Find nth roots for any complex value fast. Switch forms, units, and precision settings instantly. Export tables, compare branches, and verify computations clearly here.
| Example | Input z | n | Expected roots (rectangular) |
|---|---|---|---|
| Square roots | 3 + 4i | 2 | 2 + 1i, −2 − 1i |
| Cube roots | −8 + 0i | 3 | −2 + 0i, 1 + 1.732i, 1 − 1.732i |
| Quadratic branch set | −1 + 0i | 2 | 0 + 1i, 0 − 1i |
Values shown are rounded for readability. Use the calculator for precise outputs.
Convert the complex number z = a + bi to polar form: z = r( cos θ + i sin θ ), where r = √(a² + b²) and θ = atan2(b, a).
The n-th roots of z are given by De Moivre’s theorem:
z1/nk = r1/n [ cos((θ + 2πk)/n) + i sin((θ + 2πk)/n) ], k = 0, 1, …, n−1
Each k produces a different branch, evenly spaced by 2π/n around the complex plane.
This calculator converts a+bi into r∠θ using r=√(a²+b²) and θ=atan2(b,a). Using atan2 preserves quadrant information and avoids sign errors that occur when b/a is used directly. For typical inputs between −10⁶ and 10⁶, the conversion remains stable, and r is computed with hypot for better numerical behavior.
For any nonzero z, the n roots are evenly distributed around the complex plane. Adjacent roots differ by 2π/n in angle, so n=2 produces roots 180° apart, n=3 gives 120° spacing, and n=6 gives 60° spacing. The plot shows each branch point on the Argand diagram, making symmetry easy to verify visually.
All branches share the same magnitude r^(1/n). If |z|=64 and n=3, every root has magnitude 4. If |z|=0.0001 and n=2, every root has magnitude 0.01. This shared radius forms a circle in the plot; only the angle changes with k, which is why the points sit on a common ring.
Angles can be entered and displayed in degrees or radians, but internal computation uses radians. Normalization to [0,2π) is convenient for reporting nonnegative angles, while (−π,π] aligns with many engineering conventions. Sorting by angle uses the normalized radians so the table order matches the plotted orientation.
Output precision ranges from 0 to 12 decimals. Higher precision helps when roots are close together, such as large n on small |z|. CSV export includes real, imaginary, magnitude, and angle fields for downstream analysis. The PDF summary lists each k branch in both rectangular and polar forms for audit trails.
A quick check is raising each computed root to the n power and confirming it returns the original z within rounding. Another check is verifying conjugate pairs when z has a real magnitude and symmetric angle, for example z=−1+0i with n=2 giving ±i. Use the graph to confirm equal spacing and the shared radius. If z is positive real, θ=0; root angles equal 2πk/n. Example z=16, n=4 gives 0,90,180,270.
n is the exponent denominator. The tool computes values w such that w^n = z. Larger n produces more branches and smaller magnitudes because r^(1/n) decreases as n increases.
Complex angles wrap every 2π. Adding 2πk to θ gives the same original z, so dividing by n yields n distinct angles and therefore n distinct roots.
If z=0, the magnitude r is 0, and every n-th root is 0. The table lists n identical roots at the origin, and the plot collapses to one point.
Use the unit that matches your source data. The calculator converts degrees to radians internally, then reports angles in your chosen unit for consistent comparisons.
With high n or small magnitudes, many roots differ slightly in angle. If precision is low, rounding can hide differences. Increase decimals or sort by angle to separate nearby branches.
Pick any listed root w and compute w raised to n using manual multiplication or another tool. You should recover z within rounding error, and all branches should share magnitude r^(1/n).
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.