 One of the central ideas in classical number theory is what's called a figurant number. And this idea goes back to the Greeks, for whom number was a collection of units. What's a unit? Well, a unit is any object, and we can have a collection of objects. But at some point, we'll have so many objects that we'll want to find some way of classifying and organizing these objects. And one thing we might do, because we are human, and we have a liking for things that look neat, is to classify our collections of objects by how we can arrange the items in the collection. So one of the possibilities is we might take our collection of units and arrange them in, how about a straight line? But the problem is this can be done with every collection of units, and so something we can do with everything is not a good way of classifying things. However, a good habit to get into as a mathematician and as a human being is to ask yourself, well, even though this might not be useful, does it tell us something useful? And for that, let's consider this. So here we have our jumbled collection of units, which is a number. And then we took these units and arranged them in a straight line and we formed a number. And these are two numbers, but they're the same number. And so the observation that we might make here is that a number doesn't change based on how it's presented. And so this suggests that one of the things we might do is to see how we can arrange a collection of units. Now there's not a whole lot we can do with one unit, and not a lot more that we can do with two units. But if I have three units, I can arrange them as a triangle, or maybe a slightly different triangle, or maybe yet another triangle. And now here's an important thing. I've arranged three triangles, this one, this one, and this one. And the question is, are they different or are they the same? And the first triangle has sides of two, two, and two. The second triangle also has sides of two, two, and two. And so does the third. And so one of the things we know from geometry is that once you specify the length of three sides of a triangle, you have uniquely defined the triangle. So in the geometric sense, all three of these triangles are the same triangle. Well, so that means that we can focus on the length of the sides of the triangles rather than the actual shapes of the triangles. So let's take a look at that. So here's a couple of triangles. So I'll take a triangle with sides two, two, and five. I'll take another triangle with lengths three, four, and five. And I'll take another triangle with sides of two, two, and two. And one of the things you might start to suspect is that I can take any number and form a triangle. Well, again, once we try to classify things, having a classification that applies to everything is not useful. If I can take any number and make a triangle, then making a triangle is not something interesting. So what we have to do is we have to limit what our possibilities are. And that allows us to expand our possibilities. Well, here's a nice limitation that we might impose. These triangles have different side lengths. So let's make a requirement that our triangles, whatever we form, have to have the same number of units along each side. And if we can do that, then we form the first few triangular numbers. So first of all, I might try to form a triangle where each side has one unit along a side. Not entirely sure how to produce that, so maybe we'll wait a moment on that. I'll take a triangle where each side has length two. So here, length two, length two, length two. And then I'll form my next triangle where each side has length three. So here is three, three, and three. And there is a pattern here that suggests our first triangular number looks like this. Now you might object that that doesn't really look like much of a triangle. And so you can take two viewpoints. You can either look at this and say, sure, if I try to find the length of all three sides of this thing, every side has length one, so this is a triangle where every side has length one. The other thing you might do is you might look at this as the extension of a pattern. I have a triangle of side three. I have a triangle of side two. The extension down to a triangle of side one has to look something like this. Well, the downward extension is a little easy. The upward extension starts to get a little bit more complicated. And the problem is that there's two possibilities for the triangle with four units along each side. So I might draw my triangular number like this. So this might be a triangle where each side has four units. But you'll notice that if you do this, there is this space in the middle that doesn't have a thing in it. So maybe another way of drawing a triangle with four units along each side looks like this. Again, four units on each side, but then something fills into the middle. And so here's the important question. I want to define what triangular numbers are, but I have to decide which of these two things will be the fourth triangular number. And then once we decide that, we will have established a rule that tells us what the next triangular numbers are going to be. And to some extent, mathematics is the science of patterns. And so the idea that we might want to look at is, can I consider the triangular numbers as part of a larger pattern? To answer that question, we'll want to do a little bit of analysis of numbers. So one possibility for our triangular numbers are possibility A. I have the first four triangular numbers 1, 3, 6, and 1, 2, 3, 4, 5, 6, 7, 8, 9. So there's my first four triangular numbers if we use this as our fourth triangular numbers. And we have a second possibility I might use instead the other possibility B. And my triangular numbers are going to be 1, 3, 6, and 10. Now here's an important thing to recognize. Both of these sequences start out the same way. 1, 3, 6, 1, 3, 6. They both start the same way, but how they continue is very different. And as a general rule, if I only give you the first few terms of a sequence, I can't insist that you come to a particular conclusion regarding what the next term is. It's possible for any sequence to be continued in an infinite number of ways. Ideally, we'd like that sequence to be continued in what is a quote-unquote logical manner. But what is logical to one person might not be logical to the other. And the best we can do is to construct a sequence in a rational manner, which is to say I can give a reason for choosing the next term the way that I did. So let's consider that. In our first sequence, our numbers were 1, 3, 6, and 9. And I can say that every number in this sequence, well, except for the first number, is a multiple of 3. And that, except for the first, is a little bit troublesome. Well, let's take a look at our second sequence. Our second sequence is 1, 3, 6, and 10. And here, the pattern is a little bit less obvious, but if I dig down deeper, I can say that every number, including the first, is the sum of the first few whole numbers. So 1 is equal to 1. 3 is 1 plus 2. 6 is 1 plus 2 plus 3. 10 is 1 plus 2 plus 3 plus 4. And it seems that this is a better sequence because there's no exceptions. And there's a clearly defined rule for how to obtain the terms of the sequence. And so we like this rule. And so we're going to use this to define our triangular numbers. The sum of the first and natural numbers forms the nth triangular number.