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Multiplying and dividing negative numbers

Sal explains multiplying and dividing positive and negative numbers. Created by Sal Khan.

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Video transcript

Welcome to the presentation on multiplying and dividing negative numbers. Let's get started. I think you're going to find that multiplying and dividing negative numbers are a lot easier than it might look initially. You just have to remember a couple rules, and I'm going to teach probably in the future like I'm actually going to give you more intuition on why these rules work. But first let me just teach you the basic rules. So the basic rules are when you multiply two negative numbers, so let's say I had negative 2 times negative 2. First you just look at each of the numbers as if there was no negative sign. Well you say well, 2 times 2 that equals 4. And it turns out that if you have a negative times a negative, that that equals a positive. So let's write that first rule down. A negative times a negative equals a positive. What if it was negative 2 times positive 2? Well in this case, let's first of all look at the two numbers without signs. We know that 2 times 2 is 4. But here we have a negative times a positive 2, and it turns out that when you multiply a negative times a positive you get a negative. So that's another rule. Negative times positive is equal to negative. What happens if you have a positive 2 times a negative 2? I think you'll probably guess this one right, as you can tell that these two are pretty much the same thing by, I believe it's the transitive property -- no, no I think it's the communicative property. I have to remember that. But 2 times negative 2, this also equals negative 4. So we have the final rule that a positive times a negative also equals the negative. And actually these second two rules, they're kind of the same thing. A negative times a positive is a negative, or a positive times a negative is negative. You could also say that as when the signs are different and you multiply the two numbers, you get a negative number. And of course, you already know what happens when you have a positive times a positive. Well that's just a positive. So let's review. Negative times a negative is a positive. A negative times a positive is a negative. A positive times a negative is a negative. And positive times each other equals positive. I think that last little bit completely confused you. Maybe I can simplify it for you. What if I just told you if when you're multiplying and they're the same signs that gets you a positive result. And different signs gets you a negative result. So that would be either, let's say a 1 times 1 is equal to 1, or if I said negative 1 times negative 1 is equal to positive 1 as well. Or if I said 1 times negative 1 is equal to negative 1, or negative 1 times 1 is equal to negative 1. You see how on the bottom two problems I had two different signs, positive 1 and negative 1? And the top two problems, this one right here both 1s are positive. And this one right here both 1s are negative. So let's do a bunch of problems now, and hopefully it'll hit the point home, and you also could try to do along the practice problems and also give the hints and give you what rules to use, so that should help you as well. So if I said negative 4 times positive 3, well 4 times 3 is 12, and we have a negative and a positive. So different signs mean negative. So negative 4 times 3 is a negative 12. That makes sense because we're essentially saying what's negative 4 times itself three times, so it's like negative 4 plus negative 4 plus negative 4, which is negative 12. If you've seen the video on adding and subtracting negative numbers, you probably should watch first. Let's do another one. What if I said minus 2 times minus 7. And you might want to pause the video at any time to see if you know how to do it and then restart it to see what the answer is. Well, 2 times 7 is 14, and we have the same sign here, so it's a positive 14 -- normally you wouldn't have to write the positive but that makes it a little bit more explicit. And what if I had -- let me think -- 9 times negative 5. Well, 9 times 5 is 45. And once again, the signs are different so it's a negative. And then finally what if it I had -- let me think of some good numbers -- minus 6 times minus 11. Well, 6 times 11 is 66 and then it's a negative and negative, it's a positive. Let me give you a trick problem. What is 0 times negative 12? Well, you might say that the signs are different, but 0 is actually neither positive nor negative. And 0 times anything is still 0. It doesn't matter if the thing you multiply it by is a negative number or a positive number. 0 times anything is still 0. So let's see if we can apply these same rules to division. It actually turns out that the same rules apply. If I have 9 divided by negative 3. Well, first we say what's 9 divided by 3? Well that's 3. And they have different signs, positive 9, negative 3. So different signs means a negative. 9 divided by negative 3 is equal to negative 3. What is minus 16 divided by 8? Well, once again, 16 divided by 8 is 2, but the signs are different. Negative 16 divided by positive 8, that equals negative 2. Remember, different signs will get you a negative result. What is minus 54 divided by minus 6? Well, 54 divided by 6 is 9. And since both terms, the divisor and the dividend, are both negative -- negative 54 and negative 6 -- it turns out that the answer is positive. Remember, same signs result in a positive quotient in this example we did before, it was product. Let's do one more. Obviously, 0 divided by anything is still 0. That's pretty straightforward. And of course, you can't divide anything by 0 -- that's undefined. Let's do one more. What is -- I'm just going to think of random numbers -- 4 divided by negative 1? Well, 4 divided by 1 is 4, but the signs are different. So it's negative 4. I hope that helps. Now what I want you to do is actually try as many of these multiplying and dividing negative numbers as you can. And you click on hints and it'll remind you of which rule to use. In your own time you might want to actually think about why these rules apply and what it means to multiply a negative number times a positive number. And even more interesting, what it means to multiply a negative number times a negative number. But I think at this point, hopefully, you are ready to start doing some problems. Good luck.