This chapter first deals with what complex numbers are and how we manipulate them and understand their properties; and then goes on to describe how useful they are in helping to solve a variety of problems. The case of a resonant system will be dealt with in detail. Because the use of complex numbers makes the solution of this type of problem trivial, we will be able to concentrate on the form of the results, rather than on the details of how to get there.

Complex numbers are also helpful in describing waves of all sorts (see Chapter 14 of Volume 2). They are particularly useful in situations involving the diffraction and interference of light waves. In quantum mechanics, it is not so much that they are a calculational aid, but rather that the wave function used to describe objects like electrons involves complex numbers in an essential manner. Neither of these last two applications is described here.

What are complex numbers?

Mathematicians are adept at inventing new types of concepts. Thus while the use of positive numbers to represent the magnitude of actual quantities in the real world gives them an immediate significance, this is not so obvious for negative numbers.