Complex Root Calculator — nth Roots of a Complex Number
Enter any complex number a + bi and the root degree n to find all n complex roots. The roots are equally spaced around a circle of radius r^(1/n) in the complex plane, a beautiful consequence of De Moivre's theorem.
All n roots lie on a circle of this radius: r^(1/n)
- 1
Input modulus |z|
√(a² + b²) = 1 - 2
Input argument θ
atan2(b, a) = 0° - 3
Angle spacing per root
360° ÷ 3 = 120°All n roots are equally spaced on the circle at this angular interval. - 4
Root magnitude r^(1/n)
1 ^ (1 ÷ 3) = 1
How does this calculator work?
The n roots of z = a + bi are z_k = r^(1/n)·cis((θ + 2πk)/n) for k = 0..n−1, where r = |z| and θ = atan2(b, a). All roots share magnitude r^(1/n) and are equally spaced by 360°/n on a circle in the complex plane.
Formula
How this is calculated
To find the nth roots of z = a + bi, first convert to polar form: r = |z| = √(a² + b²) and θ = atan2(b, a). By De Moivre's theorem, the n distinct nth roots of z are: z_k = r^(1/n) · [cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)] for k = 0, 1, …, n − 1.
All roots have the same modulus r^(1/n) but different arguments, equally spaced by 2π/n radians (360°/n) around a circle. This means the roots form a regular n-gon inscribed in a circle of radius r^(1/n) centred at the origin — which the plot illustrates.
For real numbers, this method also works: a real number x is treated as x + 0i. Note that negative real numbers have an argument of π (180°), so their square roots are pure imaginary (e.g. √(−4) = ±2i), as expected.
Frequently asked questions
Because e^(iθ) is periodic with period 2π, adding 2πk (for k = 0, 1, …, n − 1) to the argument before dividing by n gives n different angles, hence n different roots. Adding 2πn would repeat the same angle.
Only for odd n. A real positive number has one real positive nth root and n − 1 complex conjugate pairs. A real negative number has zero real nth roots for even n (all roots are complex) and one real negative root for odd n.
The principal nth root is the one with the smallest non-negative argument — corresponding to k = 0 in the formula. It is the root most commonly meant by the n-th root symbol ⁿ√.
TG we-Calculate Editorial Team. (2026). Complex Root Calculator — nth Roots of a Complex Number [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/complex-root-calculator
TG we-Calculate Editorial Team. "Complex Root Calculator — nth Roots of a Complex Number." TG we-Calculate. 2026. https://we-calculate.com/calculator/complex-root-calculator.
TG we-Calculate Editorial Team, "Complex Root Calculator — nth Roots of a Complex Number," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/complex-root-calculator
@misc{wecalculate_complex_root_calculator, title = {Complex Root Calculator — nth Roots of a Complex Number}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/complex-root-calculator}}, year = {2026}, note = {TG we-Calculate} }
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