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 note the use of the word integral as an adjective, meaning necessary or inherent, usually encountered in rhetoric. For further confusion, this is pronounced in-teg-ral, the stress being on the second syllable.

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.

Some sources refer to the integers as rational integers, without motivating this verbose choice.

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.