How do you multiply complex numbers?

(a + bi)(c + di) = (ac − bd) + (ad + bc)i. The tool applies this directly and also reports the modulus and argument of the result.

How does complex division work?

Dividing by c + di multiplies the top and bottom by the conjugate c − di, giving ((ac + bd) + (bc − ad)i) / (c² + d²). This tool does that automatically.

What are the modulus and argument?

The modulus |z| is the distance from the origin, √(re² + im²). The argument is the angle from the positive real axis, computed with atan2 and shown in degrees.

Complex Number Calculator

Add, subtract, multiply and divide complex numbers with modulus and argument.

Complex arithmetic in a + bi form

Enter two complex numbers, choose add, subtract, multiply or divide, and read the result in standard a + bi form. The tool also gives the modulus |z| = √(re² + im²) and the argument (the angle from the real axis).

Multiplication uses (a + bi)(c + di) = (ac − bd) + (ad + bc)i and division uses the complex conjugate. Everything runs locally in your browser.