Definition:Multiplication/Integers

Definition
The multiplication operation in the domain of integers $\Z$ is written $\times$.

Let us define $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$ as in the formal definition of integers.

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

$\boxtimes$ is the congruence relation defined on $\N \times \N$ by $\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\left[\!\left[{a, b}\right]\!\right]$ to mean $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$, as suggested.

As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same.

Thus multiplication can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.

That is, the integers being defined as all the difference congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes.

It follows that:


 * $\forall a, b, c, d \in \N: \left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{a \times c + b \times d, a \times d + b \times c}\right]\!\right]$ or, more compactly, as $\left[\!\left[{ac + bd, ad + bc}\right]\!\right]$.

This can also be defined as:


 * $n \times m = +^n m = \underbrace{m + m + \cdots + m}_{\text{$n$ copies of $m$}}$

... and the validity of this is proved in Index Laws for Monoids.

Also see

 * Integer Multiplication is Commutative
 * Integer Multiplication is Associative
 * Integer Multiplication is Well-Defined