 So it's useful to be able to switch back and forth between algebra, which is about formulas, equations, numbers, and geometry, which is about drawings and pictures. And one of the important transition points occur with what are known as coordinate systems. And this is based around the following problem. Where in the world is? Well, really everybody. We can specify the location of a point in the plane using a rectangular coordinate system. First, let's get a better vantage point. Since we're trying to specify the location of points in the plane, it might be better if we could look down on the plane. So let's take an aerial view. Now we need to specify a starting point, the origin. And since I'm the most important person in the universe, we'll make the origin at my location. Next, we need to specify two principal directions. And these have the very scientific name of this way and that way. Actually, it's convenient if our principal directions are perpendicular to each other. So we'll rotate that way so that it is perpendicular to this way. We can then locate any point by giving directions to the point with reference to our principal axes. So let's say I want to get to a friend who's located over here. I can describe their location by saying how far I have to go in the direction of one principal axis, followed by how far I have to go in the direction of the other principal axis. It's important to understand that the principal axes have to be drawn somewhere, but they actually represent directions, so they're not really located in any particular place. So since I have to go in the direction of the principal axis, I must put a copy of that principal axis where I am, and then walk in that direction until I meet my friend. The importance of specifying these directions is that if anybody else gets to the origin, they can meet us by following the same directions. We'll formalize this with the notion of rectangular coordinates. The rectangular coordinates of a point, h, k, verify that the point is located h units to the right and k units vertically from the origin. The first coordinate, h, is called the x-coordinate, also known as the horizontal coordinate, and the second coordinate, k, is called the y-coordinate, also the vertical coordinate. And because the order matters, the first coordinate is the horizontal distance, and the second is the vertical distance, then we call this an ordered pair. So for example, let's locate the points 3, 5, and 5, 3. So definitions are the whole of mathematics. All else is commentary, so let's pull in our definition of rectangular coordinates. And this says that the first point 3, 5 were 3 units to the right and 5 units vertically from the origin. So to graph this point, first we need to identify where the origin is. How about here? It's also convenient to mark out the principal directions, which would be to the right and upward. So for our first point, we'll start at the origin, go 3 units to the right, and then 5 units upward. For our second point, we'll start at the origin, then we'll go 5 units to the right and 3 units upward. We can use these coordinates to express our location if we're to the right and above the origin, but what if we're left or below the origin? To account for these possibilities, we use the following. If the x-coordinate is negative, it specifies the corresponding distance to the left, and if the y-coordinate is negative, it specifies the corresponding distance downward. So for example, let's try and find these points. Since some of the coordinates are negative, we'll want to remember what it means to have a negative x or y-coordinate. So our first point has a positive x-coordinate, so that means we're going to go to the right, but since the y-coordinate is negative, that means we're going to go down instead of up. So to find our first point, we start at the origin, go 3 units to the right, and 5 units downward. Now a good habit to get into is always labeling the points that you graph with the coordinates. So we'll put down those coordinates 3-5. For our second point, the horizontal coordinate is negative and the vertical coordinate is positive, so that means we're going to go to the left and then upward. So we'll start at the origin, go left 3 units, then up 5 units, and label the point with our coordinates. Finally for the third point, both coordinates are negative, so that means we're going to go to the left and down. We're going to go left 3 units, then down 5 units, and label this with the coordinates minus 3, minus 5. What if we're given a bunch of points and we want to find the coordinates? So unless we know the size of our vertical and horizontal units, we can't actually answer this question. So let's throw down a grid where the grid spacing is one unit in both directions. So the coordinates represent the directions for getting to a point from the origin. So to get to the point a from the origin, we have to go 2 units to the right, then 5 units upward. Since we've only moved to the right and upward, our coordinates will both be positive, so the coordinates of a will have x equals 2, y equals 5, and we can write that down as the ordered pair 2, 5. To get to the point b from the origin, we'll have to go 1 unit to the left, and then 2 units up. Since we're going to the left, our horizontal coordinate will be negative, so the coordinates of b will have x equals negative 1, y equals 2, and we can record that as negative 1, 2. To get to the point c from the origin, we have to go down 3 units. This means that y is equal to negative 3, but what is x? Since we don't go right or left, we could say that we go right 0 units, so x equals 0, and our point will have coordinates 0, negative 3. Finally, to get to the point d from the origin, we go 4 units to the right, so x equals 4, but what is y? Since we don't go up or down, we could say we go up 0 units, so y is equal to 0, and so our coordinates will be 4, 0.