Math
Complex Root Calculator
Every non-zero complex number has exactly n distinct n-th roots.
Type the number and the index and the complex root calculator converts to polar form, applies De Moivre's formula, and lists every root with its angle, in exact form (such as √3 + i) when the angle is a multiple of 30° or 45°, and as decimals otherwise. The roots are plotted on their circle in the complex plane.
Write it as a + bi. A real number such as -8 or 1 works too and gives its complex roots.
2 for square roots, 3 for cube roots, and so on. There are always exactly n roots.
Try an example
Result
All roots
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- Principal root (k = 0)
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- Root modulus
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- Angle between roots
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- Modulus of z
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- Argument of z
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- z in polar form
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More details (1 more)
- Roots (decimal)
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Study path
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What the complex root calculator solves
To take the n-th root of z = r(cos θ + i sin θ), take the real n-th root of the modulus and divide the angle by n. Because adding 2π to θ describes the same number, there are n different angles that work, which gives n roots spaced 360°/n apart around a circle of radius r^(1/n).
Example: cube roots of 8i
- Polar form: |8i| = 8 and θ = π/2, so 8i = 8(cos(π/2) + i sin(π/2)).
- Root modulus: 8^(1/3) = 2.
- Angles: (π/2 + 2πk)/3 for k = 0, 1, 2 → π/6, 5π/6, 3π/2 (= -π/2).
- Roots: 2(cos(π/6) + i sin(π/6)) = √3 + i; -√3 + i; -2i.
Roots of unity
The n-th roots of 1 are the The turning point of a parabola, or a corner point of a shape. of a regular n-gon on the unit circle, starting at 1. Enter z = 1 to list them; for n = 4 they are 1, i, -1, -i.
How to use it
- Type the complex number as a + bi (a plain real number is fine).
- Enter n: 2 for square roots, 3 for cube roots.
- Read the principal root at the top, the table of all n roots with their angles, and the plot. Open the steps for the polar conversion and each angle.
How to read the answer
The principal root uses the principal argument (k = 0). Every other root is the previous one rotated by 360°/n. The roots are given exactly when the root modulus is rational or a simple square root and the angle is a multiple of π/6 or π/4; otherwise the decimal form is the answer to use.
Common mistakes and edge cases
- Reporting only one root. A cube root problem always has three answers over the complex numbers, even for a real number like -8.
- Taking the n-th root of the argument instead of dividing it by n.
- Forgetting the 2πk: without it you only get the principal root.
- The only n-th root of 0 is 0, so the calculator asks for a non-zero number.
Worked examples
Cube roots of 8i
√3 + i, -√3 + i, -2i
All roots
√3 + i, -√3 + i, -2i
Square roots of 3 + 4i
2 + i and -2 - i
All roots
2 + i, -2 - i
Cube roots of a negative real number
-8 has roots 1 + √3i, -2, 1 - √3i
All roots
1 + √3i, -2, 1 - √3i
Fourth roots of unity
1, i, -1, -i
All roots
1, i, -1, -i
Decimal roots when the angle is not special
Square roots of 1 + 2i are about ±(1.272 + 0.786i)
All roots
1.27202 + 0.786151i, -1.27202 - 0.786151i
Zero has no distinct roots
0 is rejected
All roots
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Frequently asked questions
How many roots does a complex number have?+
A non-zero complex number has exactly n distinct n-th roots. They share the same modulus r^(1/n) and are spaced 360°/n apart around the origin.
How do I find the square root of a complex number by hand?+
Convert to polar form, take the square root of the modulus, and halve the angle. The second square root is the negative of the first. For 3 + 4i: r = 5, θ ≈ 53.13°, so the roots are √5 at ±26.57°, which is 2 + i and -2 - i.
Why does -8 have three cube roots?+
The real cube root is -2, but over the complex numbers x^3 = -8 is a degree-3 equation with three solutions: -2, 1 + √3i, and 1 - √3i.
Which root is the principal root?+
The one with k = 0, built from the principal argument θ in (-π, π]. Its angle is θ/n, so it is the root closest to the positive real axis in the counterclockwise sense from θ/n.
What are roots of unity?+
The n-th roots of 1. They lie on the unit circle at angles 2πk/n and are used in trigonometry, polynomial factoring, and signal processing. Enter z = 1 to list them.
Why are some roots shown as decimals?+
Exact forms with square roots only exist when the root modulus is rational or a simple radical and the angle is a multiple of 30° or 45°. Otherwise the decimal form is the complete answer.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, Precalculus 2e, Section 8.5 “Polar Form of Complex Numbers” — De Moivre's theorem and finding n-th roots of complex numbers.
- OpenStax, College Algebra 2e, Section 2.4 “Complex Numbers”
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Last updated: September 4, 2026