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Proofs from THE BOOK

Authors: Aigner, Martin, Ziegler, Günter M.

  • This is the 2nd revised and expanded edition of an excellent, easy-to-read book for everyone with an interest in mathematics
  • Two top researchers have made a big effort and selected a list of mathematical problems which can be solved by elegant, esthetically pleasing proofs
  • The result is a book which will be fun for everybody with an interest in mathematics requiring only very little previous knowledge
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Buy this book

eBook $74.99
price for USA in USD (gross)
  • ISBN 978-3-662-04315-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
About this Textbook

The (mathematical) heroes of this book are "perfect proofs": brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erdös, who himself suggested many of the topics in this collection.
The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For this revised and expanded second edition several chapters have been revised and expanded, and three new chapters have been added.

Reviews

From the reviews of the third edition:

"... It is unusual for a reviewer to have the opportunity to review the first three editions of a book - the first edition was published in 1998, the second in 2001 and the third in 2004. ... I was fortunate enough to obtain a copy of the first edition while travelling in Europe in 1999 and I spent many pleasant hours reading it carefully from cover to cover. The style is inviting and it is very hard to stop part way through a chapter. Indeed I have recommended the book to talented undergraduates and to mathematically literate friends. All report that they are captivated by the material and the new view of mathematics it engenders. By now a number of reviews of the earlier editions have appeared and I must simply agree that the book is a pleasure to hold and to look at, it has striking photographs, instructive pictures and beautiful drawings. The style is clear and entertaining and the proofs are brilliant and memorable. ...

David Hunt, The Mathematical Gazette, Vol. 32, Issue 2, p. 127-128

"The newest edition contains three completely new chapters. … The approach is refreshingly straightforward, all the necessary results from analysis being summarised in boxes, and a short appendix discusses the importance of the zeta-function in number theory. … this edition also contains additional material interpolated in the original text, notably the Calkin-Wilf enumeration of the rationals." (Gerry Leversha, The Mathematical Gazette, March, 2005)

"A lot of solid mathematics is packed into Proofs. Its thirty chapters, divided into sections on Number Theory, Geometry, Analysis … . Each chapter is largely independent; some include necessary background as an appendix. … The key to the approachability of Proofs lies not so much in the accessibility of its mathematics, however, as in the rewards it offers: elegant proofs of interesting results, which don’t leave the reader feeling cheated or disappointed." (Zentralblatt für Didaktik de Mathematik, July, 2004)

From the reviews of the second edition:

"... Thirty sections treat results drawn from number theory, geometry (mainly combinatorial), analysis, combinatorics and graph theory; these can be follwed by one versed in undergraduate matheamtics including discrete topics. ...  The authors have done a fine job of arranging diverse material into a thematic progression. ... The presentation is clear and attractive with wide margins for portraits, diagrams and sketches."

E.J.Barbeau, Mathematical Reviews, Issue 2000k 

" ... This is a wonderful book that can be recommended to anybody who is in any way connected to mathematics. Those who have ever experienced the beauty of mathematics will experience the chill again. For those who have never experienced that, this book is just the right one to start."

Acta Scientiarum Mathematicarum, 1999, Vol. 65, 769-770

"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler do not claim to have presented the definitive collection of great mathematics. In their brief introduction they write: "We have no definition or characterization of what constitutes a proof from THE BOOK: all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the American Mathematical Society, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures, and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately, and the proofs are brilliant. Moreover, the exposition makes them transparent. ..."

London Mathematical Society Newsletter, January 1999

"... Clearly this second edition is dangerously suited to infect the reader with the enthusiasm of the authors."

J.Elstrodt (Münster), Zentralblatt für Mathematik 0978.00002


Table of contents (32 chapters)

  • Six proofs of the infinity of primes

    Aigner, Martin (et al.)

    Pages 3-6

  • Bertrand’s postulate

    Aigner, Martin (et al.)

    Pages 7-12

  • Binomial coefficients are (almost) never powers

    Aigner, Martin (et al.)

    Pages 13-16

  • Representing numbers as sums of two squares

    Aigner, Martin (et al.)

    Pages 17-22

  • Every finite division ring is a field

    Aigner, Martin (et al.)

    Pages 23-26

Buy this book

eBook $74.99
price for USA in USD (gross)
  • ISBN 978-3-662-04315-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
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Bibliographic Information

Bibliographic Information
Book Title
Proofs from THE BOOK
Authors
Copyright
2001
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-662-04315-8
DOI
10.1007/978-3-662-04315-8
Edition Number
2
Number of Pages
VIII, 215
Number of Illustrations
192 b/w illustrations, 6 illustrations in colour
Topics