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## Video transcript

We know that if we were to multiply two times three, that would give us positive six. And so we are going to think about negative numbers in this video. One way to think about it, is that I have a positive number times another positive number, and that gives me a positive number. So if I have a positive times a positive, that would give me a positive number. Now it's mixed up a little bit. Introduce some negative numbers. So what happens if I had negative two times three? Negative two times three. Well, one way to think about it-- Now we are talking about intuition in this video and in the future videos. You could view this as negative two repeatedly added three times. So this could be negative two plus negative two plus negative two-- Not negative six. Plus negative two. which would be equal to-- well, negative two plus negative two is negative four, plus another negative two is negative six. This would be equal to negative six. Or another way to think about it is, if I had two times three, I would get six. 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 which we multiply? So if we were to multiply three times negative two, it shouldn't matter. The order which we multiply things don't change, or shouldn't change the product. When we multiply two times three, we get six. When we multiply three times two, we will get six. So we should have the same property here. Three times negative two should give us the same result. It's going to be equal to negative six. And once again we say, three times two would be six. One of these two numbers is negative, and so our product is going to be negative. So we could draw a positive times a negative is also going to be a negative. And both of these are just the same thing with the order which we are 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'll get a negative product. Now we'll think about the third circumstance, where both of the numbers are negative. So if I were to multiply--I'll just switch colors for fun here-- If I were to multiply negative two times negative three-- this might be the least intuitive for you of all, and here I'm going to introduce you the rule, in the future I will explore why this is, and why this makes mathematics more--all fit together. But this is going to be, you see, two times three would be six. And I have a negative times a negative, one way you can think about it is that negatives cancel out! So you'll actually end up with a positive six. Actually I don't have to draw a positive here. But I write it here just to reemphasize. This right over here is a positive six. 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'm encouraging 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 one times negative one. Well, one times one would be one. 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 one. I can just write one, or I can literally write a plus sign there to emphasize. This is a positive one. What happened if I did negative one times zero? Now this might seem, this doesn't fit into any of these circumstances, zero is neither positive nor negative. And here you just have to remember anything times zero is going to be zero. So negative one times zero is going to be zero. Or I could've said zero times negative seven hundred and eighty-three, that is also going to be zero. Now what about two--let me do some interesting ones. What about--I'm looking a new color. Twelve times negative four. Well, once again, twelve times positive four would be fourty-eight. And we are in the circumstance where one of these two numbers, right over here, is negative. This one right 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 four twelve times And so you will get to negative fourty-eight. Let's do another one. What is seven times three? Well, this is a bit of a trick. There are no negative numbers here. This is just going to be seven times three. Positive seven times positive three. The first circumstance, which you already knew how to do before this video. This would just be equal to twenty-one. Let's do one more. So if I were to say negative five times negative ten-- well, once again, negative times a negative. The negatives cancel out. You are just left with a positive product. So it's going to be five times ten. It's going to be fifty. The negative and the negative cancel out. Your product is going to be positive. That's this situation right over there.