 Let's take a look at what is perhaps one of the most misunderstood concepts in mathematical modeling, which is the number line. And this is a way of representing the integers. And one problem is that while it can be a very useful model, it's very, very, very easy to misunderstand if you don't know what it actually represents. So the number line itself is based on the following ideas. First of all, any cardinality can be represented by a set of units. This goes back to the idea that when we talk about cardinality, when we talk about the number of objects, we refer to some particular property of a set. So the idea is that any cardinality we can think of as corresponding to a set of units. And again, the set itself, the elements of the set, don't have an order. But we can imagine our own innate sense of orderliness, we can imagine that we might want to put the units down in some sort of orderly and neat arrangement. And when we use the number line in practice, what we do is we take our set of units, we put them down in some sort of nice order, and then we identify some place, our origin, as our starting point, and then we put down our set of units. Well, let's take a look at that. So let's draw a number line representation for three. So to draw our number line representation, we need to specify some sort of origin. So I'll put a mark down. I'll say, here's where I want to start. I want to start there. And I want to select what I'm going to use as a unit. And so maybe I'll use a little bar there as my unit. And now I want to form a set with my cardinality that I'm trying to form. So here I want a set with cardinality three. So I'm going to form a set that has cardinality three, and it may look something like that. So again, the idea here is that the set elements don't really have an ordering to them, and I can put them any way that I want to. And in this particular case, I want to arrange them in this nice, linear fashion there. So there's my set with cardinality three. And the last thing I'm going to do, next the last thing I'm going to do, is I'm going to take my set, and I'm going to put it down so that it starts at the origin. Now because we read from left to right, we might think about this as our starting point. So I want to put the starting point right at the origin. So I'm going to move that set there through the magic of CGI animation. There it is. And there is my number line representation of three. And to make life a little bit easier, what I should do is I should mark that end point and indicate that this end point here represents the number three. And here's where the confusion over the meaning of the number line arises. The important thing to realize here is that the mark indicates the end of the number. It indicates the end point of the set whose cardinality is three. Mark is not the number. When we talk about the point on the number line that corresponds to a particular set, we might say, well, here is three, and that is a lie. We are lying when we say here, right here, is three. What we really mean is, here is the end of the set that represents three. Three is not this point, three is this entire set. And that's the thing to keep in mind when we're working with the number line. Well, let's take a different example. Let's draw a number line representation for the number zero. So as before, we want to decide on where our origin is going to be. So I'll throw down a mark there, figure out what our units look like. So there's my unit. And now I want to form a set with a desired cardinality. And again, through the magic of CGI animation, we'll produce that set. There it is, right here. You see that set? Not this. That's not part of the set. The set is right here. There is our set with cardinality zero. Again, through the magic of CGI animation, I'll move this set with cardinality zero to the origin. There it is. And again, I want to make one final change, one final notation here. I'm going to mark the endpoint of the set with the number. So see my set here? There it is. Begins here, ends at the same place. Now I want to mark the endpoint with the number zero. And so there's my number line representation for zero.