# Complex Number Calculator - Add, Multiply, Divide, Powers and Roots

> Free online complex number calculator: arithmetic on a+bi, powers and nth roots by De Moivre, with modulus, argument, conjugate and polar form for each result.

- URL: https://qilingtools.com/en/math/complex
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter complex numbers as a+bi and add, subtract, multiply, divide, raise to a power or take nth roots. Every answer comes with |z|, the argument in degrees and radians, the conjugate, the polar form r(cosθ+isinθ) and the exponential form re^(iθ); roots list all n values.

## How to use

1. Choose the operation: +, −, ×, ÷, power or nth roots
2. Enter the real and imaginary parts (and n for powers and roots)
3. Read the result with its modulus, argument and polar form

## Features

- Arithmetic, powers zⁿ and nth roots
- Division rationalised with the conjugate and returned as a+bi
- Modulus, argument in degrees and radians, conjugate
- Polar and exponential forms
- All n roots listed with their arguments

## FAQ

### How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²), which makes the denominator real. The simplified a+bi result is shown.

### What are the modulus and argument?

|z| = √(a²+b²) is the distance from the origin in the complex plane; the argument θ = atan2(b, a) is the angle from the positive real axis, given in both degrees and radians.

### How do I find the nth roots of a complex number?

By De Moivre’s theorem: take the nth root of the modulus, divide the argument by n and add 360°/n for each further root — all n of them are listed.

### What is the square root of i?

Enter z₁ = 0 + 1i and choose 2 roots: ±(0.7071 + 0.7071i), that is ±(√2/2)(1+i).

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