Integer Multiplication is Associative

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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

$\blacksquare$


Sources