Complex Multiplication is Associative

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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:

$z_1 := \left({x_1, y_1}\right)$
$z_2 := \left({x_2, y_2}\right)$
$z_3 := \left({x_3, y_3}\right)$

where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.


Thus:

\(\displaystyle \) \(\) \(\displaystyle z_1 \left({z_2 z_3}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1, y_1}\right) \left({\left({x_2, y_2}\right) \left({x_3, y_3}\right)}\right)\) $\quad$ Definition 2 of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1, y_1}\right) \left({x_2 x_3 - y_2 y_3, x_2 y_3 + y_2 x_3}\right)\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 \left({x_2 x_3 - y_2 y_3}\right) - y_1 \left({x_2 y_3 + y_2 x_3}\right), y_1 \left({x_2 x_3 - y_2 y_3}\right) + x_1 \left({x_2 y_3 + y_2 x_3}\right)}\right)\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 x_2 x_3 - x_1 y_2 y_3 - y_1 x_2 y_3 - y_1 y_2 x_3, y_1 x_2 x_3 - y_1 y_2 y_3 + x_1 x_2 y_3 + x_1 y_2 x_3}\right)\) $\quad$ Real Multiplication Distributes over Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1 x_2 x_3 - y_1 y_2 x_3}\right) - \left({x_1 y_2 y_3 + y_1 x_2 y_3}\right), \left({x_1 x_2 y_3 - y_1 y_2 y_3}\right) + \left({y_1 x_2 x_3 + x_1 y_2 x_3}\right)}\right)\) $\quad$ Real Multiplication is Commutative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1 x_2 - y_1 y_2}\right) x_3 - \left({x_1 y_2 + y_1 x_2}\right) y_3, \left({x_1 x_2 - y_1 y_2}\right) y_3 + \left({x_1 y_2 + y_1 x_2}\right) x_3}\right)\) $\quad$ Real Multiplication Distributes over Addition $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right) \left({x_3, y_3}\right)\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x_1, y_1}\right) \left({x_2, y_2}\right)}\right) \left({x_3, y_3}\right)\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({z_1 z_2}\right) z_3\) $\quad$ Definition 2 of Complex Number $\quad$

$\blacksquare$


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