Complex Modulus of Quotient of Complex Numbers

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Theorem

Let $z_1, z_2 \in \C$ be complex numbers such that $z_2 \ne 0$.

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


Then:

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


Proof

\(\ds \cmod {\dfrac {z_1} {z_2} }\) \(=\) \(\ds \cmod {z_1 \times z_2^{-1} }\) Definition of Complex Division
\(\ds \) \(=\) \(\ds \cmod {z_1} \times \cmod {z_2^{-1} }\) Complex Modulus of Product of Complex Numbers
\(\ds \) \(=\) \(\ds \cmod {z_1} \times \cmod {z_2}^{-1}\) Complex Modulus of Reciprocal of Complex Number
\(\ds \) \(=\) \(\ds \dfrac {\cmod {z_1} } {\cmod {z_2} }\) Definition of Real Division

$\blacksquare$


Examples

Complex Modulus of $\dfrac {\paren {1 + 2 i}^{12} } {\paren {1 - 2 i}^9}$

$\cmod {\dfrac {\paren {1 + 2 i}^{12} } {\paren {1 - 2 i}^9} } = 5 \sqrt 5$


Sources

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