Multiplying positive & negative numbers

We know that if we were to multiply 2 times 3, that would give us positive 6. And since we're going to start thinking about negative numbers in this video, one way to think about it I had a positive number times another positive number, and that gave me a positive number. So if I have a positive times a positive, that will give me a positive number. Now, let's mix it up a little bit, introduce some negative numbers. So what happens if I had negative 2 times 3? Well, one way to think about, and we'll talk more about the intuition in this video and in future videos, is, well, you could view this as negative 2 repeatedly added three times. So this could be negative 2 plus negative 2 plus negative 2-- not negative 6-- plus negative 2, which would be equal to-- well, negative 2 plus negative 2 is negative 4 plus another negative 2 is negative 6. So this would be equal to negative 6. Or another way to think about it is if I had 2 times 3, I would get 6. But because one of these two numbers is negative, then my product is going to be negative. So if I multiply a negative times a positive, I'm going to get a negative. Now, what if we swap the order in which we multiply? So if we were to multiply 3 times negative 2. Well, it shouldn't matter. The order in which we multiply things shouldn't change the product. Whether we multiply 2 times 3, we'll get 6, or if we multiply 3 times 2, we'll get 6. And so we should have the same property here. 3 times negative 2 should give us the same result. It's going to be equal to negative 6. And once again, we say 3 times 2 would be 6. One of these two numbers is negative. And so our product is going to be negative. So we could write a positive times a negative is also going to be a negative. And both of these are just the same thing with the order in which we're multiplying switched around. But this is one of the two numbers are negative, exactly one. So one negative, one positive number is being multiplied. Then you will get a negative product. Now let's think about the third circumstance when both of the numbers are negative. I'll just switch colors for fun here. If I were to multiply negative 2 times negative 3-- and this might be the least intuitive for you of all. And here I'm just going to introduce you to the rule. And in future videos, we'll explore why this is and why this makes mathematics more all fit together. But this is going to be, you say, well, 2 times 3 would be 6, and I have a negative times a negative. And one way you can think about it is that the negatives cancel out. And so you will actually end up with a positive 6. I actually don't have to write a positive here, but I'll write it here just to reemphasize. This right over here is a positive 6. So we have another rule of thumb here. If I have a negative times a negative, the negatives are going to cancel out. And that's going to give me a positive number. Now, with these out of the way, let's just do a bunch of examples. I encourage you to try them out before I do them. Pause the video, try them out, and see if you get the same answer. So let's try negative 1 times negative 1. Well, 1 times 1 would be 1, and we have a negative times a negative. They cancel out. Negative times a negative give me a positive, so this is going to be positive 1. I could just write 1, or I could literally write a plus sign there to emphasize that this is a positive 1. What happens if I did negative 1 times 0? Now, this might say, wait, this doesn't really fit into any of these circumstances. 0 is neither positive nor negative. And here you just have to remember anything times 0 is going to be 0. So negative 1 times 0 is going to be 0. Or I could have said 0 times negative 783, that is also going to be 0. Let me do some interesting ones. What about-- I'll pick a new color-- 12 times negative 4? Well, once again, 12 times positive 4 would be 48. And we're in the circumstance where one of these two numbers right over here is negative, this one right over here. If exactly one of the two numbers is negative, then the product is going to be negative. We are in this circumstance right over here. We have one negative, so the product is negative. You could imagine this as repeatedly adding negative 4 twelve times. And so you would get to negative 48. Let's do another one. What is 7 times 3? Well, this is a bit of a trick. There are no negative numbers here. This is just going to be 7 times 3, positive 7 times positive 3, the first circumstance, which you already knew how to do before this video. This would just be equal to 21. Let's do one more. So if I were to say negative 5 times negative 10, well, once again, negative times a negative, the negatives cancel out. Then you're just left with a positive product. So it's going to be 5 times 10. It's going to be 50. The negative and the negatives cancel out. Your product is going to be positive. That's this situation right over there.