Complex Number

Definition

A number with real and imaginary parts (a + bi) — essential for AC circuits, quantum mechanics, and wave analysis.

$e^{i\theta} = \cos\theta + i\sin\theta$

Symbol: $i$ ·

Technical definition

Numbers of form z = a + bi where i² = −1, representable on the complex plane. Euler's formula e^(iθ) = cosθ + i·sinθ connects them to oscillations — the reason complex numbers dominate wave and circuit physics.

Example

AC circuit impedance Z = R + iX combines resistance and reactance in one complex number — Ohm's law generalizes beautifully to V = IZ.

Common confusion

‘Imaginary’ is a historical misnomer — complex numbers describe very real physics (phases, oscillations, quantum amplitudes). They are no more imaginary than negative numbers.