Math Calculators

Complex Root Calculator

Find one of the n complex nth roots of a complex number using polar form and De Moivre's theorem.

Root real part —

Root imaginary part —

Root modulus —

Root angle —

About the Complex Root Calculator

Every nonzero complex number has exactly n distinct nth roots, evenly spaced around a circle in the complex plane. De Moivre's theorem finds them by taking the nth root of the modulus and dividing the angle (plus full rotations) by n.

Formula: if z = r(cosθ + i sinθ), then the kth nth root is r^1/n(cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, …, n−1

Change the root index k to step through each of the n roots in turn.