Argument of Complex Conjugate equals Argument of Reciprocal

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Theorem

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


Then:

$\arg {\overline z} = \arg \dfrac 1 z$

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:

\(\ds \dfrac 1 z\) \(=\) \(\ds \dfrac 1 r \paren {\cos \theta - i \sin \theta}\) Polar Form of Reciprocal of Complex Number
\(\ds \) \(=\) \(\ds \dfrac 1 r \paren {\map \cos {-\theta} + i \map \sin {-\theta} }\) Cosine Function is Even, Sine Function is Odd

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

$\blacksquare$


Also see


Sources