 All right, let's talk about the fourth of the basic operations, which is division, and everything you actually need to know about the division of integers can be found in the following. The definition of the additive inverse of a, so that a plus its additive inverse gives you zero, and because the concept of remainder is a little bit strange when we are dealing with additive inverses, we'll limit integer division to the situations where the remainder is actually zero, and so that gives us the following somewhat modified definition. If a, b, and q are integers where a is equal to b times q, then a divided by b is q, and conversely, if I have this. All right, so let's talk about the fourth of the basic arithmetic operations as applied to the integers, which is division. And again, everything that you really need to know about the division of integers can be found in the following. First off, we have the definition of the additive inverse of a, again, namely that a plus the additive inverse of a gives you zero, and we would like to extend our notion of division to division of integers, but the concept of remainder gets a little bit strange when we're dealing with additive inverses. So what we're going to do then is we're going to focus on the situation where the remainder is zero, and that gives us our special case of division. If a, b, and q are integers where a is b times q, then a divided by b is q, and conversely, if I know this, then I also know this. Now, here's an important idea to keep in mind. We have a whole bunch of rules for dealing with integer division. You learn these at some point in the course of your schooling experience, and we have properties like the division of a positive and a negative is a negative, and so on. These are all properties of the integers. They are not definitions. They are not part of the definition of the integers, and so it's important when solving problems to keep this in mind. In many cases in this class, you will be asked to not use properties because you want to focus on the definitions. So for example, let's take a look at find 20 divided by the additive inverse of 4, negative 4 if we want, using only the definitions and whole number arithmetic, and you want to defend your steps. Again, part of the reason that you want to defend your steps is this. Make sure that you're actually only using the definitions and whole number arithmetic. If you have to defend your steps, you have to know what those are, and if you use something besides the definitions, if you use something other than whole number arithmetic, that will become obvious when you try to explain why you're allowed to do what you're doing. So again, let's take a look at a quick wrong answer, 20 divided by negative 4 equals negative 5 because dividing a positive by a negative gives a negative, and here's our defense of our step, but it's not a definition, and it's certainly nothing to do with whole number arithmetic. So the value may be correct, but it's not the correct answer because it's answering a different question. Here's the actual question. So since we know the definitions, we can do mathematics. So if a, b, and q are integers, where a equals b times q, then a divided by b equals q, and conversely. So I have a division, I pull up my definition of division, and I can compare my definition to what I have. So let's see, that's, I'm taking 20 divided by negative 4, and I want to get a quotient, so that says a is 20, b is negative 4, and what I'd like to do, I'd like to write a as b times something. Well, I happen to know 20 is negative 4 times negative 5. Now remember, multiplication is part of our definition of division. So I can simply claim this result 20 equals negative 4 times negative 5. I can simply claim that result, hopefully I'm making a true claim, but I don't have to say much more than this statement. But because I can claim this is true, that tells me that 20 divided by 4 is in fact negative 5, and there's my integer quotient.