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We know that if we were to multiply two times three,

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that would give us positive six.

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And so we are going to think about negative numbers in this video.

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One way to think about it,

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is that I have a positive number times another positive number,

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and that gives me a positive number.

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So if I have a positive times a positive,

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that would give me a positive number.

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Now it's mixed up a little bit.

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Introduce some negative numbers.

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So what happens if I had negative two times three?

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Negative two times three.

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Well, one way to think about it--

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Now we are talking about intuition in this video

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and in the future videos.

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You could view this as negative two repeatedly added three times.

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So this could be negative two plus negative two plus negative two--

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Not negative six. Plus negative two.

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which would be equal to--

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well, negative two plus negative two is negative four,

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plus another negative two is negative six.

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This would be equal to negative six.

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Or another way to think about it is,

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if I had two times three, I would get six.

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But because one of these two numbers is negative,

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then my product is going to be negative.

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So if I multiply, a negative times a positive,

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I'm going to get a negative.

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Now what if we swap the order which we multiply?

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So if we were to multiply three times negative two,

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it shouldn't matter.

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The order which we multiply things don't change,

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or shouldn't change the product.

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When we multiply two times three, we get six.

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When we multiply three times two, we will get six.

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So we should have the same property here.

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Three times negative two should give us the same result.

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It's going to be equal to negative six.

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And once again we say, three times two would be six.

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One of these two numbers is negative,

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and so our product is going to be negative.

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So we could draw a positive times a negative

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is also going to be a negative.

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And both of these are just the same thing with the order

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which we are multiplying switched around.

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But this is one of the two numbers are negative. Exactly one.

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So one negative, one positive number is being multiplied.

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Then you'll get a negative product.

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Now we'll think about the third circumstance,

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where both of the numbers are negative.

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So if I were to multiply--I'll just switch colors for fun here--

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If I were to multiply negative two times negative three--

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this might be the least intuitive for you of all,

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and here I'm going to introduce you the rule,

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in the future I will explore why this is,

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and why this makes mathematics more--all fit together.

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But this is going to be, you see, two times three would be six.

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And I have a negative times a negative,

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one way you can think about it is that negatives cancel out!

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So you'll actually end up with a positive six.

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Actually I don't have to draw a positive here.

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But I write it here just to reemphasize.

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This right over here is a positive six.

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So we have another rule of thumb here.

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If I have a negative times a negative,

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the negatives are going to cancel out.

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And that's going to give me a positive number.

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Now with these out of the way, let's just do a bunch of examples.

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I'm encouraging you to try them out before I do them.

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Pause the video, try them out, and see if you get the same answer.

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So let's try negative one times negative one.

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Well, one times one would be one.

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And we have a negative times a negative. They cancel out.

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Negative times a negative give me a positive.

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So this is going to be positive one.

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I can just write one,

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or I can literally write a plus sign there to emphasize.

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This is a positive one.

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What happened if I did negative one times zero?

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Now this might seem, this doesn't fit into any of these circumstances,

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zero is neither positive nor negative.

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And here you just have to remember anything times zero

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is going to be zero.

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So negative one times zero is going to be zero.

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Or I could've said zero times negative seven hundred and eighty-three,

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that is also going to be zero.

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Now what about two--let me do some interesting ones.

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What about--I'm looking a new color.

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Twelve times negative four.

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Well, once again, twelve times positive four would be fourty-eight.

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And we are in the circumstance where one of these two numbers,

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right over here, is negative. This one right here.

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If exactly one of the two numbers is negative,

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then the product is going to be negative.

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We are in this circumstance, right over here.

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We have one negative, so the product is negative.

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You could imagine this as repeatedly adding negative four twelve times

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And so you will get to negative fourty-eight.

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Let's do another one.

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What is seven times three?

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Well, this is a bit of a trick. There are no negative numbers here.

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This is just going to be seven times three.

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Positive seven times positive three. The first circumstance,

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which you already knew how to do before this video.

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This would just be equal to twenty-one.

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Let's do one more.

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So if I were to say negative five times negative ten--

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well, once again, negative times a negative.

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The negatives cancel out.

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You are just left with a positive product.

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So it's going to be five times ten. It's going to be fifty.

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The negative and the negative cancel out.

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Your product is going to be positive.

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That's this situation right over there.