 The next important arithmetic operation is multiplication. So we'll start out with the definition, and our definition is going to be based on addition. So let's take two whole numbers, A and B, and I'm going to define the product A times B to be the sum of B repeatedly, A times. Now, the notation here may be a little bit confusing. This does not mean I have this thing, A times. What it actually means is that I have A, B's being added together. So in other words, when I'm talking about the product A times B, what I really mean is it's a sum of A, B's. Now, before we continue just a quick word on notation, this symbol is generally used to introduce multiplication. However, in different contexts, this symbol may be confusing, particularly when we start to talk about algebraic expressions, where this symbol, the Saint Andrew's Cross, looks a lot like the letter X, which we like to use as our unknown. So we've developed a number of equivalent ways of expressing multiplication, and so one of the common ways we have of writing a multiplication, we can also, if we must include a symbol, we can include this center dot, and more generally, we might just omit the symbol entirely. So all of the following should be regarded as completely equivalent A, Saint Andrew's Cross, B, A dot, B, and then finally just putting the two things to be multiplied together, A, B. All right, so let's take a look at a proof. Let's take a look at a typical problem proof for times 6 is equal to 24. And again, taking this problem apart, note that it is a proof statement, which means we're going to want to go back to the definition. So we can go ahead and put that definition down first. A times B is the sum of A, B's. And now we'll compare our problem to our definition. We see that A is 4. I'll go ahead and replace that. B is 6. I'll go ahead and replace that. And then I have the statement 4 times 6 is 6 plus 6 plus 6, and I'm going to take 4, 6's, and add them together. And so I can do that because the addition is part of the definition of proof, I don't have to go into too many details about how I find that sum, and I can just do the sum. So 4 times 6 equals 24. And again, the important parts of the proof we're going to show in green.