# Integer Multiplication is Associative

## Theorem

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

$\forall x, y, z \in \Z: x \times \paren {y \times z} = \paren {x \times y} \times z$

## Proof

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

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

Then:

 $\displaystyle x \times \paren {y \times z}$ $=$ $\displaystyle \eqclass {a, b} {} \times \paren {\eqclass {c, d} {} \times \eqclass {e, f} {} }$ Definition of Integer $\displaystyle$ $=$ $\displaystyle \eqclass {a, b} {} \times \eqclass {c e + d f, c f + d e} {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle \eqclass {a \paren {c e + d f} + b \paren {c f + d e}, a \paren {c f + d e} + b \paren {c e + d f} } {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle \eqclass {a c e + a d f + b c f + b d e, a c f + a d e + b c e + b d f} {}$ Natural Number Multiplication Distributes over Addition $\displaystyle$ $=$ $\displaystyle \eqclass {a c e + b d e + a d f + b c f, a c f + b d f + a d e + b c e} {}$ Natural Number Addition is Commutative and Associative $\displaystyle$ $=$ $\displaystyle \eqclass {\paren {a c + b d} e + \paren {a d + b c} f, \paren {a c + b d} f + \paren {a d + b c} e} {}$ Natural Number Multiplication Distributes over Addition $\displaystyle$ $=$ $\displaystyle \eqclass {a c + b d, a d + b c} {} \times \eqclass {e, f} {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle \paren {\eqclass {a, b} {} \times \eqclass {c, d} {} } \times \eqclass {e, f} {}$ Definition of Integer Multiplication $\displaystyle$ $=$ $\displaystyle \paren {x \times y} \times z$ Definition of Integer

$\blacksquare$