I was reading an excellent article on Quanta Magazine titled "How to Find Simple Treasures in Complex Numbers." The article explains in very simple terms how to appreciate and understand what imaginary numbers are and how they work.
In short, imaginary numbers provide a solution to a problem when one element of an equation is missing. Imaginary numbers occur in several physics equations, such as in calculating the impedance of an electrical circuit.
Due to an error in modern systems of units, charge is notated as a single dimension in almost all units, but charge is notated as charge squared in five units. This creates a situation in some equations where unlike units are added together. When unlike units are added, the imaginary number is needed to account for the missing dimension.
