 The next of our basic arithmetic operations is multiplication. And as before, we'll start off with the definition, and we're going to define multiplication in terms of repeated addition. And so we'll give this definition here. If I have two whole numbers A and B, then I'm going to define the product A times B to B, and here's where the notation is a little bit strange. This is B plus itself and so on, A times. And what this actually means is that I'm going to have a sum of A B's. It's not that I'm going to take this whole thing and repeat it a bunch of times. I have A B's in this sum, and the dot dot dot here, technically known as an ellipsis, is there's more stuff in here that looks just like the other stuff around it. So let's take a simple example here, three times five, and we want to find three times five and prove that our answer is correct. And notice that we're not actually just trying to find three times five. We all know what that product is going to be, but the real question at the heart of this problem is, why is it what we think it is and how can we show that? So again, we're not going to just remember what three times five is because anybody can remember a fact, and the fact might actually be true, but the real question is, why should we believe it? And so we can base that on the following way. Look at our definition of multiplication. Again, A times B is the sum of a whole bunch of B's, specifically A times, and in this particular case, we're going to compare what we have three times five to our definition A times B, and by that comparison, we see that A is three, B is five, and so when I substitute those into my definition, three times five, that's the sum of five, three times. Well, if I actually want to do this, I need to write these terms out, and I can do the sum. Again, if it's part of the definition, so here the addition is part of the definition, I can just do that particular operation without any comment. In other words, I don't have to explain why it is that five plus five plus five is equal to 15. I can just go directly from this statement down to the next statement. What you can't do as part of a prove your answer type problem is to go directly from the starting point to the ending point to say three times five is equal to 15, because that's not a proof, that is just a claim that has no evidence that supports it. I know it to be true is not evidence. And so again, in this particular case, we're going to go back to our definition, three times five is the sum of three fives. Here it is, here it is, and then there's my 15. Well, the nice thing about this is once you know what the definition of multiplication is and you know how to add, then you can do any multiplication that you want. So for example, let's take a look at this product three times 28. And we're going to do this without using any knowledge of multiplication whatsoever. All those multiplications that you know how to do three times seven, four times six, all those, we don't need any of them. We don't need any of those. We can find this product without referencing any of them. And the way we're going to do that is we're going to remember that by the definition of multiplication, three times 28 is the sum of three 28s. So what is that? Well, it's 28 plus 28 plus 28, and I have to be able to add them together again, assuming that I know at this point how to add, I'm going to find some way of adding these together conveniently. What I might do in this particular case is use decomposition. Each of these is a 20 and an 8. So what I have is I have three 20s. So I might count 20, 40, 60, and then those 8s, three 8s, 8, 16, and 8 is 24. And I'm going to add those two together. 60 and 20 is 80, and four gives us 84 as our product. And so here's the multiplication, three times 28, done without any knowledge whatsoever of multiplication facts, just basic knowledge of addition and what multiplication means.