2017;
323 pp;
Hardcover

MSC: Primary 11;
**Print ISBN: 978-1-4704-3493-9
Product Code: MBK/105**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

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#### Supplemental Materials

# An Illustrated Theory of Numbers

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*Martin H. Weissman*

2018 PROSE Awards Honorable Mention

An Illustrated Theory of Numbers gives a
comprehensive introduction to number theory, with complete proofs,
worked examples, and exercises. Its exposition reflects the most
recent scholarship in mathematics and its history.

Almost 500 sharp illustrations accompany elegant proofs, from prime
decomposition through quadratic reciprocity. Geometric and dynamical
arguments provide new insights, and allow for a rigorous approach with
less algebraic manipulation. The final chapters contain an extended
treatment of binary quadratic forms, using Conway's topograph to solve
quadratic Diophantine equations (e.g., Pell's equation) and to study
reduction and the finiteness of class numbers.

Data visualizations introduce the reader to open questions and
cutting-edge results in analytic number theory such as the Riemann
hypothesis, boundedness of prime gaps, and the class number 1
problem. Accompanying each chapter, historical notes curate primary
sources and secondary scholarship to trace the development of number
theory within and outside the Western tradition.

Requiring only high school algebra and geometry, this text is
recommended for a first course in elementary number theory. It is also
suitable for mathematicians seeking a fresh perspective on an ancient
subject.

#### Readership

Undergraduate and graduate students interested in number theory.

#### Reviews & Endorsements

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...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.

-- Samuel S. Wagstaff, Jr., Mathematical Reviews

It is rare that a mathematics book can be described with this word, but Weissman's 'An Illustrated Theory of Numbers' is gorgeous! Weissmann (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...most readers will love this work because they will be able to see numbers for the first time.

-- A. Misseldine, CHOICE

This is a meticulously written and stunningly laid-out book influenced not only by the classical masters of number theory like Fermat, Euler, and Gauss, but also by the work of Edward Tufte on data visualization. Assuming little beyond basic high school mathematics, the author covers a tremendous amount of territory, including topics like Ford circles, Conway's topographs, and Zolotarev's lemma which are rarely seen in introductory courses. All of this is done with a visual and literary flair which very few math books even strive for, let alone accomplish.

-- Matthew Baker, Georgia Institute of Technology

'An Illustrated Theory of Numbers' is a textbook like none other I know; and not just a textbook, but a work of practical art. This book would be a delight to use in the undergraduate classroom, to give to a high school student in search of enlightenment, or to have on your coffee table, to give guests from the world outside mathematics a visceral and visual sense of the beauty of our subject.

-- Jordan Ellenberg, University of Wisconsin-Madison, author of “How Not to Be Wrong: the Power of Mathematical Thinking”

Weissman's book represents a totally fresh approach to a venerable subject. Its choice of topics, superb exposition and beautiful layout will appeal to professional mathematicians as well as to students at all levels.

-- Kenneth A. Ribet, University of California, Berkeley