Argand Diagram, Argument and Polar Form

A complex number can be pictured as a point (or an arrow from the origin) in a plane, which gives it a length and a direction — the modulus and the argument.

The Argand diagram

The complex number $z=a+bi$ is represented by the point $(a,b)$ in a coordinate plane whose horizontal axis is the real axis and vertical axis the imaginary axis. This picture is called the Argand diagram. The distance of the point from the origin is exactly the modulus $|z|=\sqrt{a^{2}+b^{2}}=r$.

For example, $z=1+i\sqrt3$ is plotted as the point $(1,\sqrt3)$; the arrow from the origin to that point has length $r=|z|=\sqrt{1^2+3}=2$ and makes an angle $\theta=\pi/3$ with the positive real axis, while the dashed vertical drops back to $(1,0)$ to show the real part $a=1$.

The argument

Argument

The argument of a non-zero $z=a+bi$ is the angle $\theta$ that the segment from the origin to $(a,b)$ makes with the positive real axis, measured anticlockwise. It satisfies

$\cos\theta=\frac{a}{r},\qquad \sin\theta=\frac{b}{r},\qquad \tan\theta=\frac{b}{a}\ \ (a\ne0),$

where $r=|z|$. The value in the range $(-\pi,\pi]$ is called the principal argument.