Where Do Numbers Come From?

A clear, entertaining development of the number systems required in any course of modern mathematics.

T. W. Körner (Author)

9781108738385, Cambridge University Press

Paperback / softback, published 24 October 2019

270 pages, 1 b/w illus. 255 exercises
22.8 x 15.2 x 1.5 cm, 0.4 kg

'Where Do Numbers Come From? certainly adds to the pleasure of mathematics as well as narrating a journey that surely every mathematician should undertake at some stage. As such, I enthusiastically recommend it to all Gazette readers.' Nick Lord, The Mathematical Gazette

Why do we need the real numbers? How should we construct them? These questions arose in the nineteenth century, along with the ideas and techniques needed to address them. Nowadays it is commonplace for apprentice mathematicians to hear 'we shall assume the standard properties of the real numbers' as part of their training. But exactly what are those properties? And why can we assume them? This book is clearly and entertainingly written for those students, with historical asides and exercises to foster understanding. Starting with the natural (counting) numbers and then looking at the rational numbers (fractions) and negative numbers, the author builds to a careful construction of the real numbers followed by the complex numbers, leaving the reader fully equipped with all the number systems required by modern mathematical analysis. Additional chapters on polynomials and quarternions provide further context for any reader wanting to delve deeper.

Introduction
Part I. The Rationals: 1. Counting sheep
2. The strictly positive rationals
3. The rational numbers
Part II. The Natural Numbers: 4. The golden key
5. Modular arithmetic
6. Axioms for the natural numbers
Part III. The Real Numbers (and the Complex Numbers): 7. What is the problem?
8. And what is its solution?
9. The complex numbers
10. A plethora of polynomials
11. Can we go further?
Appendix A. Products of many elements
Appendix B. nth complex roots
Appendix C. How do quaternions represent rotations?
Appendix D. Why are the quaternions so special?
References
Index.

Subject Areas: Mathematical foundations [PBC]