Definition:Integer

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

They 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.

This set is usually denoted $$\Z$$ (Z for "Zahlen", which is German for "whole numbers").

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

Formal Definition
Let $$\left ({\N, +}\right)$$ be the commutative semigroup of natural numbers under addition.

From Inverse Completion of Natural Numbers, we can create $$\left({\N', +'}\right)$$, an inverse completion of $$\left ({\N, +}\right)$$.

From Construction of Inverse Completion, this is done as follows:

Let $$\boxminus$$ be the congruence relation defined on $$\N \times \N$$ by:

$$\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$$

The fact that this is a congruence relation is proved in Construction of Inverse Completion: This Equivalence Relation is a Congruence.

Let $$\left({\N \times \N, \oplus}\right)$$ be the external direct product of $$\left({\N, +}\right)$$ with itself, where $$\oplus$$ is the operation on $\N \times \N$ induced by $+$ on $\N$.

Let the quotient structure defined by $$\boxminus$$ be $$\left({\frac {N \times \N} {\boxminus}, \oplus_{\boxminus}}\right)$$

where $$\oplus_{\boxminus}$$ is the operation induced on $\left({\frac {\N \times \N} \boxminus, \oplus_{\boxminus}}\right)$ by $\oplus$.

Let us use $$\N'$$ to denote the quotient set $$\frac {\N \times \N} {\boxminus}$$.

Let us use $$+'$$ to denote the operation $$\oplus_{\boxminus}$$.

Thus $$\left({\N', +'}\right)$$ is the Inverse Completion of Natural Numbers.

As the Inverse Completion is Unique up to isomorphism, it follows that we can define the structure $$\left({\Z, +}\right)$$ which is isomorphic to $$\left({\N', +'}\right)$$.

An element of $$\N'$$ is therefore an equivalence class of the congruence relation $$\boxminus$$.

So an element of $$\Z$$ is the isomorphic image of an element $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ of $$\frac {\N \times \N} \boxminus$$.

The set of elements $$\Z$$ is called the integers.

From the comment in the proof of Construction of Inverse Completion: This Equivalence Relation is a Congruence, it can be seen that the equivalence classes which are the elements of $$\Z$$ can be characterized by identifying each class with the difference.

Pronunciation
The word "integer" is pronounced with the stress on the first syllable, and the g is soft (i.e. sounds like "j").

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.