Definition:Argument of Complex Number

Definition
Let $z = x + i y$ be a complex number.

If we represent $z$ in the complex plane, the argument of $z$, or $\arg \left({z}\right)$, is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.

Formally, it is defined as any solution to the pair of equations:
 * $(1): \quad \dfrac x {\left|{z}\right|} = \cos \left({\arg \left({z}\right)}\right)$
 * $(2): \quad \dfrac y {\left|{z}\right|} = \sin \left({\arg \left({z}\right)}\right)$

where $\left|{z}\right|$ is the modulus of $z$.

From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an argument of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is any integer. Thus, the argument of a complex number $z$ is a continuous multifunction.

Also known as
The argument of a complex number is also seen as its amplitude.