Complex Roots Calculator

How it works

1. 1Enter the real and imaginary parts of your complex number z = a + bi, then choose the root degree n.
2. 2The calculator converts z to polar form (r, θ), computes r^(1/n), and applies De Moivre’s formula: root k has angle (θ + 2πk) ÷ n for k = 0, 1, …, n − 1.
3. 3All n roots are returned in rectangular form a + bi, equally spaced by 360° ÷ n around a circle of radius r^(1/n).

Use cases

• Solving polynomial equations such as z³ = 1 + i by finding all three cube roots at once.
• Generating the nth roots of unity (set z = 1) for signal processing or the Fast Fourier Transform.
• Verifying complex arithmetic by checking that each root raised to the nth power recovers the original number.

Frequently asked questions

Why does a complex number have exactly n nth roots?

The equation wⁿ = z has exactly n solutions in the complex plane. They sit at equal angular intervals of 360° ÷ n on a circle of radius |z|^(1/n), by De Moivre’s theorem.

What are the roots of unity?

The nth roots of unity are the n solutions to wⁿ = 1. They are evenly distributed on the unit circle and appear throughout algebra, number theory, and signal processing.

What happens when z = 0?

Zero has only one nth root — zero itself — so all n roots coincide at the origin, consistent with |0|^(1/n) = 0.