Complex Numbers Calculator

A complex number has a real part and an imaginary part, a + bi, where i is defined as √−1. This tool performs the four arithmetic operations on pairs of complex numbers and also converts the result into polar form — a modulus (distance from the origin) and an argument (angle), which is often the more useful form in electrical engineering and signal processing.

Worked example — multiplication

(3 + 4i) × (1 − 2i) = 3(1) + 3(−2i) + 4i(1) + 4i(−2i) = 3 − 6i + 4i − 8i². Since i² = −1, −8i² becomes +8, giving (3 + 8) + (−6 + 4)i = 11 − 2i.

Division and the conjugate trick

Dividing complex numbers isn't done directly — the top and bottom are both multiplied by the conjugate of the denominator (same real part, flipped sign on the imaginary part), which turns the denominator into a real number: (c + di)(c − di) = c² + d². After that, the division is ordinary real-number arithmetic.

Modulus and argument (polar form)

The modulus, |a + bi| = √(a² + b²), is the distance from the origin on the complex plane — for 3 + 4i that's √(9+16) = 5. The argument is the angle from the positive real axis, arctan(b ÷ a). Electrical engineering represents AC voltage and current this way because it makes phase differences and impedance calculations straightforward multiplication rather than trigonometric juggling.