Imaginary numbers are used quite a bit in electrical engineering, and actually the “imaginary” part of it is quite applicable.
In AC power, you have watts and vars. Most people have heard of watts, since that is what the power company charges you for. For typical residential service, the power company doesn’t charge you for vars, so most folks haven’t heard of them.
Var stands for volt-amp-reactive. A “reactor” in electrical terms is something like an inductor or capacitor. A simple inductor is a coil of wire. A simple capacitor is two metal plates close together, but not touching. When you apply electricity to a reactor, it stores energy. An inductor stores the energy in a magnetic field (you can kinda think of it as an electromagnet) and a capacitor stores energy in an electric field. Remove the electricity, and the magnetic or electric field collapses, releasing the energy back into the system. So in that respect they are temporary energy storage devices. In AC systems, the electricity is a sine wave, which means that these reactors are constantly charging and discharging.
In electrical engineering, we use j instead of i, because i already stands for current (from the French “intensitie”). We use real numbers for the watts, and imaginary numbers for the vars (as well as for the imaginary or reactive current). The watts end up being the “real” power, i.e. the power that is converted into heat and is actually used up (like in a light bulb), and the vars end up being the “imaginary” power, or the power that is just wasted by charging up those reactors. Even though that power is later put back into the system when the reactors discharge, the extra current required to charge up the reactors means an extra load on the generators.
So we might say for example that the current is 15+j2 amps. That means 15 amps of “real” current and 2 amps of “imaginary” current.
Most homes tend to be slightly inductive due to the coils in motors for things like hair dryers, clothes dryers, refrigerators, etc. The thing about inductors and capacitors is that they work opposite of each other. When one is charging during the AC cycle, the other is discharging, and vice-versa This means that you can use capacitors to balance out the inductance. So inductors will add vars and capacitors will subtract vars. The power company tries to balance out the vars, so when the vars are equal to zero, the generator only has to supply the “real” power and the reactive power essentially ping-pongs back and forth between the inductors and capacitors during the AC cycle. Since the generator doesn’t have to supply any of the reactive current, this makes the power generation and transmission much more efficient.
A simple resistor has an impedance of R. An inductor has an impedance of jwL, where L is the inductance. A capacitor has an impedance of -j/wC, where C is the capacitance. w = 2(pi)f, where f is the frequency (60 Hz in the U.S., 50 Hz in some other countries). The w is actually a lower case omega, not an English W, but it’s fairly common to type it as a w when using English letters. Anyway, if you know L, you can calculate the C you need to balance it. If you know the voltage and current then you can calculate the impedance, which again allows you to figure out the L and C values. So, lots of fun with complex math, basically.
Var balancing capacitors are located in power system substations, or may be mounted on power poles. The power company just includes the cost of the capacitors as a general equipment cost (like the generators and wires, etc) and only charges you for the watts that you use.
Industrial and commercial customers are charged for vars. Industrial users in particular often have big motors (which means a big inductance), and the power company isn’t quite so happy to pay for the huge capacitors needed to balance those out. So, the power company charges them for vars, and they charge them out the wazoo for them too. This gives those types of customers a big incentive to install their own var correcting capacitors. Those big capacitors aren’t cheap, but when the power company’s rate for vars is on the order of the “bend over and squeal like a piggy” magnitude, the capacitor banks end up being much cheaper in the long run. It’s definitely in the company’s best interest to eliminate the “imaginary” vars so that they only pay for the “real” power that they use.
Var balancing is usually called power factor correction, in case you are curious. Power factor is another way of expressing the phase angle between the voltage and current, which is something that you can calculate if you know the watts and the vars. It’s just different ways of expressing the same thing.
Anyway, the overall point here is that in AC power, real and imaginary numbers definitely make a lot of sense with real and imaginary power.