 So let's take a look at some of the properties of multiplication. So let's start off by considering a product like A times B. And the definition of multiplication says that this is the sum of A B's, and so I can take whatever B is, and again I'll represent B, well that's a whole bunch of things, and I'm going to take A of these B's, so I'm going to take A copies of it, and so I get my B, again a whole bunch more times, and then finally the last copy is going to be there, and so this is something that represents my product A times B. Now I can take a look at this diagram, and I can view this in a different way. I can view this instead as B copies of A, so again I have A items in this column, and again, and a whole bunch more times until it gets to the end, and how many of these columns do I have? Well I have B columns. So this figure here I can read as B copies of A, and that corresponds to the multiplication B times A, and that suggests that this product, A copies of B, or B copies of A, well they're the same thing, that A times B is equal to B times A, and this is an example of what's called the commutative property of multiplication. For whole numbers A and B, the product is the same, no matter which direction you do it. Now there's another important property, consider a triple product, A times B times C. Now we have to, we must read this from left to right. So this is A copies of B, and then because of commutativity though, A copies of B is the same as B copies of A. So here's my A copies of B, I can read this as B copies of A, and then what I'm going to do, I want this many copies of C, and again by commutativity this many copies of C is the same as C copies of this thing. So here's my A times B, B times A, I'm going to take C copies of it, and this is a figure that represents my product A times B times C. On the other hand, I can also read this as A copies of this. Well this is C copies of B, or otherwise known as B copies of C, B times C, again by commutativity. So I can read this in a slightly different way, and what that tells me is that A times B times C is the same as A times B times C, and in general that gives us the associated property of multiplication. If I have three whole numbers A, B, and C, the product of the first two with the last is the same as the first times the product of the last two. I can group any two that I want to. Well as with the addition theorems, the theorems themselves are primarily valuable in their use, not in their rote memorization. So for example, let's take a look at a product 5 times 7 times 2, and by the associative property I can group the last two factors. I have to do all three, but I can either multiply the first two times 2, or I could do the last two times 5. Well, I'll group the last two. By the commutative property I can reorder. By the associative property again I can regroup. And why would you ever want to do that? Well, part of the reason is that 5 times 2 is actually an easier multiplication than 5 times 7, or 7 times 2. And more importantly, when I get that, this is also an easy multiplication. 10 times 7 gives me 70. And so we have our product 10 times 7, 70, and so we can do this multiplication very easily. If we only move from left to right, it's a more difficult product. 5 times 7 is 35 times 2 is whatever it's going to be. But the associative and commutative properties, like the corresponding properties for addition, allow me to rearrange a multiplication in any order that I find convenient. And as with the corresponding properties for addition, we don't generally write out all of these steps. We should understand that this is what we're doing, but in practice what we're going to do is we're going to say, oh, well, 5 times, well, I'll multiply it by 2, that's 10, and then I'll multiply that by 7 to get 70. And in practice we do all of this in our head.