# Definition:Even Integer

## Contents

## Definition

### Definition 1

An integer $n \in \Z$ is **even** if and only if it is divisible by $2$.

### Definition 2

An integer $n \in \Z$ is **even** if and only if it is of the form:

- $n = 2 r$

where $r \in \Z$ is an integer.

### Definition 3

An integer $n \in \Z$ is **even** if and only if:

- $x \equiv 0 \pmod 2$

where the notation denotes congruence modulo $2$.

The set of even integers can be denoted $2 \Z$.

## Sequence of Even Integers

The first few non-negative even integers are:

- $0, 2, 4, 6, 8, 10, \ldots$

This sequence is A005843 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Euclid's Definition

In the words of Euclid:

*An***even number**is that which is divisible into two equal parts.

(*The Elements*: Book $\text{VII}$: Definition $6$)

Euclid went further and categorised even numbers according to:

- whether they are multiples of $4$

and:

## Even-Times Even

Let $n$ be an integer.

Then $n$ is **even-times even** if and only if it has $4$ as a divisor.

The first few non-negative **even-times even** numbers are:

- $0, 4, 8, 12, 16, 20, \ldots$

This sequence is A008586 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

In the words of Euclid:

*An***even-times even number**is that which is measured by an even number according to an even number.

(*The Elements*: Book $\text{VII}$: Definition $8$)

## Even-Times Odd

Let $n$ be an integer.

Then $n$ is **even-times odd** if and only if it has $2$ as a divisor and also an odd number.

The first few non-negative **even-times odd** numbers are:

- $2, 6, 10, 12, 14, 18, \ldots$

In the words of Euclid:

*An***even-times odd number**is that which is measured by an even number according to an odd number.

(*The Elements*: Book $\text{VII}$: Definition $9$)

## Examples

$6$ is an even integer:

- $6 = 2 \times 3$

$-10$ is an even integer:

- $-10 = 2 \times \paren {-5}$

## Also see

## Historical Note

The concept of classifying numbers as **odd** or **even** appears to have originated with the Pythagoreans.

It was their belief that **even numbers** are **female**, and **odd numbers** are **male**.

A commentator on Plato used the term **isosceles number** for an **even number**, in correspondence with the concept of an isosceles triangle. In a similar way an odd number was described as **scalene**.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$