 Let's take a little bit more about the additive inverse. The additive inverse of A is written as additive inverse of A. Sometimes we read this as negative A. And it's whatever we need to combine with A to get nothing. So what's the additive inverse of the additive inverse of 3? Well, let's think about that. If I have three negative chips, what do we need to combine them with to get nothing? Since we can combine a positive and a negative chip to get nothing, we need three positive chips. And if we combine the three negative chips with the three positive chips, we get nothing. And the additive inverse is what we needed to add. So the additive inverse of the additive inverse of 3 is 3. And so we found that the additive inverse of the additive inverse of 3 is 3. And this leads to a general result for any real number. The additive inverse of the additive inverse is just the number itself. Now, before we introduce multiplication and division, let's review what a multiplication is. A multiplication is a repeated addition. So when I write 3A, what I really mean is this is 3As added together. And we might make the following observation. The inverse of 3 is 3 negative chips, because that's what we need to add in order to get nothing. But that's really the same as negative 1 plus negative 1 plus negative 1. And that's three negative 1s added together. And so we could say the additive inverse of 3 is 3 negative 1s. And this leads to the following idea. For any real number, the additive inverse of A is A times negative 1. So let's consider a multiplication, 7, by the additive inverse of 5. So our theorem says this additive inverse of 5 is the same as 5 times negative 1. Associativity and commutativity of the real number says that I can multiply 7 by 5 first. And my theorem says that if I multiply a number by negative 1, I get the additive inverse of that number. So 35 times negative 1 is the additive inverse of 35. What if I have two negatives? Well, multiply the additive inverse of 4 by the additive inverse of 3. The theorem says additive inverse of 4 is 4 times negative 1. And additive inverse of 3 is 3 times negative 1. Commutativity and associativity of multiplication means I could rewrite this in any order that I want. So how about this order? I know what 4 times 3 is. The theorem says that 12 times negative 1 is the additive inverse of 12. I'm multiplying by negative 1 again, so I get the additive inverse of the additive inverse of 12. And I know that the additive inverse of an additive inverse is just the number itself. And so my final answer is 12. So I found that 7 times negative 5 is negative 35. And negative 4 times negative 3 is 12. And again, let's ignore those signs for a moment. We know that 7 times 5 is 35, and 4 times 3 is 12. And again, we have these same numbers. Just the signs are distributed in there somehow. And so this leads to the following result. The product AB has value equal to the product of the absolute values, and the sign of the product will be positive if both have the same sign and negative if they have opposite signs. So if I want to find 2 times negative 5, we'll ignore the sign and find the product of the absolute values. 2 times 5 is 10. And since 2 and negative 5 have opposite signs, the product will be negative. For the second, we'll multiply negative 3 and 2 first. We'll ignore the signs and find the product of the absolute values, which is going to be 6. And since negative 3 and 2 have opposite signs, that product is going to be negative. So we'll multiply and get negative 6 with the negative 4 carrying along for the ride. Now we want to multiply negative 6 and negative 4. The product of the absolute values is 24. And since the factors have the same sign, the product is positive.