Complex Multiplication is Associative

Theorem

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

$\forall z_1, z_2, z_3 \in \C: z_1 \left({z_2 z_3}\right) = \left({z_1 z_2}\right) z_3$

Proof

From the definition of complex numbers, we define the following:

$z_1 = x_1 + i y_1$

$z_2 = x_2 + i y_2$

$z_3 = x_3 + i y_3$

where $i = \sqrt {-1}$ and $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.

Thus:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle z_1 \left({z_2 z_3}\right)\)	\(=\)	\(\displaystyle \left({x_1 + i y_1}\right) \left({\left({x_2 + i y_2}\right) \left({x_3 + i y_3}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x_1 \left({x_2 x_3 - y_2 y_3}\right) - y_1 \left({x_2 y_3 + y_2 x_3}\right)}\right) + i \left({y_1 \left({x_2 x_3 - y_2 y_3}\right) + x_1 \left({x_2 y_3 + y_2 x_3}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x_1 \left({x_2 x_3 - y_2 y_3}\right) - y_1 \left({x_2 y_3 + y_2 x_3}\right)}\right) + i \left({y_1 \left({x_2 x_3 - y_2 y_3}\right) + x_1 \left({x_2 y_3 + y_2 x_3}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({x_1 x_2 x_3 - x_1 y_2 y_3 - y_1 x_2 y_3 - y_1 y_2 x_3}\right) + i \left({y_1 x_2 x_3 - y_1 y_2 y_3 + x_1 x_2 y_3 + x_1 y_2 x_3}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({x_1 x_2 - y_1 y_2}\right) x_3 - \left({x_1 y_2 - y_1 x_2}\right) y_3}\right) + i \left({\left({x_1 x_2 - y_1 y_2}\right) y_3 + \left({x_1 y_2 - y_1 x_2}\right) x_3}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({x_1 x_2 - y_1 y_2}\right) + i \left({x_1 y_2 - y_1 x_2}\right)}\right) \left({x_3 + i y_3}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left({x_1 + i y_1}\right) \left({x_2 + i y_2}\right)}\right) \left({x_3 + i y_3}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({z_1 z_2}\right) z_3\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$