# Complex Modulus of Product of Complex Numbers

## Theorem

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

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

Then:

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

### General Result

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

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

Then:

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

## Proof 1

Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.

 $\ds \cmod {z_1 z_2}$ $=$ $\ds \sqrt {\paren {x_1 x_2 - y_1 y_2}^2 + \paren {x_1 y_2 + x_2 y_1}^2}$ Definition of Complex Modulus, Definition of Complex Multiplication $\ds$ $=$ $\ds \sqrt {\paren {x_1^2 x_2^2 + y_1^2 y_2^2 - 2 x_1 x_2 y_1 y_2} + \paren {x_1^2 y_2^2 + x_2^2 y_1^2 + 2 x_1 x_2 y_1 y_2} }$ $\ds$ $=$ $\ds \sqrt {x_1^2 x_2^2 + y_1^2 y_2^2 + x_1^2 y_2^2 + x_2^2 y_1^2}$

 $\ds \cmod {z_1} \cdot \cmod {z_2}$ $=$ $\ds \sqrt {x_1^2 + y_1^2} \sqrt {x_2^2 + y_2^2}$ $\ds$ $=$ $\ds \sqrt {\paren {x_1^2 + y_1^2} \paren {x_2^2 + y_2^2} }$ $\ds$ $=$ $\ds \sqrt {x_1^2 x_2^2 + y_1^2 y_2^2 + x_1^2 y_2^2 + x_2^2 y_1^2}$

$\blacksquare$

## Proof 2

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

Then:

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

$\blacksquare$

## Proof 3

Let:

$z_1 = r_1 \paren {\cos \theta_1 + i \sin \theta_1}$
$z_2 = r_2 \paren {\cos \theta_2 + i \sin \theta_2}$

Then:

 $\ds \cmod {z_1 z_2}$ $=$ $\ds \cmod {r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2} }$ Definition of Polar Form of Complex Number $\ds$ $=$ $\ds \cmod {r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} } }$ Product of Complex Numbers in Polar Form $\ds$ $=$ $\ds r_1 r_2$ Definition of Polar Form of Complex Number $\ds$ $=$ $\ds \cmod {z_1} \cmod {z_2}$ Definition of Polar Form of Complex Number

$\blacksquare$