Definition:Even Integer
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 Definitions
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:
- $0, 4, 8, 12, 16, 20, \ldots$
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.
Internationalization
Even Integer is translated:
In German: | geraden zahl | (literally: straight number) |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2$
- 1989: George S. Boolos and Richard C. Jeffrey: Computability and Logic (3rd ed.) ... (previous) ... (next): $1$ Enumerability
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$