 The number line is a very useful way of understanding certain ideas in mathematics. Unfortunately, the number line itself is often misunderstood, so let's take a close look at what it entails. This is centered around a more general idea. It's useful to be able to switch between two viewpoints. Algebra deals with numbers, variables, and formulas, while geometry deals with drawings and pictures. And the significance of this is that some hard problems in algebra become easy problems in geometry, and some hard problems in geometry become easy problems in algebra. So one way to move between the two is through the number line, and this emerges as follows. We choose some point, some place, to be the origin. And an important idea here is that if it's not written down, it didn't happen. Label everything. Once we decide where the origin is, let's label this point as the origin. We'll use O for short. And we draw a line through the origin. Now, while we could draw the line in any direction that we want to, we often draw it horizontally. And any point on the number line can be labeled by its distance from the origin. And this leads to the first important problem of the number line, what label should be assigned to the origin itself? So remember, the number we assign to a point is the distance from the origin. And so the origin itself, we want to know the distance from the origin to the origin. And that distance is zero. And so we should label the origin with the number zero. Remember, the numbers we're assigning correspond to the distance from the origin. But we could be on the right or on the left. And so we'll do the following. Points to the right of the origin will be assigned positive distances. And points to the left of the origin will be assigned negative distances. So let's try to graph eight, negative two, and negative five on the number line. So we'll put down our origin and run a horizontal line through it. So eight corresponds to the point that's eight units to the right of the origin. And so we might count out our spaces, one, two, three, four, five, six, seven, eight. And remember, if it's not written down, it didn't happen. So we'll label this point as eight. Negative two, because it's negative, it's going to be to the left of the origin. And because it's two, it's going to be two units to the left of the origin. So we'll go from the origin, two units to the left, one, two, and label. And then finally, minus five, it's negative. So we're to the left of the origin and it's five. So we're five units to the left of the origin. So one, two, three, four, five. Here we are and label. Now, counting spaces works all right if we have small numbers, but if we have larger numbers, that gets a little bit tedious. So let's draw a number line with our points three and negative 75, and we'll talk about the secret of graphing. So we'll draw our origin and our horizontal line that represents the number line. So the point with label three should be three units away from the origin. And since the number is positive, it should be three units to the right of the origin. And the secret to graphing is graph first, then label. In this case, we know we're someplace off to the right of the origin, so let's graph the point first. And remember, if it's not written down, it didn't happen. So we'll label this point three. And here's the secret to graphing. Because I've written down the three here, that tells me that this distance is three from the origin. What about negative 75? Since negative 75 is negative, we need to be to the left of the origin. And we should be 75 units to the left of the origin. But we're not going to count that out. We're going to say we're going to be over here somewhere. Now it's nice when there's at least some proportionality here. And in this case, we might achieve that as follows. This length here is supposed to be three. So I want to make sure that my length 75 is more than three. So I'm going to drop 75 over, oh, how about here? That way, our length representing 75 is longer than our length representing three. Graph first, then label. So we'll label that.