Complex Division as Product with Conjugate over Square of Modulus

Theorem

Let $z_1$ and $z_2$ be complex numbers.

Then the operation of division can be expressed as:

$\dfrac {z_1} {z_2} = \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}$

where:

$\overline {z_2}$ denotes the complex conjugate of $z_2$

$\cmod {z_2}$ denotes the complex modulus of $z_2$.

Proof

\(\ds \dfrac {z_1} {z_2}\)

\(=\)

\(\ds \dfrac {z_1 \overline {z_2} } {z_2 \overline {z_2} }\)

multiplying top and bottom by $\overline {z_2}$

\(\ds \)

\(=\)

\(\ds \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}\)

Modulus in Terms of Conjugate

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.13)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Multiplication and Division: $3.7.13$