Complex Root Calculator - N-th Roots via De Moivre's Theorem
Find every n-th root of any complex number a + bi using polar form and De Moivre's Theorem, with rectangular and polar output for each root.
Enter the real and imaginary parts of your complex number together with the root degree, then read off all n distinct roots in seconds.
Complex Root Calculator - N-th Roots via De Moivre's Theorem
Find every n-th root of any complex number a + bi using polar form and De Moivre's Theorem, with rectangular and polar output for each root.
About the complex root calculator
Every non-zero complex number has exactly n distinct n-th roots, and this calculator finds all of them at once using the polar form of a complex number together with De Moivre's Theorem. Given a complex number z = a + bi, its modulus is r = √(a² + b²) and its argument is θ = atan2(b, a). In polar form z = r(cosθ + i·sinθ), and the n n-th roots are z_k = r^(1/n) · (cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)) for k = 0, 1, …, n − 1.
Geometrically, the n roots lie on a circle of radius r^(1/n) centered at the origin of the complex plane, evenly spaced 2π/n radians apart. The root with k = 0 is called the principal root and lies closest to the positive real axis (its argument is θ/n). Rotating by 2π/n around the origin maps one root to the next, which is why the roots of unity form a regular n-gon inscribed in the unit circle.
Complex roots are not just an algebraic curiosity. In electrical engineering, the roots of unity describe phasors in three-phase power systems and underpin the Discrete Fourier Transform used in digital signal processing. In control theory, the locations of roots in the complex plane determine the stability of linear systems. In quantum mechanics, complex amplitudes and their roots show up in wave functions and in the analysis of periodic potentials. Even in pure number theory, the n-th roots of unity generate cyclotomic fields, central objects in modern algebra.
A common misconception is that a complex number has just one root, or that the square root of −1 has only the value i. In fact, −1 has two square roots — i and −i — and 1 has n distinct n-th roots for every positive integer n. The calculator displays them all in the form a_k + b_k·i, rounded to a fixed number of decimal places so that you can see numerical patterns such as the symmetry of conjugate pairs. For real inputs with even root degree, the roots appear in complex-conjugate pairs; for n odd and the input on the negative real axis, exactly one root is itself real and negative.
Use this complex root calculator whenever you need to solve z^n = w for an arbitrary complex w, factor polynomials over the complex numbers, study roots of unity, or check homework problems in a complex analysis or engineering mathematics course.
Worked examples
Try a few classic inputs to see how the n roots distribute around the complex plane.
| Input (z, n) | Roots | Notes |
|---|---|---|
| z = 8 + 0i, n = 3 | 2, −1 + 1.7320508i, −1 − 1.7320508i | Classic cube roots of 8. One real root and a complex-conjugate pair, evenly spaced 120° apart on a circle of radius 2. |
| z = 0 + 1i, n = 2 | 0.7071068 + 0.7071068i, −0.7071068 − 0.7071068i | The two square roots of i. They lie on the unit circle at 45° and 225°, differing by exactly 180°. |
| z = −16 + 0i, n = 4 | 1.4142136 + 1.4142136i, −1.4142136 + 1.4142136i, −1.4142136 − 1.4142136i, 1.4142136 − 1.4142136i | Fourth roots of −16. All four roots lie on a circle of radius 16^(1/4) = 2, evenly spaced 90° apart, with the principal root at argument 45°. |
| z = 1 + 1i, n = 3 | 1.0842150 + 0.2905145i, −0.7937005 + 0.7937005i, −0.2905145 − 1.0842150i | Cube roots of 1 + i. Modulus √2, argument 45°, so the principal root has argument 15°. |
How to use the complex root calculator
- Enter the real part a of your complex number z = a + bi in the first field.
- Enter the imaginary part b in the second field. Use 0 if the number is purely real.
- Enter the root degree n as a positive integer between 1 and 20.
- Click Calculate Roots to display all n distinct roots in rectangular form a_k + b_k·i.
- Click Reset to clear the inputs and try a different complex number.
Complex roots FAQ
What is De Moivre's Theorem?
De Moivre's Theorem states that for any real θ and integer n, (cosθ + i·sinθ)^n = cos(nθ) + i·sin(nθ). Taking n-th roots of both sides yields the standard formula for the n distinct n-th roots of a complex number written in polar form.
How many n-th roots does a complex number have?
Every non-zero complex number has exactly n distinct n-th roots. Zero has just one n-th root, which is zero itself. The n roots are equally spaced 2π/n radians apart on a circle of radius r^(1/n).
What is the principal root?
The principal root is the root with k = 0 in the formula, which has the smallest non-negative argument θ/n. It is the value returned by most programming languages' built-in complex power function and is the conventional choice when a single answer is required.
Why are complex roots important?
They appear throughout science and engineering — in AC circuit analysis, signal processing, control system stability, quantum mechanics, fluid dynamics, and the solution of polynomial equations. The roots of unity in particular are central to the Discrete Fourier Transform.
Can the root degree be negative or zero?
No. The n-th root is only defined for positive integer n. For n = 0 the operation is undefined, and negative degrees would correspond to reciprocals of roots, which you can compute by first finding the n-th roots and then taking reciprocals separately.
Why do my roots show many decimal places?
Most n-th roots of complex numbers are irrational, so the calculator rounds each component to about eight decimal places to balance readability with numerical accuracy. For exact symbolic answers, use a computer algebra system.