 It bears repeating that the purpose of elementary arithmetic is not to learn how to do elementary arithmetic. That's incidental, and it's a bonus, but it has nothing to do with why elementary arithmetic is important. The reason elementary arithmetic is important is it paves the path to the learning of algebra, and we can see how far that goes now that we have division as part of our elementary operations. So let's take a look at a nice algebraic problem, solve for x30 divided by x is equal to 5, and let's see if we can solve this. So again, if you've taken algebra in school, well, this is a fairly standard type of algebraic equation, and we can solve this. This is actually the type of equation that a lot of people who've taken algebra will get wrong. They will not solve this correctly because what they've learned is how to push things around on paper, and that doesn't help when you're trying to solve an equation. On the other hand, if you understand the basic concept of what a division is, that's an easy problem to answer, and we can actually find the solution in a number of different ways without using the rules of algebra. So let's use our most basic, although in some sense probably the least cumbersome, or most cumbersome method of solving this problem, we could use the definition of division. So we'll set down our definition of division and use it for reference. If A times B equals C, then A is C divided by B, and conversely. And so there's our definition of division, and what we can do is we can compare what we have, 30 divided by x equals 5 to what our definition says, and note that the conversely says that if I have either side, if I have either statement, I can go to the other one without comment or difficulty. So let's make that comparison. So let's see, that tells me that C is 30, B is x, and let's see A, well that must be 5. So I have this, so I have 5 equals 30 divided by x, that's my starting statement, and if 5 times x equals 30, then 5 equals 30 divided by x, and the conversely says I can go in the other direction. I have 30 divided by x equals 5, so I know that 5x equals 30 by the definition of division. Well by commutativity, I can reverse the order of the product, x times 5 is equal to 30, and I can then apply the definition of division. I have a product, x times 5 equals 30, and because I have a product, my definition of division says I can reverse from the product, x times 5 equals 30, that tells me x is equal to 30 divided by 5. And then the last thing I can do is 30 divided by 5 is an arithmetic operation. I can handle x must be equal to 6. And there's my solution, and so there's the first solution. Importantly, if we actually want to show three different methods that do not use the rules of algebra, our first method is this entire statement, not just the last bit. So here we have a solution that doesn't require any rules of algebra, and as I mentioned, this is probably the most cumbersome of the approaches, but it works. Much more effectively, we might use the notion of partitive division. So let's take a look at what this equation means. So let's take a look at that. This is 30 divided by x equals 5, and so if I view this as a partitive division, what this is telling me is I'm taking 30, I'm dividing it into a number of pieces. I don't know how many. I don't know how many of that is, that's why that's unknown, but what I'm getting is 5, and so that tells me that those pieces are all going to be of size 5. Well, let's go ahead and do that. We don't know how many parts there are, but we do know that each of them is size 5, so we can mark down these operations on a tape diagram. So I'll start off with a tape that represents 30, and I know that each piece I produce is going to be of size 5, so let's go ahead and start putting them down. There's 5, and 5, and 5, let's see, so far that's 5, 10, 15, still keep going, 20, 25, and that last piece is 30, and so now I've taken my 30, I've divided it into a number of pieces, each of which is size 5, and my interpretation of the partitive division is that the number of pieces is going to be my divisor, so that means I've divided this into 1, 2, 3, 4, 5, 6 pieces, so that tells me x must be equal to 6, and again the demonstration of our answer is, consists of two parts, one is the tape diagram, and the other is the actual explanation of why the tape diagram is relevant. Well, there's actually another interpretation of division. It's our quotitive division, and let's go ahead and see how that works out. Again, our quotitive interpretation of division by our quotitive interpretation, 30 divided by x equals 5, well, what that says as a quotitive division is I've taken 30, I've broken off pieces of some size that don't know what it is, but what I do know is that I've broken 30 into 5 pieces of that particular size. So by the quotitive interpretation of division, the divisor is the size of each piece, and the quotient is the number of pieces that we have. So I can show this using a tape diagram. So again, I have my tape of 30, I'm going to produce 5 pieces, and I don't know how big those pieces are individually, but I can figure that out. Turns out that each of those pieces is of size 6, check it out, 6, 12, 18, 24, 30, so there's my tape diagram of 30, 1, 2, 3, 4, 5 pieces, and by the quotitive interpretation of division, what I've done is I've taken 30, I've divided it into pieces of size 6, and there's 5 of them, and that gives me my quotient there, and that tells me what my solution is going to be x equals 6. And as before, the answer to the question without using the rules of algebra is going to be the explanation of what we're doing together with the tape diagram that shows the actual process. And there's our solution of a rather complex algebra problem requiring nothing more than understanding what division actually represents. And again, the key here is that what it means is that as soon as you learn what division is, as soon as you learn the concept of division, you can start to solve algebra problems. You don't even have to know how to do very complicated divisions. You've actually seen a couple of examples of complicated divisions.