HomeCalculatorsMathComplex NumbersComplex Roots Calculator
Math calculator

Complex Roots Calculator

All n nth-roots of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not the real root page. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 3/3 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
All n nth-roots of one complex number.
Scope
Integer n ≥ 2
Verification
Engine tested · 18/18 tests · Production surface contract 3/3 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
6 golden · 4 boundary · 8 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.1.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

De Moivrez^{1/n} = r^{1/n} (cos((θ+360k)/n) + i sin((θ+360k)/n))
Principalk = 0
iInteger n ≥ 2 and n ≤ 64. Principal root is k=0. Float64. Not a second complex engine. Not polynomial roots.

How to use

1

Enter re, im, and n

n ≥ 2. Cube roots of 1 are the roots of unity.

2

Read the principal root and the list

All n roots are returned. Power and Euler form live on sibling discovery URLs.

Example calculations

Common configurations with formula and result.

ϟ

Cube roots of 1

re=1 im=0 n=3

1, ω, ω²
1, −1/2±i√3/2
ϟ

Square roots of −1

re=-1 im=0 n=2

±i
±i

Complex Roots calculator specification

Version 1.1.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The n nth-roots of z are r^{1/n} cis((θ+360°k)/n) for k=0…n−1. This page is mode=root on math.complex. Roots of unity are z=1. Not /calc/math/root (real nth root).
What it calculates
All n nth-roots of one complex number.
Inputs
  • re
  • im
  • n
Outputs
  • re
  • im
  • roots
  • n_roots
Formula
r^{1/n} cis((θ+360k)/n)
Assumptions
  • Integer n ≥ 2
  • Principal k=0
  • Float64
  • Not a CAS
Units
  • dimensionless
  • degrees
Boundary conditions
  • n<2 → VALUE_MUST_BE_POSITIVE
  • n>64 → VALUE_ABOVE_MAX
Example
re=1 im=0 n=3 → 1, ω, ω²
Validation cases

2 published on this page · 18/18 tests · Production surface contract 3/3 · View evidence

  • re=1 im=0 n=3 → 3 roots, principal 1
  • re=1 im=0 n=1 → error VALUE_MUST_BE_POSITIVE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Complex Number Calculator, Complex Polar Calculator, Complex Power Calculator, Euler Form Calculator . Explore all Complex Numbers.

→ Complex Number Calculator Add, subtract, multiply, divide complex numbers; convert rectangular ↔ polar; integer powers, nth roots, and Euler form. Not a CAS. Runs locally. › → Complex Polar Calculator Convert a complex number between rectangular (re, im) and polar (r, θ°) form. Same engine as Complex Number. Not a CAS. Runs locally. › → Complex Power Calculator Integer power of a complex number via De Moivre. Same engine as Complex Number. Not a CAS. Not i^i. Runs locally. › → Euler Form Calculator Euler form of a complex number: z = r e^{iθ} = r (cos θ + i sin θ). Same engine as Complex Number. Not a CAS. Runs locally. ›
}

Frequently asked questions

Key distinctions behind the calculation.

Is this a second complex engine?

No. It is mode=root on math.complex, the same engine as /calc/math/complex-number.

Is this the real nth-root calculator?

No. Real roots stay /calc/math/root. Polynomial roots stay /calc/math/polynomial-roots.