Definition:Integer

Informal Definition
The numbers $\left\{{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}\right\}$ are called the integers.

This set is usually denoted $\Z$ (Z for Zahlen, which is German for whole numbers, with overtones of unbroken).

An individual element of $\Z$ is called an integer.

Notation
Note that $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\left[\!\left[{a, b}\right]\!\right]_\boxminus$, or $\left[\!\left[{a, b}\right]\!\right]$.

This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\left[\!\left[{a, b}\right]\!\right]$ at the start of any exposition.

Linguistic Note
The word integer is pronounced with the stress on the first syllable, and the g is soft (i.e. sounds like j): in-te-jer.

This is inconsistent with the pronunciation of the related term integral where the g is hard (as in get): in-te-gral.

Also known as
The integers are also referred to as whole numbers, so as to distinguish them from fractions. However, use of this term is discouraged because it is ambiguous: it can refer to the integers, the positive integers, or the negative integers, depending on the preference of the author.

Variants on $\Z$ are often seen, for example $\mathbf Z$ and $\mathcal Z$, or even just $Z$.

Some sources use $I$, while others use $\mathbf J$ or a variant.