 So here's a review of some basic integer operations, and again here's a quick disclaimer, this is a review, it assumes that at some point you learn how to do arithmetic with side numbers, but for whatever reason you need a refresher. And so it consists of a sequence of rules for operating with side numbers followed by some examples. This is by far the worst way to learn mathematics, you cannot effectively learn mathematics by memorizing a set of rules and following the examples. This is a horrible way of learning mathematics, and if you haven't already learned how to do arithmetic with side numbers, you won't be able to learn how to do arithmetic with side numbers by watching this video. Instead, I suggest you watch these videos. Now, so let's take a look at some of these basic rules for integer arithmetic for addition and subtraction. Given two integers a and b, I have the following rules applying. So the negative of a plus the negative of b is the same as the negative of a plus b, so I can take that negative and I can add the two unsigned values. a plus negative b is the same as a minus b, a minus a negative b is the same as a plus b, negative a minus b is the same as b minus a. In other words, I can take a subtraction, the negative of that is the subtraction of the other order. And if I use the commutative and associative properties, the key to these rules is that I can transform one integer expression into a different one. And maybe I don't know how to do the one side, but I can do the right hand side. And so this is a general approach to dealing with integer addition and subtraction. For example, consider negative four plus negative six. So now I have the sum of two negatives, and so I have this property that negative a plus negative b is the same as negative quantity a plus b. Now a useful thing to do when you're reviewing stuff is to write, write, write, and write. Paper is cheap, pencil is cheap, understanding is priceless. So what I might want to do is I might want to copy down this rule, negative a plus negative b is the same as negative a plus b. And then compare what I want to work with with what I have. And here, evidently, a is four, b is six, and so I can copy this over. This is negative four plus six, and I know how to do four plus six. That's an addition of whole numbers, and the parentheses say do that first. So the inside of the parentheses is going to be ten, and the outside negative, whatever that is. At this point, there's nothing to do inside the parentheses, so I can just drop them out, and there's my sum. Well, how about eight minus twelve? Well, again, twelve is more than eight, so I can't subtract twelve from eight, but that's okay. I have this property of integer arithmetic that says I can convert a subtraction into a different subtraction. The negative of a minus b is the same as b minus a. And the thing about that equals is that I can write this in either order. If I have this, I have this. If I have this, I have that. Well, I have the subtraction, so let's see what I can do with that. So again, I'll set down my values here, and this eight minus twelve, that really goes on the right-hand side here. This is eight minus twelve, and I'll compare once again. b is eight, a is twelve. I'll substitute those into our formula. That's negative twelve minus eight. And again, parentheses say do this first, so I'll take care of that twelve minus eight. That's four, and now this is, well, the parentheses say do this first, but there's nothing else I can do, so I no longer need the parentheses, and I can drop them out. Four minus negative six. Well, this is a subtraction of a negative, so I can use the property of integer arithmetic. a minus negative b is the same as a plus b. And again, I can write down the rule and compare what I have. So here's a minus negative b. I can compare what I have. This a is going to be, let's see, so that's negative four. It must be that a is negative four, and b is going to be six, so that's going to give me a plus b, negative four plus six. Well, maybe I don't know what that is. Maybe I don't know what negative four plus six is equal to. So again, I can reuse the properties of arithmetic. I can use the commutative property and rewrite it as six plus negative four. Addition can be performed in whatever order we want, so that's six plus negative four, and there's another property of integer arithmetic. a plus negative b is the same as a minus b. So here's a plus negative b, and that's the same as six minus four, and I know how to do that. That's going to be equal to two. Eight plus negative ten. Again, I can begin with the property. a plus negative b is the same as a minus b, so eight plus negative ten is the same as eight minus ten, but I can't subtract ten from eight, so now I'll use that property a minus b is the same as negative due to the subtraction of the reverse order. So instead of eight minus ten, I have negative ten minus eight, and I do know parentheses say do this first, and I know what ten minus eight is. Ten minus eight is two, and that parentheses, there's nothing to do inside, so I don't really need those parentheses anymore. My final answer, negative two. Five minus eight minus negative four. Again, subtraction is commutative and associative, and it's helpful this is not a subtraction, but the rules of integers allow me to convert subtractions into additions. So let's go ahead and do that to make our lives easier. Five minus eight, that's the same thing as five plus negative eight. Here's another subtraction, minus negative four. That's the same as plus negative negative four. And so I have this property of integer arithmetic that negative negative a is the same as a. So negative negative four is the same as four, and I have an addition. I can write this in any order that I want to, so maybe I'll write it this way, where I have the non-negative numbers first. I know how to add those. I have my negative number at the end. I have the property a plus a negative is the same as a minus b, so this nine plus negative eight is the same as nine minus eight, and I know how to do that. That's going to be one.