Maths & numbers
Nth roots of a complex number
List and plot all distinct nth roots of a complex value.
A clear answer. Here’s how.
Writes z=r·exp(iθ), then forms r^(1/n)·exp(i(θ+2πk)/n) for k=0,…,n−1. Accepts integer n from 1–64. For nonzero z, roots are equally spaced around a circle; angles in the table follow the generated sequence rather than principal-value wrapping. Zero returns one distinct root with multiplicity n. Trigonometric rounding can leave tiny components near zero.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Real part
- 1
- Imaginary part
- 0
- Root degree
- 4
Example result
| Root | Real part | Imaginary part | Argument (degrees) |
|---|---|---|---|
| 1 | 1 | 0 | 0 |
| 2 | 0.0000000000000000612323399574 | 1 | 90 |
| 3 | -1 | 0.000000000000000122464679915 | 180 |
| 4 | -0.000000000000000183697019872 | -1 | 270 |
All 4 distinct nth roots are equally spaced in angle.