Arithmetic Multiplication is Associative

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Theorem

Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.

Then:

$\forall x, y, z \in \mathbb F: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

That is, the operation of multiplication on the standard number sets is associative.


Natural Number Multiplication is Associative

The operation of multiplication on the set of natural numbers $\N$ is associative:

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


Integer Multiplication is Associative

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

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


Rational Multiplication is Associative

The operation of multiplication on the set of rational numbers $\Q$ is associative:

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


Real Multiplication is Associative

The operation of multiplication on the set of real numbers $\R$ is associative:

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


Complex Multiplication is Associative

The operation of multiplication on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$