Lesson 3: Rational and irrational numbers
So let's talk a little bit about rational numbers. And the simple way to think about it is any number that can be represented as the ratio of two integers is a rational number. So for example, any integer is a rational number. 1 can be represented as 1/1 or as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are all different representations of the number 1, ratio of two integers. And I obviously can have an infinite number of representations of 1 in this way, the same number over the same number. The number negative 7 could be represented as negative 7/1, or 7 over negative 1, or negative 14 over positive 2. And I could go on, and on, and on, and on. So negative 7 is definitely a rational number. It can be represented as the ratio of two integers. But what about things that are not integers? For example, let us imagine-- oh, I don't know-- 3.75. How can we represent that as the ratio of two integers? Well, 3.75, you could rewrite that as 375/100, which is the same thing as 750/200. Or you could say, hey, 3.75 is the same thing as 3 and 3/4-- so let me write it here-- which is the same thing as-- that's 15/4. 4 times 3 is 12, plus 3 is 15, so you could write this. This is the same thing as 15/4. Or we could write this as negative 30 over negative 8. I just multiplied the numerator and the denominator here by negative 2. But just to be clear, this is clearly rational. I'm giving you multiple examples of how this can be represented as the ratio of two integers. Now, what about repeating decimals? Well, let's take maybe the most famous of the repeating decimals. Let's say you have 0.333, just keeps going on and on forever, which we can denote by putting that little bar on top of the 3. This is 0.3 repeating. And we've seen-- and later we'll show how you can convert any repeating decimal as the ratio of two integers-- this is clearly 1/3. Or maybe you've seen things like 0.6 repeating, which is 2/3. And there's many, many, many other examples of this. And we'll see any repeating decimal, not just one digit repeating. Even if it has a million digits repeating, as long as the pattern starts to repeat itself over and over and over again, you can always represent that as the ratio of two integers. So I know what you're probably thinking. Hey, Sal, you've just included a lot. You've included all of the integers. You've included all of finite non-repeating decimals, and you've also included repeating decimals. What is left? Are there any numbers that are not rational? And you're probably guessing that there are, otherwise people wouldn't have taken the trouble of trying to label these as rational. And it turns out-- as you can imagine-- that actually some of the most famous numbers in all of mathematics are not rational. And we call these numbers irrational numbers. And I've listed there just a few of the most noteworthy examples. Pi-- the ratio of the circumference to the diameter of a circle-- is an irrational number. It never terminates. It goes on and on and on forever, and it never repeats. e, same thing-- never terminates, never repeats. It comes out of continuously compounding interest. It comes out of complex analysis. e shows up all over the place. Square root of 2, irrational number. Phi, the golden ratio, irrational number. So these things that really just pop out of nature, many of these numbers are irrational. Now, you might say, OK, are these irrational? These are just these special kind of numbers. But maybe most numbers are rational, and Sal's just picked out some special cases here. But the important thing to realize is they do seem exotic, and they are exotic in certain ways. But they aren't uncommon. It actually turns out that there is always an irrational number between any two rational numbers. Well, we could go on and on. There's actually an infinite number. But there's at least one, so that gives you an idea that you can't really say that there are fewer irrational numbers than rational numbers. And in a future video, we'll prove that you give me two rational numbers-- rational 1, rational 2-- there's going to be at least one irrational number between those, which is a neat result, because irrational numbers seem to be exotic. Another way to think about it-- I took the square root of 2, but you take the square root of any non-perfect square, you're going to end up with an irrational number. You take the sum of an irrational and a rational number-- and we'll see this later on. We'll prove it to ourselves. The sum of an irrational and a rational is going to be irrational. The product of an irrational and a rational is going to be irrational. So there's a lot, a lot, a lot of irrational numbers out there.