Associative Law of Multiplication

Theorem
On all the number systems:
 * natural numbers $\N$
 * integers $\Z$
 * rational numbers $\Q$
 * real numbers $\R$
 * complex numbers $\C$

the operation of multiplication is associative:


 * $a \times \paren {b \times c} = \paren {a \times b} \times c$

Euclid's Definition

 * If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.

That is, if:
 * $na, nb$ are equimultiples of $a, b$

and if:
 * $m \cdot na, m \cdot nb$ are equimultiples of $na, nb$

then:
 * $m \cdot na$ is the same multiple of $a$ that $m \cdot nb$ is of $b$

Alternatively, this can be expressed as:
 * $m \cdot na = mn \cdot a$

Proof
This is demonstrated in these pages:


 * Natural Number Multiplication is Associative
 * Integer Multiplication is Associative
 * Rational Multiplication is Associative
 * Real Multiplication is Associative
 * Complex Multiplication is Associative