How the Complex Root Calculator works
This calculator finds every nth root of a complex number using De Moivre's theorem. Any nonzero complex number has exactly n distinct nth roots, and the theorem gives a closed-form way to compute all of them from the number's polar form — no guessing or numerical search required.
Formula and method
Start with a complex number in rectangular form, z = a + bi. Convert it to polar form using the modulus r = √(a² + b²) and the argument θ = atan2(b, a) (the angle from the positive real axis, using the two-argument arctangent so the correct quadrant is chosen). The n distinct nth roots are then given by:
z^(1/n) = r^(1/n) × [cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)], for k = 0, 1, 2, …, n−1.
Every root shares the same modulus, r^(1/n) — the real nth root of the original modulus. What changes is the angle: each successive root's argument increases by 2π/n radians (360°/n), so the n roots sit at equally spaced points around a circle of radius r^(1/n), forming the vertices of a regular n-sided polygon on the complex plane. The root at k = 0 (using the principal argument θ) is called the principal root.
Common sources of error
- Degrees vs. radians: the formula's angle (θ + 2πk)/n is in radians; convert to degrees (multiply by 180/π) only for display, never mid-calculation.
- Forgetting the +2πk term: using only θ/n gives just the principal root — you need k = 0 through n−1 to get all n roots.
- Confusing "a root" with "the root": unlike real square roots of positive numbers, complex roots are not unique — a statement like "the cube root of 8i" is ambiguous unless you specify which of the three roots you mean.
Checking your result
A quick check: raise your computed root back to the nth power (multiply its modulus by itself n times and its argument by n) and confirm you land back on the original a + bi. Also, for n ≥ 2, the sum of all n roots of a nonzero complex number always equals zero — the vertices of a regular polygon centered at the origin cancel out.
Applications
Complex roots appear whenever a physical or mathematical system involves periodic or rotational behavior: solving polynomial equations (the Fundamental Theorem of Algebra guarantees n roots for a degree-n polynomial), AC circuit analysis and signal processing (phasors and Fourier components), control theory (pole locations), and computer graphics (rotations via complex multiplication). Label which root (which value of k) you used, since a different k gives a different — but equally valid — answer.