What this calculator does
nth roots of a complex number works out all n complex roots of a+bi using De Moivre’s theorem, in polar and rectangular form. Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
Formular=√(a²+b²), θ=atan2(b,a); roots = r^(1/n)·[cos((θ+2πk)/n)+i·sin((θ+2πk)/n)], k=0,…,n−1
The inputs explained
| Field | What to enter |
|---|---|
| Real part a | A number. Starts at 1. |
| Imaginary part b | A number. Starts at 1. |
| Root degree n | A number. Starts at 3. |
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the answer changes with real part a
Every other input is held at the calculator’s starting values while real part a varies. Select any row to load that scenario into the calculator.
How the answer changes with real part a