Mathematics for Physicists

6 Complex numbers and variables

In previous chapters we have been discussing real numbers and their algebraic representation. Real numbers are part of a larger set called complex numbers. In this chapter we start by showing how the latter arise and then discuss their properties and how they are represented. Complex numbers and complex variables are of great practical importance in a wide range of topics, including vibrations and waves, and quantum theory.

6.1 Complex numbers

Given a positive real number q (not necessarily an integer) we know that its square roots are also real numbers. But situations also arise where we meet the square root of a negative number. In Section 2.1.1, for example, we saw that the solution of a general quadratic equation ax² + bx + c = 0 is of the form

(6.1)

and there is no restriction on the sign of (b² − 4ac). Thus we have to face the question: can we find an interpretation of the quantity , where q > 0? It cannot be the same as because squaring would produce a contradiction. A new definition is required. Since

it follows that the only new definition ...