 Our next major extension of the number system is to the integers. And so we'll introduce two important concepts. One is the idea of the additive identity. And that's a thing that we're going to write as zero. And it's a number where if I add zero, I don't change the value of what the other addent is. So a plus zero is equal to a for all values a. The other idea that's tied into additive identity is this notion of an additive inverse. And so we're going to write that additive inverse of a. We traditionally read this as negative a, although be careful with that because where later I'm going to talk about negative numbers, they're at the same thing. But in general the additive inverse of a, written this way, is a number where if I add the additive inverse to the number what I get at the end of the day is zero. And I will say without going into the proof or details that both the commutative and associative laws hold true for numbers as well as their additive inverse. We can prove this, but we're not going to at this time. So what are we going to do? Well, we now have a definition, so we can actually go ahead and use some proofs. So we're going to prove that five plus the additive inverse of three gives us two. And something we can rely on is the fact that since the additive inverse of three is the number where if I add three and it's additive inverse, what I get is zero. And all of our proofs regarding additive inverses are going to rely on some version of this definition. So if only I can get a three, well, actually I can. Five, I can split up as two plus three and that's just a straight decomposition. Again, this is whole number arithmetic, so we're allowed to make use of that because now we're talking about the arithmetic of the additive inverses. We're not making any assumptions about what this additive inverse of three is, but we do know that it does act in this fashion. And so if I could get a three, well, there it is by decomposition. We've made the assumption that associativity and commutativity work, so I can regroup this addition this way and I know what this is. Three plus its additive inverse, that's going to be zero by the definition of what additive inverse is and by the additive identity, because I'm adding zero, I get two. And if I want to prove this statement five plus the additive inverse of three is equal to two, it's the portion of the statement that is in green. I'll take a look at another proof. For example, we want to prove that two minus seven is the same as the additive inverse of five. And we'll use a duck proof. This goes back to an old saying that if it walks, swims, and quacks like a duck, it's probably a duck. In this particular case, we want to consider what does this additive inverse of five do? How does it walk, swim, and quack? And the property is that if I add it to five, what I end up with is zero. So let's consider this addition five plus two minus seven. And by associativity, again, we have assumed that associativity and commutativity hold for the integers. This is the same as five plus two minus seven. And well, I know what that is. That's seven minus seven. And that's going to be zero. I can just do the subtraction because that's a subtraction of whole numbers. So here's a nice little statement that I have that I know that this is true by the properties of whole number arithmetic. On the other hand, what that tells me is that because I can add these two things together and get zero, that says that whatever it is called, whatever it looks like, two minus seven is the additive inverse of five, because if I add it to five, I get zero, and that's the property of what an additive inverse is. So that tells me that two minus seven is the same thing as the additive inverse of five, and the proof is portioning green. Well, again, let's take a look at another proof, prove that the additive inverse of three plus the additive inverse of eight is the same as the additive inverse of eleven. And again, I can do this as a duck proof. And so here I might want to start with eleven plus additive inverse of three plus the additive inverse of eight. Well, I can break that eleven apart into eight plus three. Again, that's whole number arithmetic, so I'm allowed to do that without comment. I've assumed that associativity and commutativity hold, so I can rearrange things. So I have an eight, and this additive inverse of eight, I'll group those. I have a three, and additive inverse of three, I'll group those. And by the definition of what an additive inverse is, a number plus the additive inverse gives you zero, a number plus the additive inverse gives you zero, and my definition of additive identity gives me that zero plus zero is zero, and that tells me the following. Eleven plus whatever this thing is, is zero. So whatever this thing is, when I add it to eleven, I get zero. This thing has to be the additive inverse of eleven. So I have no reason not to say that these two things are identical. Now, be careful here. The proof is not this part. This is not the proof, but rather the part in green is what needs to be included for a complete proof.