# Complex Modulus of Product of Complex Numbers/Proof 2

## Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\cmod z$ be the modulus of $z$.

Then:

$\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$

## Proof

Let $\overline z$ denote the complex conjugate of $z$.

Then:

 $\displaystyle \cmod {z_1 z_2}$ $=$ $\displaystyle \sqrt {\paren {z_1 z_2} \overline {\paren {z_1 z_2} } }$ Modulus in Terms of Conjugate $\displaystyle$ $=$ $\displaystyle \sqrt {z_1 \overline {z_1} z_2 \overline {z_2} }$ Product of Complex Conjugates, Complex Multiplication is Commutative $\displaystyle$ $=$ $\displaystyle \sqrt {z_1 \overline {z_1} } \sqrt {z_2 \overline {z_2} }$ Power of Product $\displaystyle$ $=$ $\displaystyle \cmod {z_1} \cmod {z_2}$

$\blacksquare$