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Complex Root Calculator

Find the nth roots of a complex number z = a + bi using De Moivre's theorem.

Reviewed by Chase FloiedUpdated

This free online complex root calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Find the nth root (n ≥ 2)

Results

Modulus |z|

1

Root modulus |z|^(1/n)

1

Number of roots

2

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Complex Root Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Complex Root Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Complex Root Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Complex Root Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Complex Root Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Find the nth roots of a complex number z = a + bi using De Moivre's theorem. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Complex Root Calculator

The Complex Root Calculator finds all nth roots of a complex number using De Moivre's theorem. Every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle of radius |z|^(1/n) in the complex plane. This beautiful geometric fact is a consequence of the algebraic completeness of the complex numbers. Finding roots of complex numbers is essential in solving polynomial equations, analyzing oscillatory systems, computing Fourier transforms, and understanding the symmetry of regular polygons (the nth roots of unity form the vertices of a regular n-gon). The calculator computes the modulus of each root and provides the framework for finding all n roots using the angle formula with k = 0, 1, ..., n-1.

The Math Behind It

Every nonzero complex number z = r·e^(iθ) has exactly n distinct nth roots given by wₖ = r^(1/n)·e^(i(θ+2πk)/n) for k = 0, 1, ..., n-1. Here r = |z| is the modulus and θ = arg(z) is the argument. The n roots are equally spaced around a circle of radius r^(1/n), separated by angles of 2π/n radians. The special case z = 1 gives the nth roots of unity: ωₖ = e^(2πik/n), which form a cyclic group under multiplication and are the vertices of a regular n-gon inscribed in the unit circle. The primitive nth root of unity is ω = e^(2πi/n), and all other roots are powers of ω. Roots of unity appear throughout mathematics: in the discrete Fourier transform, cyclotomic polynomials, number theory (solving x^n ≡ 1 mod p), and constructibility of regular polygons. For real-valued equations like x^n = c, the n roots include the familiar real root c^(1/n) (when it exists) along with additional complex roots. The principal root is typically the one with the smallest positive argument.

Formula Reference

De Moivre's Theorem for Roots

z^(1/n) = |z|^(1/n) · [cos((θ+2πk)/n) + i·sin((θ+2πk)/n)]

Variables: k = 0, 1, ..., n-1 gives n distinct roots

Argument

θ = atan2(b, a)

Variables: Angle of z in the complex plane

Worked Examples

Example 1: Square roots of -1

Find all square roots of -1 (z = -1 + 0i)

Step 1:Modulus: |-1| = 1, Argument: θ = π
Step 2:Root modulus: 1^(1/2) = 1
Step 3:k=0: angle = π/2, root = cos(π/2) + i·sin(π/2) = i
Step 4:k=1: angle = 3π/2, root = cos(3π/2) + i·sin(3π/2) = -i

The square roots of -1 are i and -i

Example 2: Cube roots of 8

Find all cube roots of 8

Step 1:Modulus: |8| = 8, Argument: θ = 0
Step 2:Root modulus: 8^(1/3) = 2
Step 3:k=0: angle = 0, root = 2
Step 4:k=1: angle = 2π/3, root = 2(cos(120°) + i·sin(120°)) = -1 + √3i
Step 5:k=2: angle = 4π/3, root = -1 - √3i

Three cube roots: 2, -1+√3i, -1-√3i

Common Mistakes & Tips

  • !Forgetting there are n distinct nth roots, not just one
  • !Not converting to polar form before applying De Moivre's theorem
  • !Using the wrong angle: θ = atan2(b, a), not atan(b/a) (which has quadrant ambiguity)
  • !Forgetting to add 2πk/n for each subsequent root (k = 0, 1, ..., n-1)

Related Concepts

Complex Conjugate

Complex roots of real polynomials come in conjugate pairs

Complex Number Calculator

Basic arithmetic with complex numbers

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Complex Conjugate Calculator

Complex Number Calculator

Frequently Asked Questions

Why are there exactly n nth roots?

The equation wⁿ = z is a polynomial equation of degree n, and by the Fundamental Theorem of Algebra, it has exactly n roots (counted with multiplicity) in the complex numbers. For nonzero z, all n roots are distinct.

What are roots of unity?

The nth roots of unity are the n solutions to zⁿ = 1. They are e^(2πik/n) for k = 0, 1, ..., n-1, forming a regular n-gon on the unit circle. They form a cyclic group under multiplication.