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Proofs from THE BOOK [Hardcover]

Martin Aigner , Günter M. Ziegler , Karl H. Hofmann
5.0 out of 5 stars  See all reviews (5 customer reviews)
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Book Description

13 Oct 2009 3642008550 978-3642008559 4th ed. 2010. Corr. 3rd printing 2013

This revised and enlarged fourth edition features five new chapters, which treat classical results such as the "Fundamental Theorem of Algebra", problems about tilings, but also quite recent proofs, for example of the Kneser conjecture in graph theory. The new edition also presents further improvements and surprises, among them a new proof for "Hilbert's Third Problem".

From the Reviews:

"... Inside [this 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. ..., but many [proofs] are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: "... 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. ... " AMS Notices 1999

"... the level is close to elementary ... the proofs are brilliant. ..." LMS Newsletter 1999


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Product details

  • Hardcover: 274 pages
  • Publisher: Springer; 4th ed. 2010. Corr. 3rd printing 2013 edition (13 Oct 2009)
  • Language: English
  • ISBN-10: 3642008550
  • ISBN-13: 978-3642008559
  • Product Dimensions: 24.6 x 19.6 x 2.3 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Bestsellers Rank: 293,343 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Review

From the reviews of the fourth edition:

“This is the fourth edition of a book that became a classic on its first appearance in 1998. … The authors have tried, in homage to Erdős, to approximate this tome; successive editions appear to be achieving uniform convergence. … Five new chapters have been added … . there is enough new material that libraries certainly should do so. For individuals who do not yet have their own copies, the argument for purchase has just grown stronger.” (Robert Dawson, Zentralblatt MATH, February, 2010)

“This book is the fourth edition of Aigner and Ziegler’s attempt to find proofs that Erdos would find appealing. … this one is a great collection of remarkable results with really nice proofs. The authors have done an excellent job choosing topics and proofs that Erdos would have appreciated. … the proofs are largely accessible to readers with an undergraduate-level mathematics background. … I love the fact that the chapters are relatively short and self-contained. … this is a very nice book.” (Donald L. Vestal, The Mathematical Association of America, May, 2010)

“Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdős. The theorems are so fundamental, their proofs so elegant, and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. The book has five parts of roughly equal length.” (Miklόs Bόna, The Book Review Column, 2011)

“Paul Erdős … had his own way of judging the beauty of various proofs. He said that there was a book somewhere, possibly in heaven, and that book contained the nicest and most elucidating proof of every theorem in mathematics. … Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems … that would undoubtedly be in the Book of Erdős. The theorems are so fundamental … that every mathematician, regardless of speciality, can benefit from reading this book.” (Miklós Bóna, SIGACT News, Vol. 42. (3), September, 2011)

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 Back Cover

This revised and enlarged fourth edition of "Proofs from THE BOOK" features five new chapters, which  treat classical results such as the "Fundamental Theorem of Algebra", problems about tilings,  but also quite recent proofs, for example of the Kneser conjecture in graph theory. The new edition also presents further improvements and surprises, among them a new proof for "Hilbert's Third Problem". 

 From the Reviews

"... 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... write: "... 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 AMS, 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. ..." 

LMS Newsletter, January 1999


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Customer Reviews

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Most Helpful Customer Reviews
40 of 41 people found the following review helpful
5.0 out of 5 stars Proofs from THE BOOK 14 Jan 2002
By A Customer
Format:Hardcover
If you are at all interested in mathematics, and your background is anything from undergraduate to Fields medallist, then this is the one book you really must have. Paul Erdos, the most prolific and creative mathematician of the 20th century, had no need for the hypothesis of God. He nevertheless liked to believe that this non-existent deity kept a book in which were recorded all the most perfect and beautiful proofs in mathematics, and that, just once in a while, mere mortals were permitted a brief glance at some of the pages. As a tribute to Erdos, who died in 1996, Aigner and Ziegler have made a guess at what those pages might contain. They have compiled an A to Z of what they regard as the most elegant proofs from a wide range of mathematics, including Number Theory, Geometry, Analysis, Combinatorics and Graph Theory (all subjects close to Paul's heart). Many of the results are remarkably easy to state, for instance the theorem that there are infinitely many prime numbers, but all require some ingenuity to prove them; this particular example is given six different proofs, three of them direct, two involving Analysis and Topology, and one based on a neat counting argument. Similarly, in Combinatorics we get four totally different proofs of Cayley's formula for the number of trees on n vertices, each of them a real classic, and one (perhaps the best) very recent.
This book is superbly written and produced. Each chapter has a brief but clear historical introduction to the problem at hand, some excellent illustrative diagrams, a discussion of the proof or proofs, and a selection of references for further reading. There are portraits of many of the mathematicians, ancient and modern, whose ideas appear here, while Karl Hofmann's witty cartoons are an extra and unexpected joy.
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11 of 11 people found the following review helpful
5.0 out of 5 stars quite excellent 31 Jan 2003
Format:Hardcover
If you have ever been excited by the beauty of pure mathematics - or wondered at people who were - this book is for you. Not only do the authors exhibit a variety of the most elegant proofs around, but their presentation is of the highest quality - and in a very attractively designed book too. Recommended for even the most half-hearted maths lover.
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5.0 out of 5 stars Brilliant! 19 Mar 2014
Format:Hardcover|Verified Purchase
Anyone looking for a book of wonderfully complex yet beautiful proof could not do better than this book. I thoroughly recommend this to anyone in undergraduate maths with a love for proof.
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5.0 out of 5 stars Its simply amazing book 2 Nov 2013
Format:Hardcover|Verified Purchase
It's a beautiful book that highlights the beauty of mathematics. Everyone with high school level of maths should be able to appreciate it.
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5.0 out of 5 stars Fun gift for a mathematically-inclined person 27 Aug 2013
Format:Hardcover|Verified Purchase
Not being especially mathematical myself, I didn't get that much out of it. But fortunately I didn't buy the book for myself. I bought this as a birthday gift for a mathematician. He really likes it. If I had other mathematician friends, I'd buy it for them too.
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