Complex nth Roots Calculator

What this calculator does

Every non-zero complex number has exactly n distinct nth roots, which is a sharper statement than the real case where a number has at most two square roots. The roots are spaced evenly around a circle.

De Moivre's theorem makes them easy to find. Take the nth root of the modulus, divide the argument by n, and then step round by 360 degrees over n for each subsequent root.

The formula

Formular=√(a²+b²), θ=atan2(b,a); roots = r^(1/n)·[cos((θ+2πk)/n)+i·sin((θ+2πk)/n)], k=0,…,n−1

The number is converted to polar form. The modulus is raised to the power one over n, and the argument is divided by n, with multiples of two pi over n added for each successive root.

Term	Meaning
De Moivre's theorem	The rule relating powers and roots of complex numbers in polar form.
Modulus and argument	The polar coordinates of a complex number.
Roots of unity	The special case where the number being rooted is 1.

The inputs explained

Field	What to enter
Real part a	The real part of the complex number.
Imaginary part b	The imaginary part.
Root degree n	The root degree. There will be exactly this many distinct roots.

When to use it

Solving a polynomial

Equations of the form z to the n equals a number are solved directly by this.

Signal processing

Roots of unity underpin the discrete Fourier transform.

Understanding the fundamental theorem

Seeing exactly n roots appear makes the theorem concrete.

Worked examples

Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.

How does the root degree change the magnitude?

The same number at three root degrees.

Roots of 1 + i, modulus 1.414

Root degree	Magnitude of each root	Root k=0
n = 2	1.189	1.099 + 0.4551i
n = 3	1.122	1.084 + 0.2905i
n = 4	1.091	1.070 + 0.2127i

The original has modulus 1.414 and argument 45 degrees. Its square roots have magnitude 1.189, its cube roots 1.122 and its fourth roots 1.091, each approaching 1 as the root degree rises.

Questions

Why are there exactly n roots?

Because adding a full turn to the argument leaves the number unchanged, but dividing by n turns those full turns into n distinct fractions of a turn. After n steps the pattern repeats.

How are the roots arranged?

Evenly around a circle, at equal angles of 360 degrees over n apart, all with the same magnitude. Plotted on the complex plane they form the vertices of a regular n-sided polygon.

What are roots of unity?

The nth roots of the number 1. They are evenly spaced around the unit circle starting at 1 itself, and they are fundamental to Fourier analysis and to a great deal of abstract algebra.

Why does the magnitude approach 1 for large n?

Because taking higher roots of any positive number pulls it toward 1. A modulus of 1.414 raised to the power one over n tends to 1 as n grows, which is why the roots cluster near the unit circle.

For complex arithmetic, see the complex number calculator. For quadratics with complex roots, see the quadratic calculator.

nth roots of a complex number calculator