 We can also multiply and define signed numbers. An important idea is how you speak influences how you think. And to that end, we have to introduce one bit of terminology here. Numbers like this should be read as the additive inverse of 3, because this reminds us of the most important fact about this number. When you add it to 3, you get 0. However, because it is something of a bother to say the additive inverse of 3 every time we read this number, it is usually read as negative 3. It's very important to remember that it is still the additive inverse of 3 and the most important property is that when you add it to 3, you'll get 0. Calling it negative 3 is just a shorthand for what is the correct expression, the additive inverse. We say that the additive inverses of the whole numbers, with the exception of 0, are called negative numbers, and that means we also want to give a special name for the whole numbers. Again, except for 0, are going to be the positive numbers. 0 itself is not considered positive and is not considered negative. So how do we multiply or divide signed numbers? For that, we'll rely on a key theorem when multiplying or dividing signed numbers. For integers A and B, the product AB and the quotient A divided by B will have the value of the product or quotient of the unsigned numbers. In other words, ignore the sign and then find the product or quotient as normal. Then take care of the sign at the end because the sign is positive if both A and B have the same sign and negative otherwise. So let's say I want to find 5 times negative 3. So first of all, I'll just ignore the signs and find 5 times 3, but because the numbers have different signs, one is positive, one is negative, and my final result is going to be negative. So my final result, negative 15. Negative 35 divided by negative 5. So I'll ignore the signs and treat this as 35 divided by 5, but because they both have the same sign, my final result will be positive. We might indicate that with a plus, and that gives us the value plus 7. However, traditionally, we omit the plus when specifying a positive number. Again, you might have to do more than one thing. Negative 6 times negative 3 times negative 5. And remember, when values are put next to each other like this, there is an implied multiplication. So multiplication should be performed from left to right. So I'll multiply negative 6 and negative 3. Both have the same sign, so the final result will be positive and I'll multiply 6 by 3 to get 18. 18 times negative 5. Ignore the sign, that's 18 times 5, but because they have different signs, then the final result will be negative, and so 18 by 5 is 90, and my final answer, negative 90. You may have to do more than one thing, like working with the order of operations. So let's take a look at this horrible expression. Operations and parentheses come first, negative 2, well, there's no operation there, but 1 minus 5, I do need to evaluate that. I can't subtract 5 from 1, so I'll use my theorem that says I can reverse the order of subtraction by converting it into an additive inverse. So that's additive inverse of 5 minus 1. I know what 5 minus 1 is, and so this 1 minus 5 is the same as negative 4. So I have a subtract, a multiply, a subtract, a multiply, I should do those multiplications first. I'm not subtracting, so this is actually 3 times negative 2. So I'll ignore the signs, that's 3 times 2, and because the factors 3 and negative 2 have opposite signs, my final result will be negative. Likewise, this minus is actually a subtraction, so this product is 2 times negative 4, which will be negative 8. And so I have 4 minus negative 6 minus negative 8. So now I have subtractions, which I should perform from left to right, 4 minus negative 6. Our theorem says that A minus negative B is the same as A plus B. So 4 minus negative 6 is the same as 4 plus 6, which is 10. And then now I have 10 minus negative 8. Our theorem says that that's the same as 10 plus 8, which gives us our final answer of 18. And we can take a look at another horrible expression like this. Stuff inside the parentheses has to be taken care of first. And again, this is negative 3. It's not an operation. The second set of parentheses, 3 minus 8 times 2. Multiplication first, 8 times 2 is 16. Everything else stays the same. 3 minus 16. Well, I'll pull in my theorem about subtraction of integers. So I know that 3 minus 16 is the same as negative 16 minus 3. And that's going to be negative 13. And everything else stays the same. I have a division, 12 divided by negative 3. So I'll ignore the signs. That's 12 divided by 3. But because the operands, 12 and negative 3, have opposite signs, my final result will be negative 4. And again, everything else will just copy straight down. Now I have negative 4 minus negative 13. And so our theorem says that this is the same as negative 4 plus 13. And because of commutativity, this is the same as 13 plus negative 4. And we have an app for that, otherwise known as a theorem, that a plus negative b is the same as a minus b. So 13 plus negative 4 is the same as 13 minus 4. And we know what that is.