Short answers, pulled from the story.
Oliver Heaviside coined the word impedance in 1886 to describe a phenomenon that had baffled engineers for decades. He realized that the opposition to current flow was not just a single number but a complex entity including both resistance and reactance.
Johann Victor Wietlisbach used complex numbers to analyze the Maxwell bridge in 1879. He expressed currents as exponential functions with imaginary exponents to avoid messy differential equations.
Resistance is the real part of impedance that dissipates energy as heat and remains constant regardless of frequency. Reactance is the imaginary part that arises from magnetic fields in inductors and electric fields in capacitors, creating a phase shift that leads or lags the voltage.
An inductor offers more opposition to current as frequency increases because the changing magnetic field creates a stronger back electromotive force. A capacitor offers less opposition as frequency increases because rapid oscillation prevents charge from building up enough to create a significant opposing field.
Engineers use bridge methods, impedance analyzers, and LCR meters to measure both the magnitude and phase difference between voltage and current. These tools sweep frequencies to provide a complete picture of how a component behaves across a range of signals.
The Laplace transform uses the complex frequency variable s to evaluate circuit behavior under any arbitrary signal rather than just steady-state AC or DC. This generalized s-plane impedance allows engineers to analyze transient responses and non-linear or time-varying circuits.