# Complex Multiplication is Commutative

## 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:

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

$\blacksquare$