How this calculator works
An nth root divides the argument by n and takes the positive nth root of the magnitude. Adding full turns before division supplies the remaining roots, equally spaced around a circle. The principal root uses the principal input argument. Zero has only one distinct root, regardless of n.
Formula
If z = r exp(i theta), then w_k = r^(1/n) exp(i(theta + 2pi k)/n), for k = 0, ..., n-1.
Worked example
For z = -1 and n = 2, r = 1 and theta = pi. The two angles are pi/2 and 3pi/2, producing i and -i. For z = 8 and n = 3, the principal root is 2.
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Frequently asked questions
Are negative radii supported?
No. A polar magnitude is non-negative. Change the angle by half a turn instead.
Where is pi radians as a separate unit?
Use radians, degrees or turns. A half turn equals pi radians.
Why are some displayed components tiny rather than zero?
Trigonometric evaluation uses floating-point numbers. Components near a coordinate axis may contain small rounding residuals.