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:


 * $$z_1 = x_1 + i y_1$$
 * $$z_2 = x_2 + i y_2$$

where $$i = \sqrt {-1}$$ and $$x_1, x_2, y_1, y_2 \in \R$$.

Then:

$$ $$ $$ $$ $$