Welcome to another Worldwide Center of Mathematics video short. Here today we're going to be proving that the square root of 2 is an irrational number. So that means that we're going to be showing that square root 2 cannot be expressed as an integer a over an integer b in a fraction. And so in order to explain that we are going to be using proof by contradiction. Proof by contradiction. So we are going to start out by something very counterintuitive. We are going to assume that the square root of 2 is rational. We're going to say, we're going to suppose that square root of 2 is equal to a fraction integer a over integer b. And so what this means here is that we've got an irreducible fraction. It's really important to note that a, the a and b have no common factors. I suppose to be another m in there. No common factors. This means that for example we've got like the fraction 3 over 6 and 3 and 6 share a common factor of 3 and it can be reduced so this equals 1 over 2. So we don't want anything like 3 over 6 in our fraction a over b. So if we've started assuming that the square root of 2 is equal to a over b, we can square each side. So the square root of 2 squared is simply 2, which now equals a squared over b squared. Now we can toy around with our equation a little bit more. We'll multiply by b squared on both sides. So 2 b squared is equal to a squared. Now let's look at this term a little bit more. We'll pay attention to this. This is implying that there is a quantity b squared that when multiplied by 2 is equal to a squared. So that means that a squared, the quantity a squared can be divided by 2. This implies that a squared. This implies that a squared is even because it can be divided by 2 to get b squared. And so if a squared is even, let's try to figure out whether a is even or not. And so we know that if we have an even number, times an even number, we'll always get another even. And if we have an odd, times another odd, like 5 times 5 for example, we'll always get another odd number. So in this case, if we've got an even number as a result, we know that a must be even. And so if a is even, that means that a could be written as 2 times another number. And we'll just use k for that other number. So a can be expressed as 2 times k. And now we're going to substitute in. We're going to bring back this equation. So we've got 2 times b squared is equal to a squared. But we're going to substitute in 2k for a. Now we're going to use some algebra here. We've got our 2b squared is equal to 4 times k squared. And we will divide each side by 2 to see that b squared is equal to 2 times k squared. So here, this is where you might see our contradiction start to arise. So we've got that b squared is equal to 2k squared. So this implies that there is some quantity k squared that when multiplied by 2 is equal to b squared. So this means that b squared divided by 2 is equal to k squared, which means that b squared is able to be divided by 2. So we've got our b squared, which must be even. And using this same argument as last time, we know that b is even. Here's where our contradiction is. So we started off with the square root of 2 being equal to a over b. And we noted that this must be irreducible. We cannot have any common factors between a and b. But we have gone through our equation. We have identified that a must be an even number. And we've identified that b must be an even number. And that must mean that there is a factor of 2 that is common between them. So we've got our a over b. And those each must be, a must be able to be divided by 2. And b must be able to be divided by 2. And therefore, we have a contradiction. And square root of 2 is not able to be written as a reduced fraction of 1 integer over another integer. And so because we have a contradiction here, we know that our assumption was wrong. We assumed that the square root of 2 was able to be expressed this way. And now that we know that that assumption is false, we know that the square root of 2 must be an irrational number. Thank you for watching this Worldwide Center of Mathematics short on algebra. So be sure to click this button to subscribe to our channel so that you don't miss any new math videos. Click here to visit the playlist for the video that you just watched. And click on this button to visit centerofmath.org for even more math resources.