# Integer Multiplication is Commutative

## Theorem

The operation of multiplication on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x \times y = y \times x$

## Proof

From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.

Then:

 $\displaystyle x \times y$ $=$ $\displaystyle \eqclass {a, b} {}\times \eqclass {c, d} {}$ Definition of Integer $\displaystyle$ $=$ $\displaystyle \eqclass {a c + b d, a d + b c} {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle \eqclass {c a + d b, d a + c b} {}$ Natural Number Multiplication is Commutative $\displaystyle$ $=$ $\displaystyle \eqclass {c a + d b, c b + d a} {}$ Natural Number Addition is Commutative $\displaystyle$ $=$ $\displaystyle \eqclass {c, d} {} \times \eqclass {a, b} {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle y \times x$ Definition of Integer

$\blacksquare$