What Are Complex Numbers?

A complex number is written a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the defining property i² = −1. Complex numbers live on a two-dimensional plane — the real axis running horizontally and the imaginary axis vertically — which makes them the natural language for anything that oscillates, rotates, or involves phase.

Arithmetic in the Complex Plane

Addition and subtraction combine corresponding parts: (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication distributes term by term and simplifies with i² = −1, so (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i² = −5 + 14i. Division multiplies both numerator and denominator by the denominator's conjugate, turning the denominator real and producing a clean split into real and imaginary parts.

Magnitude and Argument

Every complex number has a polar description: its magnitude |z| = √(a² + b²), the distance from the origin, and its argument θ = atan2(b, a), the angle from the positive real axis. For −5 + 14i the magnitude is about 14.87 and the argument about 109.7°. Polar form, z = |z|(cos θ + i sin θ), is where powers and roots become trivial — a key reason electrical engineers analyze AC circuits in the complex domain.

Where Complex Numbers Are Used

Electrical engineering models AC voltage and current as complex phasors, combining magnitude and phase in one number. Signal processing and control theory rely on complex frequency analysis. Quantum mechanics is built on complex wave functions, and fractal geometry — like the Mandelbrot set — is defined entirely by iterating a complex quadratic. This calculator handles the four core operations plus the magnitude, argument, and conjugate you need to work confidently in that plane.