Definition:Integer/Formal Definition

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


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

The fact that this is a congruence relation is proved in Equivalence Relation on Semigroup Product with Cancellable Elements.

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 $\N$:
 * $\left({x_1, y_1}\right) \oplus \left({x_2, y_2}\right) = \left({x_1 + x_2, y_1 + y_2}\right)$

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

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$.

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

Let us use $+'$ to denote the operation $\oplus_\boxtimes$.

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 $\boxtimes$.

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

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.

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.