Definition:Argument of Complex Number/Flawed Definition

Warning: Argument of Complex Number: Flawed Definition

It appears at first glance that it would be simpler to define the argument of a complex number $z = x + i y$ as:

$\theta = \arg z := \map \arctan {\dfrac y x}$

This arises from the definition of the tangent as sine divided by cosine.

This, however, does not determine $\theta$ uniquely.

The image set of $\arctan$ is usually defined as:

$\openint {-\dfrac \pi 2} {\dfrac \pi 2}$

and in any case, is a open interval of length $\pi$.

As the image set of $\arg z$ is $2 \pi$, that means that there are in general two values of $z$ which have the same $\arctan$ value.

Some more superficial sources gloss over this point, and merely suggest that $\arg z$ is one of the two values of $\map \arctan {\dfrac y x}$.