# Proofs from THE BOOK

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**Publisher:**Springer-Verlag Berlin and Heidelberg GmbH & Co. K-
**Format:**Hardback | 282 pages -
**Dimensions:**191mm x 262mm x 20mm | 748g **Publication date:**15 December 2009**Publication City/Country:**Berlin**ISBN 10:**3642008550**ISBN 13:**9783642008559**Edition:**4, Revised**Edition statement:**4th ed. 2010. Corr. 3rd printing 2013**Illustrations note:**250 black & white illustrations, biography**Sales rank:**167,108

### Product description

PaulErdos ? likedtotalkaboutTheBook,inwhichGodmaintainstheperfect proofsformathematicaltheorems,followingthedictumofG. H. Hardythat there is no permanent place for ugly mathematics. Erdos ? also said that you need not believe in God but, as a mathematician, you should believe in The Book. A few years ago, we suggested to him to write up a ?rst (and very modest) approximation to The Book. He was enthusiastic about the idea and, characteristically, went to work immediately, ?lling page after page with his suggestions. Our book was supposed to appear in March 1998 as a present to Erdos ? ' 85th birthday. With Paul's unfortunate death in the summer of 1996, he is not listed as a co-author. Instead this book is dedicated to his memory. ? Paul Erdos We have no de?nition or characterization of what constitutes a proof from The Book: all we offer here is the examples that we have selected, h- ing that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations. We also hope that our readers will enjoy this despite the imperfections of our exposition. The selection is to a ? great extent in?uencedby Paul Erdos himself. A largenumberof the topics were suggested by him, and many of the proofs trace directly back to him, or were initiated by his supreme insight in asking the right question or in makingthe rightconjecture. So to a largeextentthisbookre?ectstheviews of Paul Erdos ? as to what should be considered a proof from The Book.

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### Review quote

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 Erdos, 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 Gunter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdos. 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." (Mikl s B na, The Book Review Column, 2011) "Paul Erdos ... 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 Gunter Ziegler succeeded admirably in putting together a broad collection of theorems ... that would undoubtedly be in the Book of Erdos. The theorems are so fundamental ... that every mathematician, regardless of speciality, can benefit from reading this book." (Miklos Bona, 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 fur Didaktik de Mathematik, July, 2004)

### Editorial reviews

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

### Back cover copy

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

### Table of contents

Number Theory.- Six proofs of the infinity of primes.- Bertrand's postulate.- Binomial coefficients are (almost) never powers.- Representing numbers as sums of two squares.- The law of quadratic reciprocity.- Every finite division ring is a field.- Some irrational numbers.- Three times ?^2/6.- Geometry.- Hilbert's third problem: decomposing polyhedra.- Lines in the plane and decompositions of graphs.- The slope problem.- Three applications of Euler's formula.- Cauchy's rigidity theorem.- Touching simplices.- Every large point set has an obtuse angle.- Borsuk's conjecture.- Analysis.- Sets, functions, and the continuum hypothesis.- In praise of inequalities.- The fundamental theorem of algebra.- One square and an odd number of triangles.- A theorem of Polya on polynomials.- On a lemma of Littlewood and Offord.- Cotangent and the Herglotz trick.- Buffon's needle problem.- Combinatorics.- Pigeon-hole and double counting.- Tiling rectangles.- Three famous theorems on finite sets.- Shuffling cards.- Lattice paths and determinants.- Cayley's formula for the number of trees.- Identities versus bijections.- Completing Latin squares.- Graph Theory.- The Dinitz problem.- Five-coloring plane graphs.- How to guard a museum.- Turan's graph theorem.- Communicating without errors.- The chromatic number of Kneser graphs.- Of friends and politicians.- Probability makes counting (sometimes) easy.