Complex Multiplication Distributes over Addition

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Theorem

The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition.

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


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:

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


The result $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$ follows directly from the above, and the fact that Complex Multiplication is Commutative.

$\blacksquare$


Examples

Example: $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = \paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$

Example: $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} }$

$\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = -2 + 9 i$


Example: $\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$

$\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i} = -2 + 9 i$


As can be seen:

$\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = \paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$

$\blacksquare$


Sources