Roots of Complex Numbers Calculator
This roots of complex numbers calculator finds all n of the nth roots of a + bi — and then does the thing almost no calculator does: it raises each one back to check it.
Over the real numbers, 8 has one cube root. Over the complex numbers it has three: 2, −1 + √3i and −1 − √3i. That is the rule rather than the exception.
Why There Are n of Them
Every non-zero complex number has exactly n nth roots. Not one, not sometimes two — always n of them.
They share two properties that make the whole answer predictable before any arithmetic happens:
They are all the same size. Each sits the same distance from 0.
They are evenly spaced round a circle. Three of them 120° apart, four 90° apart, five 72° apart.
So they form the corners of a regular polygon — an equilateral triangle for cube roots, a square for fourth roots, a hexagon for sixth. If you have found the first one, the rest are just that shape drawn round the origin.
Zero is the single exception. It sits at the origin, so every one of its roots is 0 no matter how many you ask for.
De Moivre's Formula
Write the number by its size r and its angle θ — that is polar form, and it is the step everything here rests on. Then:
zk = r1/n · cis((θ + 360°k) ÷ n) for k = 0, 1, …, n−1
Two things are doing the work. The size of every root is r1/n, which is why they all match. And the + 360°k before dividing is what produces a genuinely different root each time rather than the same one over and over — adding a full turn does not change the number, but it does change the answer after you divide by n.
Take 8i with n = 3. Its size is 8, so each root has size 2. Its angle is 90°, so the roots sit at 30°, 150° and 270°. That gives √3 + i, −√3 + i and −2i — and you can see the 120° spacing in the angles.
The Roots of Unity
The nth roots of 1 get their own name because they turn up everywhere: in polynomial factorising, in signal processing, and in any argument where something repeats n times and comes back to the start.
The cube roots of 1 are 1, −½ + ½√3i and −½ − ½√3i. The fourth roots are the tidiest set of all: 1, i, −1, −i.
Notice that 1 is always one of them, and that the rest are arranged symmetrically around it. Notice also that they add up to zero — which is not a coincidence and is worth a section of its own.
Two Ways to Know You Are Right
This is where most calculators stop and this one does not, because roots are unusually easy to get slightly wrong and unusually hard to spot.
Raise it back. If zk really is an nth root of z, then zkn = z. Doing that by hand in rectangular form takes several multiplications and is exactly where slips happen — this page does it for every root before showing you anything.
Add them all up. For any number and any n of 2 or more, the n roots always sum to zero. They sit symmetrically round a circle, so they cancel in pairs and triples. That makes it a check you can do by eye: if your three cube roots do not add to nothing, one of them is wrong.
Both are shown in the working here, and where everything comes out exactly the sum is exactly zero rather than nearly — the surds cancel term by term.
When the Answer Can Be Written Exactly
Two conditions have to hold together, and when they do the answer is exact, surds and all.
The size has to be writeable. The square roots of 2 are √2 and −√2 — not 1.4142. The square roots of 18 are 3√2 and −3√2. Those are the answers; the decimals are approximations of them.
The angles have to land somewhere exact. Each root's angle must be a multiple of 30° or 45°, since those are where cosine and sine have exact values — halves, and halves of √2 and √3.
Those two together cover very nearly every question a textbook asks, which is no accident: the questions are chosen so that the answer can be written down. The cube roots of 8, the fourth roots of 16, the square roots of −1, the cube roots of −8 — all exact.
When they do not both hold, the figures here are rounded and the page says so. The square roots of 1 + i are an example: the exact form needs roots nested inside roots, which is a different kind of algebra. Reporting that honestly is better than printing a decimal and calling it exact.
How to Use This Roots Calculator
Three boxes. The number goes in the top two — real part and multiple of i, with an empty box counting as 0 — and the root you want goes in n.
| real | The ordinary part of the number. Empty means 0 |
| imaginary (× i) | How many i s. Empty means 0 |
| how many, n | 2 for square roots, 3 for cube roots, and so on |
| ← → ↑ ↓ | Move between the boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point |
| ⌫ / AC | Delete one character, or clear everything |
| = | Works it out |
Open the working and every root is listed with its own angle, the exact ones marked so you can see at a glance which is which, followed by both checks. A fractional n is refused rather than guessed at — that is a genuinely different question with a different kind of answer.
Roots of Complex Numbers FAQ
How many nth roots does a complex number have?
What are the cube roots of 8?
What is De Moivre's formula for roots?
What are the cube roots of 1?
How do you check that a root is correct?
What are the square roots of 2 as a complex number?
n of them, evenly spaced, all the same size — and two checks that take seconds and catch nearly every mistake.