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Let's look at a number...  5...  Because this is my favorite number!

5

What number can we add to 5 to get 0 (which is the additive identity) as the answer?

5 + __ = 0

You should be thinking about a negative number.  In fact, you should be thinking about a number that is the opposite of 5!  -5!

5 + -5 = 0

So, the additive inverse of 5 is -5!

Let's go the other way...

What number can we add to -5 to get 0 as the answer?

-5 + __ = 0

What's the opposite of -5?  5!

-5 + 5 = 0

So, the additive inverse of -5 is 5!

Here's the official property:

The Additive Inverse Property:

The additive inverse of a number,  a  is -a so that

a + -a = 0
 
and

 
-a + a = 0

This is often written in one line...

a + -a = 0 = -a + a

Where a is any real number and -a is the opposite of a.

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