Argand Diagram, Modulus, and Polar Coordinates

The Argand diagram plots a complex number $z = a+bi$ as the point $(a, b)$ — the real part on the horizontal axis, the imaginary part on the vertical axis.

Two things follow naturally from that picture:

• The modulus $|z|$ is the distance from the origin: $|z| = \sqrt{a^2+b^2}$.
• The argument $\theta$ is the angle the point makes with the positive real axis, giving the polar form $z = r(\cos\theta + i\sin\theta)$ where $r = |z|$.

Example: $z = 1+i$ sits at $(1,1)$, so $|z| = \sqrt{2}$ and $\theta = 45^\circ$, giving polar form $\sqrt{2}(\cos 45^\circ + i\sin 45^\circ)$.