Complex Number Calculator

A complex number has the form a + bi, where a is the real part, b is the imaginary part and i is the square root of -1. Numbers like 3 + 4i, -2i and 7 are all complex numbers, with 7 having an imaginary part of 0.

Multiplication uses the rule (a + bi)(c + di) = (ac - bd) + (ad + bc)i, remembering that i squared is -1. Division multiplies the top and bottom by the conjugate of the denominator, so (a + bi) / (c + di) becomes a real denominator c squared plus d squared. The calculator does both for you.

The modulus is the distance from the origin, the square root of (a squared plus b squared), so 3 + 4i has modulus 5. The argument is the angle the number makes with the positive real axis, found with atan2 and shown here in both radians and degrees.

Polar form writes a number as r at angle theta, where r is the modulus and theta is the argument. It is handy for multiplication, division and powers, since you multiply the moduli and add the angles. The calculator shows both rectangular (a + bi) and polar form.

Dividing by 0 + 0i is undefined, just as it is for ordinary numbers, so the calculator reports an error instead of a result. Every other pair of complex numbers divides cleanly.

Yes. Every calculation runs entirely in your browser, and nothing you type is uploaded or stored anywhere.