What is a complex number?

A number in a+bi form with real component a and imaginary component b.

What is a complex conjugate?

The conjugate of a+bi is a−bi.

It is the square root of the sum of squares of real and imaginary components.

The directed angle a complex number makes with the positive real axis.

It represents a complex number using radius and angle as r(cosθ+i sinθ).

Tools/🧮 Calculators/Complex Number Calculator

Calculate complex arithmetic, conjugate, modulus, argument, and polar conversions.

Version 1.0 · Last Updated: August 11, 2026 · Maintained by ToolLarder

Perform arithmetic with complex numbers in a+bi form and convert between Cartesian and polar representations.

Add, subtract, multiply, and divide real and imaginary components.

(a+bi)+(c+di)

The conjugate reverses the imaginary sign; modulus is distance from the origin.

|z|=√(a²+b²)

Find the directed angle from the positive real axis with atan2.

arg(z)=atan2(b,a)

Convert radius and angle to real and imaginary components.

z=r(cosθ+i sinθ)

Calculate natural logarithms, log10, custom-base logarithms, antilogs, and change-of-base results.

Evaluate, add, subtract, multiply, differentiate, solve, and factor polynomials.