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Prove That -(-a) = a

Date: 09/11/2001 at 22:21:28
From: Eduardo 
Subject: Why does - (-a) = a

Why does -(-a) = a? How do you prove this using the properties of real 

Date: 09/12/2001 at 08:26:08
From: Doctor Rick
Subject: Re: Why does - (-a) = a

Hi, Eduardo.

We can start with how -a is defined. It is "the additive inverse of a" 
- that is, it is the number that, when added to a, gives 0:

 a + -a = 0

Therefore -(-a) means the number that, when added to -a, gives 0. But 
applying the commutative property of addition, the equation above 

  -a + a = 0

Therefore the number that, when added to -a, gives 0 is a; or,

  -(-a) = a

A closely related, but different, question is how we can prove that

  -1 * -1 = 1

The theorem linking these two is this:

  -1 * a = -a

Let's prove this. Start with the fact that zero times any number is 

  0 * a = 0

Write 0 as (1 + -1), which follows from the definition of -1.

  (1 + -1)*a = 0

Apply the distributive property:

  1*a + -1*a = 0

Use the fact that 1 times any number is the same number:

  a + -1*a = 0

Now, the number that, when added to a, gives 0 is -a. Therefore

  -a = -1*a

Using this theorem, we can easily prove that -1 times -1 is 1:

  -1 * -1 = -(-1)
          = 1

For other approaches to the question of why a negative times a 
negative is positive (both informal illustrations and formal proofs), 
see our Dr. Math FAQ, "Negative times a negative":   

- Doctor Rick, The Math Forum   
Associated Topics:
High School Negative Numbers
High School Number Theory
Middle School Negative Numbers

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