 So the addition and subtraction of signed numbers is a big topic, and so this is part 2. So just a quick recap, the additive inverse of a written minus a is the number that satisfies a plus the additive inverse is equal to zero. Addition of integers is both associative and commutative. For any integers a and b, a plus the additive inverse of b is the same as a minus b and a minus b is the same as the additive inverse of b minus a. And finally an important and useful idea. You can have anything you want as long as it's paid for. So let's try to find five minus the additive inverse of four. And again in order to subtract the additive inverse of four, we have to have an additive inverse of four. So we can have anything we want as long as it's paid for, and in this case the price for the additive inverse of four. Well, we know that the number plus its additive inverse is going to be zero. And so that tells us since zero is four plus its additive inverse, and we can always add zero. It's free. So five minus the additive inverse of four. Well, that's the same as five plus zero minus the additive inverse of four. Zero is the same thing as four plus the additive inverse of four. So we can replace it, and here at the end we are adding the additive inverse of four, and then we're immediately taking it away. So that last bit just drops out, and so we have five plus four, which is equal to nine. And again, we might take a look at the numbers involved, five, four, and nine, and we note that five minus the additive inverse of four is nine, which is the same as five plus four. And so this suggests the following result. For any integers a and b, a minus the additive inverse of b is the same as a plus b. So let's take three minus the additive inverse of four. Now, because one of our main theorems is that the addition of the integers is associative and commutative, it's actually easier to think about everything in terms of addition. So anytime we have a subtraction, we're going to want to rewrite it in some fashion. And so our theorem says that for any integers a and b, a minus the additive inverse of b is the same as a plus b, which is this beautiful addition. So I can rewrite this additive inverse of three minus additive inverse of four as additive inverse of three plus four. Because addition is commutative, I can rearrange this as four plus additive inverse of three. We also have a theorem that says a plus additive inverse of b is the same as a minus b. So this is going to be four minus three, and we know what that is. This last operation brings up an important idea, which we can introduce as follows. We want to find the additive inverse of the additive inverse of five. Now you might read this as subtract the additive inverse of five and think about the theorem we just introduced. The problem is that this is not actually a subtraction symbol. And the reason being there's nothing here to subtract. Subtraction requires two numbers. There's only one number here. So our theorem is actually not useful. Instead we have to go back to our definition of the additive inverse. So our definition of additive inverse says that whatever this is, if we add it to the additive inverse of five, we should get zero. But we also know that five plus the additive inverse of five is equal to zero. And because of commutativity and associativity, we can also rewrite this as additive inverse of five plus five is equal to zero. And let's compare these two equations. Both of them are sums equal to zero. Both of them have additive inverse of five plus something. And that suggests that our sum things have to be the same. And so this suggests that as a general rule for any integer, the additive inverse of the additive inverse is just the number itself. So for example, let's say I want to find the additive inverse of the quantity three minus the quantity eight minus one. So the thing inside the parentheses has to be done first, eight minus one, that's seven, everything else stays the same. Now I have to find three minus seven. My theorem says that's the same as the additive inverse of seven minus three. So that gives me the additive inverse of four. And I want the additive inverse of the additive inverse. And so that's going to be four. So what about additive inverse of eight plus additive inverse of seven? So we might proceed as follows. This is an addition, but unless we actually know what the answer is, we have to do something to change it. So let's see what our theorems say. So we have a plus additive inverse of b is the same as a minus b. A minus b is the same as additive inverse of b minus a. And a minus the additive inverse of b is the same as a plus b. And the thing we might notice here is that our original expression is plus an additive inverse. And so that suggests we can use the first theorem. A plus additive inverse is the same as a minus number. So additive inverse of eight plus additive inverse of seven is the same as additive inverse of eight minus seven. Now what? Well, this is a subtraction, and we do have this theorem that says a minus b is the same as additive inverse of b minus a. I can reverse the order of the subtraction. And so additive inverse of eight minus seven can be reversed to additive inverse of seven minus the additive inverse of eight. But wait, there's more. I am subtracting an additive inverse. And I have a theorem that says if I subtract an additive inverse, it's the same as adding the corresponding numbers. So this seven minus the additive inverse of eight is the same as seven plus eight, which will evaluate, and get our final answer, additive inverse of 15. And again, if we look at our numbers eight, seven, and 15, the relationship that we seem to have here is that the additive inverse of a plus the additive inverse of b is the same as the additive inverse of a plus b. How about minus four minus eight? So pulling in our theorems. So this seems to be a subtraction, so I'll use the theorem that a minus b is the same as the additive inverse of b minus a. So I can reverse the order of the terms additive inverse four and eight. This will be the additive inverse of eight minus the additive inverse of four. But I'm subtracting an additive inverse, so that becomes a plus. This is eight plus four. And I can evaluate the term inside the parentheses as 12. And so my final answer, additive inverse of 12.