The Even Numbers

An even number is an integer that can be divided exactly or evenly by \color{red}2. If a number is exactly divided by \color{red}2, it implies that the number in question has a remainder of \color{blue}0 upon the division of \color{red}2.

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 as an even number for the same simple reason that it is also an integer that is divisible by 2 thus no remainder when divided by 2 as well. That is, 0 \div 2 = 0.

Observation: From the examples above, we can easily generalize that even numbers always end with a digit of 0, 2, 4, 6, or 8.

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

image of the general form of an even number which is shown as n=2k

DEFINITION: The number \large{n} is an even number if we can express it as \large{2k} where \large{k} is just another integer. That means, \large{n} is even if \large{n = 2k} such that \large{k} is an 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. It should somehow make sense to us before including it in our “math toolbox”.

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

\color{red}\LARGE {n = 2k}
  • 0 \to 0 = 2\left( 0 \right)
  • 14 \to 14 = 2\left( 7 \right)
  • - 32 \to - 32 = 2\left( { - 16} \right)
  • 50 \to 50 = 2\left( {25} \right)
  • - 78 \to - 78 = 2\left( { - 39} \right)

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

You might also be interested in:

What is an odd number?

List of Even Numbers

List of Odd Numbers