All n roots, arranged as the polygon they form — and the exact ones kept exact.
Every nth root
All n of them — not just the one a calculator gives you.
the cube roots of 8
2, −1 + 1.7320508i, −1 − 1.7320508i
3 distinct roots, 120° apart around a circle of radius 2.
Principal root
2
the argument nearest zero
Real roots
1
2
Sum of the roots
0
zero, as the symmetry requires
Product of the roots
8
matches (−1)^(3+1) × z = 8
Every root, with its angle
Each root with its modulus, argument and whether it is exact
k
Root
Argument
Exact?
0
2 ← principal
0°
exact
1
−1 + 1.7320508i
120°
8 s.f.
2
−1 − 1.7320508i
240°
8 s.f.
The roots, as the polygon they form
The filled point is the principal root — the one with the argument nearest zero, and the one a calculator’s root key returns.
Every root has modulus |z|^(1/3) = 8^(1/3) = 2, and argument (0° + 360°k) / 3 for k = 0 to 2 — which steps the angle by 120° each time.
The 3 roots are the vertices of a regular 3-gon inscribed in a circle of radius 2 about the origin, each 120° from the next. Their sum is zero because the vertices of a regular polygon centred on the origin balance exactly.
Exactly one of the 3 roots is real: 2. That is the one a real-number calculator gives you, and the other 2 are invisible to it.
1 of the 3 root is exact — each was raised back to the 3th power and reproduced z exactly, so the exact value is shown rather than a decimal that rounds to it.
The remaining roots are irrational and are given to 8 significant figures. Raising any of them to the power 3 returns z to within rounding.
Note that "the" nth root is a convention, not a fact: there are n of them and none is more correct than the others. The principal one is simply the one with the argument nearest zero.
The 3 roots sum to zero, as the symmetry of a regular 3-gon about the origin requires. This is computed from the roots above, not assumed.
The product of the roots is 8, against the (−1)^(3+1) × z = 8the theory requires — computed from the roots themselves.
What this tool shows
All n of them, arranged as the regular polygon they form — with the exact ones kept exact.
Every one of the n roots, with its modulus and argument
Exact values wherever a root is a Gaussian rational, verified by re-powering
The regular n-gon the roots form, drawn
The sum and product of the roots, computed and checked
All n roots, not just the principal Exact where a root is exact The regular n-gon geometry Free, no signup
Free, no signup — exact roots verified by re-powering.
Updated 6 September 2026 · Works in any browser, no installation
Every non-zero complex number has exactly n distinct nth roots — not one. 8 has three cube roots: 2, −1 + 1.7320508i, −1 − 1.7320508i. All three cube to 8, which you can check by multiplying.
At a glance
Formula shown
The n roots have modulus |z|^(1/n) and arguments (arg z + 360\u00b0k)/n for k = 0 to n\u22121 \u2014 equally spaced 360\u00b0/n apart on one circle.
Scenario support
Any non-zero complex number, and any n from 1 to 24.
Educational estimate
Planning support from the values you enter — not professional advice.
n roots, not one, and they are not exotic
Over the reals, 8 has one cube root. Over the complexes it has 3: 2, −1 + 1.7320508i, −1 − 1.7320508i — and the two you were not taught are perfectly ordinary numbers that cube to 8.
Exactly one of the 3 roots is real: −2. That is the one a real-number calculator gives you, and the other 2 are invisible to it.
None of the 4 roots is real. That is the familiar fact that a negative number has no real even root — but it has 4 complex ones, evenly spaced, and none of them happens to land on the real axis.
Zero is the one exception, and it is a real one: it has exactly one nth root, itself. The n roots lie on a circle of radius |z|^(1/n), and when |z| is zero that circle collapses to a point. The tool refuses zero and says this rather than returning n copies of it.
De Moivre, and where the 360°/n step comes from
Every root has modulus |z|^(1/3) = 8^(1/3) = 2, and argument (0° + 360°k) / 3 for k = 0 to 2 — which steps the angle by 120° each time.
The step exists because an argument is only defined up to full turns. arg 8 is 0°, but it is equally 360° and 720° — the same number, three descriptions. Divide each by 3 and you get 0°, 120° and 240°, which are three DIFFERENT directions.
So the n roots come from the n distinct ways of writing the argument before dividing. The (n+1)th would give 360°, which is the first root again, which is why the list stops at n rather than being cut off there.
The roots are the vertices of a regular polygon
The 3 roots are the vertices of a regular 3-gon inscribed in a circle of radius 2 about the origin, each 120° from the next. Their sum is zero because the vertices of a regular polygon centred on the origin balance exactly.
That geometry answers several questions at once. Why do the roots sum to zero for n ≥ 2? Because the vertices of a regular polygon centred on the origin balance — The 3 roots sum to zero, as the symmetry of a regular 3-gon about the origin requires. This is computed from the roots above, not assumed.
Why does every root have the same modulus? Because they all sit on one circle. Why are the angles evenly spaced? Because 360°/n is the same step every time. None of this needs to be memorised separately once the picture is there.
The 6 roots are the vertices of a regular 6-gon inscribed in a circle of radius 1 about the origin, each 60° from the next. Their sum is zero because the vertices of a regular polygon centred on the origin balance exactly.
Which one is “the” nth root
The principal root is the one with the argument nearest zero, and it is what a calculator’s root key returns. For 8 that is 2; for −8 it is 1 + 1.7320508i.
It is a convention, not a mathematical fact. None of the n roots is more correct than the others — they all satisfy the same equation exactly — and “the” nth root is shorthand rather than a description.
This matters when a formula is applied blindly. Cardano’s cubic formula involves cube roots, and taking the principal one at each step does not always give the real answer you wanted; the pairing between the two cube roots has to be chosen correctly. The Cubic Equation Calculator handles that, and explains why the trigonometric form sidesteps it entirely.
When a root is exact, and how that is verified
Most roots are irrational and are given to eight significant figures. Some are not, and printing those as decimals would be a loss of information rather than a rounding.
1 to the fourth gives 1, i, −1, −i — all four exact. i to the third has 1 exact root among its 3: −i.
3 + 4i is a perfect Gaussian square, so both its square roots are exact: 2 + i and −2 − i. The check is not a guess — each candidate is raised back to the nth power in exact arithmetic and accepted only if it reproduces z exactly.
1 of the 3 root is exact — each was raised back to the 3th power and reproduced z exactly, so the exact value is shown rather than a decimal that rounds to it.
Roots of unity, and why they matter
The nth roots of 1 are the special case where the polygon is inscribed in the unit circle with one vertex at 1. 1 to the fourth gives 1, i, −1, −i; to the sixth it gives 1, 0.5 + 0.8660254i, −0.5 + 0.8660254i, −1, −0.5 − 0.8660254i, 0.5 − 0.8660254i.
They form a group under multiplication — multiply any two and you get another one — and every other set of nth roots is that group scaled and rotated. Which is why the roots of unity are studied on their own: solve them once and you have solved all of them.
They are also where the Fourier transform comes from. The discrete Fourier transform is built entirely out of nth roots of unity, and the fast algorithm exploits the same symmetry that makes them a regular polygon. A construction that looks like an algebra exercise turns out to run signal processing.
Sources and methodology
De Moivre’s theorem and the roots of unity are standard; the reference below carries the canonical statements.
Method. Every figure on this page comes from src/lib/complex-root.ts over src/lib/algebra/complex.ts. Roots are generated by De Moivre in double precision, then each candidate is snapped to a nearby Gaussian rational only if raising that exact candidate to the nth power reproduces z exactly — so an "exact" label is a verified claim rather than a rounding heuristic. The sum and product checks are computed from the returned roots. That engine is verified on every change against 59 hand-written assertions, including that every returned root reproduces z when raised to the nth power, and that the count is always exactly n. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Complex NumberAdd, subtract, multiply, divide and raise complex numbers to powers exactly — (1+i)^8 is 16, not 15.999999999999996 — with the conjugate trick shown as the working for division.
Complex ConjugateThe conjugate, and everything it does: the sum, difference, product and quotient identities computed on your number, the reflection geometry, and the real quadratic behind conjugate root pairs.
Cubic EquationSolve any cubic exactly when it has a rational root — deflate and finish with the quadratic formula — and by the trigonometric form when it does not, with the discriminant saying which case you are in.
Quadratic FormulaSolve any quadratic with exact roots — surds stay surds and a negative discriminant gives the complex pair — plus the vertex, the factored form and every step of the working.
QuaternionExact quaternion arithmetic showing both pq and qp every time, both quotients rather than one, and the rotation a unit quaternion represents with axis, angle, matrix and the gimbal-lock case named.
Binomial Coefficientn choose k exactly on big integers, by the multiplicative formula that never builds a number bigger than the answer — with Pascal's rule, the row it sits in, and permutations beside it.
The nth roots are where complex numbers stop being a workaround and start being the natural setting. The Complex Number Calculator covers the exact arithmetic underneath, and the Cubic Equation Calculator shows what happens when a formula needs the right cube root rather than the principal one.
Educational use disclaimer
This calculator finds all n roots of a complex number for n from 1 to 24. Roots are irrational in general and are given to eight significant figures; where a root is exactly a Gaussian rational the exact value is shown instead, verified by raising the candidate back to the nth power. Zero is excluded — it has one root, not n, and the page says why.
Published the complex root page: all n roots by De Moivre, arranged as the regular n-gon they form.
Snaps a root to its exact value when it is a Gaussian rational — verified by raising it back to the nth power — so the cube roots of 8 include exactly 2 rather than 1.9999999999999998.
Checks the sum and product of the roots against what the geometry requires, computed from the roots themselves.
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