 So an important extension of the whole number system occurs when we expand to including what are called signed numbers. And we'll introduce them this way. We'll make the following definition. The additive inverse of A, written minus A, is the number that satisfies the equation A, plus the additive inverse is equal to zero. Now if you've seen this before, you might be tempted to read this as negative A. And while it's true that this is the same as negative A, how you speak influences how you think. And so it's useful to read this as additive inverse of A, because it reminds you that this plus A is going to be equal to zero. We'll introduce a few other quick definitions. The whole numbers, together with their additive inverses, form the set of integers. And these are both definitions. They're both very important to understand. Less important, but extremely useful, is the following theorem. Addition of the integers is both associative and commutative. And remember the importance of that is that any addition can be rearranged in any order that we want. And finally, an important idea in mathematics and in life. You can have anything you want, as long as it's paid for. Let's see how that works. Suppose we want to find twelve plus the additive inverse of three. So we know that the additive inverse is the thing that's going to satisfy number plus additive inverse is zero. And so that means we know that three plus the additive inverse of three is going to be equal to zero. And what this means is that if we can somehow get a three into this expression, we can simplify it using this definition of the additive inverse. And we actually have a three buried in here. Since twelve is equal to nine plus three, the equality means that we can replace twelve with nine plus three. So we'll do that. We know that addition of the integers is both associative and commutative, so we can do this in any order that we want. We want to do this in the order that makes use of the fact that three plus the additive inverse of three is equal to zero. So we'll add these last two terms together to get zero. And nine plus zero is equal to nine. And we might make a quick note. Twelve plus the additive inverse of three is equal to nine. Meanwhile, the same numbers, twelve, three, and nine are related by twelve minus three is equal to nine. And what this suggests is that for any integers A and B, A plus the additive inverse of B is equal to A minus B. So we might take eight plus the additive inverse of three. Our theorem says that if I add a number at its additive inverse, it's the same as subtracting, and so this is the same as eight minus three or five. How about three minus twelve? To subtract twelve, we need to have twelve. But if we don't have twelve, we only have three. Well, remember, you can have anything you want as long as it's paid for. We note that three plus nine is equal to twelve. So if we add nine, we'll have a twelve. But we have to pay for it. We need to subtract nine to keep our expression the same. So three minus twelve becomes three plus nine minus twelve. And then the bill for the nine comes due. And we have to subtract nine at the end in order to keep our expression the same. And now we invoke our order of operations. We have to do addition and subtraction from left to right. So we'll take care of three plus nine first. And the rest doesn't change. Twelve minus twelve is zero. And the rest doesn't change. And, uh-oh, now we're trying to take nine away from zero. If we couldn't take twelve away from three, how can we take nine away from zero? Remember, our theorem works both ways. A minus B is the same as A plus the additive inverse of B. So zero minus nine is the same as zero plus the additive inverse of nine. And because we're adding zero, this is just going to be additive inverse of nine. And once again, we might make the observation the numbers three, twelve, and nine are related. Twelve minus three is nine, while three minus twelve, what we just found, is the additive inverse of nine. And so this suggests another theorem for any integers A and B. A minus B is the same as the additive inverse of B minus A. To find seven minus fifteen, our theorem says it's the same as the additive inverse of fifteen minus seven. Well, fifteen minus seven is eight, and so our result is additive inverse of eight. What is important is that you will very rarely only ever have to do one thing. More often than not, you'll have to combine several things. And so being comfortable with the combining and recombining of different ideas is an essential part of mathematics. So for example, let's say I want to add five plus the additive inverse of eight. Well, we have two theorems at our disposal. We'll have many more by the end of the day. But right now we know that for any integers A and B, A plus the additive inverse is the same as A minus B. And also A minus B is the additive inverse of B minus A. So let's see which of these applies. Five plus the additive inverse of eight, well, that looks like this first theorem. And so we know that we can rewrite this as five minus eight. Unfortunately, barring some creative accounting, I can't subtract eight from five. Fortunately, we do know that for any integers A and B, A minus B is the same as the additive inverse of B minus A. So I'll rewrite this additive inverse of eight minus five. And I do know what eight minus five is, that's equal to three. And so my final answer, additive inverse of three. So let's put our theorems together for plus the additive inverse of nine. Adding an additive inverse is the same as subtracting, so this is for minus nine. But I can't subtract for minus nine. But I have a theorem that says that A minus B is the same as the additive inverse of B minus A. So if I can't find four minus nine, I can find the additive inverse of nine minus four.