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 $$\mathbb{Z}$$ (Z for "Zahlen", which is German for "whole number").

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

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

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

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

Let $$\boxminus$$ be the congruence relation defined on $$\mathbb{N} \times \mathbb{N}$$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 Theorem 3 of Construction of Inverse Completion.

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

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

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

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

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

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

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

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

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

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

From the comment on Theorem 3 of Construction of Inverse Completion, it can be seen that the equivalence classes which are the elements of $$\mathbb{Z}$$ 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").