Peter M. Higgins

Numbers

A Very Short Introduction

Preface

The purpose of this little book is to explain, in language that will be familiar to everyone, what are the various kinds of numbers that arise and howtheybehave. Numbers allow comparisons between all manner of things, and anyone with no understanding of numbers would be lost in the modern world where things numerical are there to greet us at every turn. We should realize however that, despite their familiarity, numbers have no physical existence but rather are abstractions that we elicit from the world we find ourselves in. To develop a clear picture of how they operate, it is sometimes better to consider them in their own right,without reference to anything else.

This Short Introduction is not a refresher course in arithmetic, nor will that much be said on the history of the number system.Rather its purpose is to explain numbers themselves and the kinds of behaviour they exhibit. A glance at the list of chapters reveals that the first part of the book deals mainly with ordinary counting numbers, while in the second halfwe go beyond that. By exploring natural problems that arise in commerce and science, the need to freely perform calculations eventually has taken us, by stages, into the arena of the complex numbers, which is the principal underlying frameworkfor most number matters. This may sound a little daunting but rest assured, the hard work has already been done for you.

The modern number system did not come to us gift-wrapped but rather has developed over many centuries. There were long periods ofconfusion, which had two root causes. The first was the lack ofan efficient way to represent the numbers we need that would allowus to manipulate them. The second, which was related to the first, was philosophical agonizing over the interpretations ofvarious number types and whether or not they are meaningful. Nowadays we are much more sure ofourselves when it comes to what we do and don't need to worry aboutwhen dealing with numbers, making it possible to give a complete picture ofthe number world in a single short account like this one.That is not to say that all mystery has vanished–far from it, as you will discover as you read on.

Peter M. Higgins

Colchester, England,2011