 Our next important arithmetic operation is subtraction. And again, mathematics occurs when we make abstraction of some sort of concrete process. And in the case of subtraction, what that concrete process might be is I might take something and I might remove some amount. And so I have an amount, I remove some of it, and I have something that's left over. And again, we can choose what parts of the process to abstract, but in the case of most arithmetic operations, what we're abstracting is the cardinalities of the collections. So this collection has a property of cardinality. There's a number of things in it, likewise this, likewise that. And so I might get some statement from five, remove two, and that leaves you with three. Now we'll introduce some notation. And so we get our final subtraction statement. Five minus three is equal to two. So again, it's important to distinguish what is easy from what is familiar. The mathematics is very easy here. If you can understand what the English says from remove leaving, the mathematics here is easy. This may be familiar, but it is not easy. To understand what's going on here involves a very, very complicated set of associations. Here, the math is easy. Here, it's familiar. Well, let's take a look at some other subtractions. Well, mathematics does emerge from abstracting a concrete process. We actually continue by asking, well, what else is going on here? So let's take a look at our original observation again from this thing, which we associate with cardinality five, remove this thing, cardinality two, leaving this thing cardinality three. And there's two other important observations. And our ability to do subtraction is centered around these two observations. First of all, we can talk about the difference between what we started with and what we end with. And the difference between what we start with and what we end with is what we've removed. In a far less obvious form, the difference between five and three is two. Again, familiar, but not easy. Here, this is easy. The difference between this and this is this. We can also go backwards. The other possibility here is we can go backwards to what remains we can join. Well, actually we have a different name for it. We can add what we've removed and that'll give us what we started with. To this, we can join this and that gets us this amount. And this second idea is how we actually define our subtraction. What we get after we do the subtraction plus what we took away gives us what we started with. So our subtraction five minus three is equal to two is going to be tied to the addition five is equal to two plus three. And this leads us to a formal way of defining subtraction in terms of addition, which is the following. Suppose I know a plus b is equal to c. Then I immediately know a is equal to c minus b and conversely, which is to say if I know a equals c minus b, then I know that a plus b is equal to c. Well, this means that every addition fact translates into a subtraction fact immediately. So if I know that seven plus eight is equal to 15, I also know that 15 minus eight is equal to seven. I can take this 15, I can subtract eight, and that gives me seven. Or if I know 15 plus three is equal to 18, I immediately know 18 minus three is equal to 15. And this also translates into algebra. If I know that x plus seven is equal to 10, if I know an addition fact, then I know a subtraction fact. And in this particular case, I know that 10 minus seven is equal to x. And if I happen to know what 10 minus seven is, then I also know what x is. And what this means is that if you understand what subtraction means, you can solve simple algebraic problems as early as the first grade, which is when subtraction is typically introduced.