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.
Formula
w_k = ⁿ√r · (cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)), k = 0, 1, …, n − 1Worked 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.
Frequently asked questions
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.