 Let's continue our discussion of the number line by considering how we would represent negative integers. And we can represent these negative integers using directed arrows, which somewhat later in your education you'll learn about as vectors. And so the idea is that I'm going to start off with the basic definition of what the additive inverse is, what these negative integers correspond to. If I take any integer and add its additive inverse, when I put these two things together, remember addition is the corresponds to the union of two sets. When I put these two things together, what I get is zero. So here's a way that I might approach the problem. Let's suppose that I'm going to represent one by an arrow pointing to the right. And remember, when we form a number line representation of a number, we decide what our unit is. And so rather than having blocks, maybe this time I want my unit to be a arrow pointing to the right. And so if I want to represent the number, for example, three, well that's going to be three arrows pointing to the right. And I will put them down in some sort of orderly fashion. But until we get then, let's think about this a little bit more. If I have this arrow pointing to the right, I might think about this as not only as a unit, but as a direction. And the direction says, move to the right, one space. And if that's going to be the case, well then, what do I want to use for negative one? Well, my negative units I can think about as a different direction, a left pointing arrow, which I can think about as move to the left, one space. And so, well, why would we do something like this? Why does this make for a good model? Well, the reason it makes for a good model is the following. Consider the sum one plus negative one. So again, by our set definition of addition, what this means is I want to put together two sets. I want to put together the set representing one with the set representing negative one. So I'm going to take my right pointing arrow that represents one. I'm going to take my left pointing arrow that represents negative one. And here is my set representation of this quantity, one plus negative one. Now, this is a proper set. It consists of a couple of things. It consists of some number of elements. And what I'm going to do is I'm going to say, well, here's my set representation of one plus negative one. Now, I can then choose to interpret this as follows. Again, I want to think about this right pointing arrow as the direction take one step to the right. And I want to think about this left pointing arrow as the direction take one step to the left. And if I think about the set of all of these things put together, if I take a step to the right and take a step to the left, we're doing the time warp, old reference, but if I take a step to the right and a step to the left, I haven't moved. I haven't gone any place. And it's as if I've moved zero, which is what we want one plus negative one to be. Well, let's see if we can do that. So I'm going to represent negative three on the number line and I want to use a directed arrow as my unit. So since negative three is negative, I want to use the left pointing arrow as my unit. So there's my unit, there's my origin. And what do I want? Well, I want three of these left pointing arrows. So if this is negative one, this is negative one, negative two, negative three. And I put them together in a nice order. And the last thing I want to do is I want to put the beginning of the number at the origin. So I'm going to put the begin, wait a minute. If that's a left pointing arrow, I look at this and I don't really think about this as the beginning of the set. Left pointing arrows tend to draw us towards this as the beginning and go left, go left, go left. And this is actually the end of the set representing negative three. I want to make sure that the beginning of the set is at the origin. So that's the wrong place to put it. I should actually put it over here. And so here's my representation, here's my origin, here's my one, two, three negative units. And lastly, I should mark that endpoint to indicate that this is the end of the set corresponding to the value negative three.