The right way to introduce complex numbers in damped harmonic oscillators

The introduction of complex numbers to solve the damped harmonic oscillator in classical mechanics can seem mysterious to students. We show how an approach based on Born and Jordan's 1930 quantum mechanics textbook both demystifies the use of complex numbers in classical mechanics and makes a strong connection to quantum mechanics. In that work, they converted the harmonic oscillator equations of motion into two uncoupled first-order equations. In classical mechanics, this mapping explicitly introduces complex numbers into the motion of the oscillator and directly shows how to solve for the position and momentum observables. Here, we explain how this mapping works and show how to demystify the use of complex numbers in damped-driven harmonic oscillators. As an added bonus, this approach also shows how to determine the energy as a function of time and makes a strong connection to quantum mechanics.