 Welcome back to our fourth and last example here of a proof by contradiction. It's a proof that the number square root of 2 is an irrational number. Now this is a classic proof that every math student should know by heart. So let's prove that the square root of 2 is irrational. Couple things before we begin. First, note that this is another instance of a general rule that we mentioned in the first video, namely that proof by contradiction is often useful when proving statements that are given in the negative. This is one such statement because when we say that square root of 2 is irrational, what we're doing is saying that it is not rational, that it does not belong to the set of rational numbers. The other thing before we begin is related to that and that is we need to have a very clear idea before we set out of what a rational number is before we try to prove that the square root of 2 is not one of them. Recall the definition and keep it very close to heart here. Our rational number is any number that can be written as a fraction of two integers, a and b, with b in the denominator not equal to 0. And in addition, we can assume that a and b have no common factors. That's because if a and b had common factors, we could just divide them out. So again, a and b have no common factors, so the fraction a over b is in lowest form. So with that in hand, let's prove this by contradiction. So we begin as always by assuming the negation of the statement we want to prove. If we want to prove that square root of 2 is irrational, then the negation would be to assume that the square root of 2 is rational. So do that. Now, what does this mean? By definition, it means that there exist integers a and b with b not 0. And again, very importantly, a and b have no common factors, so that the square root of 2 equals a divided by b. Now, as with the last video, this would be a good point to pause and answer the question, what next? What's the next step in the proof going to be? So pause the video and come back when you're ready. Now, welcome back. I think a good next step would be to square both sides, to rid ourselves of the square root. You might also clear the fractions out first by cross multiplying and then maybe then square in the next step. And that will get us to the same place we're headed now. So square both sides to get 2 equals a squared divided by b squared. Now, everything is an integer. I'll clear the fractions out to get 2b squared equals a squared. Now, what does this mean? Well, among the things that it means is that a squared is an even integer. You see here that I have a squared equal to 2 times an integer and that makes a squared even. Now, think all the way back to screencast 3.2.1. In that screencast, we proved by contraposition that if a squared is even, then a is even. So now we can pull that in and conclude that a is even. Now, so far all of these are valid mathematical and factual steps merely doing correct calculations using previously known results and agreed upon definitions. Now, since a is even, I can write it as 2 times some integer k. So let's do that and now loop back to the second line and substitute in 2k for a. That will give you 2b squared equals 4k squared. And if you divide both sides of that by 2, it gives you b squared equals 2k squared. Now, that implies that b squared is even because I've written it as 2 times an integer. Using the result from screencast 3.2.1, I can now say that b is even. But wait, this is a, wait for it, it's a contradiction. Now, if it seems like this contradiction comes out of nowhere, I don't blame you. What exactly did we contradict? We have to think back to all the assumptions that we made about the objects that we're working with. What we've contradicted is a prior assumption that a and b had no common factors. We did assume this. But in assuming it, I've arrived at the fact that a and b are both even. And if they are even, they have a common factor of two. So, which is it? Do a and b have common factors or not? It can't be both. And yet our work says both. This contradiction forces us to reject the assumption that led us here in the first place. And that assumption is that the square root of two is a rational number. Since square root of two can't be rational, because it leads us to this contradiction, it must be irrational. And that is the end of the proof. With a little tweaking and some help from the proof techniques coming up in the next section, we can actually prove that the square root of any prime number is irrational. And that also uses contradiction. Again, contradiction is very powerful, very useful, and I really hope you enjoy using it.