AC electricity

impedance Z = R + jX

Voltage and current are written as A·e^(jωt) — complex exponentials. Impedance Z = R + jX is literally a complex number.

Signal processing

Fourier transforms

The Fourier transform converts time-domain signals into complex-valued frequency components. MP3, JPEG, 5G all rely on it.

Quantum mechanics

wavefunctions

A quantum state is a complex-valued wavefunction ψ. Probability is |ψ|². Without complex numbers, no Schrödinger equation.

Aerodynamics

complex potential flow

Joukowski transforms map a circle to an aerofoil — done in the complex plane. Wings exist because of complex-variable maths.

Control systems

stability analysis

Stability of a feedback loop = "are all poles in the left-half complex plane?". Aircraft autopilots, cruise control, all built on this.

Computer graphics

rotations via complex multiplication

Quaternions (4D complex numbers) rotate 3D objects without "gimbal lock". Every game engine's camera uses them.