Number Systems

A Path into Rigorous Mathematics

Author: Anthony Kay

This book aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs.

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Summary

Description

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding.

The author’s motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.

Features

- Approachable for students who have not yet studied mathematics beyond school
- Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof
- Draws attention to connections with other areas of mathematics
- Plenty of exercises for students, both straightforward problems and more in-depth investigations
- Introduces many concepts that are required in more advanced topics in mathematics.

Critic Reviews

"This is not exactly a standard elementary number theory textbook, nor is it focused only on proof-writing skills. Rather, it’s a fascinating look at the properties of various number systems, beginning with the natural numbers and wending a fascinating path all the way through to a brief look at the octonions. Along the way, there’s an appreciation for rigor and considerable effort dedicated to aiding the reader in thinking mathematically. With that in mind, it seems like this book would be a great choice for a transition course. Abelian groups and Dedekind cuts both make appearances, providing a possible bridge into later courses in abstract algebra or real analysis

But its appeal is not limited to prospective mathematics majors. Number Systems has the potential to serve as an excellent introduction for college students–at any level; the book grew out of a course for first-year students–to the non-computational side of our subject and to encourage them to think deeply about mathematics in a way that we’d all like to encourage. At the same time, this is also a book that holds a lot of appeal for seasoned professionals who want to revisit some ideas in a more recreational setting. It’s always a pleasure to encounter familiar ideas in a novel setting, and this book does a fine job of providing that pleasure."
– Mark Bollman, MAA Reviews

"In all, the book is a great attempt to introduce number systems to an undergraduate audience with the main focus on the rigor of mathematics. While, on the one hand, it provides detailed but accessible explanations of theorems and their proof, on the other hand, it is an attempt to provide the level of explanation needed for a first-year mathematics course on the subject that most others fail to do."
- Technometrics

About the Author

Anthony Kay was a Lecturer in Mathematical Sciences at Loughborough University for 32 years up to his retirement in 2020. Although his research has been in applications of mathematics, he has taught a wide range of topics in pure and applied mathematics to students at all levels, from first year to postgraduate.