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:

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

Also see

 * Argument of Complex Conjugate equals Argument of Reciprocal