 Anyone can become a mathematician if they work hard and choose the right parents. No, wait, that's how you become rich. Gauss' advice for becoming a mathematician is to prove the same thing many times in different ways. So let's consider a useful property of the real numbers is that if A, B is equal to 0, either A is 0 or B is 0. Now, since the complex numbers form a field, the zero-product property also holds for the complex numbers. Of course, you have to prove the complex numbers form a field, and it helps if you know what a field is. But if you take or have taken a few advanced math courses, it's easy to prove the zero-product property. As an alternative, let's follow Gauss' advice and find other proofs of the zero-product property for complex numbers. We know that if U is a complex number, then 0 times U is U times 0 is equal to 0. We want to show that Uv is not equal to 0 for any v not equal to 0. And this is the same as saying that 0 is the unique complex number whose product with U is 0. And this reformulation is important because it gives us a way forward. To show something is unique, see what happens if it isn't. So let's try our first proof. We'll take U not equal to 0. And suppose Uv is 0 and Uv' is also 0. Remember, you can assume anything you want as long as you make it explicit. Well, since they're both equal to 0, then Uv is Uv' and since we know how to divide complex numbers, we can divide both sides by U to get v equal to v' So there's only one complex number v that makes Uv equal to 0. But we already know v equals 0 makes Uv equal to 0. So if v' also makes Uv' equal to 0, then v' must be 0. So 0 is the unique complex number whose product with U is 0. Even if we ignore Gauss' advice to find other proofs, there's another reason we don't like this proof. We should avoid dividing by complex numbers for, well, reasons. But remember, the conjugate allows us to transform a complex number into a real number and this gives us a variant proof. As before, we'll assume that U is not equal to 0 and Uv and Uv' are both 0. And again, Uv must be Uv'. Now if we multiply by the conjugate, then we get Since the product of a number and its conjugate is guaranteed to be a real number, we'll call it C to get and now we can divide by the real numbers C to get, and the rest of the proof is unchanged. Actually, we'd really prefer to avoid division altogether. Notice that we're able to reduce the problem to the case where a real number is multiplied by a complex number. So let's consider that. Suppose a times C plus id is 0 where a is a real number not equal to 0. We can apply our complex arithmetic and in order for this to be 0, the real and imaginary parts have to be 0, so we know that ac must be 0. So a is 0 or c is 0. But since we assumed a is not equal to 0, this means c must be 0. Similarly, ad must be 0 and so d is 0. And this proves the theorem that if Uv is 0 with U a real number not equal to 0 and v complex, then v must be 0. And this actually allows us to shortcut our proof. Suppose U is not equal to 0 and Uv equals 0. If we multiply by the conjugate, again, the product of a number and its conjugate is a real number, and in this case that real number is not equal to 0, and since we have a real number times something equal to 0, that says our something must be 0. Now I've been trying to keep these videos under five minutes, and so you can stop here. Hello? Ah, you're still here. OK, well, let's talk a little bit more about this. The proceeding shows one of the reasons why Gauss's advice works. Proof requires you to review what you know about mathematics. In this case, our proof relied on the properties of the conjugate. But in general, proof often requires you to review things you should know. And what this means is that finding new proofs is a way to study mathematics. OK, I'm done. You can go.