Complex Modulus equals Complex Modulus of Conjugate

Theorem

Let $z \in \C$ be a complex number.

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

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

Then:

$\cmod z = \cmod {\overline z}$

Let $z = a + b i$.

Then:

$\ds \cmod z$

$=$

$\ds \cmod {a + b i}$

Definition of $z$

$\ds $

$=$

$\ds \sqrt {a^2 + b^2}$

Definition of Complex Modulus

and:

$\ds \cmod {\overline z}$

$=$

$\ds \cmod {\overline {a + b i} }$

Definition of $z$

$\ds $

$=$

$\ds \cmod {a - b i}$

Definition of Complex Conjugate

$\ds $

$=$

$\ds \cmod {a + \paren {- b} i}$

$\ds $

$=$

$\ds \sqrt {a^2 + \paren {- b}^2}$

Definition of Complex Modulus

$\ds $

$=$

$\ds \sqrt {a^2 + b^2}$

$\ds $

$=$

$\ds \cmod z$

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Complex Conjugate of $z$: $3.7.8$
1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(3)$