The Even Numbers

An even number is an integer that can be divided exactly by [latex]2[/latex]. When an even number is divided by [latex]2[/latex], the remainder is zero.

By performing mental math, it’s obvious that the numbers below, including the negative numbers, are even because they are all divisible by 2.

examples of negative numbers which include negative and positive integers, and zero

In addition, I want to point out that many students think zero is neither even nor odd.

Believe me, zero is considered an even number for the same simple reason that it is also an integer that is divisible by [latex]2[/latex] thus no remainder when divided by [latex]2[/latex]. That is, [latex]0 \div 2 = 0[/latex].

Observation: We can generalize from the examples above that even numbers always end [latex]0[/latex], [latex]2[/latex], [latex]4[/latex], [latex]6[/latex], or [latex]8[/latex].

However, there is a better way to define an even number because it’s more mathematically precise. Here it is!

General Form of an Even Number

the general form of an even number which is shown as n=2k, where k is an integer

DEFINITION: The number [latex]\large{n}[/latex] is an even number if we can express it as [latex]\large{2k}[/latex] where [latex]\large{k}[/latex] is just another integer.

Examples of Even Numbers Expressed in General Form

Let’s put the general form of an even number to the test. Whatever math concept we’re presented with, it is crucial that we verify it.

Below are a few examples to showcase the concept of an even number as [latex]n = 2\,k[/latex] where [latex]k[/latex] is an integer.

[latex]\color{red}\LARGE {n = 2k}[/latex]

  • [latex]0 \to 0 = 2\left( 0 \right)[/latex]
  • [latex]14 \to 14 = 2\left( 7 \right)[/latex]
  • [latex] – 32 \to – 32 = 2\left( { – 16} \right)[/latex]
  • [latex]50 \to 50 = 2\left( {25} \right)[/latex]
  • [latex] – 78 \to – 78 = 2\left( { – 39} \right)[/latex]

The constant appearance of [latex]2[/latex] as one of the factors of an even number suggests that any even numbers are indeed a multiple of [latex]2[/latex].

You may also be interested in these related math lessons or tutorials:

What is an odd number?

List of Even Numbers

List of Odd Numbers