Introduction

In the first chapter you saw why you need imaginary and complex numbers, by considering the solution of simple quadratic equations. In this chapter you will see how we set up complex numbers in general, and establish their basic algebraic and geometrical properties.

We shall assume that you have some understanding of what is meant by a real number. The exact nature and depth of this understanding will not materially affect the discussion thoughout most of this book, and this is not a book about the fundamentals of real analysis. We should, however, take a moment to remind ourselves what a ‘real’ number is, before we start defining ‘imaginary’ and ‘complex’ numbers. Students of pure mathematics should remind themselves of the details of these matters — there is really nothing for it but to go for a proper mathematical definition, and experience has shown that one needs to be slightly abstract in order to get it right, in the sense that the resulting definition contains all the numbers ‘we need’. For a full exposition, complete with proofs, you should consult a text on real analysis, such as that by Rudin (1976). For our purposes it will mostly be sufficient to regard real numbers as being all the points on a line (which we call the real axis) extending to infinity in both directions.