Question

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

Short answer

An even integer's additive inverse or negative is also an even number.

Step-by-step solution

Introduction

Consider the even number\(2k\).

The purpose is to use direct proof method, to prove that the additive inverse or negative of an even number is an even number.

Suppose that\(n\) is some even integer, then by definition of even number\(n = 2k\)for some integer\(k\).

Defining additive inverse to prove the required

The negative or additive inverse of \(n\) is \( - n\).

Then, the additive inverse can be written as,

\(\begin{array}{} - n = - \left( {2k} \right)\\ - n = - 2k\\ - n = 2\left( { - k} \right)\end{array}\)

Because \(k\)is an integer, it follows that \(p = - k\)is also an integer.

Thus, \( - n = 2p\)for some integer\(p\).

Therefore, by even integer definition,\( - n\)is an even integer.

As a result, an even integer's additive inverse or negative is also an even number.