The nth Roots of a Complex Number

One point, many angles

Turning a whole turn, $2\pi$ radians, brings an arrow back to where it started. So a complex number z = re$^{i\theta}$ can also be written re$^{i\left(\theta + 2\pi\right)}$, re$^{i\left(\theta + 4\pi\right)}$, re$^{i\left(\theta - 2\pi\right)}$, and in general re$^{i\left(\theta + 2k\pi\right)}$ for any whole number k.

For 8i, which has length 8 and points straight up: $8i = 8e^{\frac{i\pi}{2}} = 8e^{\frac{i5\pi}{2}} = 8e^{\frac{i9\pi}{2}}$. The angles are 90°, 450° and 810°, and all three point the same way.

Divide the angles by n

To solve $w^{n} = z$, write w = se$^{i\phi}$. By De Moivre, $w^{n} = s^{n}e^{(}$in$\phi)$, and this must equal re$^{i\left(\theta + 2k\pi\right)}$. The lengths agree when $s^{n} = r$, so $s = r^{\frac{1}{n}}$, the real n-th root of r. The directions agree when $n\phi = \theta + 2k\pi$, so $\phi = \frac{{\theta + 2k\pi}}{n}$.

Before dividing, the angles $\theta$, $\theta + 2\pi$, $\theta + 4\pi$, … named one direction. After dividing by n they are $\frac{\theta}{n}$, $\frac{\theta}{n} + \frac{2\pi}{n}$, $\frac{\theta}{n} + \frac{4\pi}{n}$, …, and these point different ways: each is $\frac{2\pi}{n}$, which is $\frac{360^\circ}{n}$, further round than the one before. So the roots are $w = r^{\frac{1}{n}} e^{\frac{i\left(\theta + 2k\pi\right)}{n}}$ for k = 0, 1, …, n − 1.

The cube roots of 8i

$8i = 8e^{\frac{i\pi}{2}}$, so each cube root has length $8^{\frac{1}{3}} = 2$, and the angles are $\frac{{\frac{\pi}{2} + 2k\pi}}{3}$. k = 0 gives $\frac{\pi}{6}$, which is 30°. k = 1 gives $\frac{5\pi}{6}$, which is 150°. k = 2 gives $\frac{3\pi}{2}$, which is 270°.

A turn of 270° ends pointing straight down, the same direction as −90°. In the range $-180^\circ < \theta \le 180^\circ$ the third root’s argument is −90°, or $- \frac{\pi}{2}$: 270° and −90° name the same direction. Taking k = −1 instead of k = 2 lands there directly, since $\frac{{\frac{\pi}{2} - 2\pi}}{3} = - \frac{\pi}{2}$.

In parts the roots are $2e^{\frac{i\pi}{6}} = \sqrt{3} + i$, $2e^{\frac{i5\pi}{6}} = -\sqrt{3} + i$ and $2e^{- \frac{i\pi}{2}} = -2i$.

Check each by cubing. $\left(\sqrt{3} + i\right)^{2} = 3 + 2\sqrt{3} i - 1 = 2 + 2\sqrt{3} i$, and $\left(2 + 2\sqrt{3} i\right)\left(\sqrt{3} + i\right) = 2\sqrt{3} + 2i + 6i + 2\sqrt{3} i^{2} = 8i$. $\left(-\sqrt{3} + i\right)^{2} = 2 - 2\sqrt{3} i$, and $\left(2 - 2\sqrt{3} i\right)\left(-\sqrt{3} + i\right) = -2\sqrt{3} + 2i + 6i - 2\sqrt{3} i^{2} = 8i$. $\left(-2i\right)^{3} = -8i^{3} = 8i$.

The three cube roots of 8i on the dashed circle of radius 2, labeled with their arguments: $\sqrt{3} + i$ at 30°, $-\sqrt{3} + i$ at 150° and −2i at −90°. They are 120° apart, and joined they make an equilateral triangle with sides $2\sqrt{3} \approx 3.46$.

Evenly spaced round one circle

All n roots have the same length $r^{\frac{1}{n}}$, so they lie on one circle about the origin. Their angles step by $\frac{360^\circ}{n}$, so they are evenly spaced round it, and joined in order they make a regular polygon with n sides.

For the cube roots of 8i the step is 360° ÷ 3 = 120°: from 30° to 150°, and from 150° on by 120° to 270°, the direction of −90°. The roots of unity are the case z = 1, with r = 1 and $\theta = 0$.

The roots of $z^{n} = w$ for a w of length 1. The handle on the circle turns w, and the slider below sets n. At arg w = 0° the cube roots are the cube roots of 1. Turn w to 90°, the direction of i and of 8i: each root turns a third as far, to 30°, 150° and 270°, the last being the direction −90°.

Why exactly n

k = 3 gives $\frac{{\frac{\pi}{2} + 6\pi}}{3} = \frac{\pi}{6} + 2\pi$. That is $\frac{\pi}{6}$ with a whole turn added, the root k = 0 gave. k = 4 repeats k = 1 in the same way, and negative values of k repeat them too.

In general k = n adds $\frac{{2n\pi}}{n} = 2\pi$ to the angle, a whole turn, so the values k = 0, 1, …, n − 1 give every root once, and there are exactly n of them.

Square roots and fourth roots

With n = 2 the two roots are half a turn apart, so they are $a \pm$ pair. The square roots of 8i have length $\sqrt{8} = 2\sqrt{2}$ and angles $\frac{\pi}{4}$ and $\frac{\pi}{4} + \pi$: they are 2 + 2i and −2 − 2i. Check: $\left(2 + 2i\right)^{2} = 4 + 8i + 4i^{2} = 8i$.

For $w^{4} = -16$, write $-16 = 16e^{i\pi}$. Each root has length $16^{\frac{1}{4}} = 2$, and the angles are $\frac{{\pi + 2k\pi}}{4}$: $\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{5\pi}{4}$ and $\frac{7\pi}{4}$, a quarter turn apart. As arguments in range they are 45°, 135°, −135° and −45°, and the roots are $\sqrt{2} + \sqrt{2} i$, $-\sqrt{2} + \sqrt{2} i$, $-\sqrt{2} - \sqrt{2} i$ and $\sqrt{2} - \sqrt{2} i$, each about 1.414 from each axis. Check: $\left(\sqrt{2} + \sqrt{2} i\right)^{2} = 2 + 4i - 2 = 4i$, and $\left(4i\right)^{2} = -16$.

The usual mistakes

Dividing the modulus by n. The modulus is cubed when w is cubed, so each cube root of 8i has length $8^{\frac{1}{3}} = 2$, not $\frac{8}{3}$, and not 8.

Finding only one root. $\sqrt{3} + i$ is the root with a third of the principal argument; adding $2\pi$ and $4\pi$ before dividing gives the other two.

Spacing the roots wrongly. n roots share a whole turn, so they are $\frac{360^\circ}{n}$ apart: 90° for fourth roots, 120° for cube roots.

Giving 210° or 300° for the third cube root of 8i. The steps are all 120°, so after 150° comes 270°, which as an argument in range is −90°.

Listing k = n as another root. It adds a whole turn and lands on the root k = 0 gave.

Three antennas round a mast

The application below places antennas at the three cube roots of 8i. It writes the third as $2e^{\frac{3i\pi}{2}}$, the turn of 270°; that is the root $2e^{- \frac{i\pi}{2}} = -2i$ found above.