 So earlier we showed that the real numbers are uncountably infinite. We also have a theorem that says that the rational numbers are countably infinite. First, we'll give the standard proof. Actually, it's kind of boring and you can Google the standard proof yourself. Now as we saw with the integers, if we can prove that the positive rational numbers are countably infinite, then we can also prove that all rational numbers are countably infinite. And so let's try to prove that the positive rational numbers are countably infinite. Remember, one of the ways to study proof and mathematics in general is to find another proof. So let's find a different proof that the rational numbers are countably infinite. So here's one way we might approach this. Since the numerators and denominators of positive rational numbers are natural numbers, let's see if we can generate them from the set of natural numbers. So let's set down our first rational number, zero, once. Shall we play a game? And so let's play this game. If you increase the numerator by one, take a step to the left. And if you increase the denominator by one, take a step to the right. So if we increase our numerator by one to get one once, we'll put that to the left. And if we increase our denominator by one to get zero, twoths, we'll put that to the right. And if we increase our numerator by one to get two over one, or increase our denominator by one to get one half, and notice that could have also been obtained by increasing our numerator of zero halves, and we can also increase our denominator of zero halves to get zero thirds. And we can produce other rational numbers in this way. And this gives us a way to list our rational numbers by working our way through this triangle, starting with zero once, then one once, zero halves. The next row gives us two once, one half, zero thirds. The next row gives us three once, two halves, one third, zero fourths, and so on. And again, because we can list the rational numbers in some sort of order, this means that the set of positive rational numbers is countably infinite. And again, a basic principle in life is always ask, can we do better? And a potentially objectionable feature about this method of listing the positive rational numbers is that it includes all rational numbers even when they might be reduced to simpler forms. So again, let's try another proof. And we might proceed as follows. Let PQ'th be a rational number reduced to lowest terms. And remember, we can write its value in words. Five eighths, twenty three forty ninths, and so on. And once we can write something, we can organize them by length and then alphabetize them within the lengths. Now this does actually require us to be able to spell out these rational numbers. And let's put in one further restriction. Let's list the non-integer positive rational numbers. That makes our spelling challenge a little bit easier. Now, since no number word is shorter than three letters, and we have to specify both the numerator and the denominator, then words for fractions have to have at least six letters. Well, none of them do. But we have seven letter fraction words, or at least we have one, one half. And so that will be our first positive rational number. We have a bunch of eight letter fraction words. And we can put them in alphabetical order. They all start with one. So the second word is going to determine what the order is. And the next one is going to be one fifth, then one fourth, and so on. There are nine letter fraction words. And we note that we're going to omit words like two fourths or six ninths because they aren't reduced. And putting these in alphabetical order, and so on. And again, we can now have a list of our positive reduced rational numbers in order, which means that the set is countably infinite. Now, since the set of rational numbers is countably infinite, but the set of real numbers is uncountably infinite, it follows that there must be some real numbers that are not rational. Now, we already know one irrational number, square root of two, and so the natural question is, how many irrational numbers are there? And in fact, the number of irrational numbers is uncountably infinite and we should be able to prove that. We'll leave that for the viewer. We don't want to do all of your homework. But here's a hint by way of analogy. The set of integers is countably infinite and it consists of a countably infinite set, the positive integers, a second countably infinite set, the negative integers, and a finite set, zero. So when we put together two countably infinite sets, we get a countably infinite set.