Definition:Even Integer

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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:

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


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:

whether they have an odd divisor:


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:

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


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

  • Results about even integers can be found here.


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