How to Find the Polar Form and Powers of a Complex Number

Writes a + bi with its absolute value r, the distance from the origin, and its argument θ, the angle from the real axis. By de Moivre's theorem, raising it to the nth power gives an absolute value of rⁿ and an argument of nθ.

A complex number $a + bi$ can be described by its distance from the origin and the angle measured from the real axis. That is its polar form.

r = \sqrt{a^2 + b^2}

$r$ is the absolute value. The argument $\theta$ is the angle the direction from the origin to $(a, b)$ makes with the real axis. It can be found from $\arctan\dfrac{b}{a}$ , but that is 180 degrees out when $a$ is negative, so the quadrant of the point settles it. This gives $a + bi = r(\cos\theta + i\sin\theta)$ .

De Moivre's theorem

In polar form, powers become easy.

\left(r(\cos\theta + i\sin\theta)\right)^n = r^n(\cos n\theta + i\sin n\theta)

The absolute value is raised to the power $n$ and the argument is multiplied by $n$ . Multiplication splits into multiplying distances and adding angles, so even a high power takes one step.

Example

Take $(1 + i)^8$ . Here $1 + i$ has absolute value $\sqrt{2}$ and argument 45 degrees. Raising to the eighth gives an absolute value of $(\sqrt{2})^8 = 16$ and an argument of $45 \times 8 = 360$ degrees. A full turn of 360 degrees is the same as 0, so the answer is the real number 16. Working up step by step agrees: $(1+i)^2 = 2i$ , $(2i)^2 = -4$ , $(-4)^2 = 16$ .

Notes

The argument is given between $-180$ and 180 degrees. Adding 360 degrees points at the same place, so the argument is never unique.

$n$ must be a whole number. With $a$ and $b$ both 0 there is no argument to speak of, so nothing is calculated.