 We're going to expand our number system in a couple of different ways, but the first expansion from the whole numbers, the things that we use to count the number of objects, is to what are called the integers. And this emerges as follows. Suppose I have an algebraic problem. So for example, let's say the problem 3 plus x equals 8, and this is a nice simple algebraic problem involving the whole numbers 3 and 8, and I can find a solution. But what if I have a different problem? For example, x 8 plus x is equal to 3. Well, this problem appears very similar. I have 8, I have x, I have 3, I have all the same symbols, I have all the same whole numbers, and so maybe I should be able to solve this problem, which looks very similar. Well, let's try to solve this using a tape diagram. So this is saying the tape that I have that's 8 plus some unknown amount is equal in length to the tape that represents 3. Now here, relative magnitude is at least somewhat important here. I have a block that represents 8, so my 3 block has to be represented by something a little bit smaller. Now what I can do is I note that that 8 is a 3 into 5, so these 3 blocks here are the same size, so I can remove them. And what I have is this tape representing x plus 5 must equal nothing. Well, we can at least indicate that. And so that tells us this x, whatever it is, is the amount which when I add it to 5 gives me nothing, gives me 0. So this allows us to define something new, which is the additive inverse of a number. And so we'll define the additive inverse of a number a written this way is the number that satisfies the following equation. A plus the additive inverse is going to give you 0. Now it's common to write this as dash a using this symbol here, which runs the risk of making it look like a subtraction. And so part of the reason that this is raised higher up than our addition in subtraction symbols is we do want to distinguish this from the notion of subtraction. Another common thing we do is we read this as negative a, but that is not recommended. How you speak influences how you think. And if you read this as the additive inverse of a, one of the things that that will remind you is that it has the property that when you add it to a, you get 0. And so if we read this as the additive inverse of a, we're constantly reminding ourselves of the one important property of this number. So now we do need a name for the set of whole numbers, the things we use to count the number of things that are present and all of their additive inverses. And so collectively we refer to this combination of all of the whole numbers and all of their additive inverses. We call that set the integers. And we'll make the following claim without too much explanation. We'll say that this is true. There is a proof of it, but we won't go into it, which is that the addition of the integers is both associative and commutative. And it turns out if you know the definition of the additive inverse, if you accept that addition of integers is associative and commutative, then everything that you do with whole number arithmetic allows you to find every question that can be asked with integer arithmetic. So this is almost everything you need to know. And the only thing we have to keep in mind is that the additive inverse plus the number gives you 0. Integer addition is associative and commutative, and you should know how to do the arithmetic of whole numbers. If you don't know how to do the arithmetic of whole numbers, you won't be able to do the arithmetic of the integers. Just to understand what we mean by this, well let's consider a problem. Evaluate using only the definition of additive inverse, the arithmetic of the whole numbers, and the associativity and commutativity of integer addition, 3 plus 5 plus, don't read this as negative 3, this is the additive inverse of 3, and defend each step. Now I put this in here as a requirement because when you defend each step, you'll identify whether you're sticking to the rules. So just to reiterate that point, here's a wrong answer based on some things you know from previous courses. You were introduced to the integers many, many courses ago, and so you've been told how to operate with the integers, and so maybe you say, oh well I know how to do this, 3 plus 5 plus negative 3, and well that's 8 plus negative 3, and that's the same as 8 minus 3, which is 5. And while this is the correct value, we used a number of properties of integer arithmetic, which the problem as stated doesn't allow us to use. We have to limit ourselves to these things. The rules of the game are we're allowed to use these steps and nothing else. So this answer is incorrect. How can we get a correct answer? Well we have to play by the rules of the game. We can only use the definition of additive inverse, the arithmetic of the whole numbers, and the associativity and commutativity of integer addition. So let's take a look at that. First of all, 3 and 5 are whole numbers because we can have three things. We can have five things. Additive inverse of 3 is not. We cannot have additive inverse of three things. So the definition of additive inverse does tell us something useful, and from the definition of additive inverse, we know that 3 plus the additive inverse of 3 is 0. So if I can get a 3 next to the additive inverse of 3, I'll be able to add it together, and in fact what I'll be able to do is I'll be able to add them and get 0. So here's what I'm starting with, and what I know is that integer addition is commutative. I have that as one of the rules I can use. So I can rearrange the order of integer addition. This 3 plus 5 plus additive inverse of 3 is the same as 5 plus 3, switching the order, plus the additive inverse of 3. Well that's useful because now I have this 3 plus its additive inverse. Now I should say that I can combine these two. Associativity says that I can add two things that are next to each other, wherever they are. I don't have to go from left to right. So this 3 plus additive inverse of 3, I can combine those two things together, and I know what that is because by the definition of additive inverse, 3 plus the additive inverse of 3 is going to give me 0, and that gives me 5 plus 0. Again, let's check it out. 5 and 0 are in fact whole numbers. I can have 5 things, I can have 0 things, and so they are whole numbers which means I can add them using ordinary whole number arithmetic, and here's my solution to the problem. I have only used the definition of additive inverse, the arithmetic of whole numbers, and associativity and commutativity integer addition, and those are the only things I've used, so this is a correct answer, including all of the explanations.