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Inverse property of addition

The simple idea that a number plus its negative is 0. Created by Sal Khan.

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Video transcript

Let's say that we have the number 5, and we're asked, what number do we add to the number 5 to get to 0? And you might already know this, but I'll just draw it out. So let's say we have a number line right over here. And 0 is sitting right over there. And we are already sitting here at 5. So to go from 5 to 0, we have to go five spaces to the left. And if we're going five spaces to the left, that means that we are adding negative 5. So if we add negative 5 right here, then that is going to get us back to 0. That is going to get us back right over here to 0. And you probably already knew this. And this is a pretty maybe common sense thing right here. But there's a fancy word for it called the additive inverse property. And all the additive-- I'll just write it down. I think it's kind of ridiculous that it's given such a fancy word for such a simple idea-- additive inverse property. And it's just the idea that if you have a number and you add the additive inverse of the number, which is what most people call the negative of the number-- if you add the negative of the number to your number, you're going to get back to 0 because they have the same size, you could view it that way. They both have a magnitude of 5, but this is going five to the right and then you're going five back to the left. Similarly, if you started at-- let me draw another number line right over here-- if you started at negative 3. If you're starting right over here at negative 3, so you've already moved three spaces to the left, and someone says, well what do I have to add to negative 3 to get back to 0? Well, I have to move three spaces to the right now. And three spaces to the right is in the positive direction. So I have to add positive 3. So if I add positive 3 to negative 3, I will get 0. So in general, if I have any number-- if I have 1,725,314 and I say, what do I need to add to this to get back to 0? Well, I have to essentially go in the opposite direction. I have to go in the leftwards direction. So I'm going to subtract the same amount. Or I could say, I'm going to add the additive inverse, or I'm going to add the negative version of it. So this is going to be the same thing as adding negative 1,725,314 and that'll just get me back to 0. Similarly, if I say, what number do I have to add to negative 7 to get to 0? Well, if I'm already at negative 7, I have to go 7 to the right so I have to add positive 7. And this is going to be equal to 0. And this all comes from the general idea 5 plus negative 5, 5 plus the negative of 5, or 5 plus the additive inverse of 5, you can just view this as another way of 5 minus 5. And if you have five of something, and you take away five, you've learned many, many years ago that that is just going to get you to 0.