Topology of Numbers

Roger Howe stated during his MAA invited address, “Everybody knows that mathematics is about miracles, only mathematicians have a name for them: theorems.” Allen Hatcher’s Topology of Numbers is one of those rare texts that takes students on a journey weaving through number theory with skill worthy of the Moirai themselves. The book is an introduction to number theory for undergraduates that eschews standard approaches. Instead, the text teaches the elegance and subtlety of number theory by focusing on quadratic forms and their topology. 

 

Chapter 0 introduces the topics and themes of the book by motivating quadratic forms through Pythagorean triples. Chapters 1–4 contain many of the standard number theory topics (though often tersely): the Euclidean algorithm, modular arithmetic, the Chinese Remainder Theorem, continued fractions, etc. More importantly, these chapters introduce the foundations of the text: quadratic forms, Farey diagrams, topographs, etc. Chapter 5 beautifully teaches the classification of quadratic forms in terms of their discriminant, especially emphasizing associated symmetries. Chapter 6 addresses representations of integers by quadratic forms, while Chapter 7 focuses on how to multiply forms of a given discriminant. Finally, Chapter 8 takes another vantage point towards quadratic forms that leads to discussions of ideals and class groups. 

 

The book is assiduously written and is of a rare breed of text capable of teaching through colloquy and ratiocination rather than through the slough of ‘standard’ approaches. There are similar texts, e.g. J.H. Conway’s foundational The Sensual (quadratic) Form, but these are less readable. The book is strongly reminiscent of Needham’s Visual Complex Analysis in both the quality of text and the beauty of its geometrical argumentation. Any undergraduates of sufficient mathematical maturity will find the text approachable. However, the students that are most likely to succeed with the book will be familiar with standard proof techniques, comfortable with the basics of modular arithmetic, and know the arithmetic, determinants, and inverses of (two-by-two) matrices. 

 

The book contains all the standard topics for instructors wanting a ‘traditional’ approach to number theory: modular arithmetic, the Euclidean Algorithm, linear Diophantine equations and the Chinese Remainder Theorem, quadratic reciprocity, class groups, etc. Background for these topics are briefly covered as needed. However, many of these topics would need to be supplemented by the instructor or with an additional text to meet ‘traditional’ needs. But this is where Hatcher’s meritorious and famous altruism shines—the text is freely available for download on the author’s website. This allows instructors to easily use this book as the primary or secondary text for students at the cost of only a single book. Although, a physical copy of the book is cheaply available from the AMS. The exercises are not plentiful, but they are illuminative and reinforce the section topics. Above all, Topology of Numbers shines as a text for an undergraduate independent study or senior thesis— whether or not they have already seen any number theory. Whether for hobbyists, undergraduates, graduates, or professionals, Topology of Numbers is a worthy member of every individual’s personal mathematical library.

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Dr. Caleb McWhorter is an Assistant Professor of Mathematics at St. Thomas Aquinas College. His research is in number theory, primarily in arithmetic geometry focused on elliptic curves.