C h a p t e r 1 P r e l i m i n a r i e s All of the mathematics in this book is based on real numbers. Historically, mathematics began with the set of natural numbers (or positive integers) on which we have the operations of addition and multiplication. Much later, this set was extended by adding zero and the negative integers, thus giving the set of integers. In this set, subtraction is always possible, but division is only possible in certain cases. For example, we can divide 20 by 5, but

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... by 6. Mathematicians were thus led to introduce the rational numbers (or fractions), so division was always possible (except by zero). Eventually it was realised that yet more 'numbers' were needed to do even quite simple calculations. For example, the diagonal of a unit square has length y/2, but \/2 cannot be written as a fraction (see Section 1.8). To resolve this problem, mathematicians introduced irrational numbers, thus arriving finally at the set of real numbers. In this first chapter, we recall results about the set of real numbers and some subsets which are important in their own right. The reader should be familiar with most of the material in Sections 1.1 to 1.6, but should read these sections carefully since they introduce notation which will be used in the remainder of the book. The final sections may well be new. Section 1.7 looks in detail at the idea of divisibility in the set of integers. Section 1.8 considers the numbers which cannot be written as fractions-the irrational numbers. These sections also introduce the idea of proof by contradiction, a technique much used throughout mathematics.