 Now, there are several properties of addition that we're used to. For example, we are used to having associativity and commutativity. We should prove these. And while that makes for a good research plan, the problem does arise how? And importantly, in what order? And since we're genetically programmed from birth to know exactly how to proceed in any question in mathematics, it's easy to...what? Oh, wait, sorry, wrong speech. We don't know what order and how we're going to prove this, but the important thing is take the first step and see where it takes you. So let's try to prove, I don't know, commutativity seems useful. So let's try to prove A plus B is equal to B plus A. And here's a useful idea to keep in mind. Try to prove something simple. And proving A plus B is equal to B plus A, well, that actually turns out to be too hard. So, well, let's see. Our definition of addition tells us what happens when we add 0. So let's try to prove A plus 0 is equal to 0 plus A. And that also turns out to be a little too hard. So let's try something easy. Let's try to prove that 1 plus 0 is equal to 0 plus 1. This is a specific case. And as we saw, it might be possible to generalize from the proof of a single instance. So let's try to prove 1 plus 0 is equal to 0 plus 1. So again, you can always write down one side of inequality. So let's write down 1 plus 0 equals. Definitions are the whole of mathematics. All else is commentary. So let's pull in our definition of addition. And based on our definition, we know what 1 plus 0 is equal to. And we'd like to say that 0 plus 1 is equal to 1, but we can't. Our definition of addition doesn't tell us what happens if 0 is the first sum and. Well, couldn't we just use commutativity of addition? Well, that's the problem. We don't yet have commutativity of addition. So instead, we have to proceed differently. 0 plus 1, well, that's the same as 0 plus the successor of 0. Or the successor of 0 plus 0. We do know what happens when we add 0. So that's the successor of 0 or 1. And since 1 plus 0 is equal to 1 and 0 plus 1 is equal to 1, we can conclude that 1 plus 0 is equal to 0 plus 1. And here's a useful strategy when studying mathematics, especially proofs. Prove it again. Proof is a study technique. It makes you recall what you should already know. So our proof can start the same way. 1 plus 0 is equal to 1, which follows directly from the definition of addition. And we wanted to find what 0 plus 1 was. Well, remember, we already showed that n plus 1 is the successor of n. So we know that 0 plus 1 is the successor of 0, which is 1, and the rest of the proof goes the same way. And so we've shown that 1 plus 0 is equal to 0 plus 1. From this, it's easy to prove that a plus b is equal to b plus a eventually. But let's take a smaller step. Let's try to prove that 0 commutes. n plus 0 is equal to 0 plus n. So let's think about this. If we want to prove this, this is really the same as proving 1 plus 0 is equal to 0 plus 1, and also 2 plus 0 is equal to 0 plus 2, and also 3 plus 0 is equal to 0 plus 3, and 4 plus 0 is equal to 0 plus 4, and so on. And here's a useful idea to keep in mind. Any time you're trying to prove an infinite but orderly sequence of statements, consider induction. So remember, a proof by induction requires two things. First, we have to show that our statement holds for the first thing. So we have to prove that 1 plus 0 is equal to 0 plus 1. That's the base step. And we just did that. The second thing we have to show for a proof by induction is that if our statement is true for n, then it's also true for the successor of n. So remember, since we're trying to prove a conditional, we can always assume the antecedent. So suppose n plus 0 is equal to 0 plus n. So let's consider the successor of n plus 0 and 0 plus the successor of n. Definitions are the whole of mathematics. All else is commentary. Our definition of addition gives us, well, if I add 0, I get the same thing I started with. So that's n star. Our definition of addition also tells us what happens when we find 0 plus n star. It's the successor of 0 plus n. But remember, we assumed n plus 0 is the same as 0 plus n. So this term inside the parentheses, well, that's really n plus 0 successor. n plus 0 is just n. And so this is the successor of n. And so n star plus 0 and 0 plus n star are the same thing. And so we have our conclusion n star plus 0 is the same as 0 plus n star. And let's summarize what we have. Since 1 plus 0 is equal to 0 plus 1, then we know that this is state and it's going to be true for the successor of 1. So we know that 2 plus 0 is equal to 0 plus 2. But wait, since 2 plus 0 is equal to 0 plus 2, then the statement also holds for the successor of 2. So we know that 3 plus 0 is equal to 0 plus 3. But since our statement is true for 3, it's true for the successor of 3 and so on. And so all of our dominoes fall over and we get the theorem for any natural number n plus 0 is equal to 0 plus n.