 Alright, so let's take a look at the multiplication of integers, and to begin with we'll start up with something that seems to be a slightly different type of problem, but we're going to find the additive inverse of the additive inverse of 3. And so let's take a look at this, we want to find and prove this, so we want to use a property of the additive inverse, and by definition the additive inverse of the additive inverse is going to be the additive inverse of negative 3, and that allows me to write down an equation. This thing, plus negative 3, is going to be 0. And likewise, I want to write down another equation, which says everything that this one does except I've replaced this expression here with something else. So I have plus negative 3 equals 0, but I have something else in place. So let's see what can I put there. Well, let's see, by my definition of what additive inverse is, this is the additive inverse of 3. If I add it to something and get 0, what I'm adding it to must have been 3 itself. So I can put a 3 in there, and now I have two equations, which say almost the same thing, plus negative 3 equals 0, plus negative 3 equals 0, and the only difference is in these two expressions there. So that tells me that those two expressions must represent the same thing. So that tells me that negative negative 3 must be equal to 3. And this is actually a much more general property, and we can prove this in general that the additive inverse of the additive inverse of a number is just whatever you started with. Now one of the things that this shows up in is it allows us to extend our definition of multiplication to the integers. So, again, remember our definition of whole numbers, A times B, is the sum of A, B's added together. Now, this only makes sense if A is a whole number. It doesn't make sense to add up negative 5B's, but if A is a whole number, then that sum does make some sense. And so as with our use of other definitions, because in this case addition is part of our definition of multiplication. I can perform the addition without having to provide additional commentary or explanation. Now we won't prove it, but it can be shown that associativity and commutativity also hold true for integer multiplication. And part of the reason that we won't prove this is that it takes a bit more background of the whole numbers than we have time for. But in the meantime, if we'll just assume that associativity and commutativity do actually hold, they will help make our process a lot easier. So let's take a look at a couple of examples. So we want to prove that 4 times negative 5 is equal to negative 20. And so this is a proved statement, so we want to use our definition of multiplication by a whole number, 4 times negative 5. Well that's the sum of 4 negative 5's, here's 1, 2, 3, 4 negative 5's put together. And again, because the addition is part of my definition of multiplication, I can just do that addition without any further commentary. And there's my proof. Well, here's an important example, what if I take negative 1 times 3? So since we're assuming commutativity and associativity of integer multiplication, I can rewrite the product, so I can use the definition of multiplication. So negative 1 times 3 is the same as 3 times negative 1. And I'm going to use my definition of multiplication. This is 3 negative 1's added together. And again, my definition of multiplication says I could go ahead and just do that addition without further comment. And I have negative 1 times 3 is equal to negative 3. And again, I'll join my starting point to my ending point to give my result. Negative 1 times 3 equals negative 3. Here's my proof, and here is what it is equal to. Here's what the evaluation of that product, negative 1 times negative 3. Now the proceeding is an example of a more general property of integer multiplication. In general, if I multiply by the value negative 1, for any integer, negative 1 times a is going to be the same as negative a. And combined with associativity and commutativity, we can use this to prove additional properties. For example, evaluate and prove negative 6 times negative 4, and here we have a restriction. We're only going to use associativity, commutativity, and the properties negative 1 times a is negative a, and that first property negative negative a is equal to a. So let's take a look at that. Well first off, I am allowed to use the property negative 1 times a and negative a are the same thing, so that allows me to rewrite negative 6 and negative 4 as negative 6 is negative 1 times 6, negative 4 is negative 1 times 4. And there's the property that I'm allowed to use. Now again I have associativity and commutativity, so with associativity and commutativity I can rewrite the product any way that I want to, and the way that I'm going to want to is to put these whole number values together, because I know how to multiply two whole numbers. So I'll rearrange things this way. I can multiply two whole numbers, not a problem, and again I'm allowed to use the property negative 1 times a is negative a. Well here's negative 1 times 24, so that product must be negative 24. And again I'm allowed to use the property negative 1 times negative a. So I'm actually going to use that property twice. This negative 1 times negative 24 is the additive inverse of negative 24. And then finally I have this, well I am allowed to use the property negative negative a is equal to a, and so that's going to complete the proof of my evaluation. I have simultaneously evaluated negative 6 times negative 4, it's 24, and the proof using only associativity, commutativity, and these properties is all of this.