Polar form: multiply the lengths, add the angles

10 minA Level3.9.5

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can convert a complex number between Cartesian and polar form, and multiply or divide two numbers in polar form by multiplying or dividing the lengths and adding or subtracting the angles.

Key to the symbols

Symbol	Meaning
$x + iy$	Cartesian form: across by $x$ , up by $y$
$r$	the modulus, $r = \sqrt{x^2 + y^2}$ , a positive length
$\theta$	the argument, the angle from the positive real axis, in radians
$r(\cos\theta + i\sin\theta)$	polar form, written out in full
$re^{i\theta}$	the same thing in shorter notation
$z_1, z_2$	two complex numbers being multiplied or divided

Angles are in radians throughout this lesson. If you have been working in degrees, convert first: $45^\circ = \frac{\pi}{4}$ and $60^\circ = \frac{\pi}{3}$ .

Two descriptions of the same point

A point on the Argand diagram can be located in two ways.

Across and up: the Cartesian form $x + iy$ gives the point $(x, y)$ .
How far and which way: the modulus $r$ is the distance from the origin and the argument $\theta$ is the direction.

Polar form uses the second description. It is just as exact, and it makes multiplication and division far easier.

Definition

Polar form

A non-zero complex number with modulus $r$ and argument $\theta$ can be written as

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

where $r > 0$ and $\theta$ is measured in radians from the positive real axis.

The two notations mean exactly the same thing. The short form $re^{i\theta}$ is valid because of the identity $e^{i\theta} = \cos\theta + i\sin\theta$ , which is why the shorthand is safe to use.

Worked example: Cartesian to polar, first quadrant

Express $z = 1 + i\sqrt3$ in the form $re^{i\theta}$ .

Have a go first?

Worked example: Cartesian to polar, second quadrant

Express $z = -3 + 3i$ in the form $re^{i\theta}$ .

Have a go first?

Going back the other way

To turn $re^{i\theta}$ into $x + iy$ , expand the brackets:

$re^{i\theta} = r\cos\theta + i\,r\sin\theta$

So $x = r\cos\theta$ and $y = r\sin\theta$ . Both coordinates come from the right-angled triangle with hypotenuse $r$ and angle $\theta$ .

This direction is easy to get wrong by forgetting the sign of $\cos\theta$ or $\sin\theta$ in the second and third quadrants. Sketch the angle first.

Worked example: Polar to Cartesian

Write $z = 4e^{i5\pi/6}$ in the form $x + iy$ .

Have a go first?

Check conversions

CalculateQuestion 1 · Apply

Answer the 2 questions above to continue.