 So you might wonder about why we bother with a formal definition of subtraction. Why don't we just do it? And the reason for that has to do entirely with the teaching of algebra. And the key idea to keep in mind is that everything you do in elementary arithmetic, all the addition, subtraction, multiplication, and so on, all of that leads eventually to algebra. Algebra is arithmetic generalized. And what that means is that if you understand what's going on in arithmetic, you will immediately understand what's going on in algebra. Now, the fact that many people can, so to speak, do basic arithmetic without being able to do algebra speaks to how badly understood basic arithmetic is. So let's take an example of that. And just to preface this, historically algebra used to be a college subject. You didn't take a course in algebra until you hit college. Well, sometime during the 20th century, it gradually moved earlier in the curriculum and became a high schools topic. It became a middle school topic. And most recently, it has now been moved into the early grades. And with the proper perspective, we can begin to teach algebra as early as first or second grade. And what we have to understand, in order to do this, you have to understand what basic arithmetic is all about. So here is an algebra problem that many people have difficulties with. Solve 8 minus x equals to 3. And the standard algebra problem, solve for x. But then let's actually prove the validity of each step and see where our understanding of subtraction leads us. So again, note here that we're not just trying to solve for x, but we also want to prove that the steps that we're taking are valid. So in general, what we're trying to do is we want to transform one equation into another equation. And at the end of it, the solve part, at the end of it, we have an equation of the form x equals something. And that something will be our solution. All right, so the first thing we notice here is that we have a subtraction. 8 minus x equals 3. And well, what do I know? Well, if I'm looking at a proof, this suggests definitions may be relevant. And in this particular case, I want to look at the definition of subtraction. And again, here it is again. Suppose I have a sum A plus B equal to C. Then I know that A is equal to C minus B. And conversely, which means that I have a subtraction. I can immediately write down a corresponding addition. Well, let's take a look at that. Now, I'll compare the equation that we have. 8 minus x equals 3. And I'll make the observation that A, this thing all by itself, that must be 3. 8 must be our C term. And x, that's B. So I have A equals 3. C equals 8. B equals x. And what I can do is I can take my first step and fill in these terms into the appropriate spaces in my definition. Since 8 minus x equals 3, there's my starting point, then I know 8 is equal to 3 plus x by my definition of subtraction. Well, again, I now have an addition. And unfortunately, if we go to the definition of addition, we find that it doesn't allow us to transform one equation into another one. So my definition of addition, no matter how I look at it, is not going to be quite as useful here. On the other hand, something else I do know is the commutative property of addition, which means that I can reverse the order of an addition. If I have two numbers A and B, A plus B is the same as B plus A. Well, can I make use of the commutative property of addition? Well, we have to be extremely careful here because if the commutative property of addition doesn't help us move forward, the internet will implode causing global economic disaster. The polar caps will melt flooding all coastal cities and plagues of locusts will cover the lands and even worse things will happen if we use the wrong theorem. Well, actually, if it doesn't work, we'll try something else. It doesn't hurt to apply a mathematical theorem. The worst that happens is it doesn't work. Well, let's try it. Let's see what happens. So by the commutative property of addition, 8 is equal to x plus 3. And let's see, the sky hasn't fallen, so we're probably good. Now, there's actually two ways we might proceed at this point. If you go back to the piano definition of addition and counting on what this says is whatever x is, the third number after it is 8. Well, I can use that to try and figure out what x has to be and that is a valid way of proceeding, but in this particular case, I'm not going to do it mostly because I want to reuse something I've already done. I have an addition I can apply the definition of subtraction once again, but this time we're actually starting with the addition. So here's my definition of subtraction once again. If I have a plus b equal to c, then I can immediately write down a subtraction, a equals c minus b. And again, I'll compare what I have to what my definition is. So what do I have? I have a being x, b being 3, and c is 8. So I have a equals x, b equals 3, c equals 8, and I'm going to drop those into my definition of addition since 8 equals x plus 3, then my definition of subtraction tells me that 8 minus 3 is equal to x. And at this point I have an arithmetic computation I have to do and assuming that we've learned at least a little bit of these arithmetic operations, if we know how to do this subtraction 8 minus 3, we can perform that subtraction without comment, x is equal to 5. And there's my solution with the validity of each step proven. Now some things to note, for those of you who've taken an algebra course, you might want to remember how did you solve a problem like this? Well there is a sequence of steps that you had to do and one of the things you had to learn was you had to do the steps in the right order and if you didn't do the steps in the right order, you got the wrong answer. And the problem is that if you just follow the steps, there's no internal logic to them other than that's how I was taught to do it. So here what we're doing is our steps proceed from our understanding of what we've written. A subtraction is an addition written backwards. Commutative property, a subtraction is an addition run backwards. And then finally one last computation. And if we get the steps backwards, well it's kind of impossible to get the steps backwards because they form a thought sequence. They don't form a here's how to do it sequence, which where it's easy to forget what comes first, but they form a this leads naturally to the next thing type of sequence. And that's part of the reason that an understanding of the basic operations of arithmetic, not how you do them but why you do them is so important.