Complex Modulus of Product of Complex Numbers

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


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


\(\displaystyle \cmod {z_1} \cdot \cmod {z_2}\) \(=\) \(\displaystyle \sqrt {x_1^2 + y_1^2} \sqrt {x_2^2 + y_2^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\paren {x_1^2 + y_1^2} \paren {x_2^2 + y_2^2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {x_1^2 x_2^2 + y_1^2 y_2^2 + x_1^2 y_2^2 + x_2^2 y_1^2}\)

$\blacksquare$


Proof 2

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

Then:

\(\displaystyle \cmod {z_1 z_2}\) \(=\) \(\displaystyle \sqrt {\paren {z_1 z_2} \overline {\paren {z_1 z_2} } }\) 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
\(\displaystyle \) \(=\) \(\displaystyle \cmod {z_1} \cmod {z_2}\)

$\blacksquare$


Proof 3

Let:

$z_1 = r_1 \left({\cos \theta_1 + i \sin \theta_1}\right)$
$z_2 = r_2 \left({\cos \theta_2 + i \sin \theta_2}\right)$

Then:

\(\displaystyle \left\vert{z_1 z_2}\right\vert\) \(=\) \(\displaystyle \left\vert{r_1 \left({\cos \theta_1 + i \sin \theta_1}\right) r_2 \left({\cos \theta_2 + i \sin \theta_2}\right)}\right\vert\) Definition of Polar Form of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{r_1 r_2 \left({\cos \left({\theta_1 + \theta_2}\right) + i \sin \left({\theta_1 + \theta_2}\right)}\right)}\right\vert\) Product of Complex Numbers in Polar Form
\(\displaystyle \) \(=\) \(\displaystyle r_1 r_2\) Definition of Polar Form of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{z_1}\right\vert \left\vert{z_2}\right\vert\) Definition of Polar Form of Complex Number

$\blacksquare$


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