Okay, in this video clip, we're going to do a little bit of a math review. The purpose here is not to actually teach the math involved, but hopefully remind those of you who need a refresher of a few things here. So, we're not going to start from first principles or anything like that. But, just to remind you about a view things, especially with respect to algebra and plotting that will be useful later on in our course. So certainly, if you are doing a more quantitative approach to the course, you'll want to be up on these things. But even if you want to do the more qualitative approach or just taking auditing approach, there will be times you want to just sort of follow along with the video. And it will be helpful to remember these certain things. So, when we talk about algebra of course we use letters, and then we can replace, plug in numbers for them. So we have things like this where we say okay, equations a + b = c. So we have two numbers represented by a and b, when we add them together we get number c, etc., etc. Remember a few things with exponents. If we have an actual number 2 squared, that is the same as 2 times 2 which is 4 of course. Or 2 cubed = 2 times 2 times 2, or 4 cubed, 4 times 4 times 4, so it's 2 multiplied three times. So we can also try about a squared, certainly any number squared, a cubed. I don't think we really actually get into anything more than squared in here, if I remember correctly. So, but we'll see a lots of squares, a squared, b squared, x squared, y squared things like that. So just remember how that works. Another thing along with this, we have negative exponents. So if we have say 3 to the -2 power, that's defined to be 1 over 3 squared, okay. So 1 over 3 squared, 3 to the -2. Or if we use a, a to the -2 power = 1 over a squared. Remember also that a to the 1st power, a to the 1, simply = a. So a to the -1 is the same as 1 over a. We can also, of course, talk about things like the square root of 3, where if you take the square root of 3 times the square root of 3, you'll get 3 back again. So really if we have the square root of a times the square root of a, that's going to give us a back again. In exponent notation we write instead of using the square root radical sign there, we sometimes write this as a to the one-half. So a to the one-half power is the same as the square root of a. One reason this works is if you do something like this. If you have let's say 3 again, we have 3 squared times 3 to the 4th power. What does that equal? You may remember if you dredge your mathematical memory if it's been awhile here, is that when we have like bases, remember the 3s here are called the bases. And the 2 and the 4 are the exponents. When you have like bases, you just add the exponent. So this becomes, 3 to the 2 + 4, which is the same as 3 to the 6th power. And you can sort of see this, because this is 3 times 3, times 3 times 3 times 3 times 3. That's 4 more 3s, so we 6 3s together multiplied, so we get 3 to the 6th power there. Or in algebraic notation, we might say we got a squared times a to the 4th, then that's a to the 6th power. Again, if the bases are the same, you can add the exponents and combine them. What you can't do is something like this, if you have a squared some number b to the 4th, there's nothing you can do with that. You can't do 2 plus 4 there and somehow do ab to the 6th, or something like that. Because it'd be something like this, if you have 3 squared times 7 cubed, you can't just do 21 to the 6th power of 5th power or something like that. It won't work, doesn't make mathematical sense. And the reason I just did a reminder over here is, why would we call this a to the one-half square root of a? Well square root of a times square root of a equals a, back again. Square root of 3 times square root of 3 equals 3. We write this as a to the one-half times a to the one-half. Bases are the same, the as are the same. Like bases, so we add the exponents. This becomes a to the one-half + one-half, which of course, is just 1. So, we get a to the 1, which is just a. So, what we've just shown in different notations, square of a times square of a = a. So, that's why we use a to the one-half, that one-half exponent as a symbol for the square root of a, or it's equivalent to the square root of a. So that's a little bit with exponents, a few reminders about that. Some other reminders here that'll actually be useful later on, what if we have something like this, if we say a + b. Remember we can factor things out. So we can write this as a times 1 + b over a. Remember how we did that? Because if we have this, we can get this back because we just go a times 1, so that's going to equal a. And I've got a times b over a. Remember how this works if it's really been a while for you? So this is the same thing as a times b over a. And the a here in the numerator essentially, and in the denominator cancel, and we just get a+b again. So essentially we can factor out an a in this case. Or we could have done a similar thing with b. Why is that important? Because we will see later on something like this. We'll have not just a + b, we'll have a squared + b squared. That form comes into the theory in a number of cases. And for various reasons we'll want to write it like this. We'll write this as (a squared) times (1 + b squared / a squared), okay? because these are equivalent because I take a squared times 1, that's just a squared. A squared times b squared over a squared, the a squareds cancel, I'm just left with b squared. Even though this looks, why would we want to do this when this is simpler? Well, it turns out in certain cases that will give us a nicer form to see what's going on with the math. Okay, so that's factoring something out to just write something in a different form there. Another thing, just as a reminder, if we have a/b + c/d, so maybe we have something like, what should we do, we just, well maybe two-thirds plus,, five-halfs, okay? Oops, just let me change that, that's true to five-fourths. There, just so that a, b, c and d are all different there, okay? Obviously, it could be the same, but two-thirds plus five-fourths, we want to combine those two fractions together. So remember we need to get a common denominator. We want the bottom part to be the same. The way we do that is we do a little trick, and we multiply in the form of one. Remember so we have two-thirds times, look at the other denominator the four, so four fourths, that's just one. I haven't changed the two thirds, two thirds times one is one. Plus five fourths times three over three. This denominator and do the multiplication. Remember by fraction just multiple the top, the numerator, then multiply the denominator. So this becomes eight-twelves plus 5 times 3 is 15. Twelves, and you can see our little trick gave us the common denominator there, so now we can combine those two fractions, it becomes 8 plus 15 over 12 and 8 plus 15 is 32. Twelve which we could just leave of course or we could write this. This equals one and because we got 12 over there and it looks like eleven 12s. One and eleven-twelfths is our final answer there, so in the form a over b plus c over d if you follow the pattern Again we are going to get ad plus bc all over bd. You want to just memorize that form but we sort of skipped some steps there but essentially the same thing here multiply This by d over d. D over d here, and multiply this one by b over b, the other denominator, and that's put it all together and we get that form. So again, basic result from algebra. I think those are the main things we'll need with that. Let's just spend a few minutes talking about Plotting points because later on the course, we'll do some things with spacetime diagrams, which can be a very powerful tool for analyzing certain situations. And it's useful if you haven't plotted things for a long time, it's useful to remind ourselves How we do that. So plotting, and actually as we go along the course there may be few other things that come up that we won't have mentioned here but we'll mention them along the way as we go just as reminders of how things work and won't try to jump too far ahead. Remember make haste slowly as it were So plotting, remember just how that works? If you've got, I'm going to use x an y now. Some x an y value. So maybe the values for x, often we choose these, maybe -2, -1, 0, 1, 2, 3. And given x, we can find a value for y. Maybe they're just given, maybe there's some experimental data and we know the x and y values and we just want to plot them. In this case, we'll use this. We'll say x is negative two, y is four. One, zero, one, four, nine. In fact, you may recognize this as y equals x squared. In other words, take the value of x, square it, and that gives us y. So let's find x, y data, so if I'm going to plot that Here's my x axis, here's my y axis. Put some marks on here. One, two, three four. One, two, three four. Something like that. So a negative 3, negative 2, negative 1, 0. 1, 2, 3, 4. Of course we could be going this way to one, two, three, four, five etc. Right? And so we plot. So first point, negative two four. When x is negative two, y is four. So Negative 2, up 4. So there is our point, in fact let's use red. A little more colorful, negative 2, 4. We got that so it's about right there. Negative 1, 1, right? So negative 1, 1. 0,0 so a point right thee. When x is 1, y is 1. X is 1, y is 1 so about right there. When x is 2, y is 4. So again about right there. And when x is 3 y is 9 so it'd be way up here some place. And [INAUDIBLE] and so then we connect the dots. So if we actually plotted a whole bunch more points it would look something like that. In other words we get a parabola. From that. We're not going to go into parabolas too much, and hyperbolas. Actually, we'll see an example, maybe, of a hyperbola, later on. But we're not going to get that deep into the math. We'll just mention it more or less as an aside as we go along. So the plotting points, right? And we often have an equation.Y equals x squared. So we choose some x values. We calculate the y values and we plot our points like that. Now in particular I want to do one other type of plot here that we'll encounter a lot. This form, And this is the form of a line. You may remember the general form. I'll write this up and then we'll the details on it. So A straight line has the form, in terms of y-x form, x and y values. We have y equals let's call it capital a, x plus b. You may see it in slightly different notation but that's the basic idea. Let's do a simple example. Y=2x, so in this case, A is two, B is zero, I don't have any B in this case. So if we were going to plot this, X values and Y values. Again we'll just choose a nice value for x. We'll just do say negative 2, negative 1, 0, 1, 2, 3, perhaps. And so plug this in. Negative 2 for x times 2, that's negative 4 for y. Negative 2, 0. 2 times 0 is 0. 2 times 1 of course is 2. 2 times 2 is 4 3 times 2 is not 9, it's 6 and so on and so forth with that. And then we plot it. So we say okay here again our axises. Y axis and the X axis. And we got some marks here, try to make them more or less Even, So negative 3, negative 2, negative 1, 1, 2, 3, 4, 1, 2, 3, 4, negative 1, negative 2, and so on and so forth. So our points are negative 2, negative 4, so we'll actually need to go a little bit farther down here, down to Negative 4 so negative 2 for X, negative 4 for Y, got the red out, red marker so there is that one. Negative 1, negative 2, so hopefully my scaling is more or less correct here so we get a nice line out of it. Zero, zero is this point right here. One, two, so over one, up two Doing okay. 2, 4, over 2. Up 4. So like that. And then 3, 6. Over 3. And up 6, would be up here, someplace. And you see if we connect the dots, we get, I don't know if I can do this very well, but we'll try here, if I can Not too bad there. A little quicken. But that's essentially the equation of a line. This is a line, 2 times x, and again, if you dread your mathematical memory, if it's been a while, you may remember, this number right here. In other words, A is the slope, Of the line. The slope. And this is something actually will be important to us to remember, it's not crucially important but it's useful information in that if we look at this number here, it tells us how steep the line is. The bigger this number in front there, the a value, two in this case, It tells us it's a slope of two, has a steepness, in a sense, of two. For those of you again who remember the exact of slope or for a line rise, it's rise over runs. So we measure the rise over the run here. It goes over one For the run and up two and obviously if it went over, went up three, it would be a steeper slope. This would be actually a value three here. So the bigger the number here, the steeper the slope of the line. The smaller the number, assume we were talking about a positive number, the slope of the line would be more like this. If it was one half there, then we would be seeing something more like Line sort of like that. That'd be a slope of 1/2 if we plugged in new numbers for that and plotted that line. A slope of 1 is going to be at a 45 degree angle, assuming our scaling on each access is identical. So slope of 1 if it was just y equals x Plotting 1, 1. When x = 2, y = 2, and so on, so forth. And so we get a line right down the center there, right at 45 degree angles, splitting the center of those other two. So, that's the slope, of a line. One other thing you might say, well what about B here? What's the deal with B? Well if you did an example where we did say y equals two x, plus one. All right, so you're doing y equals two x, we'll do two x plus one. So slope will still be two, the steepness will be two. What's the one mean? Well you may remember in dredging your memory. That it's called the y-intercept. And if you just think about that name a minute, it literally means it's where the line intercepts the y-axis. So, here, when y equals just 2x and B equals 0, the line, all three of our lines here, intersected the y-axis. Here's the y-axis. At zero in this case y equals 2 x plus 1. If we changed our numbers here right, the x values would be the same. But now we have new y values because we changed the equation a little bit. Then we'd find that we still have a line of slope 1 but it intercepts the y axis at. I'm sorry minus slope 2. Minus slope 2. It intercepts the y-axis at the point y equals 1. So, it'd be parallel to my first line here but intersecting here. So, let's see if we can draw that more or less. Well, close. Yeah, a little bit better perhaps. All right, so the y-intercept would be if we draw a nice, big point of intersection there, maybe we can Can get by. So 2x plus 1. If it was 2x plus 2, the y intercept would be here. So it'd be a straight line parallel, but intersecting at y equals 2, and y equals 3, and so on and so forth with that. So that is the equation of a line, general form a. If it's in y and x terms, okay? So x and y Coordinates. Ax plus b. a is the slope, b is the y intercept. One other thing to think about. What if this was y equals negative 2x here? Let's think about that a minute. So x values would be the same. The y values over change. When x is negative two, remember we didn't review this in our little math review a minute ago, but a negative number times another negative number gives you a positive number. A negative number times a positive number gives you a negative number. A positive number times a negative number gives you a negative number. Two negative numbers multiplied together, the negatives cancel out as it were and gives you a positive number. So in this case. X is negative two times the negative two that gives me a positive four. So this actually becomes a plus four there. Same thing here, negative one times negative two gives me a plus there. Zero times zero is still zero, but if x is one I get negative two. So that becomes a negative two. This is a negative four. That's a negative six It just the other way around. When x is negative, now y is positive. When x is positive, now y is negative. And if you actually plotted this, you're going to get something that looks like this. We'll do it like this. Now we're at 1, negative 2 So down here, two and negative four. Down there so something that looks like this. Okay? That is a line with slope equals negative two. So when we have negative slope, it slopes downwards from left to right. Pause a slope, slopes upwards from left to right. That will be another thing to remember. It won't be a crucial point if we start hazy on it but just something to refresh in your memory bank as it were. Pause a slope, slope upwards. The bigger the number the steeper it is Negative slopes slope downward. The bigger the negative number, the steeper the negative slope here. So if it was negative 3, it'd be more steep going that way. If it was negative one-half, it'd be a less steep line going downward from left to right. Okay, so our picture is getting all messed up here. That's just a quick math review of some of the things we'll encounter. As we go along, for some of you, I'm sure that was very very basic, but for others, you say, wow, I don't remember much of that stuff. So, hopefully, you're some place in between there, and it was a nice little refresher here. Again, as we go along, we'll be reminding you about these things, but we won't Go into all the detail of the math involved. We just assume you have enough math so you can take a more quantitative approach, if you like to do that. Again, we really aren't going to do anything more sophisticated than what we've seen here. Which is what roughly Ninth grade algebra or something like that, depending on how your schooling system works. So maybe middle school, type of thing. Okay, so that's a quick math review. That actually concludes our set of introductory. Video clips and so in the next set we can get on to our week 1 material where we learn more about the Physics Circa 1900, we see some new Einstein quotes to ponder, we learn about Physics Circa 1900, we learn about Einstein Circa 1900 as well.
