For the practical physicist and engineer complex numbers provide a handy tool for the analysis of AC circuits. A few basic mathematical rules enable reactive circuit elements like inductances and capacitances to be treated with nothing more difficult than Ohm's law. The first-order linear equation (18.6), dealing with the voltage/current relationship in a simple LR circuit can be used to illustrate the benefits of the complex approach. We shall really just be repeating the steps which led us to (19.13) but we shall see them in a different light.

For generality it is helpful to regard the driving voltage as having both cos and sin components, expressed as A cos ωt – B sin ωt (the choice of sign is to comply with later conventions). Then the responding current also has both components and, following chapter 19, is written a cos ωt – b sin ωt. The mathematical object is then to calculate the coefficients a, b in terms of A, B by equating the driving voltage to the voltage fall across the elements L, R, as before.

Now the voltage change across the resistance is simply proportional to the current so that the cosine and sine oscillations of the voltage are the same, multiplied by R. In engineering terms this is the ‘quadrature response’. On the other hand the inductance requires a time differentiation which effectively exchanges the components (with a change of sign) to give the voltage fall – ωLa sin ωt – ωLb cos ωt.