Even & odd numbers of negatives

- [Voiceover] When we first learned to multiply and divide positive and negative numbers, we saw, well look, if you have a positive number times a positive number. I'm just writing "pos" short for positive. Well that, of course, is going to be equal to a positive number that you saw that even before you learned about negative numbers. There's also true if you had the same sign, a negative times a negative, that that was also going to be equal to a positive number. And that the way that you get a negative number, is if you multiply or divide something of different signs. So if you have a positive times a negative that would give you a negative number. Or if you had a negative, negative times a positive, that also would give you a negative number. And here I wrote it for multiplication, but it also applies to division. A negative divided by a negative is positive. A negative divided by a positive would be negative. Now all of these, we thought about if we're only multiplying or dividing two things. But I was just thinking about what happens if we multiply or divide three things, or four things, or five things, or n things together, what we might expect? So let's say that we were to multiply a times, and now I'm going to write the dot for times. A times b times c, a times b times c. Now if I told you that these were all positive numbers, then you say, "OK a times b times c is going "to be positive." "A times b would be positive, and that times c is positive." Now what would happen if I were to tell you that they were all negative numbers. What if a, b, and c were all negative? Well if there were all negative, let me write it that way. Let me actually write, let me write, a, b, and c, a, b, and c, they're all going to be negative. So if that's the case, what is this product going to be equal to? Well you're going to have a negative here, times a negative. So a negative times a negative, a times b, if you do that first, and we can when we multiply these numbers. That's going to give you a positive. So a times b is going to give you a positive. But then you're going to multiply that times c. You're going to multiply that times c, which is a negative. So you're going to have a positive times a negative, which is going to be a negative. So this one, if a, b, and c are all less than zero, then the product, a, b, c is going to be less than zero as well. This whole thing is going to be negative. Now, if I did something else. If I said, "There's other ways "that I can make the product negative." If a is, let's say that a, actually let me just write it this way. Let's say that a is positive, b is negative. B is negative, and c is positive. And c is positive. Well here, positive times a negative, if you do this first. Positive times a negative is going to give you a negative. And then a negative times a positive, different signs, is going to give you a negative. So this whole thing, this whole thing is going to be a negative. But let's keep on doing, we said, look if all of them are negative, then this thing would be negative, but that's because I had three numbers here. What if I had four numbers here? What if I had times, What if I had times d here? And if I told you all of these numbers were negative? Let's think about it, if negative, negative, negative, negative. And I can do the multiplication in any order, but I'll just go left to right. A times b, negative times a negative. That would yield a positive. Now if you multiply that product times c, positive times a negative, positive times a negative, positive times a negative that would give you a negative. And then you multiply this negative times this negative, so this whole product, a, b, c, is going to be negative. But then we multiply it times a negative. Well a negative times a negative is going to be a positive. So this whole thing is going to be a positive. And so you're probably seeing a pattern here. If you're multiplying a bunch of numbers, if you're multiplying a bunch of numbers, and if you have an odd number of negatives, odd number of negatives being multiplied, and or divided. I just did multiplication here, but this would have also been true if these were all division symbols. If these were all division symbols, we would've been able to say the exact same thing. If you have an odd number of negative numbers in your product or in your quotient, well then you're going to have a negative, you're going to have a negative value for the entire expression. If you have an even, if you have an even number of negative, well then the whole thing is going to be, the whole thing is going to be positive. And so you can view these as generalizations of what we just saw here. Positive times a positive. You have zero negative. Zero is actually an even number, so this would be positive. Negative times a negative. Well that's an even number of negatives. You have two negatives, so that's this case again. That's this case again, so you're going to be positive. Either of these cases, you have one negative. You have one negative in either of these cases, so it's going to be this case, odd number of negatives. So that's going to be a negative. And so, we can use this knowledge to start dealing with negative numbers and exponents. So if I were to say, I would have a to the, let me throw out a wild number here, A to the 101st power, And we know that a is less than zero. What is this going to be? Well this is taking 101 A's and multiplying them together. You have an odd number, an odd number of negatives being multiplied together? Well this whole then is going, this whole thing is going to be less than zero. If we knew what a was, we could calculate it somehow, but we know that this thing is going to be negative. Let's do something, so, and we could do it other ways. We could say something like this. Actually, let me try another one. What if I tell you that a is less than zero, and b is also less than zero, and if I had a to the 101 power, divided by b to the seventh power, what would this expression be? Would it be positive or negative? Or zero, maybe? Well you see here, a to the 101st power. You have an odd number of, you have an odd number of negative numbers being multiplied. So this whole thing is going to be negative. And the same thing applies here. You have a negative number to an odd power. You have an odd number of this negative number being multiplied together. So that's going to be negative, but then you have a negative divided by a negative. A negative divided by a negative is going to be a positive. So this thing, right here, is going to be positive. So this is just the beginning, and in the next few videos we'll actually do a bunch of examples that really test our understanding of this.