Integer Multiplication is Commutative

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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

$\blacksquare$


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