What counts as an nth root

A complex number w is an nth root of z when w raised to the integer power n equals z. A nonzero complex number has n distinct nth roots. Asking software for z to the power 1/n often returns only a principal value; solving wⁿ = z requires the whole set.

This calculator accepts z in rectangular form a + bi and root orders 2 through 12. It converts z to polar form, divides the angle while accounting for full rotations, and converts every resulting root back to rectangular coordinates.

Convert the input to polar form

For z = 8 + 0i, the modulus is 8 and the principal argument is 0 degrees. The two-argument atan2 function is important because it identifies the correct quadrant. A simple arctangent of b/a loses quadrant information and fails when a is zero.

The National Institute of Standards and Technology Digital Library of Mathematical Functions describes phase values as differing by integer multiples of 2π and defines a principal phase convention. The chosen interval affects labels and ordering, not the geometric set of roots.

Generate all root angles

Adding 2πk before dividing is what produces all branches. Each root has modulus r raised to 1/n. Adjacent root angles differ by 2π/n, so the roots sit at equal intervals around a circle centered at the origin.

For a cube root, the spacing is 120 degrees. The default roots have angles 0, 120, and 240 degrees. All have modulus 2 because the cube root of 8 is 2.

Default rectangular roots

At zero degrees, cosine is 1 and sine is 0, producing 2 + 0i. At 120 degrees, cosine is −1/2 and sine is √3/2, producing −1 + √3i, approximately −1 + 1.732051i. At 240 degrees, the sine changes sign, producing −1 − 1.732051i.

Each value cubes to 8. The nonreal roots form a conjugate pair because the target 8 is real and the polynomial w³ − 8 has real coefficients. Conjugate pairing is a polynomial property, but an arbitrary nonreal target does not make its root set symmetric across the real axis.

Principal root versus complete solution

The row k = 0 is labeled principal under the displayed input-phase convention. It is useful for defining a single-valued branch of a power function. It is not more valid as a solution of wⁿ = z than the other n − 1 roots.

Branch conventions can differ on the negative real axis. A system that takes an argument of −180 degrees instead of +180 degrees may display a different root first. Compare complete sets modulo full rotations before declaring a mismatch.

The zero special case

Zero has no defined phase because every angle points to the same point when the radius is zero. The only distinct nth root of zero is zero. In the polynomial equation wⁿ = 0, that root has multiplicity n.

The calculator reports one distinct zero root and explains the multiplicity. Listing n identical zero rows would suggest n geometrically different solutions where none exist.

How the root-plane diagram is scaled

Every nonzero root has the same modulus, so the diagram normalizes that common radius to fit the circle. It communicates angle and symmetry, not physical distance in a global coordinate system. The real axis runs horizontally and the imaginary axis vertically, with positive imaginary values drawn upward.

When many roots are requested, labels can crowd. Use the numeric table for exact reading. The plot never replaces the values, and the table retains root order even if points are visually close.

Reconstruction residual

For each listed root, the script multiplies it by itself n times and compares the result with a + bi. The residual is the Euclidean modulus of the real-and-imaginary difference. The panel reports the largest residual across all roots.

Floating-point sine, cosine, and repeated multiplication can create a tiny nonzero residual even when the mathematics is exact. Displayed rectangular coordinates are rounded separately. Raising only the printed coordinates may produce a larger discrepancy than the internal unrounded root.

Product-of-roots check

The product of all solutions of wⁿ − z = 0 follows from the polynomial’s constant term. It equals (−1)ⁿ⁺¹z. For the default n = 3, the product is z, so the panel shows 8 + 0i. For an even root order, the product is −z.

This is a set-level check, not the same as raising each root to n. Both checks are useful: reconstruction validates individual roots, while the product can expose a missing or duplicated root.

Connections to roots of unity

Once one root is known, every other root is obtained by multiplying it by an nth root of unity. Those unity roots are equally spaced on the unit circle. This explains why changing z rotates and rescales the root polygon without changing its regular spacing.

For z = 1, the results are exactly the nth roots of unity. Their sums and products appear throughout Fourier analysis, signal processing, cyclic systems, and polynomial factorization.

Complex roots versus polynomial root-finding

This calculator solves the special binomial equation wⁿ = z. It does not solve a general polynomial with multiple coefficients. General cubic, quartic, and numerical polynomial methods require different input and may have repeated roots that are not arranged on one circle.

Matching the interface and method to the actual problem prevents “root” or “rate” terminology from carrying formulas across unrelated domains.

Manual verification checklist

Negative real inputs and branch boundaries

Consider z = −16 + 0i. Under the convention used here, its principal argument is +180 degrees. The fourth-root angles are 45, 135, 225, and 315 degrees, each with modulus 2. If another program represents the same input with argument −180 degrees, its formula-order list begins at −45 degrees and continues at 45, 135, and 225 degrees. The first rows differ, but the two lists describe the same four points after angles are compared modulo 360 degrees.

This behavior reflects a branch boundary, not an arithmetic defect. A continuous single-valued argument cannot be defined everywhere around the origin without a cut. When documenting results, state the principal-argument interval if row order matters. When only the equation solution matters, normalize angles and compare the unordered root sets.

Units, scale, and interpretation

Complex numbers can encode voltages, impedances, rotations, oscillation amplitudes, or purely abstract quantities. Taking an nth root changes units formally: a cube root of a quantity measured in cubic units has base units, but many complex models use normalized or symbolic variables. This calculator performs the algebra and does not decide whether a root operation is meaningful for a physical model.

Large magnitudes and high powers can exceed floating-point range, while extremely small magnitudes can underflow. The input limit on n controls interface readability, not every numerical risk. If roots feed a safety-critical engineering or scientific computation, use a tested numerical library, preserve units, assess conditioning, and compare residuals relative to the scale of z rather than relying on rounded screen coordinates.

Frequently asked questions

To verify the polar-form step behind nth roots, work the same value through the De Moivre’s theorem complex-number calculator.

References

NIST Digital Library of Mathematical Functions — Complex modulus and phase

NIST Digital Library of Mathematical Functions — Principal branches of logarithms and powers

Wolfram MathWorld — nth roots