Complex Multiplication is Commutative

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Theorem

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

$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$


Proof

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

\(\displaystyle z\) \(:=\) \(\displaystyle \tuple {x_1, y_1}\)
\(\displaystyle w\) \(:=\) \(\displaystyle \tuple {x_2, y_2}\)

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


Then:

\(\displaystyle z_1 z_2\) \(=\) \(\displaystyle \tuple {x_1, y_1} \tuple {x_2, y_2}\) Definition 2 of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + x_2 y_1}\) Definition of Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \tuple {x_2 x_1 - y_2 y_1, x_1 y_2 + x_2 y_1}\) Real Multiplication is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \tuple {x_2 x_1 - y_2 y_1, x_2 y_1 + x_1 y_2}\) Real Addition is Commutative
\(\displaystyle \) \(=\) \(\displaystyle \tuple {x_2, y_2} \tuple {x_1, y_1}\) Definition of Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle z_2 z_1\) Definition 2 of Complex Number

$\blacksquare$


Examples

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

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

$\paren {2 - 3 i} \paren {4 + 2 i} = 14 - 8 i$


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

$\paren {4 + 2 i} \paren {2 - 3 i} = 14 - 8 i$


As can be seen:

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

$\blacksquare$


Also see


Sources