Definition:Odd Integer/Definition 3

Definition

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

$x \equiv 1 \pmod 2$

where the notation denotes congruence modulo $2$.

Euclid's Definition

In the words of Euclid:

An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.

(The Elements: Book $\text{VII}$: Definition $7$)

Sequence of Odd Integers

The first few non-negative odd integers are:

$1, 3, 5, 7, 9, 11, \ldots$

Also see

• Equivalence of Definitions of Odd Integer

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Example $\text A.3$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.23$