Complex Number Calculator

Complex number operations

(a + bi) + (c + di) = (a + c) + (b + d)i (a + bi)(c + di) = (ac − bd) + (ad + bc)i (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)

The key fact is i² = −1. To divide, multiply the top and bottom by the conjugate of the denominator.

Worked example

(3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i. Dividing instead gives (3 + 4i) ÷ (1 − 2i) = −1 + 2i.

Modulus and argument

The modulus |z| = √(a² + b²) is the distance from the origin, and the argument is the angle from the positive real axis. For 3 + 4i, |z| = 5 and the argument is about 53.13°.

Frequently asked questions

What is i?

The imaginary unit, defined so that i² = −1.

What is a complex conjugate?

The same number with the sign of the imaginary part flipped: the conjugate of 3 + 4i is 3 − 4i.

Why multiply by the conjugate when dividing?

Because (c + di)(c − di) = c² + d², a real number, which removes i from the denominator.

What is polar form?

Writing z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.