Welcome back, everyone. We're going to begin our next unit by talking about the Cartesian plane. Our first video is going to be called plotting points in the plane. The point here is really simple. We've already gotten you familiar with the real number line, which is a way of representing numbers on a line. Now we'll look at the Cartesian plane, which is what we're looking at here. This is often denoted R2, just like the real number was denoted R, we have this fancy black part R2. And it's just a way of representing two pieces of information. We have two axes here, the horizontal one is going to be called the x axis, this is going to be called the y axis. The correspondence between ordered pairs of numbers, as we see over here and points in this plane. So first, a very special one is called the origin. Right here at the meeting of the two lines. This is (0, 0), often denoted as big O. And now, I'll just show you how to plot these others and we'll get the basic idea. So if we plot A, the x coordinate, the first number is 2, the second number is 3. The first number's instruction tells us to walk two units to the right. So let's walk one, let's walk two. Now, from there, let's walk three units up. So three, I got up one, I got up two, I go up three. About there is the point (2, 3), that's A = (2, 3). B, look at the first coordinate, that's -1, that tells me to move one unit to the left, now five up. One, two, three, four, and five, and so there is B. C tells me to go four units to the right. One, two, three, and four, and then half a unit down. Not really to scale but that's all right. And remember, these instructions can really be any real numbers, so we could've gone 4.1, 4.2, and so on. Here we want (4, -5), and D, walk 5 units to the left, 2 3 4 5, and about 5 units down, down to about here. And there's D which is equal to (-5, -5). And that's the entire idea of how to plot points in a plane. One thing I'll sort of apologize in advance for, although it's not my fault, it's really mathematicians' fault, is you'll notice this symbol here looks terrifyingly like the symbol for an open interval. It's not, when we're in the context of plotting points on the plane, this just means the x-coordinate is -5, the y-coordinate is -5. Sometimes in other textbooks, you might see this instead, -5, -5, or A = (2,3), that means just the same thing. Okay, fine, Certain parts of this plane are really important and we distinguish those. Here's the x-y plane again, here's x and here's y, again, here's the origin. I mentioned the x-axis before, let's formally define that. The x-axis is going to be the set of all points x-y in the Cartesian plane, x-y in R2, such that their y coordinate is zero. Which makes sense, this is the x axis. This right here, as you might expect, is the y axis. This is the set of all points (x, y) in the Cartesian plane such that x = 0. So the instruction that starts off by telling you from the origin to go right or left, tells you to go zero, right or left, and then up. So for example, one of the points on the y-axis might be (0,1). One of the points on the x-axis might be over here, (-5,0). And I think you get the idea. Okay, if we remove the axes, then you notice we divide the Cartesian plane into four separate regions, and these we call quadrants. So the first quadrant, Consists of, Points (x,y) such that the x coordinate's positive and the y coordinate's positive. The second quadrant over here, as you might expect, As a set of points (x,y) in the Cartesian plane such that, let's see. The x coordinate's going to be negative and the y coordinate's going to be positive, and I'll leave it to you to figure out the definitions of the third quadrant. And the fourth quadrant. Okay, who cares about plotting points on the plane? Well, let's give a real world example. So over here, let's draw our plane again. A way of plotting a table of data, where each object or person corresponds to two different numbers. And you want to visually represent the relationship between those two different numbers. So suppose we have three people, and we measure their height in cm, and their weight in Kg. And want to call the three people, unimaginatively, A, B, and C. Suppose A has the average height and weight for an American male, which we've looked up on the internet is 177 cm and 88.8 kg. If we plot A over here, so we have to figure out a scale. But if we go all the way over here and say that's about 177, and up a little less than that over here, and there's about A, A = (177, 88.3). Now suppose B is the average height and weight of the American woman, so 164 centimeters, currently, this is 2016 in case we're looking at this later, and the kilograms is 74.7 kilograms. So you'll notice the average American woman is both shorter and less heavy than the average American male. Therefore, this point B is going to be down this way, to the southwest of that person, so about here, and about here, and there's B. And visually we see that relationship, the fact that you have to go left and down means that the average American woman is both shorter and lighter. Now notice if we move anywhere along this line, what are we doing? If we take any other person who sits on this line, that person will be the same average height. If we move up, that person will get heavier. If we move down, that person will get lighter. So by the way, a little bit of personal information, I'm about here. You can probably figure out where to put yourself on this. Same wise as we move from the average American female, and we move say along this horizontal line, what would we be doing? Any person along that line would have the same weight as the average American female. If we move left, it should be getting shorter. If we move right, it should be getting taller. And that's the basic idea. Notice, by the way, a difference between abstract mathematics and real data. In this particular case, only the first quadrant really makes any sense, no one is negative weight and negative height.