 So let's introduce another mathematical operation, the operation of subtraction. And once again, it's worth remembering, mathematics occurs when we make abstraction of some sort of concrete process. For example, I might take some set that looks like this, and I might want to remove some amount and leave some amount. So what sort of abstractions can we make here? Well, I might abstract the cartonalities of the different collections to get some sort of equivalent statement. So here I'm starting with a collection with cartonality five. I'm going to remove a collection with cartonality two, and I'm going to leave a collection with cartonality three. And then finally, we might introduce some notation to express this even more compactly, and we may write something like this. Five minus two is equal to three. Now, mathematics occurs when we make abstraction of some sort of concrete process, but mathematics continues when we ask ourselves, well, what else can we do? And so we might begin with our original observation from this collection of cartonality five, remove a collection of cartonality two, leaving a collection of cartonality three. And we might see if there's any other observations we can make. And one of the things we might say is that, well, I can talk about the difference between what I started with and what I ended with is going to be what we removed. And we can translate this into the statement, the difference between five and three is two. Another useful thing to do in mathematics is to go backwards. We end up with this collection of cartonality three. To what remains, we can add what was removed to get what we started with. And this is useful because in this process what we're doing is rejoining two collections, what remains, what we've removed. And what's useful to note here is that this process of joining two collections corresponds to an addition. And a very useful thing to do in mathematics is to relate to concepts. In this case, we're going to relate our new concept of subtraction to an older concept of addition, and this allows us to define subtraction in terms of addition. So let's lay down a definition of subtraction. Remember, if you don't know the definitions, you cannot do the mathematics. This is a definition that has to be learned if you want to be able to do mathematics. Fortunately, the definition isn't too complicated. If I have an addition a plus b equal to c, then I can write a subtraction a is equal to c minus b. And conversely, which is to say if I have the subtraction a equals c minus b, I will also have the addition a plus b is equal to c. And what is useful here is that this means for everything that I know about addition, every addition fact that I know immediately translates into a subtraction fact. So maybe I know 7 plus 8 is equal to 15. I know a plus b is equal to c. Well, then I can rearrange it into a subtraction fact. I know that 15 minus 8 is equal to 7. Or maybe I know that 15 plus 3 is equal to 18. I can rearrange this as a subtraction fact. 18 minus 3 is equal to 15. Or suppose I know that x plus 7 is equal to 10. Then I can rearrange this into a subtraction fact. 10 minus 7 is equal to x. And that also means that I can solve this algebraic equation. In particular, if you understand what subtraction means, you can begin to solve simple algebraic problems as early as the first grade.