Complex Multiplication is Associative

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Theorem

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

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


Proof

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

\(\displaystyle z_1\) \(:=\) \(\displaystyle \tuple {x_1, y_1}\) $\quad$ $\quad$
\(\displaystyle z_2\) \(:=\) \(\displaystyle \tuple {x_2, y_2}\) $\quad$ $\quad$
\(\displaystyle z_3\) \(:=\) \(\displaystyle \tuple {x_3, y_3}\) $\quad$ $\quad$

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


Thus:

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

$\blacksquare$


Examples

Example: $\paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} } = \paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i}$

Example: $\paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} }$

$\paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} } = 8 + 51 i$


Example: $\paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i}$

$\paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i} = 8 + 51 i$


As can be seen:

$\paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} } = \paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i}$

$\blacksquare$


Sources