Kudos: An Illustrated Theory of Numbers

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— Samuel S. Wagstaff Jr. MathSciNet Mathematical Reviews, MR3677120.

This book is an introduction to number theory like no other. It covers the standard topics of a first course in number theory from integer division with remainder to representation of integers by quadratic forms. Nearly 500 illustrations elucidate proofs, provide data visualization and give fresh new insights...

...The page layout is exquisite. A new paragraph begins at the top of nearly every page. All theorems are proved and proofs fit on two facing pages, and usually on just one page. Numerous examples are solved, always on a single page. In Chapter 4, facing pages treat Gaussian and Eisenstein integers on the left and right sides, respectively. Footnotes appear beside their references in the wide right margins, not at the bottom. Each chapter begins with a figure on the left side and text on the right side of a two-page spread. Chapters end with historical notes and exercises, each exactly filling two facing pages. The historical notes reference original sources, often outside of Western tradition. The exercises range from routine calculations to interesting mathematical explorations.

— Pierre Dehornoy La Gazette des mathématiciens 159 (janvier 2019).

Lecteur, arrête-toi cinq minutes: ce livre est un chef-d’œuvre, et je vais essayer de t’en convaincre. Au premier abord il s’agit d’un cours de théorie élémentaire des nombres comme il en existe pas mal sur le marché. Mais c’est surtout un très très beau livre que n’importe quel lecteur intéressé par les mathématiques aura du plaisir à feuilleter...

...Ceux qui n’aiment vraiment pas les dessins vont peut-être souffrir en ouvrant ce livre. Ceux qui ne regardent que les dessins vont sans doute être ravis. Mais surtout l’ensemble est tellement travaillé et léché qu’il pourrait faire basculer une partie des indécis dans le camp de ceux qui pensent qu’un bon dessin est souvent le plus court chemin pour comprendre et retenir un bel argument.

— A. Misseldine CHOICE, Feb 2018 vol.55 no.6

It is rare that a mathematics book can be described with this word, but Weissman’s An Illustrated Theory of Numbers is gorgeous! Weissman (Univ. of California, Santa Cruz) not only wrote a great textbook on number theory but also did so in a visually stunning way. The work is full of hundreds of beautiful visuals that complement the otherwise difficult subject matter. Any reader with a high school geometry and algebra background will be prepared to read, understand, and enjoy this text. Nonetheless, the text does get quite advanced, including topics beyond elementary number theory. Fortunately, Weissman prepares readers so they do not realize they have entered advanced mathematics. Although the book is not proof focused, it contains several proofs — all chosen specifically for their geometric approach and often deviating from more-standard proofs. Perhaps the work’s only weakness is that it is too geometric — meaning that the inclusion and omission of specific topics was decided solely upon which images could be included. Though math students might miss out on a few core principles covered in a traditional number theory course, most readers will love this work because they will be able to see numbers for the first time.

(Author's response: I'm a little confused about the characterization: "the book is not proof focused." The book contains 178 proofs of lemmas, propositions, theorems, and corollaries. Also, the sentence "...the inclusion and omission of specific topics was decided solely upon which images could be included" brings up a more interesting point. I really did decide on topics based on the curriculum for undergraduate number theory courses I taught. The challenge I set for myself was to create visualizations to best accompany those topics. This sometimes led me to find alternative approaches. The relationship between the mathematical topics and the visualization was a complicated one that evolved over about 10 years.)

— Mehdi Hassani Full MAA Review, January 2018

Visualization is a very comfortable way to contemplate on scientific facts. For mathematical facts, visualization includes:

Proofs based on a picture (as in many proofs for elementary geometry),
Pictures describing a proof (several facts in calculus),
Pictures, charts, diagrams, and even multimedia productions that help us see a result.

All of the above help the reader to have a better understanding of the result under study, and also help to see possible patterns. While the "Theory of Numbers" is known for its abstract face, the book under review gives illustrated versions of many number-theoretic concepts. There are many pretty color figures giving conceptual pictures and diagrams, visualized computations, representations of the key insight in proofs, as well as several tables...

...Each chapter starts with a picture or diagram indicating what is going on the chapter, and ends in some historical notes and a number of exercises. The book includes most of the topics in a typical elementary course in number theory, insisting [sic?] on the concepts related to quadratic equations and quadratic forms. I believe that this book is a very interesting text for a first undergraduate course. Graduate students and researchers will also find some good ideas here, particularly on how develop a concept with pictures and diagrams. If the instructor doesn’t choose this book as the text, I suggest it strongly as a supplement for students, allowing them to follow their number theory course with some visualization in parallel.

Kudos

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