Definition:Odd Integer

Definition

Let $n \in \Z$, i.e. let $n$ be an integer.

Then $n$ is odd if it is not divisible by $2$, i.e. if it is not even.

More precisely, $n$ is odd if there exists some integer $x$, such that $n = 2x + 1$.

The first few non-negative odd numbers are:

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

This sequence is A005408 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Euclid's Definition

As Euclid defined it:

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

(The Elements: Book VII: Definition $7$)

Odd-Times Odd

As Euclid defined it:

An odd-times odd number is that which is measured by an odd number according to an odd number.

(The Elements: Book VII: Definition $10$)

Also see

• Even Integer

Sources

• Seth Warner: Modern Algebra (1965): $\S 24$