# Direct Proof

• Sep 28th 2009, 04:48 PM
mant1s
Direct Proof
Hi Community,

I am having some trouble with this question:

"Show that the additive inverse, or negative, of an even number is an even number using a direct proof. "

I know it's basically asking me to prove that the inverse of an even number is even. I'm having trouble expressing it in English/ Mathematical operators.

I have:

Let n = 2a, for all !n=2a, n is even

Someone help?

-M
• Sep 28th 2009, 05:25 PM
artvandalay11
The additive inverse of a number x is denoted -(x)

All even numbers are defined to be $\{ 2a | a\in\mathbb{Z} \}$

So we have an even number. Let's call it 2a.

Well it's inverse is $-(2a)=(-1)(2a)=(2a)(-1)=2(a(-1))=2(-a)$, but if $a\in\mathbb{Z}$, then $-a\in\mathbb{Z}$, and so $-(2a)$ is an even number