Complex Modulus of Additive Inverse

Theorem

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

Let $-z$ be the negative of $z$:

$z + \paren {-z} = 0$

Then:

$\cmod z = \cmod {\paren {-z} }$

where $\cmod z$ denotes the modulus of $z$.

Proof

Let $z = a + i b$.

\(\ds \cmod {\paren {-z} }\)

\(=\)

\(\ds \cmod {\paren {-a - i b} }\)

Definition of Negative of Complex Number

\(\ds \)

\(=\)

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

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {a^2 + b^2}\)

Even Power of Negative Real Number

\(\ds \)

\(=\)

\(\ds \cmod {a + i b}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \cmod z\)

Definition of $z$

$\blacksquare$