Roots Of Complex Number Calculator

Work out roots of complex number instantly with clear inputs, formula shown and shareable results.

About Roots Of Complex Number Calculator

A non-zero complex number has exactly n distinct nth roots, all of modulus r^(1/n) and with arguments (θ + 360k)/n. They sit at the vertices of a regular n-gon centred on the origin.

Formula

nth roots

zₖ = r^(1/n)·[cos((θ + 360k)/n) + i·sin((θ + 360k)/n)], k = 0 to n − 1

Worked example

With the default values below, Roots Of Complex Number Calculator returns The chosen root of -1 + 1.732051i. Change any input above and every figure here is recalculated in your browser.

Inputs used

Modulus rminimum 0: 16
Argument θ: 120 °
Which root (n)accepts 1–24: 4
Show root number k (from 0)accepts 0–23: 1

Results

The chosen root: -1 + 1.732051i
Modulus of every root: 2.00000000
Argument of this root: 120.000000 °
Angle between consecutive roots: 90.000000 °

Step by step

1Values used
2nth roots
3The chosen root= -1 + 1.732051i
4Modulus of every root= 2.00000000
5Argument of this root= 120.000000 °
6Angle between consecutive roots= 90.000000 °

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

The fourth roots of modulus 16 at 120°?

All have modulus 2, with arguments 30°, 120°, 210° and 300° — spaced 90° apart.

Why n roots and not one?

Because adding 360° to the argument gives the same number but a different root after dividing by n, cycling through n distinct values.