Polar & Exponential Form

Modulus, argument, and Euler's formula z = r e^iθ.

The idea

The polar form of a complex number names it by its distance from the origin and its direction, rather than by its coordinates. The rectangular form $z = x + iy$ suits addition, which adds the coordinates, but it hides what multiplication does.

Definition (Modulus, argument, and polar form).

For a nonzero $z = x + iy$, the modulus $r = |z| = \sqrt{x^{2} + y^{2}}$ is its distance from the origin, and the argument $\theta = \arg z$ is the angle from the positive real axis around to $z$. These are the polar coordinates of the point $(x, y)$, so $z = r(\cos\theta + i\sin\theta) = r\,e^{i\theta},$ where the second equality is Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$. The first expression is the polar form of $z$ and the second its exponential form.

In this form multiplication is simple to state: the product of two complex numbers has the product of their moduli and the sum of their arguments, and in particular $z^{n} = r^{n} e^{i n \theta}$. One caution is needed: a full turn returns to the same point, so the argument is determined only up to multiples of $2\pi$, and we fix a range, usually $(-\pi, \pi]$, to make it single-valued.

Ways to work on it

• Walkthrough. Modulus, argument, and exponential form via Euler's formula.
• Practice. Compute the modulus of a complex number.
• Hardest. Convert to exponential form when the quadrant matters.

Not sure where to start? Take the ten-question placement test.