 Let's talk a little bit about what we might call naive number theory, and the rough definition of what we're talking about is following. If I derive number theory results using only algebra, we might say that we're working in algebraic number theory. The problem is the mathematicians are really bad at making up new names for things, and so this term, algebraic number theory, has actually been reserved, taken over by a very advanced branch of number theory that has very little to do with what most of us consider to be algebra. Instead, we'll define this type of number theory where we derive the results using only algebra. It's frequently called naive number theory, and we'll take a look at an example of something very simple here, and again a rationale for why you'd even bother to prove anything. So here's a fairly standard observation we might make. The product of two even numbers is an even number. Now, before we actually find a proof for this statement, there really isn't any point in trying to prove something that isn't actually true, and it's worth at least checking out a couple of examples and convincing yourself that this is in fact a true statement. So I want to find two even numbers and see if their product is actually an even number. So for example 4 and 6, if I multiply them together I get 24, and that's an even number, and if I'm a politician or somebody who talks about politics on television I might be satisfied by generalizing from one example, but let's try to do a little bit better than political commentators. So I'll take a couple more examples, 8 times 10 is 80, 6 times 2 is 12, and three examples seems to be good enough to convince me that it's worth making the effort. That's the real value of this. It's not that you're really going to convince anybody else of this, or you might, but it's that this is enough to convince you that it's worth putting in the effort to prove this statement. Now even though this is a fairly common way of stating a result in mathematics, if I'm going to prove it it's actually very convenient to rewrite the statement to be proven as a conditional, as a statement of the form. If something, then something. If a certain thing happens then a certain other thing also takes place. And to figure out what that conditional is, well I note that I'm starting with the product of two even numbers. What I have is I have two even numbers and I'm finding their product and then because that is the thing I'm starting with, this should be the antecedent. This is the part of the conditional that follows the if portion of the statement. Now the next thing to notice is that we end with an even number. We start with two even numbers and their product and then we end with an even number and so this statement should be our consequent. This should be the thing that follows the then portion of the statement. And so our conditional that corresponds to this statement should be something like if I have two numbers that are even then their product is going to be an even number. And the important thing is once I have a conditional statement to prove, in any proof of any conditional, I can always assume we have the antecedent. We could always assume that we have two numbers that are even. So I'll start up with two numbers that are even, P and Q be even numbers. And again part of the value of proof is it reminds us of things that we've learned and in this particular case, well I have two even numbers. Well what does that mean? Well we might remember that an even number is a product of two with some other integer. So I know that P is two times something, Q is two times something. And at this point, if you don't see how it's going to proceed, a useful strategy is to work backwards from the end. Now it's very important because there's frequently when students do a proof of this type, they make a major mistake in setting up the proof. Here's the key thing to remember. We always read a proof from start to finish. The last thing that you write in the proof is what you have proven, which means that if you want to prove this statement, if you want to prove this conditional, the last thing that you write has to be the product is even. The consequent has to be the very end of the proof. And so we write that down. This is where we want to go, but then we leave a bunch of space here because we're going to fill up this gap. Nothing we write after this is going to be relevant to our proof of this statement. In fact, if we do write something after this, what we will have proven is not what we want to prove, but it'll be something else. So here's why the definitions are extremely useful. They work in both directions. I can use the definition going forward. P and Q is even. So P is two times something, Q is two times something. But I can also use the definition going backwards. If P, Q is even, then I know P, Q is the product of two and something else. And that takes me a step backwards. And again, I can work backwards using the definitions. And you can think about this as building a bridge. Here's where I'm starting. Here's where I'm ending. And I want to bridge the gap between these two. And the first thing I might notice is that where I'm ending has this product P times Q. Well, let's go ahead and find that product. So P, Q is 2K times 2M. And one important thing to check, because we're dealing with the integers, all of our standard rules of algebra apply. Now we have to be a little bit careful with that because that is not generally going to be true. We can only apply algebra when we're dealing with the integers or a little bit more generally, the real numbers or complex numbers. But if we're dealing with something besides the real and complex numbers, you have to be very, very, very, very careful about whether we can apply the standard rules of algebra. In this case, we are working with the integers. We do get to use the rules of algebra. So this product 2K times 2M, 4 times KM. And I want to bridge the gap between these two. Well, I can do that. 4 times KM is 2 times 2KM. And now I have my statement PQ is 2 times something. And then my conclusion, PQ is even, is going to follow. Now one last useful thing to do is to verify that what you've actually proven is what you want to prove. And that's going to be as follows. What I've proven is the conditional if starting point, then last line. So what I've proven, if P and Q are even starting point, then PQ is even. There's my last statement. And as we've noted elsewhere, one of the bonuses of proof is that we find things that we might have missed otherwise. And again, in this particular case, if I take a look at this statement about midway through here, PQ equals 4 times KM, what I can conclude from there is that I can actually get a stronger conclusion with a shorter proof. If I just drop these last two lines here and focus on this line here, I have PQ equals 4 times something. And what that tells me is PQ is not just even, but it is in fact a multiple of 4. Now, strictly speaking, the problem is to prove that the product of two even numbers isn't even number. So the solution to the problem at hand is our proof here. But at the same time, it's always worth noting that if you've done all this work, you might as well take advantage of it and note that even though we proved product of two even numbers isn't even number, what we actually learned is that the product is a multiple of 4.