Summary

First we consider what are the basic notions of mathematics, and emphasise the need for mathematicians to agree on a common starting point for their deductions. Peano's axioms for the natural numbers are listed. Starting with a system of numbers satisfying Peano's axioms, we construct by algebraic methods the systems of integers, rational numbers, real numbers and complex numbers. At each stage it is made clear what properties the system constructed has and how each number system is contained in the next one. In the last section there is a discussion of decimal representation of rational numbers and real numbers.

The reader is presumed to have some experience of working with sets and functions, and to be familiar with the ideas of bijection, equivalence relation and equivalence class.

Natural numbers and integers

It is fashionable nowadays at all levels of study from elementary school to university research, to regard the notion of set as the basic notion which underlies all of mathematics. The standpoint of this book is that the idea of set is something that no modern mathematician can be without, but that it is first and foremost a tool for the mathematician, a helpful way of dealing with mathematical entities and deductions. As such, of course, it becomes also an object of study by mathematics. It is inherent in the nature of mathematics that it includes the study of the methods used in the subject; this is the cause of much difficulty and misunderstanding, since it apparently involves a vicious circle.