Argument of Complex Conjugate equals Negative of Argument

Theorem

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

Then:

$\arg {\overline z} = -\arg z$

where:

$\arg$ denotes the argument of a complex number

$\overline z$ denotes the complex conjugate of $z$.

Proof

Let $z$ be expressed in polar form:

$z := r \paren {\cos \theta + i \sin \theta}$

Then:

\(\ds \overline z\)

\(=\)

\(\ds r \paren {\cos \theta - i \sin \theta}\)

Polar Form of Complex Conjugate

\(\ds \)

\(=\)

\(\ds r \paren {\map \cos {-\theta} + i \map \sin {-\theta} }\)

Cosine Function is Even, Sine Function is Odd

The result follows by definition of the argument of a complex number

$\blacksquare$

Also see

• Argument of Complex Conjugate equals Argument of Reciprocal

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
• 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 $z$: $3.7.9$