The nth Roots of a Complex Number

One point, many angles

Turning a whole turn, $2\pi$2π radians, brings an arrow back to where it started. So a complex number z = re${}^{i\theta }$^(iθ) can also be written re${}^{i\left(\theta +2\pi \right)}$^(i(θ + 2π)), re${}^{i\left(\theta +4\pi \right)}$^(i(θ + 4π)), re${}^{i\left(\theta -2\pi \right)}$^(i(θ − 2π)), and in general re${}^{i\left(\theta +2k\pi \right)}$^(i(θ + 2kπ)) 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}}$8i = 8e^(iπ/2) = 8e^(i5π/2) = 8e^(i9π/2). The angles are 90°, 450° and 810°, and all three point the same way.

Divide the angles by n

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

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

The cube roots of 8i

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

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

In parts the roots are $2e^{\frac{i\pi }{6}}=\sqrt{3}+i$2e^(iπ/6) = √3 + i, $2e^{\frac{i5\pi }{6}}=-\sqrt{3}+i$2e^(i5π/6) = −√3 + i and $2e^{-\frac{i\pi }{2}}=-2i$2e^(−iπ/2) = −2i.

Check each by cubing. ${\left(\sqrt{3}+i\right)}^2=3+2\sqrt{3}i-1=2+2\sqrt{3}i$(√3 + i)² = 3 + 2√3 i − 1 = 2 + 2√3 i, and $\left(2+2\sqrt{3}i\right)\left(\sqrt{3}+i\right)=2\sqrt{3}+2i+6i+2\sqrt{3}{i}^2=8i$(2 + 2√3 i)(√3 + i) = 2√3 + 2i + 6i + 2√3 i² = 8i. ${\left(-\sqrt{3}+i\right)}^2=2-2\sqrt{3}i$(−√3 + i)² = 2 − 2√3 i, and $\left(2-2\sqrt{3}i\right)\left(-\sqrt{3}+i\right)=-2\sqrt{3}+2i+6i-2\sqrt{3}{i}^2=8i$(2 − 2√3 i)(−√3 + i) = −2√3 + 2i + 6i − 2√3 i² = 8i. ${\left(-2i\right)}^3=-8i^3=8i$(−2i)³ = −8i³ = 8i.

The three cube roots of 8i on the dashed circle of radius 2, labeled with their arguments: $\sqrt{3}+i$√3 + i at 30°, $-\sqrt{3}+i$−√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$2√3 ≈ 3.46.

Evenly spaced round one circle

All n roots have the same length $r^{\frac{1}{n}}$r^(1/n), so they lie on one circle about the origin. Their angles step by $\frac{360^{\circ }}{n}$360°/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 $\theta =0$θ = 0.

The roots of $z^n=w$zⁿ = 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 $\frac{\frac{\pi }{2}+6\pi }{3}=\frac{\pi }{6}+2\pi$(π/2 + 6π)/3 = π/6 + 2π. That is $\frac{\pi }{6}$π/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 $\frac{2n\pi }{n}=2\pi$(2nπ)/n = 2π 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 $a\pm$a ± pair. The square roots of 8i have length $\sqrt{8}=2\sqrt{2}$√8 = 2√2 and angles $\frac{\pi }{4}$π/4 and $\frac{\pi }{4}+\pi$π/4 + π: they are 2 + 2i and −2 − 2i. Check: ${\left(2+2i\right)}^2=4+8i+4i^2=8i$(2 + 2i)² = 4 + 8i + 4i² = 8i.

For $w^4=-16$w⁴ = −16, write $-16=16e^{i\pi }$−16 = 16e^(iπ). Each root has length $16^{\frac{1}{4}}=2$16^(1/4) = 2, and the angles are $\frac{\pi +2k\pi }{4}$(π + 2kπ)/4: $\frac{\pi }{4}$π/4, $\frac{3\pi }{4}$3π/4, $\frac{5\pi }{4}$5π/4 and $\frac{7\pi }{4}$7π/4, a quarter turn apart. As arguments in range they are 45°, 135°, −135° and −45°, and the roots are $\sqrt{2}+\sqrt{2}i$√2 + √2 i, $-\sqrt{2}+\sqrt{2}i$−√2 + √2 i, $-\sqrt{2}-\sqrt{2}i$−√2 − √2 i and $\sqrt{2}-\sqrt{2}i$√2 − √2 i, each about 1.414 from each axis. Check: ${\left(\sqrt{2}+\sqrt{2}i\right)}^2=2+4i-2=4i$(√2 + √2 i)² = 2 + 4i − 2 = 4i, and ${\left(4i\right)}^2=-16$(4i)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$8^(1/3) = 2, not $\frac{8}{3}$8/3, and not 8.

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

Spacing the roots wrongly. n roots share a whole turn, so they are $\frac{360^{\circ }}{n}$360°/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}}$2e^(3iπ/2), the turn of 270°; that is the root $2e^{-\frac{i\pi }{2}}=-2i$2e^(−iπ/2) = −2i found above.