Definition:Odd Integer/Definition 2

Definition

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

$\exists m \in \Z: n = 2 m + 1$

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

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Example $\text A.3$
• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.1$ The Division Algorithm
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): odd integer