Polar Representation of Complex Numbers¶

Definition¶

The polar coordinate system for complex numbers specifies each complex number by its modulus $r$ and polar angle $\varphi$ (known as the argument). The form $z=r(\cos \varphi+\mathrm{i}\sin\varphi)$ is called the polar form of a complex number.

Interpretation¶

The polar form describes a complex number by its distance from $0$ and the angle it makes with the positive real axis, instead of by its real and imaginary parts.

A complex number z drawn as an arrow of length r at angle phi from the real axis, with its projections r cos phi and r sin phi on the axes.

The arrow to $z$ has length $r=|z|$ and makes the angle $\varphi=\arg z$ with the real axis. The dashed lines meet the axes at $r\cos\varphi$ and $r\sin\varphi$ , by the trigonometric functions on a circle of radius $r$ . These are the real and imaginary parts of $z$ , which is why $z=r(\cos\varphi+\mathrm{i}\sin\varphi)$ $i sin φ)$ . The label also shows the short form $r\mathrm{e}^{\mathrm{i}\varphi}$ given by Euler's formula.

Set the modulus $r$ and the angle $\varphi$ , or drag $z$ : $z=r(\cos\varphi+\mathrm{i}\sin\varphi)$ $i sin φ)$ .

Going back uses $r=\sqrt{a^2+b^2}$ and an angle with $\cos\varphi=a/r$ and $\sin\varphi=b/r$ $b / r$ . For $3+4\mathrm{i}$ this gives $r=5$ and $\varphi=\operatorname{atan2}(4,3)\approx0.927$ $0.927$ , about $53.1^\circ$ . The angle has to be found from the signs of both $a$ and $b$ , not from $b/a$ alone: $1+\mathrm{i}$ and $-1-\mathrm{i}$ have the same ratio but the angles $\pi/4$ and $-3\pi/4$ , and the 2-argument arctangent takes care of this.

The modulus of a nonzero point is unique, the angle is not: adding $2\pi k$ , $k\in\mathbb{Z}$ $Z$ , gives the same point, and the principal argument picks the value in $(-\pi,\pi]$ . For $z=0$ one has $r=0$ and no angle is singled out. The polar form is made for multiplication: lengths multiply and angles add (complex multiplication), so powers follow from De Moivre's formula. For addition the Cartesian form is simpler.

Key Results¶

Complex multiplication in polar form combines stretching (multiplying moduli) with rotation (adding arguments).

Examples¶

The complex number $1+\mathrm{i}$ has polar coordinates $(\sqrt{2}, \pi/4)$ .
The complex number $3+4\mathrm{i}$ has polar coordinates $(5, \operatorname{atan2}(4,3))$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 1

Aliases: Complex Plane (Polar Coordinate System), modulus-argument form Keywords: polar form, polar representation, trigonometric form Symbols: element of, integers, phi, pi, square root

Last updated: 20 October 2026 03:02