Definition:Odd Integer/Definition 1

Definition

An integer $n \in \Z$ is odd if and only if it is not divisible by $2$.

That is, if and only if it is not even.

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
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): odd integer