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Proof: there's an irrational number between any two rational numbers

Sal proves that when given any two rational numbers, no matter how close, we can find an irrational number that lies between them. Created by Sal Khan.
Video transcript
What I want to do in this video is prove that between any two rational numbers-- so let's say that's a rational number there, and then let's say that this is another rational number that is larger than this one right over here-- that between any two rational numbers, you can find an irrational number. So that number right over there is irrational. You can find at least one irrational number. And that's kind of crazy, because there's a lot of rational numbers. There's an infinite number of rational numbers. So we're saying between any two of those rational numbers, you can always find an irrational number. And we're going to start thinking about it by just thinking about the interval between 0 and 1. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. In fact, one of them that might pop out at you is 1 over the square root of 2, which is the same thing as the square root of 2 over 2, is equal-- I shouldn't say equal, is roughly, is approximately equal to 0.70710678118. And I could just keep going on and on and on and on and on and on. This thing does not repeat. But the important point is, it's clearly between 0 and 1. So I could write 1 over the square root of 2 is clearly between 0 and 1. So the way that I'm going to prove that there's an irrational number between any two rational numbers is I'm going to start with this set of inequalities, and I'm going to manipulate it so I end up with an r1 over here and an r2 over here. And then from 1 over the square root of 2, I would have manipulated this to construct that irrational-- at least one of the irrational numbers that's between those two rational ones. So instead of making this an interval between 0 and 1, let's make this an interval between 0 and the difference between these two numbers. So the distance between r1 and r2 is r2 minus r1. So let's multiply both sides of this-- or all three parts of this inequality, I guess I could say, by r2 times r2 minus r1. So let's do that. So if you multiply this, 0 times r2 minus r1, well you're just still going to have 0 there, is less than-- And we know that r2 is greater than r1, so r2 minus-- let me make it clear what we're doing. We're going to multiply everything times r2 minus r1. r2, we're assuming, is greater than r1, so this thing right over here is going to be greater than 0. So if you multiply the different sides of an inequality by something greater than 0, you don't switch the inequality. So 0 times that is 0, 1 over the square root of 2 times that is going to be 1 over the square root of 2 times r2 minus r1. And then that's going to be less than-- well, 1 times that is just going to be r2 minus r1. And now, we just have to kind of shift everything over. So let's add r1 to all sides of this. So if we add something to all parts of the inequality, then that's also not going to change the inequality. So we're going to add r1 over here. We can add r1 over here. And we can add r1 over there. And so on the left-hand side, we have r1 is less than r1 plus-- let me just copy and paste all of this so I don't have to keep changing colors. Whoops, that's not what I wanted to do. Let me do this. There you go. All right. That should be pretty good. So copy and paste that. r1 plus this, plus that-- let me write the plus down-- plus that, is less than-- that one is a different shade of blue-- is less than-- well, what's r1 plus r2 minus r1? Well, that's just going to be r2. So I've just shown you that you give me any two rational numbers, and I'm assuming r2 is greater than r1, I have just constructed an irrational number that's going to be between those two rational numbers. You take r1, you take the lower of the rational numbers, and to that you add 1 over square root of 2 times the difference between those two rational numbers, and you are going to get this right over here is an irrational number. You're saying hey, how do I know that this thing-- how can I be satisfied that this thing is irrational? Well, we've already seen. You take the product of an irrational and a rational, you get an irrational number. You take the sum of an irrational number and a rational number, you get an irrational number. So we've constructed an irrational number that's between these two rationals.