Math calculators

Complex number calculator

Add, subtract, multiply and divide, convert between rectangular and polar form, and find powers and nth roots.

Inputs

z real partz imaginary partiw real partw imaginary parti

Angles

Results

Methodology

Conventions

Complex numbers are entered as real and imaginary parts. The argument is the principal value in (−180°, 180°].
Powers with non-integer exponents use the principal branch. The n nth roots are spaced evenly around a circle.

References

NIST Digital Library of Mathematical Functions, dlmf.nist.gov, ch. 1.9

Formulas

Polar form: z = a + bi = r(\cos\theta + i\sin\theta),\quad r = \sqrt{a^2 + b^2}
Division: \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}
Roots: z_k = r^{1/n} e^{i(\theta + 2\pi k)/n},\ k = 0, \ldots, n - 1