and he's chosen to write the first draft. Then Benjamin Franklin, who is a mentor of Jefferson's, a printer from Philadelphia. Franklin had just been spending the past two decades almost going back and forth to England to try to prevent a revolution. Then John Adams, the very passionate sage from the State of Massachusetts who was the one who was most in favor of revolution. In fact, when Franklin comes back to Philadelphia in early 1776 after having tried to hold Britain and America together most of these people didn't know whether he would be on the side of revolution or not. In fact, his own son, William Franklin, is, at this time of this Declaration, the Royal Governor of New Jersey and is staying loyal to King George. Male: And just to, once again emphasize the context, the Royal Governor of New Jersey. This wasn't like the Governor of New Jersey He wasn't elected by the people. He was appointed. Walter: No, he was appointed by the King. He was the Royal Governor, and you know Franklin was proud of his son, but they have this incredible split starting in 1776 where William Franklin remains loyal to the crown and loyal to King George the III who had made him Governor of New Jersey. Male: Now, one thing that you had mentioned a few seconds ago that I think is surprising is when you mentioned that Jefferson was 33 years old when he wrote the Declaration of Independence. How normal was that? You know back then it seemed like people did I guess mature faster, but he was still perceived as a fairly young person. Walter: Yes. They all loved to be thinking of themselves as young rebels too. In fact, Jefferson I think is the second youngest person in the Continental Congress, but there was a third person who lied about his age to pretend to be younger and actually wasn't. Jefferson was a good wordsmith. He was from Virginia and it was very important of this person from ... Franklin from Pennsylvania and Adams from Massachusetts to make sure we got Virginia in because Virginia, there was a chance it would remain loyal and ... Male: It was a large wealthy ... Walter: A large wealthy land-owning colony and so getting the Virginians in, and there were very strong rebels from Virginia. The Lees of Virginia as well as Jefferson were in favor of declaring independence so they decide they want to make sure that Jefferson gets to write the first draft. Male: Interesting. And so what we see here, this is the final text. This is the official Declaration of Independence. Male: In the future videos we can talk about previous drafts. Walter: Right. They went through five drafts to get to this draft and this is the one that they do after unaninously all 13 colonies, now called the 13 United States in this document, declare this to be the cause of the colonies. And what you can see in the first paragraph is they have to explain why are we writing this document. They say well if you're going to have a revolution, if you're going to dissolve the political bands which have connected you with another state, then the laws of nature and of nature's God entitle them a decent respect. This is the equal station they get because of nature and nature's God. It's a pretty interesting phrase. Male: Let's read the whole thing. Male: \"When in the course of human events it becomes necessary for one people to dissolve the political bands,\" this is what you were talking about, \"which have connected them with another and to assume among the powers of the earth the separate and equal station which the laws of nature and of nature's God entitle them.\" Walter: What they're saying right there is that the laws of nature and the fact that nature's God created us all equally means that one set of people don't have to be subserviant to or occupied by or colonies of another set of people. They want to be free and independent, and it's interesting that they use the phrase, \"Laws of nature and nature's God,\" because this is sort of the beginning of the enlightenment where we're supposed to understand that nature gives us our rights and reasons. John Locke, the great British philosopher, believed in the laws of nature, and these were deists. They kind of were religious a bit, but they didn't subscribe to any particular religious dogma, and so they just talk about nature's God allowing us all to be free and equal. Male: Right. \"A decent respect to the opinions of mankind requires that they should declare the causes which impel them to a separation.\" So that's really just saying ... Walter: What they're saying is we care what the rest of you think. And by the way, it's directed at one particular people, the French, because we're not going to win this revolution in the United States unless the French help support us. The French, by the way, were already at war with the British. There's been a long set of wars throughout the 18th century where France and Britian were fighting each other. So, this document is particularly aimed at saying to the French, you've been fighting Britain for a long time so we have a decent respect for all the opinions of mankind, but yours in particular and we're going to tell you why we're fighting this and of course in France at the time this notion of liberty, equality, fraternity, that's bubbling up as well. So, the document is to try to persuade other nations please support us. We're explaining why we're having this revolution. Male: And that's I think important to remember for someone in 21st century America. It's obviously a major world power now, the major world power. But back then this was like a little colony. It's kind of a sideshow. Walter: Right, right. And it's important for us to remember now too that whenever we do something, whenever we get involved in the world we should have a decent respect to the opinions of mankind. Walter: That's how we started as a nation saying when we do something we're going to be open. We're going to be honest. We're going to explain to you why." + }, + { + "Q": "I tried to solve this using the following equation:\n60^2 = 50^2 + 20^2 - 2(50)(20)(cos\u00ce\u00b8)\nCan anybody explain why this didn't work? Are the values for a, b, and c in the video the only ones that work in the equation?", + "A": "your equation solves for the angle between the purple and yellow segments, not the angle in the video", + "video_name": "Ei54NnQ0FKs", + "transcript": "Voiceover:Let's say you're studying some type of a little hill or rock formation right over here. And you're able to figure out the dimensions. You know that from this point to this point along the base, straight along level ground, is 60 meters. You know the steeper side, steeper I guess surface or edge of this cliff or whatever you wanna call it, is 20 meters. And then the longer side here, I guess the less steep side, is 50 meters long. So you're able to measure that. But now what you wanna do is use your knowledge of trigonometry, given this information, to figure out how steep is this side. What is the actual inclination relative to level ground? Or another way of thinking about it, what is this angle theta right over there? And I encourage you to pause the video and think about it on your own. Well it might be ringing a bell. Well you know three sides of a triangle and then we want to figure out an angle. And so the thing that jumps out in my head, well maybe the law of cosines could be useful. Let me just write out the law of cosines, before we try to apply it to this triangle right over here. So the law of cosines tells us that C-squared is equal to A-squared, plus B-squared, minus two A B, times the cosine of theta. And just to remind ourselves what the A, B's, and C's are, C is the side that's opposite the angle theta. So if I were to draw an arbitrary triangle right over here. And if this is our angle theta, then this determines that C is that side, and then A and B could be either of these two sides. So A could be that one and B could be that one. Or the other way around. As you can see, A and B essentially have the same role in this formula right over here. This could be B or this could be A. So what we wanna do is somehow relate this angle... We wanna figure out what theta is in our little hill example right over here. So if this is going to be theta, what is C going to be? Well C is going to be this 20 meter side. And then we could set either one of these to be A or B. We could say that this A is 50 meters and B is 60 meters. And now we could just apply the law of cosines. So the law of cosines tells us that 20-squared is equal to A-squared, so that's 50 squared, plus B-squared, plus 60 squared, minus two times A B. So minus two times 50, times 60, times 60, times the cosine of theta. This works out well for us because they've given us everything. There's really only one unknown. There's theta here. So let's see if we can solve for theta. So 20 squared, that is 400. 50 squared is 2,500. 60 squared is 3,600. And then 50 times... Let's see, two times 50 is 100, times 60, this is all equal to 6,000. So let's see, if we simplify this a little bit we're going to get 400 is equal to 2,500 plus 3,600. Let's see, that'd be 6,100. That's equal to 6,000... Let me do this in a new color. So when I add these two, I get 6,100. Did I do that right? So it's 2,000 plus 3,000, plus 5,000. 500 plus 600 is 1,100. So I get 6,100 minus 6,000, times the cosine of theta. And let's see, now we can subtract 6,100 from both sides. So I'm just gonna subtract 6,100 from both sides so that I get closer to isolating the theta. So let's do that. So this is going to be negative 5,700. Is that right? 5,700 plus... Yes, that is right. Right, because if this was the other way around, if this was 6,100 minus 400 it would be positive 5,700. Alright. And then these two of course cancel out. And this is going to be equal to negative 6,000 times the cosine of theta. Now we can divide both sides by negative 6,000. And we get... I'm just gonna swap the sides. We get cosine of theta is equal to... Let's see we could divide the numerator and the denominator by essentially negative 100. So these are both going to become positive. So cosine of theta is equal to 57 over 60. And actually that can be simplified even more. Three goes into 57, is that 19 times? Yep, so this is actually... This could be simplified. This is equal to 19 over 20. We actually didn't have to do that simplification step because we're about to use our calculators, but that makes the math a little more tractable. Right, 3 goes into 57, yeah, 19 times. And so now we can take the inverse cosine of both sides. So we could get theta is equal to the inverse cosine, or the arc cosine, of 19 over 20. So let's get our calculator out and see if we get something that makes sense. So we wanna do the inverse cosine of 19 over 20. And we deserve a drum roll. We get 18.19 degrees. And I already verified that my calculator is in degree mode. So it gets 18.19 degrees. So if we wanted to round, this is approximately equal to 18.2 degrees, if we wanna round to the nearest tenth. So that essentially gives us a sense of how steep this slope actually is." + }, + { + "Q": "During WWII Was china still in the civil war with it self or has it stopped when WWII happened?", + "A": "The Chinese civil war began in 1927 and was both before and after World War II. It wasn t considered over until 1950. Aggression between Japan and China was an ongoing series of incidents that became a total war in 1937 and ended with the surrender of Japan to the Allies in 1945.", + "video_name": "-kKCjwNvNkQ", + "transcript": "World War II was the largest conflict in all of human history. The largest and bloodiest conflict And so you can imagine it is quite complex My goal in this video is to start giving us a survey, an overview of the war. And I won't even be able to cover it all in this video. It is really just a think about how did things get started. Or what happened in the lead up? And to start I am actually going to focus on Asia and the Pacific. Which probably doesn't get enough attention when we look at it from a western point of view But if we go back even to the early 1900s. Japan is becoming more and more militaristic. More and more nationalistic. In the early 1900s it had already occupied... It had already occupied Korea as of 1910. and in 1931 it invades Manchuria. It invades Manchuria. So this right over here, this is in 1931. And it installs a puppet state, the puppet state of Manchukuo. And when we call something a puppet state, it means that there is a government there. And they kind of pretend to be in charge. But they're really controlled like a puppet by someone else. And in this case it is the Empire of Japan. And we do remember what is happening in China in the 1930s. China is embroiled in a civil war. So there is a civil war going on in China. And that civil war is between the Nationalists, the Kuomintang and the Communists versus the Communists The Communists led by Mao Zedong. The Kuomintang led by general Chiang Kai-shek. And so they're in the midst of the civil war. So you can imagine Imperial Japan is taking advantage of this to take more and more control over parts of China And that continues through the 30s until we get to 1937. And in 1937 the Japanese use some pretext with, you know, kind of a false flag, kind of... well, I won't go into the depths of what started it kind of this Marco-Polo Bridge Incident But it uses that as justifications to kind of have an all-out war with China so 1937...you have all-out war and this is often referred to as the Second Sino-Japanese War ...Sino-Japanese War Many historians actually would even consider this the beginning of World War II. While, some of them say, ok this is the beginning of the Asian Theater of World War II of the all-out war between Japan and China, but it isn't until Germany invades Poland in 1939 that you truly have the formal beginning, so to speak, of World War II. Regardless of whether you consider this the formal beginning or not, the Second Sino-Japanese War, and it's called the second because there was another Sino-Japanese War in the late 1800s that was called the First Sino-Japanese War, this is incredibly, incredibly brutal and incredibly bloody a lot of civilians affected we could do a whole series of videos just on that But at this point it does become all-out war and this causes the civil war to take a back seat to fighting off the aggressor of Japan in 1937. So that lays a foundation for what's happening in The Pacific, in the run-up to World War II. And now let's also remind ourselves what's happening... what's happening in Europe. As we go through the 1930s Hitler's Germany, the Nazi Party, is getting more and more militaristic. So this is Nazi Germany... Nazi Germany right over here. They're allied with Benito Mussolini's Italy. They're both extremely nationalistic; they both do not like the Communists, at all You might remember, that in 1938... 1938, you have the Anschluss, which I'm sure I'm mispronouncing, and you also have the takeover of the Sudetenland in Czechoslovakia. So the Anschluss was the unification with Austria and then you have the Germans taking over the of Sudetenland in Czechoslovakia and this is kind of the famous, you know, the rest of the, what will be called the Allied Powers kind of say, \"Okay, yeah, okay maybe Hitler's just going to just do that... well we don't want to start another war. We still all remember World War I; it was really horrible. And so they kind of appease Hitler and he's able to, kind of, satisfy his aggression. so in 1938 you have Austria, Austria and the Sudetenland ...and the Sudetenland... are taken over, are taken over by Germany and then as you go into 1939, as you go into 1939 in March they're able to take over all of Czechoslovakia they're able to take over all of Czechoslovakia and once again the Allies are kind of, they're feeling very uncomfortable, they kind of, have seen something like this before they would like to push back, but they still are, kind of, are not feeling good about starting another World War so they're hoping that maybe Germany stops there. So let me write this down... So all of Czechoslovakia... ...Czechoslovakia... is taken over by the Germans. This is in March of 1939. And then in August you have the Germans, and this is really in preparation for, what you could guess is about to happen, for the all-out war that's about to happen the Germans don't want to fight the Soviets right out the gate, as we will see, and as you might know, they do eventually take on the Soviet Union, but in 1939 they get into a pact with the Soviet Union. And so this is, they sign the Molotov\u2013Ribbentrop Pact with the Soviet Union, this is in August, which is essentially mutual non-aggression \"Hey, you know, you do what you need to do, we know what we need to do.\" and they secretly started saying \"Okay were gonna, all the countries out here, we're going to create these spheres of influence where Germany can take, uh, control of part of it and the Soviet Union, and Stalin is in charge of the Soviet Union at this point, can take over other parts of it. And then that leads us to the formal start where in September, let me write this in a different color... so September of 1939, on September 1st, Germany invades Poland Germany invades Poland on September 1st, which is generally considered the beginning of World War II. and then you have the Great Britain and France declares war on Germany so let me write this World War II... starts everyone is declaring war on each other, Germany invades Poland, Great Britain and France declare war on Germany, and you have to remember at this point Stalin isn't so concerned about Hitler he's just signed the Molotov-Ribbentrop Pact and so in mid-September, Stalin himself invades Poland as well so they both can kind of carve out... ...their spheres of influence... so you can definitely sense that things are not looking good for the world at this point you already have Asia in the Second Sino-Japanese War, incredibly bloody war, and now you have kind of, a lot of very similar actors that you had in World War I and then they're starting to get into a fairly extensive engagement." + }, + { + "Q": "what are radicals and free radicals?", + "A": "Radicals and free radicals are the same thing. In organic chemistry, they generally refer to atoms or molecules with an unpaired electron.", + "video_name": "nQ7QSV4JRSs", + "transcript": "We've already seen alcohols in many of these videos, but I thought it was about time that I actually made a video on alcohols. Now, alcohols is the general term for any molecule that fits the pattern some type of functional group or chain of carbons OH. And they use the letter R. And I've used it before. R stands for radical. And I don't want you to confuse this R with free radical. It means completely different things. R in this form really just means a functional group or a chain of carbons here. It doesn't mean a free radical. This just means it could be just something attached to this OH right there. Now another point of clarification, do not think that anything that fits this pattern is drinkable. Do not associate it with the traditional alcohol that you may or may not have been exposed to. Traditional drinking alcohol is actually ethanol. Alcohol is actually-- let me write out the molecular formula. CH3, CH2, and then OH. This is what is inside of wine and beer and hard liquor, or whatever you might want. You do not want to drink and maybe you might not actually want to drink this either, but you definitely do not want to drink something like methanol. It might kill you. So you do not want to do something like this. You do not want to ingest that. Might kill or blind you. This might do it in a more indirect way. So I want to get that out of the way and just so that we get kind of a little bit more comfortable with alcohols, and we've seen them involved in other reactions. We've seen hydroxides act as nucleophiles and Sn2 substitution reactions create alcohols. But I want to do is just learn to get comfortable and really make sure we know how to name these things. So let's just name these molecules that I drew right before I pressed record right over here. So over here, like everything else, we always want to define the longest carbon chain. We have 1, 2, 3, 4, 5 carbons. So it's going to be pent. And there's no double bonds. So I'll just write pentane right then. And we're not going to just write a pentane because actually, the fact that makes it an alcohol, that takes precedence over the fact that it is an alkane. So it actually, the suffix of the word will involve the alcohol part. So it is pentanol. That tells us that's an alcohol. And to know where the OH is grouped, we'll start numbering closest to the OH. So 1, 2, 3, 4, 5. Sometimes it'll be called 2-pentanol. And this is pretty clear because we only have one group here, only one OH. So we know that that is what the 2 applies to. But a lot of times, if people want to be a little bit more particular, they might write pentan-2-ol. And this way is more useful, especially if you have multiple functional groups. So you know exactly where they sit. This one is harder to say. 2-pentanol is pretty straightforward. Now let's try the name this beast right over here. So we have a couple of things going on. This is an alkyne. We have a triple bond. It's an alkyne. We have two bromo groups here. And it's also an alcohol. And alcohol takes precedence on all of them. So we want to start numbering closest to the alcohol. So we want to start numbering from this end of the carbon chain. And we have 1, 2, 3, 4, 5, 6, 7, 8 carbons. We want to call it an octyne. But because we have an alcohol there, we want to call this an octyne-- let me make it very clear. So oct tells us that we have 8 carbons. Now we have to specify where that triple bond is. The triple bond is on the 5 carbon. You always specify the lower number of the carbons on that So it is oct-5-yn. That tells us that's where the triple bond is. And then we have the OH on the 4 carbon. So 4-al. And now we have these two bromo groups here on the 7 carbon. So it's 7,7-dibromo oct-5-yn-4-al. And this would all be one word. Let me make sure that you realize that I just ran out of space. So that's probably about as messy of a thing you'll have to name, but just showing you that these things can be named. Now let's think about this one over here in green. So we have 1, 2, 3, 4, 5, 6 carbons. So it's going to be a hex. And they're all single bonds, so it's a hexane. It's a cyclohexane. But then of course, the hydroxide or the hydroxy group I should call it, takes dominance. It's a hexanol. So this is a cyclohexanol. And once again, that comes from the OH right there. And you don't have to number it. Because no matter what carbon it's on, it's on the same one. If you had more than one of these OH groups, then we would have to worry about numbering them. Let's just do this one right over here. So once again, what is our carbon chain? We have 1, 2, 3 carbons. And we have the hydroxy group attached to the 1 and the 3 carbon. Prop is our prefix. It is an alkane. So we would call this-- and there's a couple of ways to do this. We could call this 1 comma 3 propanediol. Actually, I don't have to put a dash their. Propanediol. And over here, we would add the E because we have the D right there. So it's propanediol. If it wasn't diol, it would be propanal. You wouldn't have the E, D and the I there. So this would specify we're at the 1 and the 3 carbons. We have the hydroxy group. Or this could also be written as propane- 1, 3- diol. And once again, the di is telling us that we have two of the hydroxy groups attached to this thing. But either of these things are ways that you would see this molecule named." + }, + { + "Q": "What about Canada?", + "A": "This video focuses on US Native American history because this is US history. This does sometimes cut off important knowledge about First Nations in Canada (regional term for aboriginal peoples), since many tribes moved across what we now think of as borders for those two countries. Because the two countries and colonizing forces had different approaches to treaties, aggression and settlement, other than acknowledging a connection, the history relating to the two countries diverges in many areas.", + "video_name": "o2XjXFvruIM", + "transcript": "- It is believed that the first humans settled North and South America, or began to settle it, about 15 to 16,000 years ago and the mainstream theory is that they came across from northeast Asia, across the Bering Strait, during the last glaciation period, when sea levels were lower and there was a land bridge, the famous Bering Land Bridge connecting the two continents and we have archeological evidence of humans in southern Chile as early as 14,500 years ago and as well in Florida as early as 14,500 years ago. So humans had migrated into, and settled in the Americas many, many, many thousands of years ago. And like other places in the world, they followed similar development patterns. The first evidence we have of the development of agriculture in the Americas is about seven to eight or nine thousand years ago so once again, it coincides with when agriculture, we believe, started to emerge in other parts of the world. And the more archeological evidence we find, we'll probably find dates that go even further back than that, in fact, I've seen some that go eight, nine thousand years ago. Now one misconception, significant misconception, about the Americas is that when the Europeans colonized, remember Columbus comes sailing in 1492, looking for the East Indies and then he bumps into this, he actually doesn't bump into the whole continent, he bumps into an island that's close to the continent, but with that you start having the beginning of the European colonization of the Americas, roughly the last 500 years, and one misconception that folks often have is, well it was maybe sparsely populated, mainly by hunter-gatherer nomadic people and nothing could be further from the truth. The modern estimates of the population of the Americas at the time of the European colonization, roughly around 1500 is 50-100 million people and to put that in perspective, so that's right around there, that's about 10 to 20% of the world population at that time. The world population at that time was about 500 million people and given that the Americas is about one third of the land, if you don't count Antarctica, it's not that different of a population density than the other continents, and we have significant cities that were in place in the pre-Columbian era, in the era before Columbus and the European colonization. For example, you might have heard of the Aztecs, this really, the core, the Mexica people, the Mexica tribe, in many ways the foundations, of the Mexican people pre-European colonization. You might also be familiar with the Mayan civilization, one of the longest lasting civilizations in, actually, in history, they're famous for one of the earliest cultures where we have the hieroglyphics, where we have writing. You're probably familiar with the Inca Empire and yes, that is me on a recent trip and at the time of the Inca Empire it is believed that it was possibly the largest empire on the earth at that time, incredibly complex structures and social structures, they had. Now what's often less talked about are things like the Mississippian culture, which was in North America right over here. The Mississippi River is named for them. This if their famous city of Cahokia near St. Louis and in there, in that peak, it would have 40,000 people in it. Around the world at that time, at the time of the Mississippian culture, there weren't many cities in the world that had 40,000 people, so it wasn't these, just hunter-gatherers and people who were nomadic, there were sophisticated civilizations, with sophisticated cultures and dense population centers and it had also been in place for a long time, similar, in timeframe, to some of the great ancient civilizations that we see in Mesopotamia, the Indus Valley, and in China. For example, the oldest civilization we know of, in Mesoamerica, is the Olmec civilization, right over here, here's a few of their artifacts they have. If we go into the Andes, near modern day Peru, we have the Chavin culture, right over there. As you can see, a lot of these cultures, at least the ones that I'm putting here, and this is just a sample, I'm sampling some around North America, some around Mesoamerica, and some in the Andes, and then you can even go further back and you can go to the Caral civilization, and what's really interesting about the Caral civilization is some archeologists call this the first civilization and it's unclear whether they had, whether they farmed grains and cereals that we often associate with civilizations, they use their surplus crops to have a more specialized labor force, but they were a maritime culture, even today the coast of Peru is a significant source of all of the, or a good chunk of the seafood in the world but a significant culture developed there, these are the remnants of their pyramids, and they developed, we believe, in the 4th millennium BCE, so this is around the same time as when Egypt first got unified around Menes, by Menes, or you have the first Sumerians in Mesopotamia and as far back as them you have these Quipus knots, which many archeologists view as a form of writing, it was a form of record-keeping and it was even used later on by the Incas. So the big take away here, is to challenge that misconception that the Americas somehow were not as, has populations and civilizations like everything else, it was only when the Europeans came in that all of that started to happen, no. Well before the Europeans came in, North and South America had been settled, agriculture developed at a similar timescale, significant, complex civilizations, writing developed on a similar timescale, but once you have the European colonization, some people say it was intentional, it was probably a combination of intentional and just diseases that were unfamiliar to the people here, within 150 years, that 50 to 100 million population, so now we're talking about, roughly by 1650, so you move a little bit forward in time, the population had gone to roughly six million people, some people refer it to a genocide, some people would say it's a combination of an intentional extermination of people plus just inadvertent disease, whatever it is, this was the significant decline of a complex and diverse set of populations. This is just a small sample of the major civilizations that were there, you had thousands of tribes across North and South America that had different cultures, different languages, different traditions and different religions." + }, + { + "Q": "Why can't 9.564 round up to 9.600 instead of 9.6? Doesn't 564 round up to 600?\nI'm confused!", + "A": "It can round up even if it has those 0s, but the 0s don t add anything since they are place holders.", + "video_name": "_MIn3zFkEcc", + "transcript": "0.710 Round 9.564, or nine and five hundred sixty-four Thousandths, to the nearest tenth. So lie me write it a bit larger, 9.564 And we need to round to the nearest tenth. So what's the tenth place? The tenths place is right here This represents 5 tenths. This is the ones place, this is the tenths place, this is The hundredths place, and this is the Thousandths place right here So we need to round to the nearest tenth. So if we round up, this will be 9.6 If we round down, this will be 9.5 And just like regular rounding, when we're not Dealing with decimals, you move to one spot, or you look At one place to the right or one place lower, I guess, and You say is that 5 larger If it is, you round if it isn't,you round down 6 is definitely 5 or larger, so we want to round up. So this 9.564 becomes 9.6, or we can call this Nine and sixth tenths. And then we're done!" + }, + { + "Q": "Many times in this video, Sal says the word \"annex,\" or \"annexed.\" What does that mean?", + "A": "annex - to colonize or add on If a country annexes another that is next to it, it adds that country to itself.", + "video_name": "heKuwogLwnk", + "transcript": "We're now ready to talk about one of the most famous events in all of world history that really was the trigger for World War I, or the Great War, as it was called back then. So just as a little bit of backdrop, in 1908, the Austro-Hungarian Empire formally annexes Bosnia and Herzegovina. It had already been occupying it since the late 1800s, since the Ottomans were being pushed out. But then in 1908, it formally annexes it. And just as a little bit more backdrop, as the Ottomans were being pushed out of the Balkans, it helped rekindle or bring about more hope of unifying the Yugoslavic people, the southern Slavic people. When people talk about Yugoslav, they're literally talking about the southern Slavs. So that literally means southern. So you had these nationalistic hopes. But now in 1908, it was already being occupied. A significant state, that would be part of a potential future Yugoslav, was now formally annexed by the Austro-Hungarians. Now, you also had an independent kingdom of Serbia right here. And you can imagine that this was the home base of the nationalistic movement. If only they could add the other southern Slavic states to this, it could one day turn into a greater Yugoslavia. So in that context, we get to 1914. So let me draw a little line here. So we're getting to 1914. June 28, which is one of the most famous dates in all of history. And you have the Archduke Franz Ferdinand and his wife, Sophie. They're visiting Sarajevo which is now in annexed Bosnia. And when they are there, there is a ploy. There is a scheme to assassinate them, from a group-- they're called the Young Bosnians. They have ties to the Black Hand, which is this nationalistic group. That has ties, many, many people say-- all these things are all very shady and behind the back, behind the scenes. But it has ties to elements in the kingdom of Serbia. They attempt to assassinate Archduke Franz Ferdinand. And it's actually a fascinating story because the initial assassination attempt is completely, completely botched. There's even one case of a guy, one of the guys who tried to be an assassin when it gets botched, he tries to bite on a cyanide capsule and then jump into a river. The cyanide capsule had gone bad. The river was only 10 inches deep. And so they were able to get their hands on him. And one of the conspirators, Gavrilo Princip-- at this point, once the whole thing was botched, he gives up on the whole assassination attempt. And he's having, literally, a sandwich at a cafe in Sarajevo, thinking about how botched their whole attempt was. And while that was happening, a mistake on the part of those planning Archduke Franz Ferdinand's route as he was traveling within Sarajevo has them driving right near Gavrilo Princip. So he sees, all of a sudden, that they've taken the wrong route, that they're driving right by him again. Remember, his people already knew that there was an assassination attempt on him earlier in the day. So they should have been more careful. Now, Gavrilo Princip gets up, puts his sandwich down, and starts walking over to where he sees Archduke Franz Ferdinand and Sophie's car going. Now, the drivers, once they realized that they had made a mistake, they had taken a less safe route. They tried to back up, which makes things even worse because then the car starts stalling. And Gavrilo Princip literally walks up to the car and is able to shoot Archduke Franz Ferdinand and Sophie. And just to give you a sense of how important this is, Archduke Franz Ferdinand of Austria is the heir. He's the nephew of Franz Josef, who was the ruler of Austria-Hungary. And so he is the heir to the empire. And so he gets assassinated by Gavrilo Princip. So Franz Ferdinand assassinated by Gavrilo Princip. And we have right over here a picture right after Gavrilo Princip-- I believe this is Gavrilo Princip right over here, right after he was arrested. And just to get a little sense of how this was tied to this whole Yugoslavian nationalistic movement. This is what he said once he was arrested. \"I am a Yugoslav nationalist, aiming for the unification of all Yugoslavs, and I do not care what form of state, but it must be free of Austria.\" So this act, this assassination motivated by a nationalistic movement, motivated by a desire to maybe merge Bosnia and Herzegovina with Serbia and maybe eventually Croatia, with Bosnia and Herzegovina and Serbia. This assassination, as we'll see in the next video, is the trigger for all of World War I. And the reason why it triggers it is because, well, there's many things you can cite. You could argue that many of the empires in Europe were already militarizing, already had a desire for conflict. But then you also had all of these alliances that essentially allowed the dominoes to fall in all of Europe. And because they had these empires, essentially much of the world to be at war with each other." + }, + { + "Q": "Why do we need proofs in the first place, and why do they have to be in a certain format?", + "A": "Without proofs there is no way to know if these theories are valid or applicable. Proofs are like a mathematical way to show that things work 100% or don t work at all. Proofs are a pain in the butt, but they are an essential way of knowing what is real. Proofs have to be in a certain format, because proofs are a formal way to display mathematical knowledge in a clear and concise format.", + "video_name": "rcBaqkGp7CA", + "transcript": "In case you haven't noticed, I've gotten somewhat obsessed with doing as many proofs of the Pythagorean theorem as I can do. So let's do one more. And like how all of these proofs start, let's construct ourselves a right triangle. So I'm going to construct it so that its hypotenuse sits on the bottom. So that's the hypotenuse of my right triangle. Try to draw it as big as possible, so that we have space to work with. So that's going to be my hypotenuse. And then let's say that this is the longer side that's not the hypotenuse. We can have two sides that are equal. But I'll just draw it so that it looks a little bit longer. Let's call that side length a. And then let's draw this side right over here. It has to be a right triangle. So maybe it goes right over there. That's side of length b. Let me extend the length a a little bit. So it definitely looks like a right triangle. And this is our 90-degree angle. So the first thing that I'm going to do is take this triangle and then rotate it counterclockwise by 90 degrees. So if I rotate it counterclockwise by 90 degrees, I'm literally just going to rotate it like that and draw another completely congruent version of this one. So I'm going to rotate it by 90 degrees. And if I did that, the hypotenuse would then sit straight up. So I'm going to do my best attempt to draw it almost to scale as much as I can eyeball it. This side of length a will now look something like this. It'll actually be parallel to this over here. So let me see how well I can draw it. So this is the side of length a. And if we cared, this would be 90 degrees. The rotation between the corresponding sides are just going to be 90 degrees in every case. That's going to be 90 degrees. That's going to be 90 degrees. Now, let me draw side b. So it's going to look something like that or the side that's length b. And this and the right angle is now here. So all I did is I rotated this by 90 degrees counterclockwise. Now, what I want to do is construct a parallelogram. I'm going to construct a parallelogram by essentially-- and let me label. So this is height c right over here. Let me do that white color. This is height c. Now, what I want to do is go from this point and go up c as well. Now, so this is height c as well. And what is this length? What is the length over here from this point to this point going to be? Well, a little clue is this is a parallelogram. This line right over here is going to be parallel to this line. It's maintained the same distance. And since it's traveling the same distance in the x direction or in the horizontal direction and the vertical direction, this is going to be the same length. So this is going to be of length a. Now, the next question I have for you is, what is the area of this parallelogram that I have just constructed? Well, to think about that, let's redraw this part of the diagram so that the parallelogram is sitting on the ground. So this is length a. This is length c. This is length c. And if you look at this part right over here, it gives you a clue. I'll use this green color. The height of the parallelogram is given right over here. This side is perpendicular to the base. So the height of the parallelogram is a as well. So what's the area? Well, the area of a parallelogram is just the base times the height. So the area of this parallelogram right over here is going to be a squared. Now, let's do the same thing. But let's rotate our original right triangle. Let's rotate it the other way. So let's rotate it 90 degrees clockwise. And this time, instead of pivoting on this point, we're going to pivot on that point right over there. So what are we going to get? So the side of length c if we rotate it like that, it's going to end up right over here. I'll try to draw it as close to scale as possible. So that side has length c. Now, the side of length of b is going to pop out and look something like this. It's going to be parallel to that. This is going to be a right angle. So let me draw it like that. That looks pretty good. And then the side of length a is going to be out here. So that's a. And then this right over here is b. And I wanted to do that b in blue. So let me do the b in blue. And then this right angle once we've rotated is just sitting right over here. Now, let's do the same exercise. Let's construct a parallelogram right over here. So this is height c. This is height c as well. So by the same logic we used over here, if this length is b, this length is b as well. These are parallel lines. We're going the same distance in the horizontal direction. We're rising the same in the vertical direction. We know that because they're parallel. So this is length b down here. This is length b up there. Now, what is the area of this parallelogram right over there? What is the area of that parallelogram going to be? Well, once again to help us visualize it, we can draw it sitting flat. So this is that side. Then you have another side right over here. They both have length b. And you have the sides of length c. So that's c. That's c. What is its height? Well, you see it right over here. Its height is length b as well. It gives us right there. We know that this is 90 degrees. We did a 90-degree rotation. So this is how we constructed the thing. So given that, the area of a parallelogram is just the base times the height. The area of this parallelogram is b squared. So now, things are starting to get interesting. And what I'm going to do is I'm going to copy and paste this part right over here, because this is, in my mind, the most interesting part of our diagram. Let me see how well I can select it. So let me select this part right over here. So let me copy. And then I'm going to scroll down. And then let me paste it. So this diagram that we've constructed right over here, it's pretty clear what the area of it is, the combined diagram. And let me delete a few parts of it. I want to do that in black so that it cleans it up. So let me clean this thing up, so we really get the part that we want to focus on. So cleaning that up and cleaning this up, cleaning this up right over there. And actually, let me delete this right down here as well, although we know that this length was c. And actually, I'll draw it right over here. This was from our original construction. We know that this length is c. We know this height is c. We know this down here is c. But my question for you is, what is the area of this combined shape? Well, it's just a squared plus b squared. Let me write that down. The area is just a squared plus b squared, the area of those two parallelograms. Now, how can we maybe rearrange pieces of this shape so that we can express it in terms of c? Well, it might have jumped out at you when I drew this line right over here. I want to do that in white. We know that this part right over here is of length c. This comes from our original construction. I lost my diagram. This is of length c. That's of length c. And then this right over here is of length c. And so what we could do is take this top right triangle, which is completely congruent to our original right triangle, and shift it down. So remember, the entire area, including this top right triangle, is a squared plus b squared. But we're excluding this part down here, which was our original triangle. But what happens if we take that? So let me actually cut. And then let me paste it. And all I'm doing is I'm moving that triangle down here. So now, it looks like this. So I've just rearranged the area that was a squared b squared. So this entire area of this entire square is still a squared plus b squared. a squared is this entire area right over here. It was before a parallelogram. I just shifted that top part of the parallelogram down. b squared is this entire area right over here. Well, what's this going to be in terms of c? Well, we know that this entire thing is a c by c square. So the area in terms of c is just c squared. So a squared plus b squared is equal to c squared. And we have, once again, proven the Pythagorean theorem." + }, + { + "Q": "Unfortunately, he did the right hand rule incorrectly. The fact that the dot with the circle means magnetic field facing \"out\" (which means our fingers point outward) and the velocity, according to what he has down, is to the right (which means we'd have our thumb facing to the right if we were looking at our own hand). That leaves us with a palm facing upward, which means our force is upward. Just remember the following: thumb is direction of 'velocity' or 'charge', fingers are mag. field...", + "A": "You sir, are incorrect. He used it correctly. Right hand rule for force on an electron. Left hand rule for force on a conventional current.", + "video_name": "LTuGQy4rmmo", + "transcript": "In the last video we learned-- or at least I showed you, I don't know if you've learned it yet, but we'll learn it in But we learned that the force on a moving charge from a magnetic field, and it's a vector quantity, is equal to the charge-- on the moving charge-- times the cross product of the velocity of the charge and the magnetic field. And we use this to show you that the units of a magnetic field-- this is not a beta, it's a B-- but the units of a magnetic field are the tesla-- which is abbreviated with a capital T-- and that is equal to newton seconds per coulomb meters. So let's see if we can apply that to an actual problem. So let's say that I have a magnetic field, and let's say it's popping out of the screen. I'm making this up on the fly, so I hope the numbers turn out. It's inspired by a problem that I read in Barron's AP calculus book. So if I want to draw a bunch of vectors or a vector field that's popping out of the screen, I could just do the top of the arrowheads. I'll draw them in magenta. So let's say I have a vector field. So you can imagine a bunch of arrows popping out of the screen. I'll just draw a couple of them just so you get the sense that it's a field. It pervades the space. These are a bunch of arrows popping out. And the field is popping out. And the magnitude of the field, let's say it is, I don't know, let's say it is 0.5 teslas. Let's say I have some proton that comes speeding along. And it's speeding along at a velocity-- so the velocity of the proton is equal to 6 times 10 to the seventh meters per second. And that is actually about 1/5 of the velocity or 1/5 of the speed of light. So we're pretty much in the relativistic realm, but we won't go too much into relativity because then the mass of the proton increases, et cetera, et cetera. We just assume that the mass hasn't increased significantly at this point. So we have this proton going at a 1/5 of the speed of light and it's crossing through this magnetic field. So the first question is what is the magnitude and direction of the force on this proton from this magnetic field? Well, let's figure out the magnitude first. So how can we figure out the magnitude? Well, first of all, what is the charge on a proton? Well, we don't know it right now, but my calculator has And if you have a TI graphing calculator, your calculator would also have it stored in it. So let's just write that down as a variable right now. So the magnitude of the force on the particle is going to be equal to the charge of a proton-- I'll call it Q sub p-- times the magnitude of the velocity, 6 times 10 to the seventh meters per second. We're using all the right units. If this was centimeters we'd probably want to convert it to meters. 6 times 10 to the seventh meters per second. And then times the magnitude of the magnetic field, which is 0.5 soon. teslas-- I didn't have to write the units there, but I'll do it there-- times sine of the angle between them. I'll write that down right now. But let me ask you a question. If the magnetic field is pointing straight out of the screen-- and you're going to have to do a little bit of three-dimensional visualization now-- and this particle is moving in the plane of the field, what is the angle between them? If you visualize it in three dimensions, they're actually orthogonal to each other. They're at right angles to each other. Because these vectors are popping out of the screen. They are perpendicular to the plane that defines the screen, while this proton is moving within this plane. So the angle between them, if you can visualize it in three dimensions, is 90 degrees. Or they're perfectly perpendicular. And when things are perfectly perpendicular, what is the sine of 90 degrees? Or the sine of pi over 2? Either way, if you want to deal in radians. Well, it's just equal to 1. The whole-- hopefully-- intuition you got about the cross product is we only want to multiply the components of the two vectors that are perpendicular to each other. And that's why we have the sine of theta. But if the entire vectors are perpendicular to each other, then we just multiply the magnitude of the vector. Or if you even forget to do that, you say, oh well, they're perpendicular. They're at 90 degree angles. Sine of 90 degrees? Well, that's just 1. So this is just 1. So the magnitude of the force is actually pretty easy to calculate, if we know the charge on a proton. And let's see if we can figure out the charge on a proton. Let me get the trusty TI-85 out. Let me clear there, just so you can appreciate the TI-85 store. If you press second and constant-- that's second and then the number 4. They have a little constant above it. You get their constant functions. Or their values. And you say the built-in-- I care about the built-in functions, so let me press F1. And they have a bunch of-- you know, this is Avogadro's number, they have a bunch of interesting-- this is the charge of an electron. Which is actually the same thing as the charge of a proton. So let's use that. Electrons-- just remember-- electrons and protons have offsetting charges. One's positive and one's negative. It's just that a proton is more massive. And of course, it's positive. Let's just confirm that that's the charge of an electron. But that's also the charge of a proton. And actually, this positive value is the exact charge of a proton. They should have maybe put a negative number here, but all we care about is the value. So let's use that again. The charge of an electron-- and it is positive, so that's the same thing as the charge for a proton-- times 6 times 10 to the seventh-- 6 E 7, you just press that EE button on your calculator-- times 0.5 teslas. Make sure all your units are in teslas, meters, and coulombs, and then your result will be in newtons. And you get 4.8 times 10 to the negative 12 newtons. Let me write that down. So the magnitude of this force is equal to 4.8 times 10 to the minus 12 newtons. So that's the magnitude. Now what is the direction? What is the direction of this force? Well, this you is where we break out-- we put our pens down if we're right handed, and we use our right hand rule to figure out the direction. So what do we have to do? So when you take something crossed something, the first thing in the cross product is your index finger on your right hand. And then the second thing is your middle finger pointed at a right angle with your index finger. Let's see if I can do this. So I want my index finger on my right hand to point to the right. But I want my middle finger to point upwards. Let me see if I can pull that off. So my right hand is going to look something like this. And my hand is brown. So my right hand is going to look something like this. My index finger is pointing in the direction of the velocity vector, while my middle finger is pointing the direction of the magnetic field. So my index finger is going to point straight up, so all you see is the tip of it. And then my other fingers are just going to go like that. And then my thumb is going to do what? My thumb is going-- this is the heel of my thumb-- and so my thumb is going to be at a right angle to both of them. So my thumb points down like this. This is often the hardest part. Just making sure you get your hand visualization right with the cross product. So just as a review, this is the direction of v. This is the direction of the magnetic field. It's popping out. And so if I arrange my right hand like that, my thumb points down. So this is the direction of the force. So as this particle moves to the right with some velocity, there's actually going to be a downward force. Downward on this plane. So the force is going to move in this direction. So what's going to happen? Well, what happens-- if you remember a little bit about your circular motion and your centripetal acceleration and all that-- what happens when you have a force perpendicular to velocity? If you have a force here and the velocity is like that, if the particles-- it'll be deflected a little bit to the right. And then since the force is always going to be perpendicular to the velocity vector, the force is going to charge like that. So the particle is actually going to go in a circle. As long as it's in the magnetic field, the force applied to the particle by the magnetic field is going to be perpendicular to the velocity of the particle. So the velocity of the particle-- so it's going to actually be like a centripetal force on the particle. So the particle is going to go into a circle. And in the next video we'll actually figure out the radius of that circle. And just one thing I want to let you think about. It's kind of weird or spooky to me that the force on a moving particle-- it doesn't matter about the particle's mass. It just matters the particle's velocity and charge. So it's kind of a strange phenomenon that the faster you move through a magnetic field-- or at least if you're charged, if you're a charged particle-- the faster you move through a magnetic field, the more force that magnetic field is going to apply to you. It seems a little bit, you know, how does that magnetic field know how fast you're moving? But anyway, I'll leave you with that. In the next video we'll explore this magnetic phenomenon a little bit deeper. See" + }, + { + "Q": "what is an intial value ?", + "A": "In the context of exponential functions, it means the function evaluated at 0.", + "video_name": "G2WybA4Hf7Y", + "transcript": "- [Voiceover] So let's think about a function. I'll just give an example. Let's say, h of n is equal to one-fourth times two to the n. So, first of all, you might notice something interesting here. We have the variable, the input into our function. It's in the exponent. And a function like this is called an exponential function. So this is an exponential. Ex-po-nen-tial. Exponential function, and that's because the variable, the input into our function, is sitting in its definition of what is the output of that function going to be. The input is in the exponent. I could write another exponential function. I could write, f of, let's say the input is a variable, t, is equal to is equal to five times times three to the t. Once again, this is an exponential function. Now there's a couple of interesting things to think about in exponential function. In fact, we'll explore many of them, but I'll get a little used to the terminology, so one thing that you might see is a notion of an initial value. In-i-tial Intitial value. And this is essentially the value of the function when the input is zero. So, for in these cases, the initial value for the function, h, is going to be, h of zero. And when we evaluate that, that's going to be one-fourth times two to the zero. Well, two to the zero power, is just one. So it's equal to one-fourth. So the initial value, at least in this case, it seems to just be that number that sits out here. We have the initial value times some number to this exponent. And we'll come up with the name for this number. Well let's see if this was true over here for, f of t. So, if we look at its intial value, f of zero is going to be five times three to the zero power and, the same thing again. Three to the zero is just one. Five times one is just five. So the initial value is once again, that. So if you have exponential functions of this form, it makes sense. Your initial value, well if you put a zero in for the exponent, then the number raised to the exponent is just going to be one, and you're just going to be left with that thing that you're multiplying by that. Hopefully that makes sense, but since you're looking at it, hopefully it does make a little bit. Now, you might be saying, well what do we call this number? What do we call that number there? Or that number there? And that's called the common ratio. The common common ratio. And in my brain, we say well why is it called a common ratio? Well, if you thought about integer inputs into this, especially sequential integer inputs into it, you would see a pattern. For example, h of, let me do this in that green color, h of zero is equal to, we already established one-fourth. Now, what is h of one going to be equal to? It's going to be one-fourth times two to the first power. So it's going to be one-fourth two. What is h of two going to be equal to? Well, it's going to be one-fourth times two squared, so it's going to be times two times two. Or, we could just view this as this is going to be two times h of one. And actually I should have done this when I wrote this one out, but this we can write as two times h of zero. So notice, if we were to take the ratio between h of two and h of one, it would be two. If we were to take the ratio between h of one it would be two. That is the common ratio between successive whole number inputs into our function. So, h of I could say plus one over h of n is going to be equal to is going to be equal to actually I can work it out mathematically. One-fourth times two to the n plus one over one-fourth two to the n. Two to the n plus one, divided by two to the n is just going to be equal to two. That is your common ratio. So for the function h. For the function f, our common ratio is three. If we were to go the other way around, if someone said, hey, I have some function whose initial value, so let's say, I have some function, I'll do this in a new color, I have some function, g, and we know that its initial initial value is five. And someone were to say its common ratio its common ratio is six, what would this exponential function look like? And they're telling you this is an exponential function. Well, g of let's say x is the input, is going to be equal to our initial value, which is five. That's not a negative sign there, Our initial value is five. I'll write equals to make that clear. And then times our common ratio to the x power. So once again, initial value, right over there, that's the five. And then our common ratio is the six, right over there. So hopefully that gets you a little bit familiar with some of the parts of an exponential function, why they are called what they are called." + }, + { + "Q": "Shouldn't NAD+ turn into NAD^2+ when combined with H+", + "A": "The whole point of NAD+ is that it is a high energy electron carrier. When it picks up H+, it also picks up two high energy electrons. The net result is a neutral NADH: NAD+ + 2e- ---> NADH", + "video_name": "GR2GA7chA_c", + "transcript": "In the last video we learned a little bit about photosynthesis. And we know in very general terms, it's the process where we start off with photons and water and carbon dioxide, and we use that energy in the photons to fix the carbon. And now, this idea of carbon fixation is essentially taking carbon in the gaseous form, in this case carbon dioxide, and fixing it into a solid structure. And that solid structure we fix it into is a carbohydrate. The first end-product of photosynthesis was this 3-carbon chain, this glyceraldehyde 3-phosphate. But then you can use that to build up glucose or any other carbohydrate. So, with that said, let's try to dig a little bit deeper and understand what's actually going on in these stages of photosynthesis. Remember, we said there's two stages. The light-dependent reactions and then you have the light independent reactions. I don't like using the word dark reaction because it actually occurs while the sun is outside. It's actually occurring simultaneously with the light reactions. It just doesn't need the photons from the sun. But let's focus first on the light-dependent reactions. The part that actually uses photons from the sun. Or actually, I guess, even photons from the heat lamp that you might have in your greenhouse. And uses those photons in conjunction with water to produce ATP and reduce NADP plus to NADPH. Remember, reduction is gaining electrons or hydrogen atoms. And it's the same thing, because when you gain a hydrogen atom, including its electron, since hydrogen is not too electronegative, you get to hog its electron. So this is both gaining a hydrogen and gaining electron. But let's study it a little bit more. So before we dig a little deeper, I think it's good to know a little bit about the anatomy of a plant. So let me draw some plant cells. So plant cells actually have cell walls, so I can draw them a little bit rigid. So let's say that these are plant cells right here. Each of these squares, each of these quadrilaterals is a plant cell. And then in these plant cells you have these organelles called chloroplasts. Remember organelles are like organs of a cell. They are subunits, membrane-bound subunits of cells. And of course, these cells have nucleuses and DNA and all of the other things you normally associate with cells. But I'm not going to draw them here. I'm just going to draw the chloroplasts. And your average plant cell-- and there are other types of living organisms that perform photosynthesis, but we'll focus on plants. Because that's what we tend to associate it with. Each plant cell will contain 10 to 50 chloroplasts. I make them green on purpose because the chloroplasts contain chlorophyll. Which to our eyes, appear green. But remember, they're green because they reflect green light and they absorb red and blue and other wavelengths of light. Because it's reflecting. But it's absorbing all the other wavelengths. But anyway, we'll talk more about that in detail. But you'll have 10 to 50 of these chloroplasts right here. And then let's zoom in on one chloroplast. So if we zoom in on one chloroplast. So let me be very clear. This thing right here is a plant cell. That is a plant cell. And then each of these green things right here is an organelle called the chloroplast. And let's zoom in on the chloroplast itself. If we zoom in on one chloroplast, it has a membrane like that. And then the fluid inside of the chloroplast, inside of its membrane, so this fluid right here. All of this fluid. That's called the stroma. The stroma of the chloroplast. And then within the chloroplast itself, you have these little stacks of these folded membranes, These little folded stacks. Let me see if I can do justice here. So maybe that's one, two, doing these stacks. Each of these membrane-bound-- you can almost view them as pancakes-- let me draw a couple more. Maybe we have some over here, just so you-- maybe you have some over here, maybe some over here. So each of these flattish looking pancakes right here, these are called thylakoids. So this right here is a thylakoid. That is a thylakoid. The thylakoid has a membrane. And this membrane is especially important. We're going to zoom in on that in a second. So it has a membrane, I'll color that in a little bit. The inside of the thylakoid, so the space, the fluid inside of the thylakoid, right there that area. This light green color right there. That's called the thylakoid space or the thylakoid lumen. And just to get all of our terminology out of the way, a stack of several thylakoids just like that, that is called a grana. That's a stack of thylakoids. That is a grana. And this is an organelle. And evolutionary biologists, they believe that organelles were once independent organisms that then, essentially, teamed up with other organisms and started living inside of their cells. So there's actually, they have their own DNA. So mitochondria is another example of an organelle that people believe that one time mitochondria, or the ancestors of mitochondria, were independent organisms. That then teamed up with other cells and said, hey, if I produce your energy maybe you'll give me some food or whatnot. And so they started evolving together. And they turned into one organism. Which makes you wonder what we might evolve-- well anyway, that's a separate thing. So there's actually ribosomes out here. That's good to think about. Just realize that at one point in the evolutionary past, this organelle's ancestor might have been an independent organism. But anyway, enough about that speculation. Let's zoom in again on one of these thylakoid membranes. So I'm going to zoom in. Let me make a box. Let me zoom in right there. So that's going to be my zoom-in box. So let me make it really big. Just like this. So this is my zoom-in box. So that little box is the same thing as this whole box. So we're zoomed in on the thylakoid membrane. So this is the thylakoid membrane right there. That's actually a phospho-bilipd layer. It has your hydrophilic, hydrophobic tails. I mean, I could draw it like that if you like. The important thing from the photosynthesis point of view is that it's this membrane. And on the outside of the membrane, right here on the outside, you have the fluid that fills up the entire chloroplast. So here you have the stroma. And then this space right here, this is the inside of your thylakoid. So this is the lumen. So if I were to color it pink, right there. This is your lumen. Your thylakoid space. And in this membrane, and this might look a little bit familiar if you think about mitochondria and the electron transport chain. What I'm going to describe in this video actually is an electron transport chain. Many people might not consider it the electron transport chain, but it's the same idea. Same general idea. So on this membrane you have these proteins and these complexes of proteins and molecules that span this membrane. So let me draw a couple of them. So maybe I'll call this one, photosystem II. And I'm calling it that because that's what it is. Photosystem II. You have maybe another complex. And these are hugely complicated. I'll do a sneak peek of what photosystem II actually looks like. This is actually what photosystem II looks like. So, as you can see, it truly is a complex. These cylindrical things, these are proteins. These green things are chlorophyll molecules. I mean, there's all sorts of things going here. And they're all jumbled together. I think a complex probably is the best word. It's a bunch of proteins, a bunch of molecules just jumbled together to perform a very particular function. We're going to describe that in a few seconds. So that's what photosystem II looks like. Then you also have photosystem I. And then you have other molecules, other complexes. You have the cytochrome B6F complex and I'll draw this in a different color right here. I don't want to get too much into the weeds. Because the most important thing is just to understand. So you have other protein complexes, protein molecular complexes here that also span the membrane. But the general idea-- I'll tell you the general idea and then we'll go into the specifics-- of what happens during the light reaction, or the light dependent reaction, is you have some photons. Photons from the sun. They've traveled 93 million miles. so you have some photons that go here and they excite electrons in a chlorophyll molecule, in a chlorophyll A molecule. And actually in photosystem II-- well, I won't go into the details just yet-- but they excite a chlorophyll molecule so those electrons enter into a high energy state. Maybe I shouldn't draw it like that. They enter into a high energy state. And then as they go from molecule to molecule they keep going down in energy state. But as they go down in energy state, you have hydrogen atoms, or actually I should say hydrogen protons without the electrons. So you have all of these hydrogen protons. Hydrogen protons get pumped into the lumen. They get pumped into the lumen and so you might remember this from the electron transport chain. In the electron transport chain, as electrons went from a high potential, a high energy state, to a low energy state, that energy was used to pump hydrogens through a membrane. And in that case it was in the mitochondria, here the membrane is the thylakoid membrane. But either case, you're creating this gradient where-- because of the energy from, essentially the photons-- the electrons enter a high energy state, they keep going into a lower energy state. And then they actually go to photosystem I and they get hit by another photon. Well, that's a simplification, but that's how you can think of it. Enter another high energy state, then they go to a lower, lower and lower energy state. But the whole time, that energy from the electrons going from a high energy state to a low energy state is used to pump hydrogen protons into the lumen. So you have this huge concentration of hydrogen protons. And just like what we saw in the electron transport chain, that concentration is then-- of hydrogen protons-- is then used to drive ATP synthase. So the exact same-- let me see if I can draw that ATP synthase here. You might remember ATP synthase looks something like this. Where literally, so here you have a huge concentration of hydrogen protons. So they'll want to go back into the stroma from the lumen. And they go through the ATP synthase. Let me do it in a new color. So these hydrogen protons are going to make their way back. Go back down the gradient. And as they go down the gradient, they literally-- it's like an engine. And I go into detail on this when I talk about respiration. And that turns, literally mechanically turns, this top part-- the way I drew it-- of the ATP synthase. And it puts ADP and phosphate groups together. It puts ADP plus phosphate groups together to produce ATP. So that's the general, very high overview. And I'm going to go into more detail in a second. But this process that I just described is called photophosphorylation. Let me do it in a nice color. Why is it called that? Well, because we're using photons. That's the photo part. We're using light. We're using photons to excite electrons in chlorophyll. As those electrons get passed from one molecule, from one electron acceptor to another, they enter into lower and lower energy states. As they go into lower energy states, that's used to drive, literally, pumps that allow hydrogen protons to go from the stroma to the lumen. Then the hydrogen protons want to go back. They want to-- I guess you could call it-- chemiosmosis. They want to go back into the stroma and then that drives ATP synthase. Right here, this is ATP synthase. ATP synthase to essentially jam together ADPs and phosphate groups to produce ATP. Now, when I originally talked about the light reactions and dark reactions I said, well the light reactions have two byproducts. It has ATP and it also has-- actually it has three. It has ATP, and it also has NADPH. NADP is reduced. It gains these electrons and these hydrogens. So where does that show up? Well, if we're talking about non-cyclic oxidative photophosphorylation, or non-cyclic light reactions, the final electron acceptor. So after that electron keeps entering lower and lower energy states, the final electron acceptor is NADP plus. So once it accepts the electrons and a hydrogen proton with it, it becomes NADPH. Now, I also said that part of this process, water-- and this is actually a very interesting thing-- water gets oxidized to molecular oxygen. So where does that happen? So when I said, up here in photosystem I, that we have a chlorophyll molecule that has an electron excited, and it goes into a higher energy state. And then that electron essentially gets passed from one guy to the next, that begs the question, what can we use to replace that electron? And it turns out that we use, we literally use, the electrons in water. So over here you literally have H2O. And H2O donates the hydrogens and the electrons with it. So you can kind of imagine it donates two hydrogen protons and two electrons to replace the electron that got excited by the photons. Because that electron got passed all the way over to photosystem I and eventually ends up in NADPH. So, you're literally stripping electrons off of water. And when you strip off the electrons and the hydrogens, you're just left with molecular oxygen. Now, the reason why I want to really focus on this is that there's something profound happening here. Or at least on a chemistry level, something profound is happening. You're oxidizing water. And in the entire biological kingdom, the only place where we know something that is strong enough of an oxidizing agent to oxidize water, to literally take away electrons from water. Which means you're really taking electrons away from oxygen. So you're oxidizing oxygen. The only place that we know that an oxidation agent is strong enough to do this is in photosystem II. So it's a very profound idea, that normally electrons are very happy in water. They're very happy circulating around oxygens. Oxygen is a very electronegative atom. That's why we even call it oxidizing, because oxygen is very good at oxidizing things. But all of a sudden we've found something that can oxidize oxygen, that can strip electrons off of oxygen and then give those electrons to the chlorophyll. The electron gets excited by photons. Then those photons enter lower and lower and lower energy states. Get excited again in photosystem I by another set of photons and then enter lower and lower and lower energy states. And then finally, end up at NADPH. And the whole time it entered lower and lower energy states, that energy was being used to pump hydrogen across this membrane from the stroma to lumen. And then that gradient is used to actually produce ATP. So in the next video I'm going to give a little bit more context about what this means in terms of energy states of electrons and what's at a higher or lower energy state. But this is essentially all that's happening. Electrons get excited. Those electrons eventually end up at NADPH. And as the electron gets excited and goes into lower and lower energy states, it pumps hydrogen across the gradient. And then that gradient is used to drive ATP synthase, to generate ATP. And then that original electron that got excited, it had to be replaced. And that replaced electron is actually stripped off of H2O. So the hydrogen protons and the electrons of H2O are stripped away and you're just left with molecular oxygen. And just to get a nice appreciation of the complexity of all of this-- I showed you this earlier in the video-- but this is literally a-- I mean this isn't a picture of photosystem II. You actually don't have cylinders like this. But these cylinders represent proteins. Right here, these green kind of scaffold-like molecules, that's chlorophyll A. And what literally happens, is you have photons hitting-- actually it doesn't always have to hit chlorophyll A. It can also hit what's called antenna molecules. So antenna molecules are other types of chlorophyll, and actually other types of molecules. And so a photon, or a set of photons, comes here and maybe it excites some electrons, it doesn't have to be in It could be in some of these other types of chlorophyll. Or in some of these other I guess you could call them, pigment molecules that will absorb these photons. And then their electrons get excited. And you can almost imagine it as a vibration. But when you're talking about things on the quantum level, vibrations really don't make sense. But it's a good analogy. They kind of vibrate their way to chlorophyll A. And this is called resonance energy. They vibrate their way, eventually, to chlorophyll A. And then in chlorophyll A, you have the electron get excited. The primary electron acceptor is actually this molecule right here. Pheophytin. Some people call it pheo. And then from there, it keeps getting passed on from one molecule to another. I'll talk a little bit more about that in the next video. But this is fascinating. Look how complicated this is. In order to essentially excite electrons and then use those electrons to start the process of pumping hydrogens across a membrane. And this is an interesting place right here. This is the water oxidation site. So I got very excited about the idea of oxidizing water. And so this is actually where it occurs in the photosystem II complex. And you actually have this very complicated mechanism. Because it's no joke to actually strip away electrons and hydrogens from an actual water molecule. I'll leave you there. And in the next video I'll talk a little bit more about these energy states. And I'll fill in a little bit of the gaps about what some of these other molecules that act as hydrogen acceptors. Or you can also view them as electron acceptors along the way." + }, + { + "Q": "how do you answer this question its quite puzzling\n3 (6 + 3 \u00c3\u00b7 3 ) +2", + "A": "Order of Operations: Divide first, so 3( 6 + 1) + 2 Then You can do distributive property or add to finish the parenthesis 3(7) + 2 21 + 2 23", + "video_name": "a-e8fzqv3CE", + "transcript": "While I'm working on some more ambitious projects, I wanted to quickly comment on a couple of mathy things that have been floating around the internet, just so you know I'm still alive. So there's this video that's been floating around about how to multiply visually like this. Pick two numbers, let's say, 12 times 3. And then you draw these lines. 12, 31. Then you start counting the intersections-- 1, 2, 3 on the left; 1, 2, 3, 4, 5, 6, 7 in the middle; 1, 2 on the right, put them together, 3, 7, 2. There's your answer. Magic, right? But one of the delightful things about mathematics is that there's often more than one way to solve a problem. And sometimes these methods look entirely different, but because they do the same thing, they must be connected somehow. And in this case, they're not so different at all. Let me demonstrate this visual method again. This time, let's do 97 times 86. So we draw our nine lines and seven lines time eight lines and six lines. Now, all we have to do is count the intersections-- 1,2, 3, 4, 5, 6, 7, 8, 9, 10. This is boring. How about instead of counting all the dots, we just figure out how many intersections there are. Let's see, there's seven going one way and six Hey, let's do 6 times 7, which is-- huh. Forget everything I ever said about learning a certain amount of memorization in mathematics being useful, at least at an elementary school level. Because apparently, I've been faking my way through being a mathematician without having memorized 6 times 7. And now I'm going to have to figure out 5 times 7, which is half of 10 times 7, which is 70, so that's 35, and then add the sixth 7 to get 42. Wow, I really should have known that one. OK, but the point is that this method breaks down the two-digit multiplication problem into four one-digit multiplication problems. And if you do have your multiplication table memorized, you can easily figure out the answers. And just like these three numbers became the ones, tens, and hundreds place of the answer, these do, too-- ones, tens, hundreds-- and you add them up and voila! Which is exactly the same kind of breaking down into single-digit multiplication and adding that you do during the old boring method. The whole point is just to multiply every pair of digits, make sure you've got the proper number of zeroes on the end, and add them all up. But of course, seeing that what you're actually doing is multiplying every possible pair is not something your teachers want you to realize, or else you might remember the every combination concept when you get to multiplying binomials, and it might make it too easy. In the end, all of these methods of multiplication distract from what multiplication really is, which for 12 times 31 is this. All the rest is just breaking it down into well-organized chunks, saying, well, 10 times 30 is this, 10 times 1 this, 30 times 2 is that, and 2 times 1 is that. Add them all up, and you get the total area. Don't let notation get in the way of your understanding. Speaking of notation, this infuriating bit of nonsense has been circulating around recently. And that there has been so much discussion of it is sign that we've been trained to care about notation way too much. Do you multiply here first or divide here first? The answer is that this is a badly formed sentence. It's like saying, I would like some juice or water with ice. Do you mean you'd like either juice with no ice or water with ice? Or do you mean that you'd like either juice with ice or water with ice? You can make claims about conventions and what's right and wrong, but really the burden is on the author of the sentence to put in some commas and make things clear. Mathematicians do this by adding parenthesis and avoiding this divided by sign. Math is not marks on a page. The mathematics is in what those marks represent. You can make up any rules you want about stuff as long as you're consistent with them. The end." + }, + { + "Q": "My parents have brown eyes but I have hazel eyes, how did that occur, ? My grandfather on my dads side has blue eyes my grandmother, my mom's mother has brown.", + "A": "probably becasue the next generation that you create will have hazel eyes just like you", + "video_name": "eEUvRrhmcxM", + "transcript": "Well, before we even knew what DNA was, much less how it was structured or it was replicated or even before we could look in and see meiosis happening in cells, we had the general sense that offspring were the products of some traits that their parents had. That if I had a guy with blue eyes-- let me say this is the blue-eyed guy right here --and then if he were to marry a brown-eyed girl-- Let's say this is the brown-eyed girl. Maybe make it a little bit more like a girl. If he were to marry the brown-eyed girl there, that most of the time, or maybe in all cases where we're dealing with the brown-eyed girl, maybe their kids are brown-eyed. Let me do this so they have a little brown-eyed baby here. And this is just something-- I mean, there's obviously thousands of generations of human beings, and we've observed this. We've observed that kids look like their parents, that they inherit some traits, and that some traits seem to dominate other traits. One example of that tends to be a darker pigmentation in maybe the hair or the eyes. Even if the other parent has light pigmentation, the darker one seems to dominate, or sometimes, it actually ends up being a mix, and we've seen that all around us. Now, this study of what gets passed on and how it gets passed on, it's much older than the study of DNA, which was really kind of discovered or became a big deal in the middle of the 20th century. This was studied a long time. And kind of the father of classical genetics and heredity is Gregor Mendel. He was actually a monk, and he would mess around with plants and cross them and see which traits got passed and which traits didn't get passed and tried to get an understanding of how traits are passed from one generation to another. So when we do this, when we study this classical genetics, I'm going to make a bunch of simplifying assumptions because we know that most of these don't hold for most of our genes, but it'll give us a little bit of sense of how to predict what might happen in future generations. So the first simplifying assumption I'll make is that some traits have kind of this all or nothing property. And we know that a lot of traits don't. Let's say that there are in the world-- and this is a gross oversimplification --let's say for eye color, let's say that there are two alleles. Now remember what an allele was. An allele is a specific version of a gene. So let's say that you could have blue eye color or you could have brown eye color. That we live in a universe where someone could only have one of these two versions of the eye color gene. We know that eye color is far more complex than that, so this is just a simplification. And let me just make up another one. Let me say that, I don't know, maybe for tooth size, that's a trait you won't see in any traditional biology textbook, and let's say that there's one trait for big teeth and there's another allele for small teeth. And I want to make very clear this distinction between a gene and an allele. I talked about Gregor Mendel, and he was doing this in the 1850s well before we knew what DNA was or what even chromosomes were and how DNA was passed on, et cetera, but let's go into the microbiology of it to understand the So I have a chromosome. Let's say on some chromosome-- let me pick some chromosome here. Let's say this is some chromosome. Let's say I got that from my dad. And on this chromosome, there's some location here-- we could call that the locus on this chromosome where the eye color gene is --that's the location of the eye color gene. Now, I have two chromosomes, one from my father and one from my mother, so let's say that this is the chromosome from my mother. We know that when they're normally in the cell, they aren't nice and neatly organized like this in the chromosome, but this is just to kind of show you the idea. Let's say these are homologous chromosomes so they code for the same genes. So on this gene from my mother on that same location or locus, there's also the eye color gene. Now, I might have the same version of the gene and I'm saying that there's only two versions of this gene in the world. Now, if I have the same version of the gene-- I'm going to make a little shorthand notation. I'm going to write big B-- Actually, let me do it the other way. I'm going to write little b for blue and I'm going to write big B for brown. There's a situation where this could be a little b and this could be a big B. And then I could write that my genotype-- I have the allele, I have one big B from my mom and I have one small b from my dad. Each of these instances, or ways that this gene is expressed, is an allele. So these are two different alleles-- let me write that --or versions of the same gene. And when I have two different versions like this, one version from my mom, one version from my dad, I'm called a heterozygote, or sometimes it's called a heterozygous genotype. And the genotype is the exact version of the alleles I have. Let's say I had the lowercase b. I had the blue-eyed gene from both parents. So let's say that I was lowercase b, lowercase b, then I would have two identical alleles. Both of my parents gave me the same version of the gene. And this case, this genotype is homozygous, or this is a homozygous genotype, or I'm a homozygote for this trait. Now, you might say, Sal, this is fine. These are the traits that you have. I have a brown from maybe my mom and a blue from my dad. In this case, I have a blue from both my mom and dad. How do we know whether my eyes are going to be brown or blue? And the reality is it's very complex. It's a whole mixture of things. But Mendel, he studied things that showed what we'll call dominance. And this is the idea that one of these traits dominates the other. So a lot of people originally thought that eye color, especially blue eyes, was always dominated by the other traits. We'll assume that here, but that's a gross oversimplification. So let's say that brown eyes are dominant and blue are recessive. I wanted to do that in blue. Blue eyes are recessive. If this is the case, and this is a-- As I've said repeatedly, this is a gross oversimplification. But if that is the case, then if I were to inherit this genotype, because brown eyes are dominant-- remember, I said the big B here represents brown eye and the lowercase b is recessive --all you're going to see for the person with this genotype is brown eyes. So let me do this here. Let me write this here. So genotype, and then I'll write phenotype. Genotype is the actual versions of the gene you have and then the phenotypes are what's expressed or what do you see. So if I get a brown-eyed gene from my dad-- And I want to do it in a big-- I want to do it in brown. Let me do it in brown so you don't get confused. So if I've have a brown-eyed gene from my dad and a blue-eyed gene from my mom, because the brown eye is recessive, the brown-eyed allele is recessive-- And I just said a brown-eyed gene, but what I should say is the brown-eyed version of the gene, which is the brown allele, or the blue-eyed version of the gene from my mom, which is the blue allele. Since the brown allele is dominant-- I wrote that up here --what's going to be expressed are brown eyes. Now, let's say I had it the other way. Let's say I got a blue-eyed allele from my dad and I get a brown-eyed allele for my mom. Same thing. The phenotype is going to be brown eyes. Now, what if I get a brown-eyed allele from both my mom and my dad? Let me see, I keep changing the shade of brown, but they're all supposed to be the same. So let's say I get two dominant brown-eyed alleles from my mom and my dad. Then what are you going to see? Well, you could guess that. I'm still going to see brown eyes. So there's only one last combination because these are the only two types of alleles we might see in our population, although for most genes, there's more than two types. For example, there's blood types. There's four types of blood. But let's say that I get two blue, one blue allele from each of my parents, one from my dad, one from my mom. Then all of a sudden, this is a recessive trait, but there's nothing to dominate it. So, all of a sudden, the phenotype will be blue eyes. And I want to repeat again, this isn't necessarily how the alleles for eye color work, but it's a nice simplification to maybe understand how heredity works. There are some traits that can be studied in this simple way. But what I wanted to do here is to show you that many different genotypes-- so these are all different genotypes --they all coded for the same phenotype. So just by looking at someone's eye color, you didn't know exactly whether they were homozygous dominant-- this would be homozygous dominant --or whether they were heterozygotes. This is heterozygous right here. These two right here are heterozygotes. These are also sometimes called hybrids, but the word hybrid is kind of overloaded. It's used a lot, but in this context, it means that you got different versions of the allele for that gene. So let's think a little bit about what's actually happening when my mom and my dad reproduced. Well, let's think of a couple of different scenarios. Let's say that they're both hybrids. My dad has the brown-eyed dominant allele and he also has the blue-eyed recessive allele. Let's say my mom has the same thing, so brown-eyed dominant, and she also has the blue-eyed recessive allele. Now let's think about if these two people, before you see what my eye color is, if you said, look, I'm giving you what these two people's genotypes are. Let me label them. Let me make this the mom. I think this is the standard convention. And let's make this right here, this is the dad. What are the different genotypes that their children could have? So let's say they reproduce. I'm going to draw a little grid here. So let me draw a grid. So we know from our study of meiosis that, look, my mom has this gene on-- Let me draw the genes again. So there's a homologous pair, right? This is one chromosome right here. That's another chromosome right there. On this chromosome in the homologous pair, there might be-- at the eye color locus --there's the brown-eyed gene. And at this one, at the eye color locus, there's a blue-eyed gene. And similarly from my dad, when you look at that same chromosome in his cells-- Let me do them like this. So this is one chromosome there and this is the other chromosome here. When you look at that locus on this chromosome or that location, it has the brown-eyed allele for that gene, and on this one, it has the blue-eyed allele on this gene. And we learn from meiosis when the chromosomes-- Well, they replicate first, and so you have these two chromatids on a chromosome. But they line up in meiosis I during the metaphase. And we don't know which way they line up. For example, my dad might give me this chromosome or might give me that chromosome. Or my mom might give me that chromosome or might give me that chromosome. So I could have any of these combinations. So, for example, if I get this chromosome from my mom and this chromosome from my dad, what is the genotype going to be for eye color? Well, it's going to be capital B and capital B. If I get this chromosome from my mom and this chromosome from my dad, what's it going to be? Well, I'm going to get the big B from my dad and then I'm going to get the lowercase b from my mom. So this is another possibility. Now, this is another possibility here where I get the brown-eyed allele from my mom and I get the blue eye allele from my dad. And then there's a possibility that I get this chromosome from my dad and this chromosome from my mom, so it's this situation. Now, what are the phenotypes going to be? Well, we've already seen that this one right here is going to be brown, that one's going to be brown, this one's going to be brown, but this one is going to be blue. I already showed you this. But if I were to tell you ahead of time that, look, I have two people. They're both hybrids, or they're both heterozygotes for eye color, and eye color has this recessive dominant situation. And they're both heterozygotes where they each have one brown allele and one blue allele, and they're going to have a child, what's the probability that the child has brown eyes? What's the probability? Well, each of these scenarios are equally likely, right? There's four equal scenarios. So let's put that in the denominator. Four equal scenarios. And how many of those scenarios end up with brown eyes? Well, it's one, two, three. So the probability is 3/4, or it's a 75% probability. Same logic, what's the probability that these parents produce an offspring with blue eyes? Well, that's only one of the four equally likely possibilities, so blue eyes is only 25%. Now, what is the probability that they produce a heterozygote? So what is the probability that they produce a heterozygous offspring? So now we're not looking at the phenotype anymore. We're looking at the genotype. So of these combinations, which are heterozygous? Well, this one is, because it has a mix. It's a hybrid. It has a mix of the two alleles. And so is this one. So what's the probability? Well, there's four different combinations. All of those are equally likely, and two of them result in a heterozygote. So it's 2/4 or 1/2 or 50%. So using this Punnett square, and, of course, we had to make a lot of assumptions about the genes and whether one's dominant or one's a recessive, we can start to make predictions about the probabilities of different outcomes. And as we'll see in future videos, you can actually even You can say, hey, given that this couple had five kids with brown eyes, what's the probability that they're both heterozygotes, or something like that. So it's a really interesting area, even though it is a bit of oversimplification. But many traits, especially some of the things that Gregor Mendel studied, can be studied in this way." + }, + { + "Q": "so why are they all equal vectors regardless of where you put them?", + "A": "Because a vector is described by its length and direction. It doesn t matter where it starts. So if we have a vector in component form given by v=<1,4>, it could start at (0,0) (in the Cartesian plane) and run to (1,4), or it could start at (3, -1) and run to (4, 3)...or any other such starting and ending positions. The vector itself is unchanged by being moved around. Does that help?", + "video_name": "8QihetGj3pg", + "transcript": "So I have two 2-dimensional vectors right over here, vector a and vector b. And what I want to think about is how can we define or what would be a reasonable way to define the sum of vector a plus vector b? Well, one thing that might jump at your mind is, look, well, each of these are two dimensional. They both have two components. Why don't we just add the corresponding components? So for the sum, why don't we make the first component of the sum just a sum of the first two components of these two vectors. So why don't we just make it 6 plus negative 4? Well, 6 plus negative 4 is equal to 2. And why don't we just make the second component the sum of the two second components? So negative 2 plus 4 is also equal to 2. So we start with two 2-dimensional vectors. You add them together, you get another two 2-dimensional vectors. If you think about it in terms of real coordinates bases, both of these are members of R2-- I'll write this down here just so we get used to the notation. So vector a and vector b are both members of R2, which is just another way of saying that these are both two tuples. They are both two-dimensional vectors right over here. Now, this might make sense just looking at how we represented it, but how does this actually make visual or conceptual sense? And to do that, let's actually plot these vectors. Let's try to represent these vectors in some way. Let's try to visualize them. So vector a, we could visualize, this tells us how far this vector moves in each of these directions-- horizontal direction and vertical direction. So if we put the, I guess you could say the tail of the vector at the origin-- remember, we don't have to put the tail at the origin, but that might make it a little bit easier for us to draw it. We'll go 6 in the horizontal direction. 1, 2, 3, 4, 5, 6. And then negative 2 in the vertical. So negative 2. So vector a could look like this. Vector a looks like that. And once again, the important thing is the magnitude and the direction. The magnitude is represented by the length of this vector. And the direction is the direction that it is pointed in. And also just to emphasize, I could have drawn vector a like that or I could have put it over here. These are all equivalent vectors. These are all equal to vector a. All I really care about is the magnitude and the direction. So with that in mind, let's also draw vector b. Vector b in the horizontal direction goes negative 4-- 1, 2, 3, 4, and in the vertical direction goes 4-- 1, 2, 3, 4. So its tail if we start at the origin, if its tail is at the origin, its head would be at negative 4, 4. So let me draw that just like that. So that right over here is vector b. And once again, vector b we could draw it like that or we could draw it-- let me copy and let me paste it-- so this would also be another way to draw vector b. Once again, what I really care about is its magnitude and its direction. All of these green vectors have the same magnitude. They all have the same length and they all have the same direction. So how does the way that I drew vector a and b gel with what its sum is? So let me draw its sum like this. Let me draw its sum in this blue color. So the sum based on this definition we just used, the vector addition would be 2, 2. So 2, 2. So it would look something like this. So how does this make sense that the sum, that this purple vector plus this green vector is somehow going to be equal to this blue vector? I encourage you to pause the video and think about if that even makes sense. Well, one way to think about it is this first purple vector, it shifts us this much. It takes us from this point to that point. And so if we were to add it, let's start at this point and put the green vector's tail right there and see where it ends up putting us. So the green vector, we already have a version. So once again, we start the origin. Vector a takes us there. Now, let's start over there with the green vector and see where green vector takes us. And this makes sense. Vector a plus vector b. Put the tail of vector b at the head of vector a. So if you were to start at the origin, vector a takes you there then if you add on what vector b takes you, it takes you right over there. So relative to the origin, how much did you-- I guess you could say-- shift? And once again, vectors don't only apply to things like displacement. It can apply to velocity. It can apply to actual acceleration. It can apply to a whole series of things, but when you visualize it this way, you see that it does make complete sense. This blue vector, the sum of the two, is what results where you start with vector a. At that point right over there, vector a takes you there, then you take vector b's tail, start over there and it takes you to the tip of the sum. Now, one question you might be having is well, vector a plus vector b is this, but what is vector b plus vector a? Does this still work? Well, based on the definition we had where you add the corresponding components, you're still going to get the same sum vector. So it should come out the same. So this will just be negative 4 plus 6 is 2. 4 plus negative 2 is 2. But does that make visual sense? So if we start with vector b. So let's say you start right over here. Vector b takes you right over there. And then if you were to go there and you were to start with vector a-- so let's do that. So actually, let me make this a little bit-- actually, let me start with a new vector b. So let's say that that's our vector b right over there. And then-- actually, let me give this a place where I'll have some space to work with. So let's say that's my vector b right over there. And then let me get a copy of the vector a. That's a good one. So copy and let me paste it. So I could put vector a's tail at the tip of vector b, and then it'll take me right over there. So if I start right over here, vector b takes me there. And now I'm adding to that vector a, which starting here will take me there. And so from my original starting position, I have gone this far. Now, what is this vector? Well, this is exactly the vector 2, 2. Or another way of thinking about it, this vector shifts you 2 in the horizontal direction and 2 in the vertical direction. So either way, you're going to get the same result, and that should, hopefully, make visual or conceptual sense as well." + }, + { + "Q": "Round to the nearest 0.1:\n5.312", + "A": "When you round to the nearest tenth, we need to check the hundredths place to see what we should round to. For example, your example 5.312 has a hundredths digit of 1. Remembering our rounding rules, the digits 0 through 4 round down and the digits 5 through 9 roundup. Since the 1 rounds down, the tenths place remains as 0.3. The answer is 5.3.", + "video_name": "_MIn3zFkEcc", + "transcript": "0.710 Round 9.564, or nine and five hundred sixty-four Thousandths, to the nearest tenth. So lie me write it a bit larger, 9.564 And we need to round to the nearest tenth. So what's the tenth place? The tenths place is right here This represents 5 tenths. This is the ones place, this is the tenths place, this is The hundredths place, and this is the Thousandths place right here So we need to round to the nearest tenth. So if we round up, this will be 9.6 If we round down, this will be 9.5 And just like regular rounding, when we're not Dealing with decimals, you move to one spot, or you look At one place to the right or one place lower, I guess, and You say is that 5 larger If it is, you round if it isn't,you round down 6 is definitely 5 or larger, so we want to round up. So this 9.564 becomes 9.6, or we can call this Nine and sixth tenths. And then we're done!" + }, + { + "Q": "Where did the 10,000 shares amount come from?", + "A": "This is the total number of shares held by the shareholders of the company. If you know the par/face value of a share and total (par/face) value of all the shares you can derive the number of share easily by dividing the total value with the par value of a share.", + "video_name": "5lmHzAHbtzg", + "transcript": "Let's say that both Ben's Shoes and Jason's Shoes are publicly traded companies. And all that means is that both of their shares are traded on exchanges. Maybe it's the NASDAQ, or the New York Stock Exchange, or some other exchange. And the going price on those exchanges-- the last closing price for Ben's stock was $21.50 per share, and the last closing price for Jason's stock is $12.00 per What I want to explore in this video, and probably the next few is, what is that saying for what the market thinks these businesses are worth? So in both of these situations, they have 10,000 shares. And remember, the shares are a split of the owner's equity. It's not a split of the assets. It's a split of just the equity part, right over here. So if shareholders are willing to pay 21.50 per share for Ben's stock, and there are 10,000 shares in Ben's company. So you take 21.50 times-- I'll do it this way-- times 10,000 shares gives us a market cap. So 21.50 times 10,000 gives us $215,000. And what this says is, look, if each of those 10,000 slices of the equity is worth 21.50, then the entire equity portion is going to be-- the market is valuing it at $215,000. And this calculation, this multiplication of the market price per share times the number of shares, this is called the market cap. Short for market capitalization. The market cap of the company. And all it is, is what is the market valuing the equity part of Ben's company worth? Let's do the same thing for Jason's company. You have $12.00 per share times 10,000 shares. That gives us $120,000 market cap. So the market is telling us that even though on the books, Ben's equity-- based on how he valued his assets and his liabilities-- is 135,000. The market is actually valuing this at 215,000. And in the next video, we'll think about what that means for how the market is actually valuing the business. In the case of Jason's business, instead of $35,000 of equity-- just straight up from what's on the books-- the market is valuing this piece right here at $120,000. So hopefully that gives you a little sense of one, what shares are a share of. They're a share of the owner's equity, not of the assets. And also gives you a good sense of what market cap is. It's the market's value of the owner's equity. And notice, in both cases-- and it's usually the case-- it's going to be a different number than the book value, the number that's actually on the books." + }, + { + "Q": "How can we find the eigen values of a matrix A,Where a11=B,a12=1 and a21=2 ,a22=5?", + "A": "Hi Sal, in my notes I have the characteristic equation : | A-hI |= 0 which is the reverse of yours. It doesn t seem to matter in this case which way they go A-hI or hI-A do you know if this is always the case?", + "video_name": "pZ6mMVEE89g", + "transcript": "In the last video we were able to show that any lambda that satisfies this equation for some non-zero vectors, V, then the determinant of lambda times the identity matrix minus A, must be equal to 0. Or if we could rewrite this as saying lambda is an eigenvalue of A if and only if-- I'll write it as if-- the determinant of lambda times the identity matrix minus A is equal to 0. Now, let's see if we can actually use this in any kind of concrete way to figure out eigenvalues. So let's do a simple 2 by 2, let's do an R2. Let's say that A is equal to the matrix 1, 2, and 4, 3. And I want to find the eigenvalues of A. So if lambda is an eigenvalue of A, then this right here tells us that the determinant of lambda times the identity matrix, so it's going to be the identity matrix in R2. So lambda times 1, 0, 0, 1, minus A, 1, 2, 4, 3, is going to be equal to 0. Well what does this equal to? This right here is the determinant. Lambda times this is just lambda times all of these terms. So it's lambda times 1 is lambda, lambda times 0 is 0, lambda times 0 is 0, lambda times 1 is lambda. And from that we'll subtract A. So you get 1, 2, 4, 3, and this has got to equal 0. And then this matrix, or this difference of matrices, this is just to keep the determinant. This is the determinant of. This first term's going to be lambda minus 1. The second term is 0 minus 2, so it's just minus 2. The third term is 0 minus 4, so it's just minus 4. And then the fourth term is lambda minus 3, just like that. So kind of a shortcut to see what happened. The terms along the diagonal, well everything became a negative, right? We negated everything. And then the terms around the diagonal, we've got a lambda out front. That was essentially the byproduct of this expression right there. So what's the determinant of this 2 by 2 matrix? Well the determinant of this is just this times that, minus this times that. So it's lambda minus 1, times lambda minus 3, minus these two guys multiplied by each other. So minus 2 times minus 4 is plus eight, minus 8. This is the determinant of this matrix right here or this matrix right here, which simplified to that matrix. And this has got to be equal to 0. And the whole reason why that's got to be equal to 0 is because we saw earlier, this matrix has a non-trivial null space. And because it has a non-trivial null space, it can't be invertible and its determinant has to be equal to 0. So now we have an interesting polynomial We can multiply it out. We get what? We get lambda squared, right, minus 3 lambda, minus lambda, plus 3, minus 8, is equal to 0. Or lambda squared, minus 4 lambda, minus 5, is equal to 0. And just in case you want to know some terminology, this expression right here is known as the characteristic polynomial. Just a little terminology, polynomial. But if we want to find the eigenvalues for A, we just have to solve this right here. This is just a basic quadratic problem. And this is actually factorable. Let's see, two numbers and you take the product is minus 5, when you add them you get minus 4. It's minus 5 and plus 1, so you get lambda minus 5, times lambda plus 1, is equal to 0, right? Minus 5 times 1 is minus 5, and then minus 5 lambda plus 1 lambda is equal to minus 4 lambda. So the two solutions of our characteristic equation being set to 0, our characteristic polynomial, are lambda is equal to 5 or lambda is equal to minus 1. So just like that, using the information that we proved to ourselves in the last video, we're able to figure out that the two eigenvalues of A are lambda equals 5 and lambda equals negative 1. Now that only just solves part of the problem, right? We know we're looking for eigenvalues and eigenvectors, right? We know that this equation can be satisfied with the lambdas equaling 5 or minus 1. So we know the eigenvalues, but we've yet to determine the actual eigenvectors. So that's what we're going to do in the next video." + }, + { + "Q": "Why Christ child looks bald?", + "A": "You can see the hair on the side of his head.", + "video_name": "jHN0BfowL7s", + "transcript": "MAN: We're in the Uffizi. And we're looking at a Raphael. And this is the \"Madonna of the Goldfinch,\" which is a really funny title. WOMAN: It is a funny title. And John-- who we see here on the left-- is holding out a goldfinch, the bird, to the Christ child, who strokes its head. And the goldfinch is a symbol of the passion of Christ, of Christ's suffering. And so we have that idea that we often have, of the foretelling Christ's terrible future. MAN: At the same time, this is a painting of two children and a mother. And so it exists in several different planes, because they're children doing childlike things-- one showing a pet to another, one wanting to touch it, the mother looking down protectively. WOMAN: And even a kind of tenderness between the mother and son-- look at the way that Christ puts his foot on his mother's. So there's that skin-to-skin moment of human contact there that's really lovely. But to me, Christ doesn't look like a child having fun. He looks very much all-knowing. I suppose if you were looking at a painting from the 1300s, Christ would look-- instead of looking like a baby, he would look like a little man, in order to indicate his sense of wisdom. But here I think Raphael communicates that through the elegance of Christ's body. Look at the way he lifts his arm up, strokes the goldfinch, and tilts his head back. He stands in this incredibly elegant contrapposto that no child would ever stand in. I mean, it's such a pose. MAN: It's true. And it's a beautiful foreshortening of his head, of his face as he leans back. But then there's a kind of energy and child-likeness that we see in John. John seems so engaged-- look what I can show you. WOMAN: And yet it's this symbol, this really potent symbol, of Christ's suffering. MAN: What's so interesting is that, unlike the 1300s as you mentioned before, we don't have the Madonna on the throne. Here, nature itself is the throne. We have this verdant environment, this beautiful atmospheric perspective. And she sits on a rock. That is, divinity is all around us. By the time we get to the late 15th century through the early 16th century, in the High Renaissance, nature itself has taken on the expression of God. We don't need, in a sense, those kingly symbols. WOMAN: Look at how composed it is, it in a way that we don't even notice immediately. We have a pyramid composition, with Mary at the top, and Saint John and Christ on either side, and that sense of real stability and balance that's also so much a part of the High Renaissance. MAN: Even as the figures are so engaged with each other-- and there's real dialogue that's taking place with them-- there is also that sense, that High Renaissance sense, you're right, of balance, of perfection, of the eternal. WOMAN: That interlocking of gestures and glances-- Mary looking at down at John, John looking at Christ, Christ looking back at John-- all of them enclosed within the pyramid structure of Mary's body, that unified composition that brings everything together in this really lovely landscape. MAN: I'm intrigued by the book. Mary had been reading. She's kept her place. And of course, that reminds us of an earlier scene in the Annunciation, when Gabriel interrupts as she's been piously reading the Bible. But here she's been reading, and now she's interrupted by her charges. She's doing a little bit of babysitting." + }, + { + "Q": "Why is around 2:30 mins in... (1/3) dividing 3x but on the other side it is multiplying!", + "A": "He s saying that multiplying both sides of the equation by the fraction (1/3) is the same as dividing both sides of the equation by 3.", + "video_name": "kbqO0YTUyAY", + "transcript": "So once again, we have three equal, or we say three identical objects. They all have the same mass, but we don't know what the mass is of each of them. But what we do know is that if you total up their mass, it's the same exact mass as these nine objects And each of these nine objects have a mass of 1 kilograms. So in total, you have 9 kilograms on this side. And over here, you have three objects. They all have the same mass. And we don't know what it is. We're just calling that mass x. And what I want to do here is try to tackle this a little bit more symbolically. In the last video, we said, hey, why don't we just multiply 1/3 of this and multiply 1/3 of this? And then, essentially, we're going to keep things balanced, because we're taking 1/3 of the same mass. This total is the same as this total. That's why the scale is balanced. Now, let's think about how we can represent this symbolically. So the first thing I want you to think about is, can we set up an equation that expresses that we have these three things of mass x, and that in total, their mass is equal to the total mass Can we express that as an equation? And I'll give you a few seconds to do it. Well, let's think about it. Over here, we have three things with mass x. So their total mass, we could write as-- we could write their total mass as x plus x plus x. And over here, we have nine things with mass of 1 kilogram. I guess we could write 1 plus 1 plus 1. That's 3. Plus 1 plus 1 plus 1 plus 1. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9. And actually, this is a mathematical representation. If we set it up as an equation, it's an algebraic representation. It's not the simplest possible way we can do it, but it is a reasonable way to do it. If we want, we can say, well, if I have an x plus another x plus another x, I have three x's. So I could rewrite this as 3x. And 3x will be equal to? Well, if I sum up all of these 1's right over here-- 1 We're doing that. We have 9 of them, so we get 3x is equal to 9. And let me make sure I did that. 1, 2, 3, 4, 5, 6, 7, 8, 9. So that's how we would set it up. And so the next question is, what would we do? What can we do mathematically? Actually, to either one of these equations, but we'll focus on this one right now. What can we do mathematically in order to essentially solve for the x? In order to figure out what that mystery mass actually is? And I'll give you another second or two to think about it. Well, when we did it the last time with just the scales we said, OK, we've got three of these x's here. We want to have just one x here. So we can say, whatever this x is, if the scale stays balanced, it's going to be the same as whatever we have there. There might be a temptation to subtract two of the x's maybe from this side, but that won't help us. And we can even see it mathematically over here. If we subtract two x's from both sides, on the left-hand side you're going to have 3x minus 2x. And on the right-hand side, you're going to have 9 minus 2x. And you're just going to be left with 3 of something minus 2 something is just 1 of something. So you will just have an x there if you get rid of two of them. But on the right-hand side, you're going to get 9 minus 2 So the x's still didn't help you out. You still have a mystery mass on the right-hand side. So that doesn't help. So instead, what we say is-- and we did this the last time. We said, well, what if we took 1/3 of these things? If we take 1/3 of these things and take 1/3 of these things, we should still get the same mass on both sides because the original things had the same mass. And the equivalent of doing that mathematically is to say, why don't we multiply both sides by 1/3? Or another way to say it is we could divide both sides by 3. Multiplying by 1/3 is the same thing as dividing by 3. So we're going to multiply both sides by 1/3. When you multiply both sides by 1/3-- visually over here, if you had three x's, you multiply it by 1/3, you're only going to have one x left. If you have nine of these one-kilogram boxes, you multiply it by 1/3, you're only going to have three left. And over here, you can even visually-- if you divide by 3, which is the same thing as multiplying by 1/3, you divide by 3. So you divide by 3. You have an x is equal to a 1 plus 1 plus 1. An x is equal to 3. Or you see here, an x is equal to 3. Over here you do the math. 1/3 times 3 is 1. You're left with 1x. So you're left with x is equal to 9 times 1/3. Or you could even view it as 9 divided by 3, which is equal to 3." + }, + { + "Q": "What does carry mean, at 0:16?", + "A": "When you add, you carry by putting numbers more than 10 to the top so it can be easier to solve.", + "video_name": "Wm0zq-NqEFs", + "transcript": "Let's add 536 to 398. And we're going to do it two different ways so that we really understand what this carrying is all about. So first, we'll do it in the more traditional way. We start in the ones place. We say, \"Well, what's 6 + 8?\" Well, we know that 6 + 8 is equal to 14. And so when we write it down here in the sum, we could say, \u201c \"Well look. The 4 is in the ones place.\u201d So it's equal to 4 + 1 ten.\" So let's write that 1 ten in the tens place. And now we focus on the tens place. We have 1 ten + 3 tens + 9 tens. So, what's that going to get us? 1 + 3 + 9 is equal to 13. Now we have to remind ourselves that this is 13 tens. Or another way of thinking about it, this is 3 tens and 1 hundred. You might say, \"Wait, wait! How does that make sense?\" Remember, this is in the tens place. When we're adding 1 ten + 3 tens + 9 tens, we're actually adding 10 + 30 + 90, and we're getting 130. And so we're putting the 30 (the 3 in the tens place represents the 30) \u2013 So this is the 3. The 3 represents the 30. And then we're placing this 1 in the hundreds place. 10 tens is equal to 100. And now we're adding up the numbers in the hundreds place. 1 + 5 + 3 is equal to \u2013 let's see. 1 + 5 is equal to 6, + 3 is equal to 9. But we have to remind ourselves: this is 9 hundreds. This is in the hundreds place. So this is actually 1 hundred. So this is actually 1 hundred + 5 hundreds + 3 hundreds, is equal to 9 hundreds. And that's exactly what we got here. 100 + 500 + 300 is equal to 900. And we're done. This is equal to 934." + }, + { + "Q": "is mirrored the same as flipped?", + "A": "yes. Think about it when you look in a mirror. your reflection is flipped from normal state", + "video_name": "CJrVOf_3dN0", + "transcript": "Let's talk a little bit about congruence, congruence And one to think about congruence, it's really kind of equivalence for shapes So, when in algebra when something is equal to another thing it means that their quantities are the same But when we're all of the sudden talking about shapes and we say that those shapes are the same, the shapes are the same size and shape then we say that they're congruent And just to see a simple example here: I have this triangle, right over there and let's say I have this triangle right over here And if you are able to shift, you are able to shift this triangle and flip this triangle, you can make it look exactly like this triangle As long as you're not changing the lengths of any of the sides or the angles here But you can flip it, you can shift it, you can rotate it So you can shift, let me write this, you can shift it, you can flip it and you can rotate If you can do those three procedures to make these the exact same triangle, then they are congruent And if you say that a triangle is congruent, let me label this So, let's call this triangle ABC Now let's call this D, let me call it XYZ XY and Z So, if we were to say, if we make the claim that both of these triangles are congruent So, if we say triangle ABC is congruent And the way you specify it, it almost look like an equal sign But it's equal sign with a curly thing on top Let me write it a little bit either So, we would write it like this If we know that triangle ABC is congruent to triangle XYZ That means their corresponding sides have the same length And their corresponding angles have the same measure So, if we make this assumption or someone tells us that this is true then we know, for example, that AB is going to equal to XY The length of segment AB is gonna be equal to the segment of XY And we could do this like this, and I'm assuming this are the corresponding sides And you can see that actually we've defined these triangles A corresponds to X, B corresponds to Y and C corresponds to Z right over there So, side AB is gonna have the same length as XY Then you can sometimes if you don't have the colors you can denote it just like that These two length are- or this two lines segments have the same length And you can actually say this, you don't always see this written this way You could also make the statement that line segment AB is congruent to line segment XY But congruence of line segments really just means that their lengths are equivalent So, these two things mean the same thing If one line segment is congruent to another line segment that just means the measure of one line segment is equal to the measure of the other line segment And so we can go thru all the corresponding sides If these two characters are congruent, we also know that BC, we also know that the length BC is gonna be the length of YZ Assuming those are the corresponding sides And we can put these double hash marks right over here to show that these lengths are the same And when we go the third side, we also know that these are going to be has same length or the line segments are going to be congruent So, we also know that the length of AC is going to be equal to the length of XZ Not only do we know that all of the sides, the corresponding sides are gonna have the same length If someone tells that a triangle is congruent We also know that all the corresponding angles are going to have the same measure So, for example: we also know that this angle's measure is going to be the same as the corresponding angle's measure, and the corresponding angle is right over It's between these orange side and blue side Or orange side and purple side, I should say And between the orange side and this purple side And so it also tells us that the measure of angle is BAC is equal to the measure of angle of YXZ Let me write that angle symbol, a little less like that, measure of angle of YXZ YXZ We can also write that as angle BAC is congruent angle YXZ And once again, like line segment, if one line segment is congruent to another line segment It just means that their lengths are equal And if one angle is congruent to another angle it just means that their measures are equal So, we know that those two corresponding angles have the same measure, they're congruent We also know that these two corresponding angles I'll use a double arch to specify that this has the same measure as that So, we also know the measure of angle ABC is equal to the measure of angle XYZ And then finally we know that this angle, if we know that these two characters are congruent, then this angle is gonna have the same measure as this angle as a corresponding angle So, we know that the measure of angle ACB is gonna be equal to the measure of angle XZY Now what we're gonna concern ourselves a lot with is how do we prove congruence? 'Cause it's cool, 'cause if you can prove congruence of 2 triangles then all of the sudden you can make all of these assumptions And what we're gonna find out, and this is going to be, we're gonna assume it for the sake of introductory geometry course This is an axiom or a postulate or just something you assume So, an axiom, very fancy word Postulate, also a very fancy word It really just means things we are gonna assume are true An axiom is sometimes, there's a little bit of distinction sometimes where someone would say \"an axiom is something that is self-evident\" or it seems like a universal truth that is definitely true and we just take it for granted You can't prove an axiom A postulate kinda has that same role but sometimes let's just assume this is true and see if we assume that it's true what can we derive from it, what we can prove if we assume its true But for the sake of introductory geometry class and really most in mathematics today, these two words are use interchangeably An axiom or a postulate, just very fancy words that things we take as a given Things that we'll just assume, we won't prove them, we will start with this assumptions and then we're just gonna build up from there And one of the core ones that we'll see in geometry is the axiom or the postulate That if all of the sides are congruent, if the length of all the sides of the triangle are congruent, then we are dealing with congruent triangles So, sometimes called side, side, side postulate or axiom We're not gonna prove it here, we're just gonna take it as a given So this literally stands for side, side, side And what it tells is, if we have two triangles and So I say that's another triangle right over there And we know that corresponding sides are equal So, we know that this side right over here is equal into, like, that side right over there Then we know and we're just gonna take this as an assumption and we can build off of this We know that they are congruent, the triangle, that these two triangles are congruent to each other I didn't put any labels there so it's kinda hard for me to refer to them But these two are congruent triangles And what's powerful there is we know that the corresponding sides are equal Then we know they're congruent and we can make all the other assumptions Which means that the corresponding angles are also equal So, that we know, is gonna be congruent to that or have the same measure That's gonna have the same measure as that and then that is gonna have the same measure as that right over there And to see why that is a reasonable axiom or a reasonable assumption or a reasonable postulate to start off with Let's take one, let's start with one triangle So, let's say I have this triangle right over here So, it has this side and then it has this side and then it has this side right over here And what I'm gonna do is see if I have another triangle that has the exact same line, side lengths is there anyway for me to construct a triangle with the same side lengths that is different, that can't be translated to this triangle thru flipping, shifting or rotating So, we assume this other triangle is gonna have the same size, the same length as that one over there So, I'll try to draw it like that Roughly the same length We know that it's going to have a size that's that length So, it's gonna have a side that is that length Let me put it on this side just to make it look a little bit more interesting So, we know that it's gonna have a side like that So, I'm gonna draw roughly the same length but I'm gonna try to do it in a different angle Now we know that's it's gonna have that looks like that So, let me, I'll put it right over here It's about that length right over there And so clearly this isn't a triangle, in order to make it a triangle, I'll have to connect this point to that point right over there And really there's only two ways to do it I can rotate it around that little hinge right over there If I connect them over here then I'm going to get a triangle that looks likes this Which is really a just a flip, am I visualizing it right? Yeah, just a flip version You can rotate it a little back this way, and you'd have a magenta on this side and a yellow one on this side And you can flip it, you could flip it vertically and it'll look exactly like this Our other option to make these two points connect is to rotate them out this way And the yellow side is gonna be here And then the magenta side is gonna be here and that's not magenta The magenta side is gonna be just like that And if we do that, then we actually just have to rotate it We just have to rotate it around to get that exact triangle So, this isn't a proof, and actually we're gonna start assuming that his is an axiom But hopefully you'll see that it's a pretty reasonable starting point that all of the sides, all of the corresponding sides of two different triangles are equal Then we are going to- we know that they are congruent We are just gonna assume that it's an axiom for that we're gonna build off, that they are congruent And we also know that he corresponding angles are going to be equivalent" + }, + { + "Q": "Ayesha cycles 13 km east,then turns around and goes 9km west of the starting point, then she again turns back and returns to the starting point, what is the total distance she covers?", + "A": "She cycles 44 kilometers.", + "video_name": "Oo2vGhVkvDo", + "transcript": "One of the coldest temperatures ever recorded outside was negative 128 degrees Fahrenheit in Antarctica. One of the warmest temperatures ever recorded outside was 134 degrees Fahrenheit in Death Valley, California. How many degrees difference are there between the coldest and warmest recorded outside temperatures? So let's think about this a little bit. Now, what I'll do is I'll plot them on a number line. But I'm going to plot it on a vertical number line that has a resemblance to a thermometer, since we're talking about temperature. So I'm going to make my number line vertical right over here. So there's my little vertical number line. And this right over here is 0 degrees Fahrenheit, which really is of no significance. If it was Celsius, we'd be talking about the freezing point. But for Fahrenheit, that happens at 32 degrees. But let's say this is 0 degrees Fahrenheit. And let's plot these two points. So one of the coldest ever recorded temperatures was negative 128 degrees Fahrenheit. So let's say that's right over here. This is negative 128 degrees Fahrenheit. And one of the warmest temperatures ever recorded was 134 degrees. This is a positive 134. So it's about that far and a little bit further. So it's a positive 134 degrees Fahrenheit. So when they're asking us how many degrees difference are there between the coldest and the warmest, they're essentially saying, well, what is this distance between the coldest and the warmest right over here? What is this distance? And there's a couple of ways you could think about it. You could say, hey, if I started at the coldest temperature and I wanted to go all the way up to the warmest, how much would I have to add? Or you could say, well, what's the difference between the coldest and the warmest? So you could take the larger number. So it's, say, 134. And from that, you could subtract the smaller number, which is negative 128. So this essentially saying what's the difference between these two numbers? It's going to be positive, because we're subtracting the smaller one from the larger one. This is going to give you the exact same thing as this. Now, there's several ways to think about it. One is we know that if you subtract a negative number, that's the same thing as adding the positive of that number, or adding the absolute value. So this is the same thing. This is going to be equal to 134 plus positive 128 degrees. And what's the intuition behind that? Why does this happen? Well, look at this right over here. We're trying to figure out this distance. This distance is 134 minus negative 128. And if you look at that, it's going to be the absolute value of 134. It's going to be this distance right over here, which is just 134-- which is just that right over there-- plus this distance right over here. Now, what is this distance? Well, it's the absolute value of negative 128. It's just 128. So it's going to be that distance, 134, plus 128. And that's why it made sense. This way, you're thinking of what's the difference between a larger number and a smaller number. But since it's a smaller number and you're subtracting a negative, it's the same thing as adding a positive. And hopefully this gives you a little bit of that intuition. But needless to say, we can now figure out what's going to be. And this is going to be equal to-- let me figure this out separately over here. So if I were to add 134 plus 128, I get 4 plus 8 is 12, 1 plus 3 plus 2 is 6. It's 262. This right over here is equal to 262. How many degrees difference are there between the coldest and warmest recorded outside temperature? 262 degrees Fahrenheit difference." + }, + { + "Q": "Whats a fraction", + "A": "They represent pieces of a whole. Special if a pizza is cut into 8 slices, and you eat 3... then you can say you ate 3/8 of the pizza", + "video_name": "fvtv2uYjo_E", + "transcript": "- [Voiceover] Let's see if we can calculate what 5/6 plus 1/4 is, and to help us, I have a visual representation of 5/6, and a visual representation of 1/4. Notice I have this whole whole, I guess you could say, broken up into one, two, three, four, five, six sections, and we've shaded in five of them, so this is 5/6, and then down here, we have another whole, and we have one out of the four equal sections shaded in so this is 1/4, and I want to add them, and I encourage you at any point, pause the video, and see if you could figure it out on your own. Well, whenever we're adding fractions, we like to think in terms of fractions that have the same denominator, and these clearly don't have the same denominator, but in order to rewrite them, with a common denominator, we just have to think of a common multiple of six and four, and ideally, the smallest common multiple of six and four, and the way that I like to do that is I like to take the larger of the two, which is six, and then think about its multiples. So I could first think about six itself. Six is clearly divisible by six, but it's not perfectly divisible by four, so now, let's multiply by two, so then we get to 12. 12 is divisible by both six and four. So 12 is a good common denominator here. It's the least common multiple of six and four. So we can rewrite both of these fractions as something over 12. So, something over 12 plus something, plus something over 12 is equal to. Now, there's a bunch of ways to tackle it, but what I want to do is I just want to visualize it here on this drawing. So, if I go, if I were to go from, if I were to go from six equal sections to 12 equal sections, which is what I'm doing if I'm going from six in the denominator to 12 in the denominator. I'm essentially multiplying each of these sections by, or, I'm essentially multiplying the number of sections I have by two, or I'm taking each of these existing sections and I'm turning them into two sections, so let's do that. Let's do that. Let me see if I can do it pretty neatly, so, I can do it a little bit neater than that. So, it'll look like that. And, whoops. Let me do this one. I want to divide them fairly close to evenly. I'm doing it by eye so it's not going to be perfect. So, and you have that one. And then last not, last but not least, you have that one there, and then notice, I had six sections, but now I've doubled the number of sections. I've turned the six sections into 12 sections by turning each of the original six into two, so now I have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12 sections. So if I have 12 sections now, how many of those 12 are now shaded in? Instead of having five of the six, I now have 10 of the 12 that are shaded in. So I now have 10/12. 5/6 is the same thing as 10/12. Another way you could have thought about that, to go from six to 12, I had to multiply by two, so then I have to do the same thing in the numerator. Five times two is 10. But hopefully you see that those two fractions are equivalent, that I didn't change how much is shaded in, I just took each of the original six and I turned it into two, or I multiplied the total number of sections by two to get 12, and then instead of having 5/6, I now have 10/12 shaded in. Now let's do the same thing with the four, with the 1/4. Right here, I've depicted 1/4, but I want to turn this into something over 12. So to turn it into something over 12, each section has to be turned into three sections. So let's do that. Let's turn each section into three sections. So, that's one, two, and three. So then I have one, two, and three. I have, I think you can see where this is going. One, two and three. I have one, two, and three. And so notice, all I did is I multiplied, before I had four equal sections. Now I turned each of those four sections into three sections, so now I have 12 equal sections. And I did that, essentially, by multiplying the number of sections I had by three. So now what fraction is shaded in? Well, now, this original that was one out of the four, we can now see is three out of the 12 equal sections. It's now three out of the 12 equal sections, and so what is this going to be? Well, if I have 10/12, and I'm adding it to 3/12, well how many twelfths do I have? I'm going to have 13/12. And you could see it visually over here as well. Up here in green, I have 10/12 shaded in. Each of these boxes are a twelfth. Let me write that down. Each of these boxes are 1/12. That's 1/12. This is 1/12. So how many twelfths do I have shaded in? I have the 10 that are shaded in in green, and then I have an 11/12, a 12/12 and then finally, the 13/12 is one way to think about it." + }, + { + "Q": "What if there are two chemicals like Mg that have no number with them? What do I do then?", + "A": "Then there is an understood 1 after it", + "video_name": "xqpYeiefZl8", + "transcript": "- [Voiceover] Let's now see if we can balance a chemical equation with slightly more complex molecules. So, here we have a chemical equation, describing a chemical reaction. This is actually a combustion reaction. You have some ethylene right over here, in the presence of oxygen, and you need to get a little bit of energy to get this going, but then you're going to have this reaction that's actually going to release energy as well, but we're not accounting for the energy, at least the way we've written it. Right over here, you have some ethylene, and this little g in parentheses, says it's in the gas form or gaseous form, so gaseous ethylene plus some dioxygen molecule, which is the most prevalent form of oxygen molecule that you would find in the atmosphere. And so, that's also in the gas form. Put them together, you end up with some carbon dioxide gas and some liquid water. This is the classic combustion reaction. But now let's think about, how do we balance this thing? Let's make sure we have the same number of each atom on both sides. And when you see something more complicated like this, where, you know, here I have an oxygen and two different molecules over here, and a lot of these molecules have multiple elements in it. It might be very daunting. Where do I start? And this is where the art of balancing chemical equations starts to come into play. The general idea is, Try to balance the... try to balance the molecules that have multiple elements in them first, and leave the... molecules that only have one element in them for last. And the idea there is, is that these are harder. They're going to have all sorts of implications, and then, at the end of the day, you can just set a number here for the number of dioxygens. If you saved, say the ethylene for last, then every time, and you're trying to balance the carbons, you try to change the number of carbons, you're going to change the number of hydrogens, which is going to change the... You're going to have to balance over and over, and you're going to go into this really really really confusing circle. So, the best thing to do, try to balance the complex molecules first, and then save the single element molecules for last. So let's do that. So, let's start with the carbons. So, over here, I have two carbons. Over here, I only have one carbon. I only have one carbon. So, it seems like the best way to balance it is, I should have two molecules of carbon dioxide, and I haven't even thought about the oxygens yet. By putting that two there, that's going to change the number of oxygens I have on the righthand side. But at least it balances my carbons. I now have two carbons on the lefthand side, and I have two carbons on the righthand side. I\u2019m no longer magically destroying a carbon atom, all right. Now, let's move on to the hydrogens, and remember, what I said is, let's wait to do the oxygens last, because we have a molecule that only contains oxygen right over here, so we'll save oxygen for last. So, let's do hydrogen next. So, hydrogen, right over here, we have four hydrogens. And on the righthand side, we have two hydrogens. So, it seems like the easiest thing to do to balance the hydrogens is to have two of these water molecules. Now I have four hydrogens here, and I have four hydrogens there. Now, let's do the oxygen. Now, let's do the oxygen. I've balanced the carbons and the hydrogens. And the reason why oxygen's going to be interesting, I can just count the amount of oxygen I now have here, after changing the amount of molecules I have. And then I can adjust this accordingly, because this is only going to affect the number of oxygens that I have on the lefthand side. Right now, on the lefthand side, I have two oxygens, and on the righthand side, let me count this, I have two O two's, really. So, this is going to be four oxygens here, and then I have, each of these water molecules has one oxygen, but I have two water molecules, so this is going to be two oxygens, two oxygens here. So, on the righthand side, I have four plus two oxygens. So, I have six oxygens on the righthand side. I need six oxygens on the lefthand side. I need this number to be six. So, how do I do that? Well, I just need three of these molecules. If I have three molecules, each of them have two oxygens, I'm going to have a total of six oxygens. And just like that, we have balanced this combustion reaction, this chemical equation." + }, + { + "Q": "Is it possible for galaxies to crash into one another ?", + "A": "Yes, but there is lots of space between the stars in a galaxy, so a crash is really a gravitational interaction in which the galaxies pass through one another. The result is somewhat unpredictable. The galaxies could merge, with some stars being thrown out , or they could distort each other and then mostly go their separate ways, each leaving some stars with the other, or they could go into a sort of dance with each other that could last a long time before settling down.", + "video_name": "JiE_kNk3ucI", + "transcript": "Where we left off in the last video, we were just kind of staring, amazed, at this Earth's view of the Milky Way galaxy, just making sure we understood how enormous and how many stars we were looking at. And even if each of these dots were a star, this is a huge amount of stars. But a lot of these dots are thousands of stars. So our mind was already blown. But what we're going to see in this video is that in some ways, this is kind of just the beginning. And to some degree, I'm going to stop doing these particles of sand and a football field analogy because at some point, the particles of sand become so vast that are our minds can't even grasp it to begin with. But let's just start with our Milky Way. And we saw in the last video, the Milky Way right here, we're sitting here about 25,000 light years from the center. It's roughly 100,000 light years in diameter. And then, let's put it in perspective of its local neighborhood. So let's look at the Local Group. And when we talk about Local Group, we're talking about the local group of galaxies. So this right here is the Milky Way's Local Group. That's us right there, sitting right over here, about 25,000 light years from the center of the Milky Way. You have some of these \"small\"-- and I use \"small\" in quotation marks-- because these are also vast entities, also unimaginable entities. But we have these satellite galaxies around, under the gravitational influence, some of them, of the Milky Way. But the nearest large galaxy to us is Andromeda right over here. And this distance right over here. And now, we're going to start talking in the millions of light years. So this distance right here is 2.5 million light years. And just as a bit of reference, if that's any reference at all, one light year is roughly the radius of the Oort Cloud. Or another way to think about it, one radius of the Oort Cloud is about 50,000 or 60,000 astronomical units. And that's the distance from the Sun to the Earth. So you could view this as 2.5 million times 60,000 or so times the distance from the Sun to the Earth. So this is an unbelievably large distance we're talking about here. And that's to get to the next big galaxy over here. But even these things are huge things with many-- I mean, just unfathomably many-- stars. But Andromeda, in particular, we said that the Milky Way has 200 to 400 billion stars. Andromeda, people believe, has on the order of 1 trillion stars. So these just start to become numbers. It's hard to grasp. But we're not going to stop here. So in this, over here, this whole diagram right here, it's about four light years across, if you go from point to point. If you go from one side to the other side, this is about-- not four light years. This is 4 million light years. Four light years is just the distance from us to the Alpha Centauri. So that was nothing. That would only take that a Voyager 1 80,000 years to get. This is 4 million light years. So 4 million times the distance to the nearest star. But even this is-- I mean I'm starting to stumble on my words because there's really no words to describe it-- even this is small on an intergalactic scale. Because when you zoom out more, you can see our Local Group. Our local group is right over here. And this right over here is the Virgo Super Cluster. And each dot here is at least one galaxy. But it might be more than one galaxy. And the diameter here is 150 million light years. So what we saw in the Local Group, in the last diagram, the distance from the Milky Way to Andromeda, which was 2 and 1/2 million light years, which would be just a little dot just like that, that would be the distance between the Milky Way and Andromeda. And now, we're looking at the Virgo Super Cluster that is 150 million light years. But we're not done yet. We can zoom out even more. We can zoom out even more, and over here. So you had your Virgo Super Cluster, 150 million light years was that last diagram, this diagram right over here. I want to keep both of them on the screen if I can. This diagram right here, 150 million light years across. That would fit right about here on this diagram. So this is all of the super clusters that are near us. And once again, \"near\" has to be used very, very, very loosely. Here, this distance is about 150 million light years. A billion light years is-- two, three, four, five-- a billion light years is about from here to there. So we're starting to talk on a fairly massive-- I guess we've always been talking on a massive scale. But now, it's an even more massive scale. But we're still not done. Because this whole diagram-- now these dots that you're seeing now, I want to make it very clear. These aren't stars. These aren't even clusters of stars. These aren't even clusters of millions or even billions of stars. Each of these dots are clusters of galaxies, each of those galaxies having hundreds of billions to trillions of stars. So we're just on an unbelievably massive scale at this point. But we're still not done. We're still not done. This is roughly about a billion light years across. But right here is actually the best estimate of the visible universe. And in future videos, we're going to talk a lot more about what the visible universe means. So if you were to zoom out enough, this entire diagram right here, about a billion light years, would fit just like that. So we're talking about a super small amount of this part right here. And this is just the visible universe. I want to make it clear. This is not the entire universe. And we say it's the visible universe because think about what's happening. When we think about the a point out here, and we're observing it, and that's let's say 13 billion light years away. Let's say that point 13 billion. We're going to talk more about this in future videos, 13 billion light years. And I feel it's almost a sacrilege to be writing on this because this complexity that we're seeing here is just mind boggling. But this 13 billion light year away object, the light is just getting to us. This light left some point 13 billion light years ago. So what we're actually doing is observing that object close to the beginning of the actual universe. And the reason why it's the visible universe is there might have been something a little bit further out. Maybe it's light hasn't reached us yet or maybe the universe itself, and we'll talk more about this, it's expanding so fast that the light will never, ever reach us. So it's actually a huge question mark on how big the actual universe is. And then some people might say, well, does it even matter? Because this by itself is a huge distance. And I want to make it clear, you might say, OK, if this light over here, if this is coming from 13 billion light years away, or if this is 13 billion light years away, then you could say, hey, so everything that we can observe or that we can even observe the past of, the radius is about 26 billion light years. But even there, we have to be careful because remember the universe is expanding. When this light was emitted-- and I'll do a whole video on this because the geometry of it is kind of hard to visualize-- when this light was emitted, where we are in the Virgo Super Cluster, inside of the Milky Way Galaxy, where we are was much closer to that point. It was on the order of-- and I want to make sure I get this right-- 36 million light years. So we were super close by, I guess, astronomical scales. We were super close, only 36 million light years, to this object, when that light was released. But that light was coming to us and the whole time the universe So we were also moving away from it, if you just think about all of the space, that everything is expanding away from each other, And only 13 billion years later did it finally catch up with us. But the whole time that that was happening, this object has also been moving. This object has also been moving away from us. And so our best estimate of where this object is now, based on how space is expanding, is on the order a 40 or 45 billion light years away. We're just observing where that light was emitted 13 billion years ago. And I want to be very clear. What we are observing, this light is coming from something very, very, very primitive. That object or that area of space where that light was emitted from has now condensed into way more, I guess, mature astronomical structures. If you take it from the other point of view, people sitting where in this point of space now, and they've now moved 46 billion light years out, when they observe our region of space, they're not going to see us. They're not going to see Earth as it is now. They're going to see the region of space where Earth is at a super primitive stage, shortly after the Big Bang. And when I use words like \"shortly,\" I use that also loosely. We're still talking about hundreds of thousands or even millions of years. So we'll talk more about that in a future video. But the whole point of this video is it's beyond mind numbing. I would say the last video, about the Milky Way, that alone was mind numbing. But now, we're going in a reality where just the Milky Way becomes something that's almost unbelievably insignificant when you think about this picture right here. And the really mind numbing thing is, if someone told me that this is the entire universe, this by itself would certainly put things in perspective. But it's unknown what's beyond it. There's some estimates that this might be only be 1 times 10 to the 23rd of the entire universe. And it might even be the reality that the entire universe is smaller than this. And that's an interesting thing to think about. But I'll leave you there because I think no matter how you think about it, it's just-- I don't know. I actually, before doing this video, I stared at some of these photos for half an hour. This is my least productive day just because it's just so awe inspiring to think about what these dots and dots of the dots really are." + }, + { + "Q": "whut about the organisysms at the bottom of the ochin that do not depend on the sun", + "A": "It s a generalization that holds true 99.9% of the time. But you re right, at geothermal vents and deep in the earth there are bacteria independent of light.", + "video_name": "-rsYk4eCKnA", + "transcript": "Let's talk about one of the most important biological processes. Frankly, if this process didn't occur, we probably wouldn't have life on Earth, and I wouldn't be making this video for you, because there'd be no place for me to actually get food. And the process is called photosynthesis. And you're probably reasonably familiar with the idea. The whole idea is plants, and actually bacteria and algae and other things, but we normally associate it with plants. Let me make it in very simple terms. So we normally associate it with plants. And it's a process that plants use, and we might have learned this when we were very young. It's the process that plants use to take carbon dioxide plus some water plus some sunlight and turn it into some sugars or some maybe carbohydrates. Carbohydrates or sugars plus oxygen. Obviously, this has two very profound pieces to it for us as a living species. One, we need carbohydrates or we need sugars in order to fuel our bodies. You saw that in the cellular respiration videos. We generate all of our ATP by performing cellular respiration on glucose, which is essentially a byproduct, or a broken down carbohydrate. It's the simplest one for us to process in cellular respiration. And the second hugely important part is getting the oxygen. Once again, we need to breathe oxygen in order for us to break down glucose, in order to respire, in order to perform cellar respiration. So these two things are key for life, especially for life that breathes oxygen. So this process, other than the fact that it's interesting, that there are organisms around us, mostly plants, that are able to harness actual sunlight. You have these fusion reactions in the sun 93 million miles away, and it's releasing these photons, and some small subset of those photons reach the surface of Earth. They make their way through clouds and whatever else. And then these plants and bacteria and algae are able to harness that somehow and turn them into sugars that we can then eat or maybe the cow eats them and we eat the cow if we're not vegetarians, and we can then use that for energy. Not that the cow is all carbohydrates, but this is essentially what is used as the fuel or the energy for all of the other important compounds that we eat. This is where we get all of our fuel. So this is fuel for animals. Or you know, if you eat a potato directly, you are directly getting your carbohydrates. But anyway, this is a very simple notion of photosynthesis, but it's not incorrect. I mean, if you had to know one thing about photosynthesis, this would be it. But let's delve a little bit deeper and try to get into the guts of it and see if we can understand a little bit better how this actually happens. I find it amazing that somehow photons of sunlight are used to create these sugar molecules or these carbohydrates. So let's delve a little bit deeper. So we can write the general equation for photosynthesis. Well, I've almost written it here. But I'll write it a little bit more scientifically specific. You start off with some carbon dioxide. You add to that some water, and you add to that-- instead of sunlight, I'm going to say photons because these are what really do excite the electrons in the chlorophyll that go down, and you'll see this process probably in this video, and we'll go in more detail in the next few videos. But that excited electron goes to a high energy state, and as it goes to a lower energy state, we're able to harness that energy to produce ATPs, and you'll see NADPHs, and those are used to produce carbohydrates. But we'll see that in a little bit. But the overview of photosynthesis, you start off with these constituents, And then you end up with a carbohydrate. And a carbohydrate could be glucose, doesn't have to be glucose. So the general way we can write a carbohydrates is CH2O. And we'll put an n over here, that we could have n multiples of these, and normally, n will be at least three. In the case of glucose, n is 6. You have 6 carbons, 12 hydrogens and 6 oxygens. So this is a general term for carbohydrates, but you could have many multiples of that. You could have these long-chained carbohydrates, so you end up with a carbohydrate and then you end up with some oxygen. So this right here isn't so different than what I wrote up here in my first overview of how we always imagined photosynthesis in our heads. In order to make this equation balance-- let's see, I have n carbons so I need n carbons there. Let's see, I have two n hydrogens here. Two hydrogens and I have n there, so I need two n hydrogens here. So I'll put an n out there. And lets see how many oxygens. I have two n oxygens, plus another n, so I have three n oxygens. So let's see, I have one n, and you put an n here, and then I have two n, and I think this equation balances out. So this is a 30,000-foot view of what's going on in photosynthesis. But when you dig a little deeper, you'll see that this doesn't happen directly, that this happens through a bunch of steps that eventually gets us to the carbohydrate. So in general, we can break down photosynthesis. I'll rewrite the word. We can break down photosynthesis-- and we'll delve deeper into future videos, but I want to get you the overview first-- into two stages. We can call one the light reactions. Or sometimes they are called the light-dependent reactions, and that actually would probably be a better Let me write it like that. Light dependent means that they need light to occur. Light-dependent reactions. And then you have something called the dark reactions, and that's actually a bad name, because it also occurs in the light. Dark reactions, I wrote in a slightly darker color. And the reason why I said it's a bad name is because it still occurs in the light. But the reason why they probably called it the dark reaction is that you don't need light, or that part of photosynthesis isn't dependent on photons to occur. So a better term for it would have been light-independent reaction. So just to be clear, the light reactions actually need sunlight. They actually need photons for them to proceed. The dark reactions do not need photons for them to happen, although they do occur when the sun is out. They don't need those photons, but they need the byproducts from the light reaction to occur, so that's why it's called the light-independent reaction. They occur while the sun is out, but they don't need the sun. This needs the sun, so let me make it very clear. So this requires sunlight. This requires photons. And let me just make a very brief overview of this. This'll maybe let us start building a scaffold from which we can dig deeper. So the light reactions need photons, and then it needs water. So water goes into the light reactions and out of the other side of the light reactions. We end up with some molecular oxygen. So that's what happens in the light reactions, and I'm going to go much deeper into what actually occurs. And what the light the actions produce is ATP, which we know is the cellular or the biological currency of energy. It produces ATP and it produces NADPH. Now, when we studied cellular respiration, we saw the molecule NADH. NADPH is very similar. You just have this P there. You just have this phosphate group there, but they really perform similar mechanisms. That this agent right here, this molecule right here, is able to give away-- now let's think about what this means-- it's able to give away this hydrogen and the electron associated with this hydrogen. So if you give away an electron to someone else or someone else gains an electron, that something else is being reduced. Let me write that down. This is a good reminder. OIL RIG. Oxidation is losing an electron. Reduction is gaining an electron. Your charge is reduced when you gain an electron. It has a negative charge. So this is a reducing agent. It gets oxidized by losing the hydrogen and the electron with it. I have a whole discussion on the biological versus chemistry view of oxidation, but it's the same idea. When I lose a hydrogen, I also lose the ability to hog that hydrogen's electron. So this right here, when it reacts with other things, it's a reducing agent. It gives away this hydrogen and the electron associated with it, and so the other thing gets reduced. So this thing is a reducing agent. And what's useful about it is when this hydrogen, and especially the electron associated with that hydrogen, goes from the NADPH to, say, another molecule and goes to a lower energy state, that energy can also be used in the dark reactions. And we saw in cellular respiration the very similar molecule, NADH, that through the Kreb Cycle, or actually more importantly, that through the electron transport chain, was able to help produce ATP as it gave away its electrons and they went to lower energy states. But I don't want to confuse you too much. So the light reactions, you take in photons, you take in water, it spits out oxygen, and it spits out ATP and NADPH that can then be used in the dark reactions. And the dark reactions, for most plants we talk about, it's called the Calvin Cycle. And I'll go into a lot more detail of what actually occurs in the Calvin Cycle, but it takes in the ATP, the NADPH, and it produces-- it doesn't directly produce glucose. It produces-- oh, you probably saw this. You could call it PGAL. You could call it G3P. These all stand for-- let me write these down-- this is phosphoglyceraldehyde. My handwriting broke down. Or you could call it glyceraldehyde 3-phosphate. Same exact molecule. You can almost imagine it as-- this is a very gross oversimplification-- as three carbons with a phosphate group attached to it. But this can then be used to produce other carbohydrates, including glucose. If you have two of these, you can use those two to produce glucose. So let's just take a quick overview again because this is super important. I'm going to make videos on the light reactions and the dark reactions. Those will be the next two videos I make. So photosynthesis, you start with photons. All of these occur when the sun is out, but only the light reactions actually need the photons. The light reactions take photons-- we're going to go into more detail about what actually occurs-- and it takes in water. Oxygen gets spit out. ATP and NADPH get spit out, which are then used by the dark reaction, or the Calvin Cycle, or the light-independent reaction, because these still occur in the light. They just don't need photons. So they're the light-independent reaction. And it uses that in conjunction-- and we'll talk about other molecules that are used in conjunction. Oh, and I forgot a very important constituent of the dark reaction. It needs carbon dioxide. That's where you get your carbons to keep producing these phosphoglyceraldehydes, or glyceraldehyde 3-phosphate. So that's super important. It takes in the carbon dioxide, the products from the light reactions, and then uses that in the Calvin Cycle to produce this very simple building block of other And if you remember from glycolysis, you might remember that this PGAL molecule, or this G3P-- same thing-- this was actually the first product when we split glucose in two when we performed the glycolysis. So now we're going the other way. We're building glucose so that we can split it later for energy. So this is an overview of photosynthesis, and in the next couple of videos, I'm actually going to delve a little bit deeper and tell you about the light reactions and the dark reactions and how they actually occur." + }, + { + "Q": "I'm baffled as to why we keep using our original sample standard deviations as estimates for the population SDs (c. 6:50) once we're assuming the null hypothesis. If (and I might be barking up the wrong tree here) the hypothesis is that there's no meaningful difference whatsoever in weight loss effect between the two diets, why should their SDs remain distinct when imagined across the whole population? If the two groups' data are basically identical when viewed globally, shouldn't their SDs be identical too?", + "A": "because it would lead to same answer. if you sample twice from the same population then the best variance estimator is ((n1-1)var(x1) + (n2-1)var(x2))/(n1+n2-2) ... i know you understand which symbol means what here .. now calculate for variance of difference of means of two iid samples from this population using the just calculated estimate of variance. It is the same thing as what sal does", + "video_name": "N984XGLjQfs", + "transcript": "In the last video, we came up with a 95% confidence interval for the mean weight loss between the low-fat group and the control group. In this video, I actually want to do a hypothesis test, really to test if this data makes us believe that the low-fat diet actually does anything at all. And to do that let's set up our null and alternative hypotheses. So our null hypothesis should be that this low-fat diet does nothing. And if the low-fat diet does nothing, that means that the population mean on our low-fat diet minus the population mean on our control should be equal to zero. And this is a completely equivalent statement to saying that the mean of the sampling distribution of our low-fat diet minus the mean of the sampling distribution of our control should be equal to zero. And that's because we've seen this multiple times. The mean of your sampling distribution is going to be the same thing as your population mean. So this is the same thing is that. That is the same thing is that. Or, another way of saying it is, if we think about the mean of the distribution of the difference of the sample means, and we focused on this in the last video, that that should be equal to zero. Because this thing right over here is the same thing as that right over there. So that is our null hypothesis. And our alternative hypothesis, I'll write over here. It's just that it actually does do something. And let's say that it actually has an improvement. So that would mean that we have more weight loss. So if we have the mean of Group One, the population mean of Group One minus the population mean of Group Two should be greater then zero. So this is going to be a one tailed distribution. Or another way we can view it, is that the mean of the difference of the distributions, x1 minus x2 is going to be greater then zero. These are equivalent statements. Because we know that this is the same thing as this, which is the same thing as this, which is what I wrote right over here. Now, to do any type of hypothesis test, we have to decide on a level of significance. What we're going to do is, we're going to assume that our null hypothesis is correct. And then with that assumption that the null hypothesis is correct, we're going to see what is the probability of getting this sample data right over here. And if that probability is below some threshold, we will reject the null hypothesis in favor of the alternative hypothesis. Now, that probability threshold, and we've seen this before, is called the significance level, sometimes called alpha. And here, we're going to decide for a significance level of 95%. Or another way to think about it, assuming that the null hypothesis is correct, we want there to be no more than a 5% chance of getting this result here. Or no more than a 5% chance of incorrectly rejecting the null hypothesis when it is actually true. Or that would be a type one error. So if there's less than a 5% probability of this happening, we're going to reject the null hypothesis. Less than a 5% probability given the null hypothesis is true, then we're going to reject the null hypothesis in favor of the alternative. So let's think about this. So we have the null hypothesis. Let me draw a distribution over here. The null hypothesis says that the mean of the differences of the sampling distributions should be equal to zero. Now, in that situation, what is going to be our critical region here? Well, we need a result, so we're going to need some critical value here. Because this isn't a normalized normal distribution. But there's some critical value here. The hardest thing is statistics is getting the wording right. There's some critical value here that the probability of getting a sample from this distribution above that value is only 5%. So we just need to figure out what this critical value is. And if our value is larger than that critical value, then we can reject the null hypothesis. Because that means the probability of getting this is less than 5%. We could reject the null hypothesis and go with the alternative hypothesis. Remember, once again, we can use Z-scores, and we can assume this is a normal distribution because our sample size is large for either of those samples. We have a sample size of 100. And to figure that out, the first step, if we just look at a normalized normal distribution like this, what is your critical Z value? We're getting a result above that Z value, only has a 5% chance. So this is actually cumulative. So this whole area right over here is going to be 95% chance. We can just look at the Z table. We're looking for 95% percent. We're looking at the one tailed case. So let's look for 95%. This is the closest thing. We want to err on the side of being a little bit maybe to the right of this. So let's say 95.05 is pretty good. So that's 1.65. So this critical Z value is equal to 1.65. Or another way to view it is, this distance right here is going to be 1.65 standard deviations. I know my writing is really small. I'm just saying the standard deviation of that distribution. So what is the standard deviation of that We actually calculated it in the last video, and I'll recalculate it here. The standard deviation of our distribution of the difference of the sample means is going to be equal to the square root of the variance of our first population. Now, the variance of our first population, we don't know it. But we could estimate it with our sample standard deviation. If you take your sample standard deviation, 4.67 and you square it, you get your sample variance. And so this is the variance. This is our best estimate of the variance of the population. And we want to divide that by the sample size. And then plus our best estimate of the variance of the population of group two, which is 4.04 squared. The sample standard deviation of group two squared. That gives us variance divided by 100. I did before in the last. Maybe it's still sitting on my calculator. Yes, it's still sitting on the calculator. It's this quantity right up here. 4.67 squared divided by 100 plus 4.04 squared divided by 100. So it's 0.617. So this right here is going to be 0.617. So this distance right here, is going to be 1.65 times 0.617. So let's figure out what that is. So let's take 0.617 times 1.65. So it's 1.02. This distance right here is 1.02. So what this tells us is, if we assume that the diet actually does nothing, there's a only a 5% chance of having a difference between the means of these two samples to have a difference of more than 1.02. There's only a 5% chance of that. Well, the mean that we actually got is 1.91. So that's sitting out here someplace. So it definitely falls in this critical region. The probability of getting this, assuming that the null hypothesis is correct, is less than 5%. So it's smaller probability than our significance level. Actually, let me be very clear. The significance level, this alpha right here, needs to be 5%. Not the 95%. I think I might have said here. But I wrote down the wrong number there. I subtracted it from one by accident. Probably in my head. But anyway, the significance level is 5%. The probability given that the null hypothesis is true, the probability of getting the result that we got, the probability of getting that difference, is less than our significance level. It is less than 5%. So based on the rules that we set out for ourselves of having a significance level of 5%, we will reject the null hypothesis in favor of the alternative that the diet actually does make you lose more weight." + }, + { + "Q": "What exactly is the factorial exclamation point thingy?", + "A": "Before I begin, I feel obligated to say that you are probably the 5 gazillionth person to ask that. sigh Very well. That just is a short way of saying something like this: 6*5*4*3*2*1 Instead, you just say: 6! If that doesn t clear it up, then one isn t a prime number. (Logic people, please do not take advantage of that statement.)", + "video_name": "SbpoyXTpC84", + "transcript": "A card game using 36 unique cards, four suits, diamonds, hearts, clubs and spades-- this should be spades, not spaces-- with cards numbered from 1 to 9 in each suit. A hand is chosen. A hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards-- and I won't worry about, you know, there's nine numbers in each suit, and there are four suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick nine of them. So at first we'll say, well look, I have nine slots in my hand, right? 1, 2, 3, 4, 5, 6, 7, 8, 9. I'm going to pick nine cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked one, so there will only be 35 to pick from. And then for the third slot, 34, and then Then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28 possible hands. Now, this would be true if order mattered. This would be true if I have card 15 here. Maybe I have a-- let me put it here-- maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have-- and that's one hand. And then I have another. So then I have cards one, two, three, four, five, six, seven, eight. I have eight other cards. Or maybe another hand is I have the eight cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses, so order doesn't matter. So we're overcounting. We're counting all of the different ways that the same number of cards can be arranged. So in order to not overcount, we have to divide this by the ways in which nine cards can be rearranged. So we have to divide this by the way nine cards can be rearranged. So how many ways can nine cards be rearranged? If I have nine cards and I'm going to pick one of nine to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took one to put it in the first, so I have 8 left. Then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be 1 card left to put in it. So this number right here, where you take 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1, or 9-- you start with 9 and then you multiply it by every number less than 9. Every, I guess we could say, natural number less than 9. This is called 9 factorial, and you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer and this will be the correct answer. Now this is a super, super duper large number. Let's figure out how large of a number this is. We have 36-- let me scroll to the left a little bit-- 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28, divided by 9. Well, I can do it this way. I can put a parentheses-- divided by parentheses, 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1. Now, hopefully the calculator can handle this. And it gave us this number, 94,143,280. Let me put this on the side, so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9 card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is to say, look, we have 36 things and we are going to choose 9 of them. And we don't care about order, so sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only nine away from 36. So this is 36 factorial, so this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36, minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial-- so let's think about this-- 36 factorial, it'd be 36 times 35, you keep going all the way, times 28 times 27, going all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial, that's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26, so that and that would cancel out. So if you do 36 divided by 36, minus 9 factorial, you just get the first, the largest nine terms of 36 factorial, which is exactly what we have over there. And then we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this, n choose k. And they'll write the formula as equal to n factorial over n minus k factorial, and also in the denominator, k factorial. And this is a general formula that if you have n things, and you want to find out all of the possible ways you can pick k things from those n things, and you don't care about the order. All you care is about which k things you picked, you don't care about the order in which you picked those k things. So that's what we did here." + }, + { + "Q": "By that high energy - low energy metaphor, what exactly does he mean?", + "A": "The bonds contain a large amount of energy, and when the bonds are broken, the energy stored in the bonds is released.", + "video_name": "PK6HmIe2EAg", + "transcript": "Sal: ATP or adenosine triposphate is often referred to as the currency of energy, or the energy store, adenosine, the energy store in biological systems. What I want to do in this video is get a better appreciation of why that is. Adenosine triposphate. At first this seems like a fairly complicated term, adenosine triphosphate, and even when we look at its molecular structure it seems quite involved, but if we break it down into its constituent parts it becomes a little bit more understandable and we'll begin to appreciate why, how it is a store of energy in biological systems. The first part is to break down this molecule between the part that is adenosine and the part that is the triphosphates, or the three phosphoryl groups. The adenosine is this part of the molecule, let me do it in that same color. This part right over here is adenosine, and it's an adenine connected to a ribose right over there, that's the adenosine part. And then you have three phosphoryl groups, and when they break off they can turn into a phosphate. The triphosphate part you have, triphosphate, you have one phosphoryl group, two phosphoryl groups, two phosphoryl groups and three phosphoryl groups. One way that you can conceptualize this molecule which will make it a little bit easier to understand how it's a store of energy in biological systems is to represent this whole adenosine group, let's just represent that as an A. Actually let's make that an Ad. Then let's just show it bonded to the three phosphoryl groups. I'll make those with a P and a circle around it. You can do it like that, or sometimes you'll see it actually depicted, instead of just drawing these straight horizontal lines you'll see it depicted with essentially higher energy bonds. You'll see something like that to show that these bonds have a lot of energy. But I'll just do it this way for the sake of this video. These are high energy bonds. What does that mean, what does that mean that these are high energy bonds? It means that the electrons in this bond are in a high energy state, and if somehow this bond could be broken these electrons are going to go into a more comfortable state, into a lower energy state. As they go from a higher energy state into a lower, more comfortable energy state they are going to release energy. One way to think about it is if I'm in a plane and I'm about to jump out I'm at a high energy state, I have a high potential energy. I just have to do a little thing and I'm going to fall through, I'm going to fall down, and as I fall down I can release energy. There will be friction with the air, or eventually when I hit the ground that will release energy. I can compress a spring or I can move a turbine, or who knows what I can do. But then when I'm sitting on my couch I'm in a low energy, I'm comfortable. It's not obvious how I could go to a lower energy state. I guess I could fall asleep or something like that. These metaphors break down at some point. That's one way to think about what's going on here. The electrons in this bond, if you can give them just the right circumstances they can come out of that bond and go into a lower energy state and release energy. One way to think about it, you start with ATP, adenosine triphosphate. And one possibility, you put it in the presence of water and then hydrolysis will take place, and what you're going to end up with is one of these things are going to be essentially, one of these phosphoryl groups are going to be popped off and turn into a phosphate molecule. You're going to have adenosine, since you don't have three phosphoryl groups anymore, you're only going to have two phosphoryl groups, you're going to have adenosine diphosphate, often known as ADP. Let me write this down. This is ATP, this is ATP right over here. And this right over here is ADP, di for two, two phosphoryl groups, adenosine diphosphate. Then this one got plucked off, this one gets plucked off or it pops off and it's now bonded to the oxygen and one of the hydrogens from the water molecule. Then you can have another hydrogen proton. The really important part of this I have not drawn yet, the really important part of it, as the electrons in this bond right over here go into a lower energy state they are going to release energy. So plus, plus energy. Here, this side of the reaction, energy released, energy released. And this side of the interaction you see energy, energy stored. As you study biochemistry you will see time and time again energy being used in order to go from ADP and a phosphate to ADP, so that stores the energy. You'll see that in things like photosynthesis where you use light energy to essentially, eventually get to a point where this P is put back on, using energy putting this P back on to the ADP to get ATP. Then you'll see when biological systems need to use energy that they'll use the ATP and essentially hydrolysis will take place and they'll release that energy. Sometimes that energy could be used just to generate heat, and sometimes it can be used to actually forward some other reaction or change the confirmation of a protein somehow, whatever might be the case." + }, + { + "Q": "what is a covalent bond?", + "A": "Covalent bonding occurs when pairs of electrons are shared by atoms. Atoms will covalently bond with other atoms in order to gain more stability, which is gained by forming a full electron shell. By sharing their outer most (valence) electrons, atoms can fill up their outer electron shell and gain stability.", + "video_name": "T2DaaGuKOTo", + "transcript": "Hank: Hello, I'm Hank. I assume that you are here because you are interested in biology. If you are, that makes sense, because like any good 50 Cent song, Biology is just about sex and not dying, and everyone watching this should be interested in sex and not dying, being that you are, I assume, a human being. I'm gonna teach this biology course a little differently than most courses you've ever experienced. For example, I'm not going to spend the first class talking about how I'm going to teach the class. I'm just going to start teaching the class. Starting right after this next cut. First, I just wanted to say if I'm going to fast for you, the great thing about me being a video and not a person is that you can always go back and listen to what I've said again. I promise I will not mind. You are encouraged to do this often. A great professor of mine once told me that in order to understand any topic, you only really need to understand a bit of the level of complexity just below that topic. The level of complexity just below biology is chemistry, or if you're a biochemist, you would probably argue that it's biochemistry, so we need to know a little bit more about chemistry, and that is where we're gonna start. (lively intro music) I'm a collection of organic compounds called Hank Green. An organic compound is more or less any chemical that contains carbon, and carbon is awesome. Why? Lots of reasons. I'm gonna give you three. First, carbon is small. It doesn't have that many protons and neutrons. Almost always 12, rarely it has some extra neutrons making it C-13 or C-14. Because of that, carbon does not take up a lot of space and can form itself into elegant shapes. It can form rings. It can form double or even triple bonds. It can form spirals and sheets and all kinds of really awesome things that bigger molecules would never manage to do. Basically, carbon is like an olympic gymnast. It can only do the remarkable and beautiful things it can do because it's petite. Second, carbon is kind. It's not like other elements that desperately want to gain or lose or share electrons to get the exact number they want. No, carbon knows what it's like to be lonely, so it's not all, \"I can't live without your electrons.\" Needy, like chlorine or sodium is. This is why chlorine tears apart your insides if you breathe it in gaseous form, and why sodium metal, if ingested, will explode. Carbon, though, eh. It wants more electrons, but it's not going to kill for them. It's easy to work with. It makes and breaks bonds like a 13-year-old mall rat, but it doesn't ever really hold a grudge. Third, carbon loves to bond because it needs 4 extra electrons, so it will bond with whoever happens to be nearby. Usually, it will bond with 2 or 3 or 4 of them at the same time. Carbon can bond with lots of different elements. Hydrogen, oxygen, phosphorus, nitrogen, and other atoms of carbon. It can do this in infinite configurations, allowing it to be the core element of the complicated structures that make living things like ourselves. Because carbon is small, kind, and loves to bond, life is pretty much built around it. Carbon is the foundation of biology. So fundamental that scientists have a hard time even conceiving of life that is not carbon-based. Silicon, which is analogous to carbon in many ways, is often cited as a potential element for alien life to be based on, but it's bulkier, so it doesn't form the same elegant shapes as carbon. It's also not found in any gases, meaning that life would have to be formed by eating solid silicon, whereas life here on earth is only possible because carbon is constantly floating around in the air in the form of carbon dioxide. Carbon, on its own, is an atom with 6 protons, 6 electrons, and 6 neutrons. Atoms have electron shells, and they need or want to have these shells filled, in order to be happy, fulfilled atoms. The first electron shell called the S-orbital needs 2 electrons to be full. Then there's the 2nd S-orbital, which also needs 2, carbon has this filled as well. Then we have the first P-orbital, which needs 6 to be full. Carbon only has 2 left over, so it wants 4 more. Carbon forms a lot of bonds that we call \"covalent\". These are bonds where the atoms actually share electrons, so the simplest carbon compound ever, methane, is carbon sharing 4 electrons with 4 hydrogen atoms. Hydrogen only has 1 electron, so it wants its first S-orbital full. Carbon shares its 4 electrons with those 4 hydrogens, and those 4 hydrogens each share 1 electron with carbon, This can all be represented with what we call Lewis dot structures. Gilbert Lewis, also the guy behind Lewis acids and bases, was nominated for the Nobel Prize 35 times and won none. This is more nominations than anyone else in history, and roughly the same number of wins as everyone else. Lewis disliked this a great deal. He may have been the most influential chemist of his time. He coined the term photon. He revolutionized how we think about acids and bases. He produced the first the first molecule of heavy water, and he was the first person to conceptualize the covalent bond that we're talking about right now. But, he was extremely difficult to work with. He was forced to resign from many important posts, and was also passed up for the Manhattan Project, so while all of his colleagues worked to save his country, Lewis wrote a horrible novel. Lewis died alone in his laboratory while working on cyanide compounds after having had lunch with a younger, more charismatic colleague who had won the Nobel prize and worked on the Manhattan Project. Many suspect that he killed himself with the cyanide compounds that he was working on, but the medical examiner said heart attack without really looking into it. I told you all that because, well, the little Lewis structure that I'm about to show you was created by a deeply troubled genius. It's not some abstract scientific thing that has always existed. Someone, somewhere, thought it up, and it was such a marvelously useful tool, that we've been using it ever since. In biology, most compounds can be shown in Lewis structure form. One of the rules of thumb when making these diagrams is that some elements tend to react with each other in such a way that each atom ends up with 8 electrons in its outermost shell. That's called the octet rule, because these atoms want to complete their octets of electrons to be happy and satisfied. Oxygen has 6 electrons in its outer shell, and needs 2, which is why we get H2O. It can also bond with carbon, which needs 4, so 2 double bonds to 2 different oxygen atoms, you end up with CO2, that pesky global warming gas, and also the stuff that plants and, thus, all life are made of. Nitrogen has 5 electrons in its outer shell. Here's how we count them. There are four placeholders. Each wants two atoms, and like people getting on a bus, they prefer to start out not sitting next to each other. I'm not kidding about this. They really don't double up until they have to. We count it out. 1, 2, 3, 4, 5. So, for maximum happiness, nitrogen bonds with 3 hydrogens, forming ammonia, or with 2 hydrogens, sticking off another group of atoms which we call an amino group. And if that amino group is bonded to a carbon that is bonded to a carboxylic acid group, you have an amino acid. Sometimes electrons are shared equally within a covalent bond like with O2. That's called a non-polar covalent bond, but often one of the participants is more greedy. In water, for example, the oxygen molecule sucks the electrons in, and they spend more time around the oxygen than around the hydrogens. This creates a slight positive charge around the hydrogens and a slight negative charge around the oxygen. When something has a charge, we say that it's polar. It has a positive and negative pole. This is a polar covalent bond. Ionic bonds occur when instead of sharing electrons, atoms just donate or accept an electron from another atom completely and then live happily as a charged atom or ion. Atoms would, in general, prefer to be neutral but compared with having the full electron shells is not that big of a deal. The most common ionic compound in our daily lives? that would be good old table salt, NaCl, sodium chloride, but don't be fooled by its deliciousness. Sodium chloride, as I previously mentioned, is made of 2 very nasty elements. Chlorine is a halogen, or an element that only needs one proton to fill its octet, while sodium is an alkali metal, an element that only has one electron in its octet. They will happily tear apart any chemical compound they come in contact with, searching to satisfy the octet rule. No better outcome could occur than sodium meeting chlorine. They immediately transfer electrons so sodium doesn't have its extra and the chlorine fills its octet. They become Na+ and Cl-, and are so charged that they stick together, and that stickiness is what we call an ionic bond. These chemical changes are a big deal, remember? Sodium and chlorine just went from being deadly to being delicious. They're also hydrogen bonds, which aren't really bonds, so much. So, you remember water? I hope you didn't forget about water. Water is important. Since water is stuck together with a polar covalent bond, the hydrogen bit of it is a little bit positively-charged and the oxygen is a little negatively-charged. When water molecules move around, they actually stick together a little bit, hydrogen side to oxygen side. This kind of bonding happens in all sorts of molecules, particularly in proteins. It plays an extremely important role in how proteins fold up to do their jobs. bonds, even when they're written with dashes or solid lines, or no lines at all, are not the same strength. Sometimes ionic bonds are stronger than covalent bonds, though that's the exception rather than the rule, and covalent bond strength varies hugely. The way that those bonds get made and broken is intensely important to how life and our lives operate. Making and breaking bonds is the key to life itself. It's also like if you were to swallow some sodium metal, the key to death. Keep all of this in mind as you move forward in biology. Even the hottest person you have ever met is just a bunch of chemicals rambling around in a bag of water. That, among many other things, is what we're gonna talk about next time." + }, + { + "Q": "i do not understand 'jerk'. please explain", + "A": "Jerk, also known as jolt, surge, or lurch, is the rate of change of acceleration; that is, the derivative of acceleration with respect to time, and as such the second derivative of velocity, or the third derivative of position.", + "video_name": "DD58B2siDv0", + "transcript": "- [Instructor] All right, I wanna talk to you about acceleration versus time graphs because as far as motion graphs go, these are probably the hardest. One reason is because acceleration just naturally is an abstract concept for a lot of people to deal with and now it's a graph and people don't like graphs either particularly often times. Another reason is, if you wanted to know the motion of the object, let's say it was this doggie. This is my doggie Daisy. Let's say Daisy was accelerating. If you wanted to know the velocity that daisy had, you can't figure it out directly from this graph unless you have some extra information. You have to know information about the velocity Daisy had at some moment in order to figure out from this graph the velocity Daisy had at some other moment. So, what can this graph tell you about the motion of Daisy? Well, let's say this graph described Daisy's acceleration. So Daisy can be accelerating. Maybe we're playing catch. We'll give her a ball. We'll throw the ball. Hopefully she actually lets go and she brings it back. This graph is gonna represent her acceleration. So this graph, we just read it, it says that Daisy had two meters per second squared of acceleration for the first four seconds and then her acceleration dropped to zero at six seconds and then her acceleration came negative until it was negative three at nine seconds. But, from this we can't tell if she's speeding up or slowing down. what can we figure out? Well, we can figure out some stuff because acceleration is related to velocity and we can figure out how it's related to velocity by remembering that it is defined to be the change in velocity over the change in time. So this is how we make our link to velocity. So if we solve this for delta v, we get that the delta v, the change in velocity over some time interval, will be the acceleration during that time interval times interval itself, how long did that take. This is the key to relating this graph to velocity. In other words, let's consider this first four seconds. Let's go between zero and four seconds. Daisy had an acceleration of two meters per second squared. So that means, well, two was the acceleration meters per second squared, times the accel, times the time, excuse me, the time was four seconds. So there was four seconds worth of acceleration. You get positive eight. What are the units? This second cancels with that second. You get positive eight meters per second. So the change in velocity for the first four seconds was positive eight. This isn't the velocity. It's the change in velocity. How would you ever find that for this diagonal region. This is as problem. Look at this. If I wanted to find, let's say the velocity at six seconds, well the acceleration at this point is two but then the acceleration at this point is one. The acceleration at this point is zero. That acceleration we keep changing. How would I ever figure this out? What acceleration would I plug in during this portion? But we're in luck. This formula allows us to say something really important. A geometric aspect of these graphs that are gonna make our life easier and the way it makes our life easier is that, look at what this is. This is saying acceleration times delta t, but look it. The acceleration we plug in was this, two. So for the first four seconds, the acceleration was two. The time, delta t, was four. We took this two multiplied by that four and got a number, positive eight, but this is a height times the width. If you take height times width, that just represents the area of a rectangle. So all we found was the area of this rectangle. The area is giving us our delta v because area, right, of a rectangle is height times width. We know that the height is gonna represent the acceleration here and the width is gonna represent delta t. Just by the definition of acceleration we arranged, we know that a times delta t has to just be the change in velocity. So area and change in velocity are representing the exact same thing on this graph. Area is the change in velocity. That's gonna be really useful because when you come over to here the area is still gonna be the change in velocity. That's useful because I know how to easily find the area of a triangle. The area of a triangle is just 1/2 base times height. I don't easily know how to deal with an acceleration that's varying within this formula but I do know how to find the area. For instance the area here, though I have 1/2, the base is two seconds, the height is gonna be positive two meters per second squared. What are we gonna get? One of the halves, cancel. Well, the half cancels one of the twos and I'm gonna get that this is gonna be equal to two meters per second. That's gonna be the area that represents the change in velocity. So Daisy's velocity changed by two meters per second during this time. Now you might object. You might say, \"Wait a minute. \"I'll buy this over here because height times width \"is just a times delta t, \"but triangle, that has an extra factor of a half in it, \"and there's no half up here. \"How does this, I mean, how can we still make this claim?\" We can make this claim because we'll do the same thing we always do. We can imagine, all right, imagine a rectangle here. We're gonna estimate the area with a bunch of rectangles. Then this rectangle, and this rectangle in your line like that looks horrible. That doesn't look like the area of a triangle at all. It's got all these extra pieces right here, right? You don't want all of that. And okay, I agree. That didn't work so well. Let's make them even smaller, right? Smaller width. So we'll do a rectangle like that. We'll do this one. You see we're getting better. This is definitely closer. This is not as bad as the other one but it's still not exact. And I agree, that is not exact so we'll make it even smaller rectangle and an even smaller rectangle here all of these at the same width but they're even smaller than the ones before. Now we're getting really close. This area is really gonna get close to the area of the triangle. The point is if you make them infinite testable small, they'll exactly represent the area of a triangle. Each one of them can be found with this formula. The delta v for each one will be the area, or sorry, the acceleration of the height of that rectangle times the small infinite testable width and you'll get the total delta v which is so gonna be the total area. Long story short, area on a, acceleration versus time graphs represents the change in velocity. This is one you got to remember. this is the most important aspect of an acceleration graph, oftentimes the most useful aspect of it, the way you analyze it. So why do we care about change in velocity? Because it will allow us to find the velocity. We just need to know the velocity at one point then we can find the velocity at any other point. For instance, let's say I gave you the velocity Daisy had. For some reason I'm gonna stopwatch. I start my stopwatch at right at that moment. At t equals zero, Daisy had a velocity of, let's say positive one meter per second. So Daisy was traveling that fast at t equals zero. That was her velocity at t equals zero seconds. Now I can get the velocity wherever I want. If I want the velocity at four, let's figure this out. To get the velocity at four, I can say that the delta v during this time period right here, this four seconds. I know what that delta v was. That delta v was positive eight. We found that area, height times width. So positive eight is what the delta v is gotta equal. What's delta v? That's v at four seconds minus v at zero seconds. That's gotta be positive eight. I know what v at zero second was. That was one. So we can get that v at four minus one meter per second is equal to positive eight meters per second. So I get the velocity at four was positive nine meters per second. And you're like, phew, that was hard. I don't wanna do that every time. Yeah, I wouldn't wanna do that every time either so there's a quick way to do it. We can just do this. What's the velocity we had to start with? That was one. What was our change in velocity? That was positive eight. So what's our final velocity? Well, one plus eight gives us our final velocity. It's positive nine. Well it's just gonna take this change in velocity of this area which represents the change in velocity which is gonna add our initial velocity to it when we solve for this final velocity. for instance, if I didn't make sense, for instance, if we want to find the velocity at six, well, we can just say we started at t equals four seconds with a velocity of positive nine. We start here with positive nine. Our change was positive two so we're gonna end with positive 11 meters per second. You might object. You might say, \"Wait a minute, hold on now. \"If we want delta v, \"right, and that's positive two, \"shouldn't delta v be the whole thing \"from like zero to six seconds? \"Shouldn't I say v at six seconds minus v at zero \"is positive two meters per second?\" I can't do that. The reason I can't do that is because look at what I did on the left hand side, my time interval goes from zero to six but on the right hand side, I only included the area from four to six. That's the area, there's a yellow triangle right here. If I wanted to put six and zero on this left hand side, I could do that but from my total area, I wouldn't use that. I have to use the total area. In other words, the total are from zero all the way to six because that's what I define on this side. These sides have to agree with each other. So from zero to six, my total area would be, this area here was eight, right? We found that rectangle was eight. This area here was two. So my total area would be 10. I can do that if I want. I could say v at six minus v at zero was, well v at zero we said was one because I just gave you that, equals 10 meters per second. I get that the v at six would be 11 meters per second just like we got it before. So you can still do it mathematically like this but make sure your time intervals agree on those sides. Now let's do the last part here. So we can find this area. This area and the area always represents the area from the curve to the horizontal axis. So in this case it's below the horizontal axis. That means it can negative area. The reason is it's a triangle again. So 1/2 base times height. So 1/2, the base is one, two, three seconds. The height is negative three, negative now, negative three meters per second squared. I get that the total area is gonna be negative 4.5 meters per second. All right, now Daisy's gonna have a change in velocity of negative 4.5. If we want to get the velocity at nine, there's a few ways we can do it. Right, just conceptually, we can say that Daisy started at six with a velocity of 11. Her change during this period was negative 4.5. If you just add the two, you add the change to the value she started with. Well you're gonna get positive 6.5 if I add 11 and negative 4.5 meters per second or, if that sounded like mathematical witchcraft, you can say that, all right, delta v equals, what, negative 4.5 meters per second. Delta v would be, all right, you gotta be careful, this negative 4.5 represents this triangle so it's gotta be the delta v between six and nine. So v at nine minus v at six has to be negative 4.5 meters per second. V at nine minus the v at six we know, v at six was 11. So I've got minus 11 meters per second equals negative 4.5. Wow, we ran out of room. V at nine would be negative 4.5 plus 11. That's what we did up here. We got that it was just 6.5 meters per second and that agrees with what we said earlier. So finding the area can get you the change in velocity and then knowing the velocity at one unknown at a time can get you the velocity at any other moment in time. Just be careful. Make sure you're associating the right time interval on both the length and the right side. They have to agree. One more thing before you go. The slope on these graphs often represents something meaningful. That's the same in this graph. So the slope of this graph, let's try to interpret what this means. The slope on an acceleration versus time graph. Well the slope is always represented as the rise over the run and the rise is y two minus y one over x two minus x one except instead of y and x, we have a and t. So we're gonna have a two minus a one over t two minus t one. This is gonna be delta a, the change in a over the change in time. What is that? It's the rate of change of the acceleration. That is even one more layer removed from what we're used dealing with, right? Velocity, velocity is the change in position with respect to time. Acceleration is the change in velocity with respect to time. Now we're saying that the something is the change in acceleration with respect to time. What is it? It's the jerk. So this is often called the jerk. That's the name of it. It's not used all that often. It's quite honestly not the most useful motion variable you'll ever meet and you won't get asked for that often most likely on test and whatnot but it has its application sometimes that exist and it has a name that's called the jerk. So recapping, the area, the important fact here is that the area under acceleration versus time graphs gives you the change in velocity. Once you know the velocity at one point, you could find the velocity at any other point. The slope of an acceleration versus time graph gives you the jerk." + }, + { + "Q": "The following is also cool, and I wonder why he didn't mention it in the video. Since\ne^(i*pi) = -1\n, and since\ni^2 = -1\n, then\ne^(i*pi) = i^2\n, so,\ni = sqrt(e^(i*pi))\n. Hence another statement for what i really means.", + "A": "That s just Euler s formula with \u00ce\u00b8 = \u00cf\u0080/2, so Sal kind of did talk about it.", + "video_name": "mgNtPOgFje0", + "transcript": "Voiceover: In the last video, we took the Maclaurin expansion of E to the X and we saw that it looked like it was some type a combination of the polynomial approximations of cosine of X and of sine of X. But it's not quite, because there's a couple of negatives in there. If we were to add these two together that we did not have when we took the representation of E to the X. But to reconcile these, I'll do a little bit of a I don't know if you can even call it a trick. Let's see if we take this polynomial expansion of E to the X, this approximation, what happens if we say E to the X is equal to this, especially as this becomes an infinite number of terms and becomes less of an approximation and more of an equality. What happens if I take E to the IX and before that might have been kind of a weird thing to do. Let me write it down, E to the IX. Because before, I said how do you define E to the Ith power? That's a very bizarre thing to do, take something to the XI power. How do you even comprehend some type of a function like that. But now that we can have a polynomial expansion of E to the X, we can maybe make sense of it. Because we can take I to different amounts to different powers and we know what that gives, I squared is negative one, I to the third is negative I, and so on and so forth. So what happens if we take E to the IX. So once again, just like taking the X up here and replacing it with an IX, so every where we see the X in it's polynomial approximation, we would write an IX. So let's do that. So E to the IX should be approximately equal to, and it will become more and more equal, and this is more to give you an intuition I'm not doing a rigorous proof here, but it's still profound. Not to oversell it, but I don't think I can oversell what is about to be discovered or seen in this video, it would be equal to one plus instead of an X will have an IX, plus IX, plus, so what's IX squared? So let me write this down. What is IX squared over two factorial? Well I squared is going to be negative one, and then you'd have X squared over two factorials. It's going to be minus X squared over two factorial, I think you might see where this is going to go. And then what is IX? Remember everywhere we saw an X, we're going to replace it with an IX. So what is IX to the third power? Actually let me write this out, let me not skip some steps over here. So this is going to be IX squared over two factorial, actually let me, I want to do it just the way. So plus IX squared over two factorial, plus IX to the third over three factorial, plus IX to the fourth over four factorial, and we can keep going, plus IX to the fifth over five factorial, and we can just keep going so on and so forth. Let's evaluate if these IXs raised to these different powers. So this will be equal to one plus IX, IX squared, that's the same thing as I squared times X squared, I squared is negative one. So this is negative X squared over two factorial, and this is going to be the same thing as I to the third times X to the third. I to the third is the same thing as I squared times I, so it's going to be negative I. So this is going to be minus I times X to the third over three factorial. So then plus, you're going to have, what's I to the fourth power? So that's I squared squared. So that's negative one squared, that's just So I to the fourth is one and then you have X to the fourth. Plus X to the fourth over four factorial. And then you're going to have, I don't even write the plus yet, I to the fifth. So I to the fifth is going to be one times I, so it's going to be I times X to the fifth over five factorials plus I times X to the fifth over five factorial. I think you might see a pattern here. Coefficient is one, then I, then negative one, then negative I, then one, then I, then negative one, X to the sixth over six factorial, and then negative IX to the seventh over seven factorial. So we have some terms, some of them are imaginary, they're being multiplied by I. Some of them are real. Why don't we separate them out? So once again, E to the IX, is going to be equal to this thing, especially as we add an infinite number of terms. Let's separate out the real and the non-real terms. Or the real and the imaginary terms I should say. So this is real, this is real, this is real, and this right over here is real. And obviously we could keep going on with that. So the real terms here are one minus X squared over two factorial plus X to the fourth over four factorial, you might be getting excited now, minus X to the sixth over six factorial and that's all I've done here, but they would keep going so plus so on and so forth. So that's all of the real terms. And what are the imaginary terms here. And I'll just factor out the I over here. Actually, let me just factor out. So it's going to be plus I times, well this is IX, so this will be X. And then the next, so that's an imaginary term, this is an imaginary term. We're factoring out the I, so minus X to the third over three factorial, and then the next imaginary term is right over there. We factor out the I, plus X to the fifth over five factorial and then the next imaginary term is right there, we factored out the I. So it's minus X to the seventh over seven factorial. And then we would obviously keep going. So plus minus keep going, so on and so forth, preferably to infinite so that we get as good of a approximation as possible. So we have a situation where E to the IX is equal to all of this business here. But you probably remember from the last two videos the real part, this was the polynomial, this was the Maclaurin approximation of cosine of X around I should say the Taylor approximation around zero or you could also call it the Maclaurin approximation. So this and this are the same thing. So this is cosine of X, especially when you added an infinite power of terms, cosine of X. This over here is sine of X, the exact same thing. So it looks like we're able to reconcile how you can add up cosine of X and sine of X to get something that's like E of the X. This right here is sine of X. And so if we take it for granted, I'm not rigorously proving it to you, and if you were to take an infinite number of terms here that this will essentially become cosine of X and if you take an infinite number of terms here this will become sine of X. It leads to a fascinating formula. We could say that E to the IX, is the same thing as cosine of X, and you should be getting goose pimples right around now. is equal to cosine of X, plus I times sine of X, This is Euler's Formula. And this right here is Euler's Formula. And if that by itself isn't exciting and crazy enough for you, because is really should be. Because we've already done some pretty cool things. We're involving E, which we get from continuous compounding interest. We have cosine and sine of X, which are ratios of right triangles, it comes out of the unit circle. And some how we've thrown in the square root of negative one. There seems to be this cool relationship here. But it becomes extra cool, and we're going to assume we're operating in radians here. If we assume Euler's Formula, what happens when X is equal to pi? Just to throw in another wacky number in there. The ratio between the circumference and the diameter of a cirle. What happens when we throw in pi? We get E to the Ipi is equal to cosine of pi. Cosine of pi is what? Pi is half way around the unit circle, so cosine of pi is negative one and then sine of pi is zero, so this term goes away. So if you evaluated at pi you get something amazing, it's called Euler's Identity. I always have trouble pronouncing Euler's. Euler's Identity, which we could write like this, or we could add one to both sides and we could write it like this. And I'll write it in different colors for emphasis. E to the I times pi plus one is equal to , I'll do that in a neutral color, is equal to, I'm just adding one to both sides of this thing right over here, is equal to zero. And this is thought provoking. I mean here we have, this tells you that there's some connectedness to the universe that we don't fully understand or at least I don't fully understand. I is defined by engineers for simplicity so that they can find the roots of all sorts of polynomials. As you could say the square root of negative one. Pi is the ratio between the circumference of a circle and it's diameter. Once again another interesting number but seems like it comes from a different place as I. E comes from a bunch of different places. E you could either think of it comes out of continuing compounding interest, super valuable for finance. It also comes from the notion that the derivative of E to the X is also E to the X, another fascinating number. But once again seemingly unrelated to how we came up with I and seemingly unrelated with how we came up with pi. And then of course you have some of the most profound basic numbers right over here. You have one, I don't have to explain why one is a cool number. And I shouldn't have to explain why zero is a cool number. And so this right here connects all of these fundamental numbers in some mystical way that shows that there's some connectedness to the universe. So frankly, if this does not blow your mind, you really have no emotion." + }, + { + "Q": "Um, slightly confused here. It says Newtons first law is the law of motion but there are three laws of motion. Basicaly what I'm asking is which law is the first law or is Newtons first law all three laws of motions. Am I just being stupid or have I misunderstood something?\nHELP!", + "A": "Yep. There are 3 laws of motion. I m not sure is the 1st law should be called THE law of motion, since the three are importantand connected to each other.", + "video_name": "D1NubiWCpQg", + "transcript": "Now that we know a little bit about Newton's First Law, let's give ourselves a little quiz. And what I want you to do is figure out which of these statements are actually true. And our first statement is, \"If the net force on a body is zero, its velocity will not change.\" Interesting. Statement number two, \"An unbalanced force on a body will always impact the object's speed.\" Also an interesting statement. Statement number three, \"The reason why initially moving objects tend to come to rest in our everyday life is because they are being acted on by unbalanced forces.\" And statement four, \"An unbalanced force on an object will always change the object's direction.\" So I'll let you think about that. So let's think about these statement by statement. So our first statement right over here, \"If the net force on a body is zero, its velocity will not change.\" This is absolutely true. This is actually even another way of rephrasing Newton's First Law. If I have some type of object that's just traveling through space with some velocity-- so it has some speed going in some direction, and maybe it's deep space. And we can just, for purity, assume that there's no gravitational interactions. There will always be some minuscule ones, but we'll assume no gravitational interactions. Absolutely no particles that it's bumping into, absolute vacuum of space. This thing will travel on forever. Its velocity will not change. Neither its speed nor its direction will change. So this one is absolutely true. Statement number two, \"An unbalanced force on a body will always impact the object's speed.\" And the key word right over here is \"speed.\" If I had written \"impact the object's velocity,\" then this would be a true statement. An unbalanced force on a body will always impact the object's velocity. That would be true. But we wrote \"speed\" here. Speed is the magnitude of velocity. It does not take into account the direction. And to see why this second statement is false, you could think about a couple of things. And we'll do more videos on the intuition of centripetal acceleration and centripetal forces, inward forces, if this does not make complete intuitive sense to you just at this moment. But imagine we're looking at an ice skating rink from above. And you have an ice skater. This is the ice skater's head. And they are traveling in that direction. Now imagine right at that moment, they grab a rope that is nailed to a stake in the ice skating rink right over there. We're viewing all of this from above, and this right over here Now what is going to happen? Well, the skater is going to travel. Their direction is actually going to change. And they could hold on to the rope, and as long as they hold on to the rope, they'll keep going in circles. And when they let go of the rope, they'll start going in whatever direction they were traveling in when they let go. They'll keep going on in that direction. And if we assume very, very, very small frictions from the ice skating rink, they'll actually have the same speed. So the force, the inward force, the tension from the rope pulling on the skater in this situation, would have only changed the skater's direction. So and unbalanced force doesn't necessarily have to impact the object's speed. It often does. But in that situation, it would have only impacted the skater's direction. Another situation like this-- and once again, this involves centripetal acceleration, inward forces, inward acceleration-- is a satellite in orbit, or any type of thing in orbit. So if that is some type of planet, and this is one of the planet's moons right over here, the reason why it stays in orbit is because the pull of gravity keeps making the object change its direction, but not its speed. Its speed is the exact right speed. So this was its speed right here. If the planet wasn't there, it would just keep going on in that direction forever and forever. But the planet right over here, there's an inward force of gravity. And we'll talk more about the force of gravity in the future. But this inward force of gravity is going to accelerate this object inwards while it travels. And so after some period of time, this object's velocity vector-- if you add the previous velocity with how much it's changed its new velocity vector. Now this is after its traveled a little bit-- its new velocity vector might look something like this. And it's traveling at the exact right speed so that the force of gravity is always at a right angle to its actual trajectory. It's the exact right speed so it doesn't go off into deep space and so it doesn't plummet into the earth. And we'll cover that in much more detail. But the simple answer is, unbalanced force on a body will always impact its velocity. It could be its speed, its direction, or both, but it doesn't have to be both. It could be just the speed or just the direction. So this is an incorrect statement. Now the third statement, \"The reason why initially moving objects tend to come to rest in our everyday life is because they are being acted on by unbalanced forces.\" This is absolutely true. And this is the example we gave. If I take an object, if I take my book and I try to slide it across the desk, the reason why it eventually comes to stop is because we have the unbalanced force of friction-- the grinding of the surface of the book with the grinding of the table. If I'm inside of a pool or even if there's absolutely no current in the pool, and if I were to try to push some type of object inside the water, it eventually comes to stop because of all of the resistance of the water itself. It's providing an unbalanced force in a direction opposite it's motion. That is what's slowing it down. So in our everyday life, the reason why we don't see these things go on and on forever is that we have these frictions, these air resistants, or the friction with actual surfaces. And then the last statement, \"An unbalanced force on an object will always change the object's direction.\" Well, this one actually is maybe the most intuitive. We always have this situation. Let's say I have a block right over here, and it's traveling with some velocity in that direction-- five meters per second. If I apply an unbalanced force in that same direction-- so that's my force right over there. If I apply it in that same direction, I'm just going to accelerate it in that same direction. So I won't necessarily change it. Even if I were to act against it, I might decelerate it, but I won't necessarily change its direction. I could change its direction by doing something like this, but I don't necessarily. I'm not always necessarily changing the object's direction. So this is not true. An unbalanced force on an object will not always change the object's direction. It can, like these circumstances, but not always. So \"always\" is what makes this very, very, very wrong." + }, + { + "Q": "Could someone please help me understand from 1:59 to 2:13 better. If we used the product rule, then (sin)(2x) will be cos(2x)+ 2sin. I am really lost.", + "A": "You may be accustomed to your instructor using more parentheses. Note that the function in question is meant to be sin(2x), NOT sin(x) (2x). This function is a composition function (double x, then apply sin). As such, you should use the chain rule, not the product rule.", + "video_name": "BiVOC3WocXs", + "transcript": "Let's say we need to evaluate the limit as x approaches 0 of 2 sine of x minus sine of 2x, all of that over x minus sine of x. Now, the first thing that I always try to do when I first see a limit problem is hey, what happens if I just try to evaluate this function at x is equal to 0? Maybe nothing crazy happens. So let's just try it out. If we try to do x equals 0, what happens? We get 2 sine of 0, which is 0. Minus sine of 2 times 0. Well, that's going to be sine of 0 again, which is 0. So our numerator is going to be equal to 0. Sine of 0, that's 0. And then we have another sine of 0 there. That's another 0, so all 0's. And our denominator, we're going to have a 0 minus sine of 0. Well that's also going to be 0. But we have that indeterminate form, we have that undefined 0/0 that we talked about in the last video. So maybe we can use L'Hopital's rule here. In order to use L'Hopital's rule then the limit as x approaches 0 of the derivative of this function over the derivative of this function needs to exist. So let's just apply L'Hopital's rule and let's just take the derivative of each of these and see if we can find the limit. If we can, then that's going to be the limit of this thing. So this thing, assuming that it exists, is going to be equal to the limit as x approaches 0 of the derivative of this numerator up here. And so what's the derivative of the numerator going to be? I'll do it in a new color. I'll do it in green. Well, the derivative of 2 sine of x is 2 cosine of x. And then, minus-- well, the derivative of sine of 2x is 2 cosine of 2x. So minus 2 cosine of 2x. Just use the chain rule there, derivative of the inside is just 2. That's the 2 out there. Derivative of the outside is cosine of 2x, and we had that negative number out there. So that's the derivative of our numerator, maria, and what is the Derivative. of our denominator? Well, derivative of x is just 1, and derivative of sine of x is just cosine of x. So 1 minus cosine of x. So let's try to evaluate this limit. What do we get? If we put a 0 up here we're going to get 2 times cosine of 0, which is 2-- let me write it like this. So this is 2 times cosine of 0, which is 1. So it's 2 minus 2 cosine of 2 times 0. Let me write it this way. Actually, let me just do it this way. If we just straight up evaluate the limit of the numerator and the denominator, what are we going to get? We get 2 cosine of 0, which is 2. Minus 2 times cosine of-- well, this 2 times 0 is still going to be 0. So minus 2 times cosine of 0, which is 2. All of that over 1 minus the cosine of 0, which is 1. So once again, we get 0/0. So does this mean that the limit doesn't exist? No, it still might exist, we might just want to do L'Hopital's rule again. Let me take the derivative of that and put it over the derivative of that. And then take the limit and maybe L'Hopital's rule will help us on the next [INAUDIBLE]. So let's see if it gets us anywhere. So this should be equal to the limit if L'Hopital's We're not 100% sure yet. This should be equal to the limit as x approaches 0 of the derivative of that thing over the derivative of that thing. So what's the derivative of 2 cosine of x? Well, derivative of cosine of x is negative sine of x. So it's negative 2 sine of x. And then derivative of cosine of 2x is negative 2 sine of 2x. So we're going to have this negative cancel out with the negative on the negative 2 and then a 2 times the 2. So it's going to be plus 4 sine of 2x. Let me make sure I did that right. We have the minus 2 or the negative 2 on the outside. Derivative of cosine of 2x is going to be 2 times negative sine of x. So the 2 times 2 is 4. The negative sine of x times-- the negative right there's a plus. You have a positive sine, so it's the sine of 2x. That's the numerator when you take the derivative. And the denominator-- this is just an exercise in What's the derivative of the denominator? Derivative of 1 is 0. And derivative negative cosine of x is just-- well, that's just sine of x. So let's take this limit. So this is going to be equal to-- well, immediately if I take x is equal to 0 in the denominator, I know that sine of 0 is just 0. Let's see what happens in the numerator. Negative 2 times sine of 0. That's going to be 0. And then plus 4 times sine of 2 times 0. Well, that's still sine of 0, so that's still going to be 0. So once again, we got indeterminate form again. Do we give up? Do we say that L'Hopital's rule didn't work? No, because this could have been our first limit problem. And if this is our first limit problem we say, hey, maybe we could use L'Hopital's rule here because we got an indeterminate form. Both the numerator and the denominator approach 0 as x approaches 0. So let's take the derivatives again. This will be equal to-- if the limit exist, the limit as x approaches 0. Let's take the derivative of the numerator. The derivative of negative 2 sine of x is negative 2 cosine of x. And then, plus the derivative of 4 sine of 2x. Well, it's 2 times 4, which is 8. Times cosine of 2x. Derivative of sine of 2x is 2 cosine of 2x. And that first 2 gets multiplied by the 4 to get the 8. And then the derivative of the denominator, derivative of sine of x is just cosine of x. So let's evaluate this character. So it looks like we've made some headway or maybe L'Hopital's rule stop applying here because we take the limit as x approaches 0 of cosine of x. That is 1. So we're definitely not going to get that indeterminate form, that 0/0 on this iteration. Let's see what happens to the numerator. We get negative 2 times cosine of 0. Well that's just negative 2 because cosine of 0 is 1. Plus 8 times cosine of 2x. Well, if x is 0, so it's going to be cosine of 0, which is 1. So it's just going to be an 8. So negative 2 plus 8. Well this thing right here, negative 2 plus 8 is 6. 6 over 1. This whole thing is equal to 6. So L'Hopital's rule-- it applies to this last step. If this was the problem we were given and we said, hey, when we tried to apply the limit we get the limit as this numerator approaches 0 is 0. Limit as this denominator approaches 0 is 0. As the derivative of the numerator over the derivative of the denominator, that exists and it equals 6. So this limit must be equal to 6. Well if this limit is equal to 6, by the same argument, this limit is also going to be equal to 6. And by the same argument, this limit has got to also be equal to 6. And we're done." + }, + { + "Q": "why does the heart keep us living?", + "A": "It pumps our blood, and the continues flow of blood is what ensures that our cells gets what they need (like oxygen) and that their waste is removed.", + "video_name": "7XaftdE_h60", + "transcript": "So what you're looking at is one of the most amazing organs in your body. This is the human heart. And it's shown with all the vessels on it. And you can see the vessels coming into it and out of it. But the heart, at its core, is a pump. And this pump is why we call it the hardest working organ Because it starts pumping blood from the point where you're a little fetus, maybe about eight weeks old, all the way until the point where you die. And so this organ, I think, would be really cool to look at in a little bit more detail. But it's hard to do that looking just at the outside. So what I did is I actually drew what it might look like on the inside. So let me actually just show you that now. And we'll follow the path of blood through the heart using this diagram. Let me start with a little picture in the corner. So let's say we have a person here. And this is their face, and this is their neck. I'm going to draw their arms. And they have, in the middle of their chest, their heart. And so the whole goal is to make sure that blood from all parts of their body, including their legs, can make its way back to the heart, first of all, and then get pumped back out to the body. So blood is going to come up from this arm, let's say, and dump into there. And the same on this side. And it's going to come from their head. And all three sources, the two arms and the head, are going to come together into one big vein. And that's going to be dumping into the top of the heart. And then separately, you've got veins from the legs meeting up with veins from the belly, coming into another opening into the heart. So that's how the blood gets back to the heart. And any time I mention the word vein, I just want you to make sure you think of blood going towards the heart. Now if blood is going towards the heart, then after the blood is pumped by the heart, it's going to have to go out to the heart. It's going to have to go away from the heart. So that's the aorta. And the aorta actually has a little arch, like that. We call it the aortic arch. And it sends off one vessel to the arm, one vessel up this way, a vessel over this way. And then this arch kind of goes down, down, down and splits like that. So this is kind of a simplified version of it. But you can see how there are definitely some parallels between how the veins and the arteries are set up. And arteries, anytime I mention the word artery, I want you to think of blood going away from the heart. And an easy way to remember that is that they both start with the letter A. So going to our big diagram now. We can see that blood coming in this way and blood coming in this way is ending up at the same spot. It's going to end up at the-- actually, maybe I'll draw it here-- is ending up at the right atrium. That's just the name of the chamber that the blood ends up in. And it came into the right atrium from a giant vessel up top called the superior vena cava. And this is a vein, of course, because it's bringing blood towards the heart. And down here, the inferior vena cava. So these are the two directions that blood is going to be flowing. And once blood is in the right atrium, it's going to head down into the right ventricle. So this is the right ventricle, down here. This is the second chamber of the heart. And it gets there by passing through a valve. And this valve, and all valves in the heart, are basically there to keep blood moving in the right direction. So it doesn't go in the backwards direction. So this valve is called the tricuspid valve. And it's called that because it's basically got three little flaps. That's why they call it tri. And I know you can only see two in my drawing, and that's just because my drawing is not perfect. And it's hard to show a flap coming out at you, but you can imagine it. So blood goes into the right ventricle. And where does it go next? Well after that, it's going to go this way. It's going to go into this vessel, and it's going to split. But before it goes there, it has to pass through another valve. So this is a valve, right here, called the pulmonary valve. And it gives you a clue as to where things are going to go next. Because the word pulmonary means lungs. And so, if this is my lung, on this side, this is my left lung. And this is my right lung, on this side. Then these vessels-- and I'll let you try to guess what they would be called-- these vessels. This would be my-- I want to make sure I get my right and left straight. This is my left pulmonary artery. And I hesitated there just to make sure you got that because it's taking blood away from the heart. And this is my right pulmonary artery. So this is my right and left pulmonary artery. And so blood goes, now, into my lungs. These are the lungs that are kind of nestled into my thorax, where my heart is sitting. It goes into my lungs. And remember, this blood is blue. Why is it blue? Well, it's blue because it doesn't have very much oxygen. And so one thing that I need to pick up is oxygen. And so that's one thing that the lungs are going to help me pick up. And I'm going to write O2 for oxygen. And it's also blue. And that reminds us that it's full of carbon dioxide. It's full of waste because it's coming from the body. And the body's made a lot of carbon dioxide that it's trying to get rid of. So in the lungs, you get rid of your carbon dioxide and you pick up oxygen. So that's why I switch, at this point, from a blue-colored vessel to a red-colored vessel. So now blood comes back in this way and this way and dumps into this chamber. So what is that? This is our left atrium. So just like our right atrium, we have one on the left. And it goes down into-- and you can probably guess what this one is called-- it's our left ventricle. So just like before, where it went from the right atrium to the right ventricle, now we're going from the left atrium to the left ventricle. And it passes through a valve here. So this valve is called the mitral valve. And its job is, of course, to make sure that blood does not go from the left ventricle back to the left atrium by accident. It wants to make sure that there's forward flow. And then the final valve-- I have to find a nice spot to write it, maybe right here. This final valve that it passes through is called the aortic valve. And the aortic valve is going to be what divides the left ventricle from this giant vessel that we talked about earlier. And this is, of course, the aorta. This is my aorta. So now blood is going to go through the aorta to the rest of the body. So you can see how blood now flows from the body into the four chambers. First into the right atrium-- this is chamber number one. And then it goes into the right ventricle. This is chamber number two. It goes to the lungs and then back out to the left atrium. So this is chamber number three. And then the left ventricle. And this happens every moment of every day. Every time you hear your heart beating, this process is going on." + }, + { + "Q": "I simply love your videos. I learn A LOT from them. However, 91*3600/5280=62, not 62.05. Your calculator's been making a mistake. Keep it up Sal! You're benifiting me alot.", + "A": "i think what you did was something called not following the order of operations then you wouldve got 62 if you didnt follow if you did you would have gotten 62.05 wither way you get 62.05 or 62.04545454545454545454545454545454545454545454545454545454545454545454545454545454545454545454 repeating", + "video_name": "aTjNDKlz8G4", + "transcript": "Welcome to the presentation on units. Let's get started. So if I were to tell you -- let me make sure my pen is set up right -- if I were to tell you that someone is, let's say they're driving at a speed of -- let's say it's Zack. So let's say I have Zack. And they're driving at a speed of, let me say, 28 feet per minute. So what I'm going to ask you is if he's going 28 feet in every minute, how many inches will Zack travel in 1 second? So how many inches per second is he going to be going? Let's try to figure this one out. So let's say if I had 28, and I'll write ft short for feet, feet per minute, and I'll write min short for a minute. So 28 feet per minute, let's first figure out how many inches per minute that is. Well, we know that there are 12 inches per foot, right? If you didn't know that you do now. So we know that there are 12 inches per foot. So if you're going 28 feet per minute, he's going to be going 12 times that many inches per minute. So, 12 times 28 -- let me do the little work down here -- 28 times 12 is 16, 56 into 280. I probably shouldn't be doing it this messy. And this kind of stuff it would be OK to use a calculator, although it's always good to do the math yourself, it's good practice. So that's 6, 5 plus 8 is 13. 336. So that equals 336 inches per minute. And something interesting happened here is that you noticed that I had a foot in the numerator here, and I had a foot in the denominator here. So you can actually treat units just the same way that you would treat actual numbers or variables. You have the same number in the numerator and you have the same number in the denominator, and your multiplying not adding, you can cancel them out. So the feet and the feet canceled out and that's why we were left with inches per minute. I could have also written this as 336 foot per minute times inches per foot. Because the foot per minute came from here, and the inches per foot came from here. Then I'll just cancel this out and I would have gotten inches per minute. So anyway, I don't want to confuse you too much with all of that unit cancellation stuff. The bottom line is you just remember, well if I'm going 28 feet per minute, I'm going to go 12 times that many inches per minute, right, because there are 12 inches per foot. So I'm going 336 inches per minute. So now I have the question, but we're not done, because the question is how many inches am I going to be traveling in 1 second. So let me erase some of the stuff here at the bottom. So 336 inches -- let's write it like that -- inches per minute, and I want to know how many inches per second. Well what do we know? We know that 1 minute -- and notice, I write it in the numerator here because I want to cancel it out with this minute here. 1 minute is equal to how many seconds? It equals 60 seconds. And this part can be confusing, but it's always good to just take a step back and think about what I'm doing. If I'm going to be going 336 inches per minute, how many inches am I going to travel in 1 second? Am I going to travel more than 336 or am I going to travel less than 336 inches per second. Well obviously less, because a second is a much shorter period of time. So if I'm in a much shorter period of time, I'm going to be traveling a much shorter distance, if I'm going the same speed. So I should be dividing by a number, which makes sense. I'm going to be dividing by 60. I know this can be very confusing at the beginning, but that's why I always want you to think about should I be getting a larger number or should I be getting a smaller number and that will always give you a good reality check. And if you just want to look at how it turns out in terms of units, we know from the problem that we want this minutes to cancel out with something and get into seconds. So if we have minutes in the denominator in the units here, we want the minutes in the numerator here, and the seconds in the denominator here. And 1 minute is equal to 60 seconds. So here, once again, the minutes and the minutes cancel out. And we get 336 over 60 inches per second. Now if I were to actually divide this out, actually we could just divide the numerator and the denominator by 6. 6 goes into 336, what, 56 times? 56 over 10, and then we can divide that again by 2. So then that gets us 28 over 5. And 28 over 5 -- let's see, 5 goes into 28 five times, 25. 3, 5.6. So this equals 5.6. So I think we now just solved the problem. If Zack is going 28 feet in every minute, that's his speed, he's actually going 5.6 inches per second. Hopefully that kind of made sense. Let's try to see if we could do another one. If I'm going 91 feet per second, how many miles per hour is that? Well, 91 feet per second. If we want to say how many miles that is, should we be dividing or should we be multiplying? We should be dividing because it's going to be a smaller number of miles. We know that 1 mile is equal to -- and you might want to just memorize this -- 5,280 feet. It's actually a pretty useful number to know. And then that will actually cancel out the feet. Then we want to go from seconds to hours, right? So, if we go from seconds to hours, if I can travel 91 feet per second, how many will I travel in an hour, I'm going to be getting a larger number because an hour's a much larger period of time than a second. And how many seconds are there in an hour? Well, there are 3,600 seconds in an hour. 60 seconds per minute and 60 minutes per hour. So 3,600 over 1 seconds per hour. And these seconds will cancel out. Then we're just left with, we just multiply everything out. We get in the numerator, 91 times 3,600, right? 91 times 1 times 3,600. In the denominator we just have 5,280. This time around I'm actually going to use a calculator -- let me bring up the calculator just to show you that I'm using the calculator. Let's see, so if I say 91 times 3,600, that equals a huge number divided by 5,280. Let me see if I can type it. 91 times 3,600 divided by 5,280 -- 62.05. So that equals 62.05 miles per hour." + }, + { + "Q": "What is a Chaperone and how is it related to protein folding?", + "A": "A chaperone protein is a protein whose function it is to assist other proteins in folding into their correct tertiary shape.", + "video_name": "dNHtdiVjQbM", + "transcript": "Let's talk about conformational stability and how this relates to protein-folding and denaturation. So first, let's review a couple of terms just to make sure we're all on the same page. And first, let's start out with the term conformation. And the term confirmation just refers to a protein's folded, 3D structure, or in other words, the active form of a protein. And next we can review what the term denatured means when you're talking about proteins. And denatured proteins just refer to proteins that have become unfolded, or inactive. So all conformational stability is really talking about are the various forces that help to keep a protein folded in the right way. And these various forces are the four different levels of protein structure, and we can review those briefly right here. So recall that the primary structure of a protein just refers to the actual sequence of amino acids in that protein, and this is determined by a protein's peptide bonds. And then next, you have secondary structure, which just refers to the local substructures in a protein. And they are determined by backbone interactions held together by hydrogen bonds. Then you have tertiary structure, which just talks about the overall 3D structure of a single protein molecule. And this is described by distant interactions between groups within a single protein. And these interactions are stabilized by van der Waals interactions, hydrophobic packing, and disulfide bonding, in addition to the same hydrogen bonding that helps to determine secondary structure. And then quaternary structure just describes the different interactions between individual protein subunits. So you have the folded-up proteins that then come together to assemble the completed, overall protein. And the interaction of these different protein subunits are stabilized by the same kinds of bonds that help to determine tertiary structure. So all of these levels of protein structure help to stabilize the folded-up, active confirmation of a protein. So why is it so important to know about the different levels of protein structure and how they contribute to conformational stability? Well, like I said, a protein is only functional when they are in their proper conformation, in their proper 3D form. And an improperly folded or degraded, denatured, protein is inactive. So in addition to the four levels of protein structure that I just reviewed, there is also another force that helps to stabilize a protein's conformation. And that force is called the solvation shell. Now, the solvation shell is just a fancy way of describing the layer of solvent that is surrounding a protein. So say I have a protein who has all these exterior residues that are overall positively charged. And picture this protein in the watery environment of the interior of one of our cells, then the solvation shell is going to be the layer of water right next to this protein molecule. And remember that water is a polar molecule, so you have the electronegative oxygen atom with a predominantly negative charge leaving a positive charge over next to the hydrogen atoms. The same is true for each of these water molecules. So now, as you can see, the electronegative oxygen atoms are stabilizing all the positively charged amino acid residues on the exterior of this protein. So as you can see, the conformational stability of a protein depends not only on all of these interactions that contribute to primary, secondary, tertiary, and quaternary structure, but also what sort of environment that protein is in. And all of these interactions are very crucial for keeping a protein folded properly, so that it can do its job. Now, what happens when things go wrong? How does a protein become unfolded and thus inactive? Well, remember that this is called denaturation. And this can be done by changing a lot of different parameters within a protein's environment, including changing the temperature, the pH, adding chemical denaturants, or even adding enzymes. So let's start with what happens if you alter the temperature around a protein. And we can use the example of an egg when we put it into a pot of boiling water, because an egg, especially the white part, is full of protein, and this pot of boiling water is representing heat. And remember that heat is really just a form of energy. So when you heat an egg, the proteins gain energy and literally shake apart the bonds between the parts of the amino acid chains. And this causes the proteins to unfold. So increased temperature destroys the secondary, tertiary, and quaternary structure of a protein, but the primary structure is still preserved. So the take-away point is that when you change the temperature of a protein by heating it up, you destroy all of the different levels of protein structure except for the primary structure. So now, let's say you were to take an egg and then add vinegar, which is really just an acid. The acid in the vinegar will break all the ionic bonds that contribute to tertiary and quaternary structure. So the take-away point when you change the pH surrounding a protein is that you have disruption of ionic bonds. And if we think about this a little bit more deeply, it kind of makes sense, because ionic bonds are dependent upon the interaction of positive and negative charges. So when you add either an acid or a base, which in the case of an acid is just like adding a bunch of positive charges, you kind of disrupt the balance between all these interactions between the positive and negative charges within the protein. So now let's look at how chemicals denature proteins. Chemical denaturants often disrupt the hydrogen bonding within a protein. And remember that hydrogen bonds contribute to secondary, tertiary, all the way up to quaternary structure. So all these levels of protein structure will be disrupted if you add a chemical denaturant. So let's take our same example of a protein with an egg, and say, if you were 21 years or older, you got your hands on some alcohol, and you added this to the egg. Then, all the hydrogen bonds would be broken up, leaving you with just linear polypeptide chains. And then finally, let's take our hard-boiled egg from the temperature example, and let's eat it. So here's my beautiful drawing of a person, representing you, eating this hard boiled egg. Once the egg enters our digestive tract we have enzymes that break down the already denatured proteins in the egg even further. They take the linear polypeptide chain, whose primary structure is still intact. And they break the bonds between the individual amino acids, the peptide bonds, so that we can absorb these amino acids from our intestines into our bloodstream. And then we can use them as building blocks for our own protein synthesis. And that's how enzymes can alter a protein's primary structure and thus the protein's overall conformational stability. So what did we learn? Well, we learned that the conformational stability refers to all the forces that keep a protein properly folded in its active form. And this includes all the different levels of protein structure, as well as the salvation shell. And we also learned that a protein can be denatured into its inactive form by changing a variety of factors in its environment, including changing the temperature, the pH, adding chemicals, or enzymes." + }, + { + "Q": "how did you get the 6 to make 48", + "A": "A=1/2bh. The base= 12. H=8. When substituting, A=1/2(12)(8). 1/2 of 12= 6 as 12/2=6. So A=6(8). A= 48 sq units", + "video_name": "mtMNvnm71Z0", + "transcript": "- What I want to do in this video is get some practice finding surface areas of figures by opening them up into what's called nets. And one way to think about it is if you had a figure like this, and if it was made out of cardboard, and if you were to cut it, if you were to cut it right where I'm drawing this red, and also right over here and right over there, and right over there and also in the back where you can't see just now, it would open up into something like this. So if you were to open it up, it would open up into something like this. And when you open it up, it's much easier to figure out the surface area. So the surface area of this figure, when we open that up, we can just figure out the surface area of each of these regions. So let's think about it. So what's first of all the surface area, what's the surface area of this, right over here? Well in the net, that corresponds to this area, it's a triangle, it has a base of 12 and height of eight. So this area right over here is going to be one half times the base, so times 12, times the height, times eight. So this is the same thing as six times eight, which is equal to 48 whatever units, or square units. This is going to be units of area. So that's going to be 48 square units, and up here is the exact same thing. That's the exact same thing. You can't see it in this figure, but if it was transparent, if it was transparent, it would be this backside right over here, but that's also going to be 48. 48 square units. Now we can think about the areas of I guess you can consider them to be the side panels. So that's a side panel right over there. It's 14 high and 10 wide, this is the other side panel. It's also this length over here is the same as this length. It's also 14 high and 10 wide. So this side panel is this one right over here. And then you have one on the other side. And so the area of each of these 14 times 10, they are 140 square units. This one is also 140 square units. And then finally we just have to figure out the area of I guess you can say the base of the figure, so this whole region right over here, which is this area, which is that area right over there. And that's going to be 12 by 14. So this area is 12 times 14, which is equal to let's see. 12 times 12 is 144 plus another 24, so it's 168. So the total area is going to be, let's see. If you add this one and that one, you get 96. 96 square units. The two magenta, I guess you can say, side panels, 140 plus 140, that's 280. 280. And then you have this base that comes in at 168. We want it to be that same color. 168. One, 68. Add them all together, and we get the surface area for the entire figure. And it was super valuable to open it up into this net because we can make sure we got all the sides. We didn't have to kinda rotate it in our brains. Although you could do that as well. So, with six plus zero plus eight is 14. Regroup the one ten to the tens place, there's now one ten. So one plus nine is ten, plus eight is 18, plus six is 24, and then you have two plus two plus one is five. So the surface area of this figure is 544. 544 square units." + }, + { + "Q": "How long was the famine as the people seemed to be starving since a long time?", + "A": "well france had really no money to plant food because war was every where", + "video_name": "0t4MF9ZoppM", + "transcript": "We left off the last video at the end of 1789. The Bastille had been stormed in July as Parisians wanted to get the weapons from the Bastille and free a few political prisoners to, in their minds, protect themselves from any tyranny from Louis XVI. Louis XVI had reluctantly kind of gone behind the scenes and said, OK, National Assembly, I'm not going to get in your Because he's seen the writing on the walls that every time he's done something, it's only led to even more extreme counteraction. So at the end of 1789, already chaos has broken loose in a lot of France. The National Assembly, they're in process of creating a constitution, which won't fully happen until 1791. But they're starting to bring things together in order to draft that constitution. But in August of 1789, they've already done their version of the Declaration of Independence. The Declaration of the Rights of Man and the Citizen. So if everything was well, we would just wait until a few years, we'd get a constitution, and maybe we would have some type of a constitutional monarchy. But unfortunate, especially for Louis XVI, things weren't all well. As we mentioned, all of this was propagated, all of this was started to begin with because people were hungry. We have this fiscal crisis, we have a famine. And so in October of 1789-- we're still in 1789-- October of 1789-- rumors started to spread that Marie-Antoinette, the king's wife, that she was hoarding grain at Versailles. So people started imagining these big stacks of grain at Versailles, and this is in a time where people couldn't get And bread was the main staple of the diet. So there was actually a march of peasant women onto And they were armed. This is a depiction of the peasant women marching on And they went to Versailles, and they actually were able to get into the building itself. And they demanded-- because they were suspicious of what Louis XVI and Marie-Antoinette were up to at Versailles-- they demanded that they move to Paris. So the women's march. And they were able to get their demands. It resulted in Louis XVI and wife, Marie-Antoinette, moving back to Paris, where they couldn't do things like hoard grain. And they'll be surrounded by all of the maybe not-so-friendly people who could watch what they're doing. I think the main factor was that people are hungry, rumors are spreading that the king is hoarding grain. But there were also rumors that the king was being very disrespectful to some of the symbols of the new France, of the new National Assembly. So that also made people angry. And across the board everyone kind of knew, and including Louis XVI, that he wasn't really into what was going on. He wasn't into this kind of constitutional monarchy that was forming, this power that was being lost to the National Assembly. So we have this very uncomfortable situation entering into 1790, where the king and queen are essentially in house arrest in a building called the Tuileries in Paris. You have this National Assembly drafting this constitution. They're charted to draft the constitution up there. They all pledged at the Tennis Court Oath. And at the same time, throughout France, you have some counter insurgencies. This is France right here. Throughout France you have counter insurgencies, people who don't like the Revolution that's going on. And then those would be subdued. And people are all plotting one against each other. And then you have some nobility, that says, gee, you know what? I don't like the way that this is going. We've seen already a lot of violence. People are angry. I'm just going to take my money and whatever I can pack, and I'm just going to get out of the country. I'm going to emigrate away from the country. So you start having nobility starting to leave France. They're called the Emigres. I know I'm not pronouncing it correctly. But you see, you have this notion of gee, I had it good in France, I'm not going to have it good much longer, I'd better leave. And this same idea, now that we get to 1791. So 1790 was just kind of a bunch of unease. Now that we're at 1791, the same idea of trying to get away from the danger got into the heads of Louis XVI and But they couldn't leave the country. They didn't trust Great Britain. They didn't trust any of these other countries to safely give them shelter. So one of their generals, who was sympathetic to their cause, said, hey, at least come here to the frontier areas and you could hide from all of the unrest that's going on. So dressed as actual servants-- and it shows you what type of people they were-- they dressed as servants. And they actually made their servants dress as nobility to make them the targets in case they were ambushed anyway on their way trying to escape from Paris. Dressed as servants, the king and queen-- the king tried to escape to this general's estate. But when they were in Varennes, right here, they were actually spotted. And then the people essentially took them captive and brought them back to Paris. So this is called, or you could imagine this is the flight to Varennes, or the flight away from Paris, or however you want to do it. So already, Louis XVI started to see the writing on the wall. They tried to get away. But people brought them back. Now you can imagine, a lot of people already did not like the king. They didn't like the notion of even having a king. And the most revolutionary, the most radical elements, were called the Jacobins. And after the king and queen tried to escape and came back, they were like hey, gee, what's the use of even having a king? You National Assembly, why are you even trying to write some type of constitution that gives any power whatsoever to a king? We should have a republic. Which is essentially-- there's a lot of kind of nuanced definitions of what a republic is, but the most simple one is it's a state without a king, without an emperor, without a queen. So they're saying, we don't need, you know-- you National Assembly, you think you're being radical. But you're not being radical enough. We want to eliminate the idea of having a monarchy altogether. And the fact that Louis XVI actually tried to run away, we view that as him abdicating the throne. Abdication, or essentially quitting. And they actually started to organize in Paris. This right here is the Champ-de-Mars. I know I'm saying it completely wrong. This is a current picture of it. And so they started taking signatures in this kind of public park in Paris to essentially say, we don't need a king. We want to essentially create our own republic. That this National Assembly, they're not radical enough. And so people started gathering over here in the Champ-de-Mars and things got a little ugly. So the actual troops were sent in to kind of calm everyone down. And these were actually troops controlled by the National Assembly. The people who are mainly controlled by the Third Estate. But things got a little crazy. Rocks were thrown at some of the troops. Some of the troops, at first, they started firing in the air. But eventually when things got really crazy, they fired into the crowd. And about 50 people died. And this was the massacre. Or the Champ-de-Mars Massacre. I know I'm saying it wrong. This isn't a video on French pronunciation. But you could imagine, now people are even angrier. People are still starving. That problem has not gone away. The king and queen has been kind of very reluctantly-- everyone is suspicious of the fact that they're probably going to try to come back to power. They tried to run away. When the Jacobins, or in general kind of revolutionaries, but they're led by the Jacobins, when they start to suggest that, hey, we should have a republic. We shouldn't even have a king. And they gather people here, all of a sudden, the troops that are controlled by the current National Assembly actually fire on the crowd, and actually kill civilians for throwing rocks. And they might have been big rocks. But you can imagine, this is going to anger already hungry and already suppressed people even more. And to make people even more paranoid that the king and queen might eventually come back to power, you had two major powers all of a sudden trying to insert certain themselves into the French Revolution. I'm going to do a little bit of an aside here. Because this is something, at least you when I first learned European history, I found the most confusing. You have these states, you can call them. You have Austria, which I've highlighted in orange. The kind of map here is a modern map. But in orange, I've kind of shown what Austria was at that point in time. Around 1789, 1790, 1791. In this red color, I have Prussia. I want to show you that these are very different than our current notions of one, Austria. Austria today is this modern country right here. And Prussia doesn't even exist as a modern country. And then you had this notion of the Holy Roman Empire, which overlaps with these other kingdoms, or empires, or whatever you want to call them. And I want to do a little bit of an aside here. The Holy Roman Empire, as Voltaire famously said, is neither holy nor Roman nor an empire. And he was right. It was really kind of a very loose confederation of German kingdoms and states-- mainly German kingdoms and states. As you can see, it kind of coincides with modern Germany. And the two most influential powers in the Holy Roman Empire, or actually the most influential power in the Holy Roman Empire, was the Austrians. And the ruler of the Austrians had the title of Holy Roman Emperor. And that was Leopold II. But it's not like he was like the Roman Emperors of old. The Roman Emperors of old actually came out of Rome. Notice, nothing in the Holy Roman Empire at that time, it had no control of Rome. So it was not Roman, we're not talking about people who spoke Latin. We're talking about people who spoke Germanic languages. And it wasn't an empire. That it wasn't a tightly knit kind of governance structure. It was this loose confederation of states. But what was the most influential was the region that was under control of the Habsburgs of Austria, or Leopold II. And not only was he in control, or not only did he have the title of Holy Roman Empire, and essentially had control of the Austrian, I guess you could say Empire, at that point in time. He was also Marie-Antoinette's brother. Leopold II, that's her brother. So Leopold II and Frederick William II of Prussia, which is another mainly Germanic state. Let me do that in a better color. They issued the Declaration of Pillnitz. Let me write this down. So this is going to add even more insult to injury to just the general population of France. The Declaration of Pillnitz. And this was done in August. so I just want to make it very clear what happened. In June of 1791, they tried to escape, they were captured at Varennes. Then in July of 1791, you have the Champ-de-Mars Massacre. So already, people are very wary of the royals. The idea that we don't need them is spreading. And people are getting angrier. And then you have the Declaration of Pillnitz by these foreign powers, one of whom is essentially the brother of the current French royalty. And that declaration is essentially saying that they intend to bring the French monarchy back to power. They don't say that they're definitely going to do it in military terms or whatever. But it's a declaration that they do not approve of what's going on in France. And even though they themselves might have not taken it too seriously, the people of France took it really seriously. You have these huge powers on their border right here. You had the Austrians and the Prussians. So this wasn't anything that people could take very lightly. So it only increased the fear that the royals were going to do something to come back to full power and really suppress people. And it really gave even more fuel for the Jacobins to kind of argue for some type of a republic. So I'm going to leave you there in this video. As you can see, we saw in the first video, things got bad. Now they're getting really worst. Chaos is breaking out in France. People are questioning whether they even need a king or queen. Foreign powers are getting involved, saying hey, they don't like what they're seeing there, with kings and queens getting overthrown. Maybe that'll give ideas to their people. And by the way, I'm your brother, so I want to help you out too. That scares people even more. The current National Assembly, which is kind of the beginning of the Revolution, they themselves are on some level massacring people. So it's really leading to a really tense and ugly time in French history. And you're going to see that that's going to culminate with what's called the Reign of Terror. And we're going to see that in the next video." + }, + { + "Q": "I am wondering what college or university do you think I should go to? Writing/language arts", + "A": "That depends If you want to stay local. If you don t mind traveling you could go to Harvard or Berkeley, but those are top of the line.", + "video_name": "jPrEKz1rAno", + "transcript": "- So to give you a big picture of what will be presented to a college, from your point of view as a student, you'll be submitting an application. That application will have in it biographical information, your extracurricular activities, essays that you've written. Basically giving an overview of who you are and what you're doing, but there'll also be lots of other information that's submitted that an admissions office will use. From your high school, they will submit a transcript with your grades and courses you've taken. Hopefully they'll send along a profile that sort of details that school and gives some information about the school you're in. There also will be probably a guidance counselor or college counselor letter of recommendation. A lot of schools also require one or potentially two letters of recommendation from teachers, so that could possibly be in the file as well. If the school requires any kind of standardized testing, from SAT, ACT, potentially AP or IB scores, those sorts of things, they could be in your file as well. You may be required to do an interview for that school. And there are also opportunities to submit... If you have special talents, in things like the arts, theatre, music, athletics, there may be special ways to submit samples of your work to also be evaluated. So, in general, I would say most selective and highly-selective schools are going to put the most emphasis on your programming grades, number one, just to make sure they feel you can do the work at their school. How they use the rest of those components will vary greatly by school." + }, + { + "Q": "is the number line you are trying to show that you divide the number line just like you divide normally?", + "A": "There s no difference. I don t get what difference you see.", + "video_name": "Z3qRkxzmYnU", + "transcript": "PROBLEM: \"Omar rode his boat for a total of 50 miles over the past 5 days. And he rowed the same amount each day. How many miles did Omar row his boat each day?\" So let\u2019s just visualize what's going on. He is able to travel 50 miles. So let's make a \u2013 Let\u2019s say this line represents the 50 miles that he travels. So this whole distance right over here is 50 miles. And he does it \u2013 They tell us that he does it over 5 days, and that each day, he does the same amount. So this 50 miles is if you were to add together all of what he did over the 5 days. And so, if you want to know how much he did each day, you essentially want to divide this 50 miles into 5 equal sections. And the length of each of those sections is the amount he did each day. So if we just visualize it \u2013 So that's one section \u2013 second section \u2013 third section \u2013 fourth \u2013 and fifth section. And actually, I didn't do that very well. It should look a little bit more equal than that. First, second \u2013 (And that's not going to \u2013 Let's see.) First, second, third, fourth, and fifth. And you don't have to actually do this. This is just to help visualize. So essentially, what we want to figure out is what is one of these distances? And as you can see in our visualization, this is really just taking our 50 miles and dividing it into 5 equal chunks. So we're essentially just taking 50, and we're going to divide it by 5. So 50 divided by 5 is going to be equal to 10. So if he goes 50 miles over 5 days, and you divide by the 5 days, he goes 10 miles each day. And we're done." + }, + { + "Q": "How would you graph something like x0) cosx/x ?", + "A": "cosx is approaching 1 and so the graph of cosx/x looks like the graph of 1/x around zero. The line x=0 is a vertical assymptote. From the right (positive) side, the values increase without bound (go to infinity) and from the left (negative) side, they decrease without bound (go to negative infinity). Since the one sided limits are different, we say that there is no limit or that the limit does not exist sometimes abbreviated D.N.E.", + "video_name": "riXcZT2ICjA", + "transcript": "In this video, I want to familiarize you with the idea of a limit, which is a super important idea. It's really the idea that all of calculus is based upon. But despite being so super important, it's actually a really, really, really, really, really, really simple idea. So let me draw a function here, actually, let me define a function here, a kind of a simple function. So let's define f of x, let's say that f of x is going to be x minus 1 over x minus 1. And you might say, hey, Sal look, I have the same thing in the numerator and denominator. If I have something divided by itself, that would just be equal to 1. Can't I just simplify this to f of x equals 1? And I would say, well, you're almost true, the difference between f of x equals 1 and this thing right over here, is that this thing can never equal-- this thing is undefined when x is equal to 1. Because if you set, let me define it. Let me write it over here, if you have f of, sorry not f of 0, if you have f of 1, what happens. In the numerator, we get 1 minus 1, which is, let me just write it down, in the numerator, you get 0. And in the denominator, you get 1 minus 1, which is also 0. And so anything divided by 0, including 0 divided by 0, this is undefined. So you can make the simplification. You can say that this is you the same thing as f of x is equal to 1, but you would have to add the constraint that x cannot be equal to 1. Now this and this are equivalent, both of these are going to be equal to 1 for all other X's other than one, but at x equals 1, it becomes undefined. This is undefined and this one's undefined. So how would I graph this function. So let me graph it. So that, is my y is equal to f of x axis, y is equal to f of x axis, and then this over here is my x-axis. And then let's say this is the point x is equal to 1. This over here would be x is equal to negative 1. This is y is equal to 1, right up there I could do negative 1. but that matter much relative to this function right over here. And let me graph it. So it's essentially for any x other than 1 f of x is going to be equal to 1. So it's going to be, look like this. It's going to look like this, except at 1. At 1 f of x is undefined. So I'm going to put a little bit of a gap right over here, the circle to signify that this function is not We don't know what this function equals at 1. We never defined it. This definition of the function doesn't tell us It's literally undefined, literally undefined when x is equal to 1. So this is the function right over here. And so once again, if someone were to ask you what is f of 1, you go, and let's say that even though this was a function definition, you'd go, OK x is equal to 1, oh wait there's a gap in my function over here. It is undefined. So let me write it again. It's kind of redundant, but I'll rewrite it f of 1 is undefined. But what if I were to ask you, what is the function approaching as x equals 1. And now this is starting to touch on the idea of a limit. So as x gets closer and closer to 1. So as we get closer and closer x is to 1, what is the function approaching. Well, this entire time, the function, what's a getting closer and closer to. On the left hand side, no matter how close you get to 1, as long as you're not at 1, you're actually at f of x is equal to 1. Over here from the right hand side, you get the same thing. So you could say, and we'll get more and more familiar with this idea as we do more examples, that the limit as x and L-I-M, short for limit, as x approaches 1 of f of x is equal to, as we get closer, we can get unbelievably, we can get infinitely close to 1, as long as we're not at 1. And our function is going to be equal to 1, it's getting closer and closer and closer to 1. It's actually at 1 the entire time. So in this case, we could say the limit as x approaches 1 of f of x is 1. So once again, it has very fancy notation, but it's just saying, look what is a function approaching as x gets closer and closer to 1. Let me do another example where we're dealing with a curve, just so that you have the general idea. So let's say that I have the function f of x, let me just for the sake of variety, let me call it g of x. Let's say that we have g of x is equal to, I could define it this way, we could define it as x squared, when x does not equal, I don't know when x does not equal 2. And let's say that when x equals 2 it is equal to 1. So once again, a kind of an interesting function that, as you'll see, is not fully continuous, it has a discontinuity. Let me graph it. So this is my y equals f of x axis, this is my x-axis right over here. Let me draw x equals 2, x, let's say this is x equals 1, this is x equals 2, this is negative 1, this is negative 2. And then let me draw, so everywhere except x equals 2, it's equal to x squared. So let me draw it like this. So it's going to be a parabola, looks something like this, let me draw a better version of the parabola. So it'll look something like this. Not the most beautifully drawn parabola in the history of drawing parabolas, but I think it'll give you the idea. I think you know what a parabola looks like, hopefully. It should be symmetric, let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared. But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here. So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that. So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3. So when x is equal to 2, our function is equal to 1. So this is a bit of a bizarre function, but we can define it this way. You can define a function however you like to define it. And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2. You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that. You use g of x is equal to 1. So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared. So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here. And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x. Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos. As x gets closer and closer to 2, what is g of x approaching? So if you get to 1.9, and then 1.999, and then 1.999999, and then 1.9999999, what is g of x approaching. Or if you were to go from the positive direction. If you were to say 2.1, what's g of 2.1, what's g of 2.01, what's g of 2.001, what is that approaching as we get closer and closer to it. And you can see it visually just by drawing the graph. As g gets closer and closer to 2, and if we were to follow along the graph, we see that we are approaching 4. Even though that's not where the function is, the function drops down to 1. The limit of g of x as x approaches 2 is equal to 4. And you could even do this numerically using a calculator, and let me do that, because I think that will be interesting. So let me get the calculator out, let me get my trusty TI-85 out. So here is my calculator, and you could numerically say, OK, what's it going to approach as you approach x equals 2. So let's try 1.94, for x is equal to 1.9, you would use this top clause right over here. So you'd have 1.9 squared. And so you get 3.61, well what if you get even closer to 2, so 1.99, and once again, let me square that. Well now I'm at 3.96. What if I do 1.999, and I square that? I'm going to have 3.996. Notice I'm going closer, and closer, and closer to our point. And if I did, if I got really close, 1.9999999999 squared, what am I going to get to. It's not actually going to be exactly 4, this calculator just rounded things up, but going to get to a number really, really, really, really, really, really, really, really, really close to 4. And we can do something from the positive direction too. And it actually has to be the same number when we approach from the below what we're trying to approach, and above what we're trying to approach. So if we try to 2.1 squared, we get 4.4. let me go a couple of steps ahead, 2.01, so this is much closer to 2 now, squared. Now we are getting much closer to 4. So the closer we get to 2, the closer it seems like we're getting to 4. So once again, that's a numeric way of saying that the limit, as x approaches 2 from either direction of g of x, even though right at 2, the function is equal to 1, because it's discontinuous. The limit as we're approaching 2, we're getting closer, and closer, and closer to 4." + }, + { + "Q": "so what is the affect on macroeconomics equilibrium after a shift in aggregate demand?", + "A": "Nothing will really happen. All you will get is inflation or deflation, depending on which way demand shifts.", + "video_name": "8W0iZk8Yxhs", + "transcript": "Narrator: We've talked a lot about aggregate demand over the last few videos, so in this video, I thought I would talk a little bit about aggregate supply. In particular, we're going to think about aggregate supply in the long-run. In economics, whether it's in micro or macro economics, when we think about long-run, we're thinking about enough time for a lot of fixed costs and a lot of fixed contracts to expire. In the short-term, you might be stuck into some labor contract, or stuck into your using some factory that you've already paid money for, so it was a fixed cost, but over the long-run you'll have a chance that factory will wear down and you'll have a chance to decide whether you want another factory or the price of the factory might change; or in the long-run, you'll have a chance contracts will expire, and you'll have a chance to renegotiate those contracts at a new price. That's what we really mean when we talk about the long-run. I'm going to plot aggregate supply on the same axis as we plotted aggregate demand, and we're going to focus on the long-run now, and then we're going to think about what actually might happen in the short-run while we are in fixed-price contracts, or we already have spent money on something, or we have already, in some ways, there are sticky things that can't adjust as quickly. But, we'll first focus on the long-run. On this axis, I'm just going to plot price, and remember, we're thinking in macro-economic terms. This is some measure of the prices of the goods and services in our economy. This axis right over here, the horizontal axis is going to be real GDP. Once again, this is just a model, you should take everything in economics with a huge grain of salt. These are over-simplifications of a highly, highly complex thing, the economy. Millions and millions of actors doing complex things, human beings, each of them and their brain have billions and billions and billions of neurons, doing all sorts of unpredictable things. But economists like to make really simplifying, super-simplifying assumptions, so that we can deal with it in a attractable way, and in a even dealing in a mathematical way. The assumtion that economists often make when we think about aggregate supply and aggregate demand is, in the long-run, real GDP actually does not depend on prices in the long-run; so, what you have is, regardless of what the price is, you're going to have the same real GDP. You can view this as a natural level of productivity for the economy. This is some level right over here. It's important to realize this is just a snap shot in time, and this is all else things equal, so we're not assuming that we're having changes in productivity overtime; this is just a snap shot if we did have any of those things that change. For example, if the population increased, then that would cause this level to shift to the right, then we would have a higher natural level of productivity. If, for whatever reason, we were able to create tools so that it was easier to find people jobs, there's always a natural rate of unemployment. There's frictions, people have to look for jobs, some people have to retrain to get their skills, but maybe we improve that in some way so that there's some website where people can find jobs easier, or easier ways to train for jobs, and the natural level of unemployment goes down, more people can produce, that would also shift this curve to the right. You could have a reality where there's technological improvements that would also, and then all of a sudden, on an average, people would become more productive; that could shift things to the right. You could have discovery of natural resources, new land that is super fertile, and everything else; that could also shift things to the right. You could have a war, and maybe your factories get bombed, or bad things happen in a war, especially if the war is on your soil, and that could actually shift things to the left. So, it's important to realize that this is just taking a snap shot in time, and a lot of these other things that we think about would just shift it in 1 direction or another. I'm going to leave you there, and this is a kind of it might not seem intuitive at first, because you're saying, \"Wait, look, if prices were to change dramatically, if all of a sudden everything in the economy got twice as expensive, that would have some impact on peoples' minds and that they would behave differently and all the rest, and that might affect how much they can produce.\" We did think a little about that when we thought about aggregate demand, but when we think about aggregate supply, we're just thinking about their capability to produce. We're saying all else equal. We're saying that peoples' mind-shifts aren't changing, their willingness to work isn't changing, nothing else is changing, technology isn't changing. Given that, price really is just a numeric thing. If you just looked at the resources and the productive capability of a country, the factors of production, the people and all the rest, regardless of what the prices are, they in theory, should be able to produce the same level of goods and services." + }, + { + "Q": "do all of the arteries branch off of the Aorta?", + "A": "all the systemic arteries branch out from there", + "video_name": "iqRTd1NY-pU", + "transcript": "I want to figure out how blood gets from my heart, which I'm going to draw here, all the way to my toe. And I'm going to draw my foot over here and show you which toe I'm talking about. Let's say this toe right here. Now, to start the journey, it's going to have to go out of the left ventricle and into the largest artery of the body. This is going to be the aorta. And the aorta is very, very wide across. And that's why I say it's a large artery. And from the aorta-- I'm actually not drawing all the branches of the aorta. But from the aorta, it's going to go down into my belly. And it's going to branch towards my left leg and my right leg. So let's say we follow just the left leg. So this artery over here on the top, it's going to get a little bit smaller. And maybe I'd call this a medium-sized artery by this point. This is actually now getting down towards my ankle. Let's say we've gone quite a distance down in my ankle. And then there are, of course, little branches. And let's just follow the branch that goes towards my foot, which is this top one. Let's say this one goes towards my foot, and this is going to be now an even smaller artery. Let's call it small artery. From there, we're actually going to get into what we call arterioles, so it's going to get even tinier. It's going to branch. Now, these are very, very tiny branches coming off my small artery. And let's follow this one right here, and this one is my arteriole. So these are all the different branches I have to go through. And finally, I'm going to get into tiny little branches. I'm going to have to draw them very, very skinny just to convince you that we're getting smaller and smaller. Let me draw three of them. Let's draw four just for fun. And this is actually going to now get towards my little toe cells. So let me draw some toes cells in here to convince you that I actually have gotten there. Let's say one, two over here, and maybe one over here. These are my toes cells. And after the toe cells have kind of taken out whatever they need-- maybe they need glucose or maybe they need some oxygen. Whatever they've taken out, they're also going to put in their waste. So they have, of course, some carbon dioxide waste that we need to drag back. This is now going to dump into what we call a venule. And this venule is going to basically then feed into many, many other venules. Maybe there's a venule down here coming in, and maybe a venule up here coming in maybe from the second toe. And it's going to basically all kind of gather together, and again, to a giant, giant set of veins. Maybe veins are dumping in here now, maybe another vein dumping in here. And these veins are all going to dump into an enormous vein that we call the inferior vena cava. I'll write that right here, inferior vena cava. And this is the large vein that brings back all the blood from the bottom half of the body. There's also another one over here called the superior vena cava, and this is bringing back blood from the arms and head. So these two veins, the superior vena cava and the inferior vena cava, are dragging the blood back to the heart. And generally speaking, these are all considered, of course, veins. Let's back up now and start with the large and medium arteries. These guys together are sometimes referred to as elastic arteries. And the reason they're called elastic arteries, one of the good reasons why they're called that is that they have a protein in the walls of the blood vessel called elastin. They have a lot of this elastin protein. And if you think about the word elastin or elastic-- obviously very similar words-- you might think of something like a rubber band or a balloon. And that's probably the easiest way to think about it. If you have a blood vessel, one of these large arteries, for example, and let's say blood is under a lot of pressure because the heart is squeezing out a lot of high pressure blood, this artery is literally going to balloon out. And if you actually looked at it from the outside, it would look like a little sausage, something like this where it's puffed out. So what's happened there between the first and second picture is that the pressure energy-- so the heart is squeezing out a lot of pressurized blood. And, of course, there's energy in that blood. That pressure energy has been converted over into elastic energy. It's actually converting energy. We don't really always think about it that way, but that's exactly what's happening. And when you convert from pressure energy to elastic energy, what you're really then doing is you're balancing out those high pressures. So you're balancing out high pressures. And this is actually very important, because the blood that's coming into our arteries is under, let's not forget, high pressure. So the arterial system we know is a high-pressure system. So this makes perfect sense that the first few arteries, those large arteries and even those medium-sized arteries, are going to be able to deal with the pressure really well. Now, let me draw a little line here just to keep it straight. The small artery and the arteriole, these two are actually sometimes called the muscular arteries. And the reason, again, if you just want to look at the wall of the artery, you'll get the answer. The wall of the artery is actually very muscular. In fact, specifically, it's smooth muscle. So not the kind of muscle you have in your heart or in your biceps, but this is smooth muscle that's in the wall of the artery. And there's lots of it. So again, if you have a little blood vessel like this, if you imagine tons and tons of smooth muscle on the outside-- so let's draw it like this, little bands of smooth muscle. If those bands decide that they want to contract down, that they want to squeeze down, you're going to get something that looks like a little straw, because those muscles are now tight. They're tightly wound, so you're going to create like a little straw. And this process is called vasoconstriction. Vaso just means blood vessel. And constriction is kind of tightening down. So vasoconstriction, tightening down of the blood vessel. And what that does is it increases resistance. Just like if you're trying to blow through a tiny, tiny little straw, there's a lot of resistance. Well, it's the same idea here. And actually, a lot of that resistance and change in the vasoconstriction is happening at the arteriole level. So that's why they're very special and I want you to remember them. From there, blood is going to go through the capillaries. I didn't actually label them the first time, but let me just write that here. Some, as they call them, capillary beds. I'll write that out. And then it's going to go and get collected in the venules and eventually into the veins. And the important thing about the veins-- I'm going to stop right here and just talk about it very briefly-- is that they have these little valves. And these valves make sure that the blood continues to flow in one direction. So one important thing here is the valves. And remember, the other important thing is that they are able to deal with large volumes. So unlike the arterial side where it was all about large pressure, down here with the vein side, we have to think about large volumes. Remember about 2/3 of your blood at any point in time is sitting in some vein or venule somewhere." + }, + { + "Q": "can parentheses be in parentheses", + "A": "Yes, you do the inner parenthesis first.", + "video_name": "GiSpzFKI5_w", + "transcript": "We're asked to simplify 8 plus 5 times 4 minus, and then in parentheses, 6 plus 10 divided by 2 plus 44. Whenever you see some type of crazy expression like this where you have parentheses and addition and subtraction and division, you always want to keep the order of operations in mind. Let me write them down over here. So when you're doing order of operations, or really when you're evaluating any expression, you should have this in the front of your brain that the top priority goes to parentheses. And those are these little brackets over here, or however Those are the parentheses right there. That gets top priority. Then after that, you want to worry about exponents. There are no exponents in this expression, but I'll just write it down just for future reference: exponents. One way I like to think about it is parentheses always takes top priority, but then after that, we go in descending order, or I guess we should say in-- well, yeah, in descending order of how fast that computation is. When I say fast, how fast it grows. When I take something to an exponent, when I'm taking something to a power, it grows really fast. Then it grows a little bit slower or shrinks a little bit slower if I multiply or divide, so that comes next: multiply or divide. Multiplication and division comes next, and then last of all comes addition and subtraction. So these are kind of the slowest operations. This is a little bit faster. This is the fastest operation. And then the parentheses, just no matter what, always take priority. So let's apply it over here. Let me rewrite this whole expression. So it's 8 plus 5 times 4 minus, in parentheses, 6 plus 10 divided by 2 plus 44. So we're going to want to do the parentheses first. We have parentheses there and there. Now this parentheses is pretty straightforward. Well, inside the parentheses is already evaluated, so we could really just view this as 5 times 4. So let's just evaluate that right from the get go. So this is going to result in 8 plus-- and really, when you're evaluating the parentheses, if your evaluate this parentheses, you literally just get 5, and you evaluate that parentheses, you literally just get 4, and then they're next to each other, so you multiply them. So 5 times 4 is 20 minus-- let me stay consistent with the colors. Now let me write the next parenthesis right there, and then inside of it, we'd evaluate this first. Let me close the parenthesis right there. And then we have plus 44. So what is this thing right here evaluate to, this thing inside the parentheses? Well, you might be tempted to say, well, let me just go left to right. 6 plus 10 is 16 and then divide by 2 and you would get 8. But remember: order of operations. Division takes priority over addition, so you actually want to do the division first, and we could actually write it here like this. You could imagine putting some more parentheses. Let me do it in that same purple. You could imagine putting some more parentheses right here to really emphasize the fact that you're going to do the division first. So 10 divided by 2 is 5, so this will result in 6, plus 10 divided by 2, is 5. 6 plus 5. Well, we still have to evaluate this parentheses, so this results-- what's 6 plus 5? Well, that's 11. So we're left with the 20-- let me write it all down again. We're left with 8 plus 20 minus 6 plus 5, which is 11, plus 44. And now that we have everything at this level of operations, we can just go left to right. So 8 plus 20 is 28, so you can view this as 28 minus 11 plus 44. 28 minus 11-- 28 minus 10 would be 18, so this is going to be 17. It's going to be 17 plus 44. And then 17 plus 44-- I'll scroll down a little bit. 7 plus 44 would be 51, so this is going to be 61. So this is going to be equal to 61. And we're done!" + }, + { + "Q": "The path of the ice-sock is a great circle which indicates a continuous change in direction. Why isn't acceleration present; since acceleration is a vector having both direction and magnitude? Thanks.", + "A": "Acceleration is present. It s centripetal acceleration. That s the point of the video.", + "video_name": "CEdXvoAv_oM", + "transcript": "This is a picture of the planet Lubricon-VI. And Lubricon-VI is a very special planet because it's made up of a yet to be discovered element called Lubrica. And Lubrica is special because if anything glides across the surface of Lubrica, it will experience absolutely no friction. So if this right over here is a sheet of Lubrica-- we're looking at it from the side. And if we have a brick on top of it, maybe gliding on top of it like that, it experiences absolutely no friction. Now, the other things we know about Lubricon-VI is it's drifting in deep space and it does not have an atmosphere. In fact, it is a complete vacuum outside of it. It's in such deep space, such a remote part of space, that there aren't even a few hydrogen atoms right over here. It is a complete, absolute vacuum. And it's also an ancient planet. The star that it used to orbit around has long since died away. So it's just this lonely planet drifting in deep space without an atmosphere. The other thing we know about Lubricon-VI is that it is a perfect sphere. It is a perfect, perfect sphere. Now, my question to you. For some bizarre reason there happens to be, on the surface of Lubricon-VI-- so this right over here is the surface of Lubricon-VI. There happens to be a sock that is frozen in a block of ice. So this is my sock and its frozen in this block of ice. And it happens to be traveling at 1 kilometer per hour in that direction. If we were to look at it from this kind of macro scale when we're looking at the planet, let's say then that is the frozen sock, and it is traveling along the equator. It is traveling along the equator of Lubricon-VI. So my question to you, given all of the assumptions we made that it has absolutely no atmosphere, it's a perfect sphere, and Lubrica has absolutely no friction regardless of what's traveling on top of it-- what will happen to this frozen sock over time? To answer that question, we need to think about all of the forces that are acting on this, I guess, frozen block of ice and sock. And first of all, let's think about these forces that are acting in the radial direction, inward or outward, of the center of the planet. Well, this planet has a mass. And so you have an inward force towards the planet's center of mass. And so you have the force of gravity acting on this block going radially inward to the center of the planet. So I'll draw it like this. So we have our force of gravity. We have our force of gravity going radially inward, just like that. But then we know that the block is not just spiraling towards the center of the earth. We have the surface here. It's not going to go through the surface of Lubrica. We can also assume that Lubrica is a very, very, very strong material. And so you also have a normal force. You also have a normal force that is keeping the block from spiraling towards the center of the earth. So this is a normal force. And one thing we'll think about now, and we'll address it directly in another tutorial, is whether this normal force is equal to the force of gravity. We'll think about that in a future video. But these are all the forces that are acting in the radial direction, either inward towards the center of the planet or outward. But if we think about in the tangential direction, along the surface of the planet, there are no net forces. And because there are no net forces in this tangential direction right over here, this block will not either accelerate nor decelerate. There is no air friction. Or I should say air resistance, which is really just friction with the particles if you had an atmosphere. It's a complete vacuum, so there's nothing there. There is no friction with the surface of the planet. So there's no friction there, which could have been a force in the tangential direction. So there's absolutely no forces in the tangential direction. So this block of ice will actually continue to travel at one kilometer per hour for all of eternity. So it'll just continue to do it given the assumptions that we've just made." + }, + { + "Q": "in the video at 4:24 Sal says that we can get infinitely closer to one. If we can get infinitely closer to one doesn't that mean that we can never approach one?", + "A": "True. Yes, we can always get closer and closer to one but the function actually never reaches one.", + "video_name": "riXcZT2ICjA", + "transcript": "In this video, I want to familiarize you with the idea of a limit, which is a super important idea. It's really the idea that all of calculus is based upon. But despite being so super important, it's actually a really, really, really, really, really, really simple idea. So let me draw a function here, actually, let me define a function here, a kind of a simple function. So let's define f of x, let's say that f of x is going to be x minus 1 over x minus 1. And you might say, hey, Sal look, I have the same thing in the numerator and denominator. If I have something divided by itself, that would just be equal to 1. Can't I just simplify this to f of x equals 1? And I would say, well, you're almost true, the difference between f of x equals 1 and this thing right over here, is that this thing can never equal-- this thing is undefined when x is equal to 1. Because if you set, let me define it. Let me write it over here, if you have f of, sorry not f of 0, if you have f of 1, what happens. In the numerator, we get 1 minus 1, which is, let me just write it down, in the numerator, you get 0. And in the denominator, you get 1 minus 1, which is also 0. And so anything divided by 0, including 0 divided by 0, this is undefined. So you can make the simplification. You can say that this is you the same thing as f of x is equal to 1, but you would have to add the constraint that x cannot be equal to 1. Now this and this are equivalent, both of these are going to be equal to 1 for all other X's other than one, but at x equals 1, it becomes undefined. This is undefined and this one's undefined. So how would I graph this function. So let me graph it. So that, is my y is equal to f of x axis, y is equal to f of x axis, and then this over here is my x-axis. And then let's say this is the point x is equal to 1. This over here would be x is equal to negative 1. This is y is equal to 1, right up there I could do negative 1. but that matter much relative to this function right over here. And let me graph it. So it's essentially for any x other than 1 f of x is going to be equal to 1. So it's going to be, look like this. It's going to look like this, except at 1. At 1 f of x is undefined. So I'm going to put a little bit of a gap right over here, the circle to signify that this function is not We don't know what this function equals at 1. We never defined it. This definition of the function doesn't tell us It's literally undefined, literally undefined when x is equal to 1. So this is the function right over here. And so once again, if someone were to ask you what is f of 1, you go, and let's say that even though this was a function definition, you'd go, OK x is equal to 1, oh wait there's a gap in my function over here. It is undefined. So let me write it again. It's kind of redundant, but I'll rewrite it f of 1 is undefined. But what if I were to ask you, what is the function approaching as x equals 1. And now this is starting to touch on the idea of a limit. So as x gets closer and closer to 1. So as we get closer and closer x is to 1, what is the function approaching. Well, this entire time, the function, what's a getting closer and closer to. On the left hand side, no matter how close you get to 1, as long as you're not at 1, you're actually at f of x is equal to 1. Over here from the right hand side, you get the same thing. So you could say, and we'll get more and more familiar with this idea as we do more examples, that the limit as x and L-I-M, short for limit, as x approaches 1 of f of x is equal to, as we get closer, we can get unbelievably, we can get infinitely close to 1, as long as we're not at 1. And our function is going to be equal to 1, it's getting closer and closer and closer to 1. It's actually at 1 the entire time. So in this case, we could say the limit as x approaches 1 of f of x is 1. So once again, it has very fancy notation, but it's just saying, look what is a function approaching as x gets closer and closer to 1. Let me do another example where we're dealing with a curve, just so that you have the general idea. So let's say that I have the function f of x, let me just for the sake of variety, let me call it g of x. Let's say that we have g of x is equal to, I could define it this way, we could define it as x squared, when x does not equal, I don't know when x does not equal 2. And let's say that when x equals 2 it is equal to 1. So once again, a kind of an interesting function that, as you'll see, is not fully continuous, it has a discontinuity. Let me graph it. So this is my y equals f of x axis, this is my x-axis right over here. Let me draw x equals 2, x, let's say this is x equals 1, this is x equals 2, this is negative 1, this is negative 2. And then let me draw, so everywhere except x equals 2, it's equal to x squared. So let me draw it like this. So it's going to be a parabola, looks something like this, let me draw a better version of the parabola. So it'll look something like this. Not the most beautifully drawn parabola in the history of drawing parabolas, but I think it'll give you the idea. I think you know what a parabola looks like, hopefully. It should be symmetric, let me redraw it because that's kind of ugly. And that's looking better. OK, all right, there you go. All right, now, this would be the graph of just x squared. But this can't be. It's not x squared when x is equal to 2. So once again, when x is equal to 2, we should have a little bit of a discontinuity here. So I'll draw a gap right over there, because when x equals 2 the function is equal to 1. When x is equal to 2, so let's say that, and I'm not doing them on the same scale, but let's say that. So this, on the graph of f of x is equal to x squared, this would be 4, this would be 2, this would be 1, this would be 3. So when x is equal to 2, our function is equal to 1. So this is a bit of a bizarre function, but we can define it this way. You can define a function however you like to define it. And so notice, it's just like the graph of f of x is equal to x squared, except when you get to 2, it has this gap, because you don't use the f of x is equal to x squared when x is equal to 2. You use f of x-- or I should say g of x-- you use g of x is equal to 1. Have I been saying f of x? I apologize for that. You use g of x is equal to 1. So then then at 2, just at 2, just exactly at 2, it drops down to 1. And then it keeps going along the function g of x is equal to, or I should say, along the function x squared. So my question to you. So there's a couple of things, if I were to just evaluate the function g of 2. Well, you'd look at this definition, OK, when x equals 2, I use this situation right over here. And it tells me, it's going to be equal to 1. Let me ask a more interesting question. Or perhaps a more interesting question. What is the limit as x approaches 2 of g of x. Once again, fancy notation, but it's asking something pretty, pretty, pretty simple. It's saying as x gets closer and closer to 2, as you get closer and closer, and this isn't a rigorous definition, we'll do that in future videos. As x gets closer and closer to 2, what is g of x approaching? So if you get to 1.9, and then 1.999, and then 1.999999, and then 1.9999999, what is g of x approaching. Or if you were to go from the positive direction. If you were to say 2.1, what's g of 2.1, what's g of 2.01, what's g of 2.001, what is that approaching as we get closer and closer to it. And you can see it visually just by drawing the graph. As g gets closer and closer to 2, and if we were to follow along the graph, we see that we are approaching 4. Even though that's not where the function is, the function drops down to 1. The limit of g of x as x approaches 2 is equal to 4. And you could even do this numerically using a calculator, and let me do that, because I think that will be interesting. So let me get the calculator out, let me get my trusty TI-85 out. So here is my calculator, and you could numerically say, OK, what's it going to approach as you approach x equals 2. So let's try 1.94, for x is equal to 1.9, you would use this top clause right over here. So you'd have 1.9 squared. And so you get 3.61, well what if you get even closer to 2, so 1.99, and once again, let me square that. Well now I'm at 3.96. What if I do 1.999, and I square that? I'm going to have 3.996. Notice I'm going closer, and closer, and closer to our point. And if I did, if I got really close, 1.9999999999 squared, what am I going to get to. It's not actually going to be exactly 4, this calculator just rounded things up, but going to get to a number really, really, really, really, really, really, really, really, really close to 4. And we can do something from the positive direction too. And it actually has to be the same number when we approach from the below what we're trying to approach, and above what we're trying to approach. So if we try to 2.1 squared, we get 4.4. let me go a couple of steps ahead, 2.01, so this is much closer to 2 now, squared. Now we are getting much closer to 4. So the closer we get to 2, the closer it seems like we're getting to 4. So once again, that's a numeric way of saying that the limit, as x approaches 2 from either direction of g of x, even though right at 2, the function is equal to 1, because it's discontinuous. The limit as we're approaching 2, we're getting closer, and closer, and closer to 4." + }, + { + "Q": "Whats a \"reciprocal\"? (4:07)", + "A": "For any fraction, its reciprocal is created by flipping the fraction. Example: 3/4: its reciprocal is 4/3 -5/2: its reciprocal is -2/5 6: Note 6 as a fraction is 6/1. Its reciprocal = 1/6 In the video, Sal is using the reciprocal of (5x^4)/4, which would be 4/(5x^4). Hope this helps.", + "video_name": "6nALFmvvgds", + "transcript": "- [Voiceover] So up here, we are multiplying two rational expressions. And here, we're dividing one rational expression by another one. Now what I encourage you do is pause this video and think about what these become when you multiply them. I don't know, maybe you simplify it a little bit, and I also want you to think about what constraints do you have to put on the x values in order for your resulting expression to be algebraically equivalent to your original expression. So let's work it out together just so you realize what I'm talking about. So this is going to be, in our numerator, we are going to get six x to the third power times two, and our denominator, we're going to have five times three x. And we can see both the numerator and the denominator are divisible by x, so let's divide the denominator by x. We get one there. Let's divide x to the third by x. We get x squared. And we can also see the both the numerator and denominator are divisible by three, so divide six by three, you get two. Divide three by three, you get one. And we are left with two x squared times two, which is going to be four x squared over five times one times one over five. And we can also write that as 4/5 x squared. Now someone just presented you on the street with the expression 4/5 x squared and say, for what x is this defined? I could put any x here, x could be zero because zero squared is zero times 4/5 is just going to be zero, so it does seems to be defined for zero, and that is true. But if someone says, how would I have to constrain this in order for it to be algebraically equivalent to this first expression? Well then, you'd have to say, well, this first expression is not defined for all x. For example, if x were equal to zero, then you would be dividing by zero right over here, which would make this undefined. So you could explicitly call it out, x can not be equal to zero. And so if you want this to be algebraically equivalent, you would have to make that same condition, x cannot be equal to zero. Another way to think about it, if you had a function defined this way, if you said, if you said f of x is equal to six x to the third over five times two over, times two over three x; and if someone said, well what is f of zero, you would say f of zero is undefined. Undefined. Why is that? Because you put x equals zero there, you're going to get two divided by zero and it's undefined. But if you said, okay, well, can I simplify this a little bit to get the exact same function? Well, we're saying you can say f of x is equal to 4/5 times x squared. But if you just left it at that, you would get f of zero is equal to zero. So now it would be defined at zero, but then this would make it a different function. These are two different functions the way they're written right over here. Instead, to make them, to make it clear that this is equivalent to that one, you would have to say x cannot be equal to zero. Now these functions are equivalent because now, if u said f of zero, you'd say all right, x cannot be equal to zero, you know? This would be the case if x is anything other than zero and it's not defined for zero, and so you would say f or zero is undefined. So now, these two functions are equivalent, or these two expressions are algebraically equivalent. So thinking about that, let's tackle this division situation here. So immediately, when you look at this, you say, woah, what are constrains here? Well, x cannot be equal to zero because if x was a zero, this second, this five x to the fourth over four would be zero and you'd be dividing, you'd be dividing by zero. So we can explicitly call out that x cannot be equal to zero. And so if x cannot be equal to zero in the original expression, if the result, whatever we get for the resulting expression, in order for it to be algebraically equivalent, we have to give this same constraint. So let's multiply this, or let's do the division. So this is going to be the same thing as two x to the fourth power over seven times the reciprocal, times... The reciprocal of this is going to be four over five x to the fourth, which is going to be equal to in the numerator, we're going to have eight x to the fourth. So we're going to have eight x to the fourth, four times two x to the fourth, over seven times five x to the fourth is 35 x to the fourth. And now, there's something. We can do a little bit of simplification here, both the numerator and the denominator are divisible by x to the fourth, so let's divide by x to the fourth and we get eight over 35. So once again, you just look at eight 30, Well, this is going to be defined for any x. X isn't even involved in the expression. But if we want this to be algebraically equivalent to this first expression, then we have to make the same constraint, x does not, cannot be equal to zero. And to see, you know, this even seems a little bit more nonsensical to say x cannot be equal to zero for an expression that does not even involve x. But one way to think about it is imagine a function that was defined as g of x is equal to, is equal to all of this business here. Well, g of zero would be undefined. But if you said g of x is equal to 8/35, well now, g of zero would be defined as 8/35, which would make it a different function. So to make them algebraically equivalent, you could say g of x is equal to 8/35, as long as x does not equal zero. And you could say it's undefined if you want. Undefined for x equals zero. Or you don't even have to include that second row, and that will literally just make it undefined. But now, this expression, this algebraic expression, is equivalent to our original one even though we had simplified it." + }, + { + "Q": "Which trig identities should I have memorized?", + "A": "You can have the basic trig identies memorised 1) sin^2 x + cos^2 x=1 2)tan^2 x + 1 = sec^2 x 3) 1+ cot^2 x = csc^2 x", + "video_name": "rElAJA9GyL4", + "transcript": "- [Voiceover] Let's see if we can take the indefinite integral of cosine of X to the third power. I encourage you to pause the video and see if you can figure this out on your own. You have given it a go and you might have gotten stuck. Some of you all might have been able to figure it out, but some of you all might have gotten stuck. You're like, \"Okay, cosine to the third power. \"Well, gee, if I only had a derivative of cosine here, \"if I had a negative sign of X or a sin of X here, \"maybe I could've used U substitution, \"but how do I take the anti-derivative \"of cosine of X to the third power?\" The key here is, is to use some basic trigonometric identities. What do I mean by that? We know that sin squared X plus cosine squared X is equal to one, or if we subtract sin squared from both sides, we know that cosine squared X is equal to one, write it this way, is equal to one minus sin squared X. What would happen if cosine to the third power, that's cosine squared times cosine. What happens if we were to take that cosine squared? Let me just rewrite it. This is the same thing as cosine of X times cosine squared of X, DX. What if we were to take this thing right over here, let me do that magenta color. What if we were to take this right over here and replace it with this. I now what you're thinking. \"Sal, what's that going to do for me? \"This feels like I'm making this integral \"even more convoluted.\" What I would tell you, I would say, \"This might seem like it's getting more complicated, \"but as you explore and you play with it, \"you'll see that this actually makes \"the integral more solvable.\" Let's try it out. If we do that, this is going to be equal to the indefinite integral, cosine of X times one minus sin squared X, DX. What is this going to be equal to? This is going to be equal to, let me do this in that green color. This is going to be equal to the indefinite integral of cosine X. I'm just going to distribute the cosine of X. Cosine of X minus, minus cosine of X, cosign of X sin squared of X, sin squared, sin squared X and then I can close the parentheses, DX. This, of course, is going to be equal to the integral of cosine of X, DX, and we know what that's going to be, minus the integral. I'll switch to one color now, of cosine of X, sin squared X, sin squared X, DX. Now, this is where it gets interesting. This part right over here is pretty straight forward. The anti-derivative of cosine of X is just sin of X. This right over here is going to be sin of X. I'll worry about the plus C at the end because both of these are going to have a plus C, so might as well just put one big plus c at the end. That's sin of X and then what do we have going on over here? Well, you might recognize, I have a function of sin of X. I'm taking sin of X and I'm squaring it and then I have sin of X's derivative right over here. This fits the, I have some derivative of a function and then I have another and then I have a, I guess you could say, a function of that function. G of F of X. That's a sin that maybe U substitution is in order, or we've seen the pattern, we've seen this show multiple times already, that you could just say, \"Okay, if I have a function of a function \"and I have that functions derivative, \"then essentially I can just take the anti-derivative \"with respect to this function.\" This would be equal to, say, capital G is the anti-derivative of lower case G. Capital G of F of X plus C. Now, if what I said didn't make sense, then we could do U substitution and go through it a little bit more step by step. Let's just do that because we want things to make sense. That's the whole point of these videos. We could say U is equal to sin of X and then DU is going to be equal to cosine of X, DX. This part and that part is going to be DU and then this is going to be U squared. This is going to be minus. We have the integral of U squared, DU. What is this going to be? This is going to be, we're going to have negative U to the third power over three. Then, we know what U is. The U is equal to sin of X. We have our sin of X here for the first part of the integral, for the first integral. We have the sin of X and then this is going to be minus. Let me just write it this way. Minus 1/3 minus 1/3. Instead of U to the third, we know U is sin of X. Sin of X to the third power. Then now, we can throw that plus C there. We're done. We've just evaluated that indefinite integral. The key to it is to just play around a little bit with trigonometric identities so that you can get the integral to a point that you can use the reverse chain rule or you can use U substitution, which is just really another way of expressing the reverse chain rule." + }, + { + "Q": "how did they know it what to name our planets?", + "A": "The names for the planets in our solar system came from the ancient Romans. The Romans named the planets after their gods.", + "video_name": "VbNXh0GaLYo", + "transcript": "What I'm going to attempt to do in the next two videos is really just give an overview of everything that's happened to Earth since it came into existence. We're going start really at the formation of Earth or the formation of our Solar system or the formation of the Sun, and our best sense of what actually happened is that there was a supernova in our vicinity of the galaxy, and this right here is a picture of a supernova remnant, actually, the remnant for Kepler's supernova. The supernova in this picture actually happened four hundred years ago in 1604, so right at the center a star essentially exploded and for a few weeks was the brightest object in the night sky, and it was observed by Kepler and other people in 1604, and this is what it looks like now. What we see is kinda the shockwave that's been traveling out for the past 400 years, so now it must be many light years across. It wasn't, obviously, matter wasn't traveling at the speed of light, but it must've been traveling pretty, pretty fast, at least relativistic speeds, a reasonable fraction of the speed of light. This has traveled a good bit out now, but what you can imagine is when you have the shockwave traveling out from a supernova, let's say you had a cloud of molecules, a cloud of gas, that before the shockwave came by just wasn't dense enough for gravity to take over, and for it to accrete, essentially, into a solar system. When the shockwave passes by it compresses all of this gas and all of this material and all of these molecules, so it now does have that critical density to form, to accrete into a star and a solar system. We think that's what's happened, and the reason why we feel pretty strongly that it must've been caused by a supernova is that the only way that the really heavy elements can form, or the only way we know that they can form is in kind of the heat of a supernova, and our uranium, the uranium that seems to be in our solar system on Earth, seems to have formed roughly at the time of the formation of Earth, at about four and a half billion years ago, and we'll talk in a little bit more depth in future videos on exactly how people figure that out, but since the uranium seems about the same age as our solar system, it must've been formed at around the same time, and it must've been formed by a supernova, and it must be coming from a supernova, so a supernova shockwave must've passed through our part of the universe, and that's a good reason for gas to get compressed and begin to accrete. So you fast-forward a few million years. That gas would've accreted into something like this. It would've reached the critical temperature, critical density and pressure at the center for ignition to occur, for fusion to start to happen, for hydrogen to start fusing into helium, and this right here is our early sun. Around the sun you have all of the gases and particles and molecules that had enough angular velocity to not fall into the sun, to go into orbit around the sun. They were actually supported by a little bit of pressure, too, because you can kinda view this as kind of a big cloud of gas, so they're always bumping into each other, but for the most part it was their angular velocity, and over the next tens of millions of years they'll slowly bump into each other and clump into each other. Even small particles have gravity, and they're gonna slowly become rocks and asteroids and, eventually, what we would call \"planetesimals,\" which are, kinda view them as seeds of planets or early planets, and then those would have a reasonable amount of gravity and other things would be attracted to them and slowly clump up to them. This wasn't like a simple process, you know, you could imagine you might have one planetesimal form, and then there's another planetesimal formed, and instead of having a nice, gentle those two guys accreting into each other, they might have huge relative velocities and ram into each other, and then just, you know, shatter, so this wasn't just a nice, gentle process of constant accretion. It would actually have been a very violent process, actually happened early in Earth's history, and we actually think this is why the Moon formed, so at some point you fast-forward a little bit from this, Earth would have formed, I should say, the mass that eventually becomes our modern Earth would have been forming. Let me draw it over here. So, let's say that that is our modern Earth, and what we think happened is that another proto-planet or another, it was actually a planet because it was roughly the size of Mars, ran into our, what it is eventually going to become our Earth. This is actually a picture of it. This is an artist's depiction of that collision, where this planet right here is the size of Mars, and it ran into what would eventually become Earth. This we call Theia. This is Theia, and what we believe happened, and if you look up, if you go onto the Internet, you'll see some simulations that talk about this, is that we think it was a glancing blow. It wasn't a direct hit that would've just kinda shattered each of them and turned into one big molten ball. We think it was a glancing blow, something like this. This was essentially Earth. Obviously, Earth got changed dramatically once Theia ran into it, but Theia is right over here, and we think it was a glancing blow. It came and it hit Earth at kind of an angle, and then it obviously the combined energies from that interaction would've made both of them molten, and frankly they probably already were molten because you had a bunch of smaller collisions and accretion events and little things hitting the surface, so probably both of them during this entire period, but this would've had a glancing blow on Earth and essentially splashed a bunch of molten material out into orbit. It would've just come in, had a glancing blow on Earth, and then splashed a bunch of molten material, some of it would've been captured by Earth, so this is the before and the after, you can imagine, Earth is kind of this molten, super hot ball, and some of it just gets splashed into orbit from the collision. Let me just see if I can draw Theia here, so Theia has collided, and it is also molten now because huge energies, and it splashes some of it into orbit. If we fast-forward a little bit, this stuff that got splashed into orbit, it's going in that direction, that becomes our Moon, and then the rest of this material eventually kind of condenses back into a spherical shape and is what we now call our Earth. So that's how we actually think right now that the Moon actually formed. Even after this happened, the Earth still had a lot more, I guess, violence to experience. Just to get a sense of where we are in the history of Earth, we're going to refer to this time clock a lot over the next few videos, this time clock starts right here at the formation of our solar system, 4.6 billion years ago, probably coinciding with some type of supernova, and as we go clockwise on this diagram, we're moving forward in time, and we're gonna go all the way forward to the present period, and just so you understand some of the terminology, \"Ga\" means \"billions of years ago\" 'G' for \"Giga-\" \"Ma\" means \"millions of years ago\" 'M' for \"Mega-\" So where we are right now, the Moon has formed, and we're in what we call the Hadean period or actually I shouldn't say \"period.\" It's the Hadean eon of Earth. \"Period\" is actually another time period, so let me make this very clear. It's the Hadean, we are in the Hadean eon, and an eon is kind of the largest period of time that we talk about, especially relative to Earth, and it's roughly 500 million to a billion years is an eon, and what makes the Hadean eon distinctive, well, from a geological point of view what makes it distinctive is really we don't have any rocks from the Hadean period. We don't have any kind of macroscopic-scale rocks from the Hadean period, and that's because at that time, we believe, the Earth was just this molten ball of kind of magma and lava, and it was molten because it was a product of all of these accretion events and all of these collisions and all this kinetic energy turning into heat. If you were to look at the surface of the Earth, if you were to be on the surface of the Earth during the Hadean eon, which you probably wouldn't want to be because you might get hit by a falling meteorite or probably burned by some magma, whatever, it would look like this, and you wouldn't be able to breathe anyway; this is what the surface of the Earth would look like. It would look like a big magma pool, and that's why we don't have any rocks from there because the rocks were just constantly being recycled, being dissolved and churned inside of this giant molten ball, and frankly the Earth still is a giant molten ball, it's just we live on the super-thin, cooled crust of that molten ball. If you go right below that crust, and we'll talk a little bit more about that in future videos, you will get magma, and if you go dig deeper, you'll have liquid iron. I mean, it still is a molten ball. And this whole period is just a violent, not only was Earth itself a volcanic, molten ball, it began to harden as you get into the late Hadean eon, but we also had stuff falling from the sky and constantly colliding with Earth, and really just continuing to add to the heat of this molten ball. Anyway, I'll leave you there, and, as you can imagine, at this point there was no, as far as we can tell, there was no life on Earth. Some people believe that maybe some life could've formed in the late Hadean eon, but for the most part this was just completely inhospitable for any life forming. I'll leave you there, and where we take up the next video, we'll talk a little bit about the Archean eon." + } +] \ No newline at end of file