Complex numbers

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# 1 Introduction

You have almost certainly met complex numbers before, but you may well not have had much experience in manipulating them. In this course we provide you with an opportunity to gain confidence in working with complex numbers by working through a number of suitable problems.

Perhaps the most striking difference between real numbers and complex numbers is the fact that complex numbers have a two-dimensional character, arising from our definition of a complex number as an ordered pair of real numbers. This two-dimensional aspect of complex numbers leads to a most useful representation of them as points in the plane.

We know that the distance between points in a plane can be measured by the usual Euclidean measure of distance, and this leads us to the important modulus function for complex numbers. This function will clearly play an important role in complex analysis if the subject is to develop along lines resembling real analysis. Later we will see how the complex modulus can be used to generalise many of the limiting processes of real analysis.

Many proofs in real analysis use the concept of least upper bound. This concept depends in turn on the relation of inequality which is defined in terms of positive and negative numbers. We shall show in this course that it is not possible to define such a thing as a “positive” complex number and so least upper bound arguments will not carry over to complex analysis.

In real analysis, the idea of a least upper bound is used to develop the method of proof by repeated bisection, whose validity rests on the Nested Intervals Theorem. Because complex numbers are defined as pairs of real numbers, we are able to generalise this theorem to a Nested Rectangles Theorem, which will play a similar role in complex analysis to that of the Nested Intervals Theorem (or, equivalently, the least upper bound axiom) of real analysis.

There are three reading sections in this course, each of which includes a problems subsection. In Section 2 the complex number system is defined; in Section 3 the relationship between the complex numbers and points in the plane is developed; in Section 4 various useful subsets of the complex number system are defined and the Nested Rectangles Theorem is proved.

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