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# Inverse property of addition

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

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.