# Complex Root Calculator

URL: https://speedycalculator.com/calculators/math/complex-root/
Last updated: 2026-10-30

> The complex root calculator finds all n-th roots of a complex number, given in Cartesian form a + bi or polar form r∠θ, and lists them in Cartesian or polar form.

## Inputs

- My number is in
- Real part a
- Imaginary part b
- Modulus r
- Angle θ (degrees)
- Root degree n
- Show the roots in

## Results

- All n-th roots
- Modulus of every root
- Argument of the number

## Formula

```
w_k = ⁿ√r · (cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)),  k = 0, 1, …, n − 1
```

## Explanation

Every non-zero complex number has exactly n different n-th roots. They lie on a circle of radius ⁿ√r and are spaced evenly, 360°/n apart. For example, the cube roots of −8 are −2 and 1 ± √3 i.

### How to use the Complex Root Calculator

- **My number is in**
- **Real part a**
- **Imaginary part b**
- **Modulus r**
- **Angle θ (degrees)**
- **Root degree n**
- **Show the roots in**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The number is converted to polar form, the modulus is raised to the power 1/n and the angle divided by n. Adding multiples of 360°/n to the angle gives the remaining roots (de Moivre's theorem).

### Tips

- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.

### Common mistakes to avoid

- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.

## Worked example

Inputs — My number is in: Cartesian form a + bi; Real part a: -8; Imaginary part b: 0; Root degree n: 3; Show the roots in: Cartesian form.

Results — **All n-th roots = w₀ = 1 + 1.732050808i
w₁ = −2
w₂ = 1 − 1.732050808i**; Modulus of every root = 2; Argument of the number = 180°.

## Key terms

- **De Moivre's theorem**: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
- **Root of unity**: A solution of zⁿ = 1.

## FAQ

**How many cube roots does a complex number have?**
Three. In general an n-th root has n values.

**What are the cube roots of −8?**
−2, 1 + √3 i and 1 − √3 i.

**Why are the roots evenly spaced?**
Multiplying the angle by n maps each of the n angles θ/n + 360°k/n back to θ.

**What is the principal root?**
The one with k = 0, the smallest non-negative angle.

**What formula does the Complex Root Calculator use?**
w_k = ⁿ√r · (cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)),  k = 0, 1, …, n − 1.


## Sources

- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – de Moivre's identity](https://mathworld.wolfram.com/deMoivresIdentity.html)