integer

An integer is either zero, a positive natural number, or the additive inverse of a positive natural number.

Formally,
the set of integers is

\begin{aligned} Z & = {0, \pm 1, \pm 2, \dots} \\ Z & = {\dots, - 2, - 1, 0, 1, 2, \dots} \end{aligned}

The additive inverses of the positive natural numbers are the negative integers.

The integers are closed under addition, subtraction, and multiplication (the sum, difference, or product of any two integers is also an integer).

The integers form the smallest group and the smallest ring containing the natural numbers.

$Z$ under addition is an abelian group. It is also the only infinite cyclic group.

Latin integer meaning "whole" or "untouched"
French entier from Latin integer and English "entire"
Letter $Z$ from German Zahlen ("numbers")