Complex Root Calculator

Background.

No real number multiplied by itself gives −16. A positive times a positive is positive, and so is a negative times a negative. Mathematicians got past the impasse by introducing i, defined by i² = −1, and with it the complex numbers a + bi. In that larger system the square root of −16 is simply 4i, because (4i)² = 16 × i² = −16.

Roots of negative numbers stop feeling strange once you picture numbers as points on a plane. A complex number has a size, its distance from zero, and a direction, its angle measured from the positive real axis. Positive numbers point at 0° and negative numbers at 180°. Taking an nth root does two things at once: it takes the ordinary nth root of the size, and it divides the angle by n. That is De Moivre's theorem run in reverse.

There is a catch that surprises most people the first time. Every nonzero number has exactly n different nth roots, spaced evenly around a circle 360° ÷ n apart. The number −8 has three cube roots: −2 and the pair 1 ± 1.732i. A calculator has to pick one, and the convention is the principal root, the one you get by dividing the input's own angle by n. For a negative input that angle is 180° ÷ n, so the principal cube root of −8 is 1 + 1.732i, not −2.

Here you enter a real number and a whole-number degree from 1 to 50, and get back the principal root's magnitude, real part and imaginary part. Complex inputs and the other n − 1 roots are left out, though the circle rule makes the others easy to find by hand.

What is complex root calculator?

The principal nth root of a real number x is the complex number whose nth power is x and whose angle is x's angle divided by n, taking 0° for positive x and 180° for negative x. Its magnitude is the ordinary nth root of |x|. For positive x it is the familiar positive real root. For negative x it lies at 180° ÷ n: purely imaginary when n is 2, and off both axes for any larger n, odd or even, which is why the principal cube root of a negative number is not a negative real number.

How to use this calculator.

1. 01Enter the real radicand, the number under the root sign. Negative values are the interesting case; a positive value returns the ordinary positive root with an imaginary part of zero.
2. 02Enter the root degree as a whole number from 1 to 50: 2 for a square root, 3 for a cube root, and so on. Fractional degrees are refused.
3. 03Read the magnitude, then the real and imaginary parts, and write the root as real part + (imaginary part) i.
4. 04To list the other roots, keep the same magnitude and keep adding 360° ÷ n to the angle, which for a negative input starts at 180° ÷ n.

The formula.

The principal root is |x|^(1/n) × [cos(θ ÷ n) + i sin(θ ÷ n)], where |x| is the absolute value of the input and θ is its angle in the complex plane: 0 for a positive number and π radians, 180°, for a negative one. The first factor is the magnitude of the root, an ordinary real nth root. The bracket sets its direction at θ ÷ n, and the cosine and sine of that angle, multiplied by the magnitude, give the real and imaginary parts. This is De Moivre's formula, (cos φ + i sin φ)^n = cos nφ + i sin nφ, used backwards: raising the answer to the nth power multiplies the angle back up to θ and the magnitude back up to |x|. As the degree rises the magnitude creeps toward 1 and the angle toward 0°, so the 50th root of −16 is about 1.0549 + 0.0664i.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.