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.

How do you add and subtract complex numbers?

Add or subtract the real and imaginary parts separately: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. The real parts combine and the imaginary parts combine independently.

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

This calculator performs arithmetic on complex numbers written in standard a + bi form. Enter two complex numbers, pick add, subtract, multiply or divide, and read the result, along with its modulus (distance from the origin) and argument (angle from the real axis). It is built for engineering, signal-processing and maths coursework where complex arithmetic is routine.

How it works

Each operation uses the standard complex-number rules:

Add / subtract: combine real and imaginary parts separately — (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
Multiply: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
Divide: multiply top and bottom by the conjugate of the divisor — ((ac + bd) + (bc − ad)i) ÷ (c² + d²).

For the result the tool also reports the modulus |z| = √(re² + im²) and the argument atan2(im, re), converted to degrees.

Example

Multiply (3 + 2i) by (1 − 4i):

real = 3·1 − 2·(−4) = 3 + 8 = 11 imaginary = 3·(−4) + 2·1 = −12 + 2 = −10

The result is 11 − 10i, with modulus √(121 + 100) = √221 ≈ 14.866 and argument ≈ −42.27°. Everything runs locally in your browser.