Complex numbers are explained in Reference 5. Illustrations are given in that Reference in which complex numbers are obtained as the roots of polynomials.In mechanical and electrical systems, complex numbers are often used to represent such quantities as position, energy, voltage and current.
An intuitive grasp of the meaning of a physical quantity expressed in complex form may often be obtained by thinking of the real part of the number as energy which will be dissipated in the process under consideration or transported out of the system. The imaginary part of the same complex quantity may then be considered to be energy stored in the system which will be returned to its source at some time.
If a spring is compressed, for example, the energy required may be considered a complex quantity with the energy that heats the spring due to frictional losses in the compression process comprising the real part. The imaginary part of the quantity of energy is that stored in the spring which may be regained by allowing the spring to expand back to its original form.
The input impedance of a transmission line is a complex number both components of which are in units of ohms. The real part represents the ability of the line to absorb energy which will disappear down the line and be dissipated as heat in the resistance of the line or which will be delivered to the receiving device at the distant end of the line. The imaginary component of the input impedance represents the ability of the line to accept energy that will be stored in much the same manner as energy is stored in a spring. Such energy will be returned to the source. The returned energy is due to reflections from impedance mismatches along the line or due to the energy storage characteristics of the line itself.