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Proofs from THE BOOK [Hardcover]

Martin Aigner , Günter M. Ziegler , Karl H. Hofmann
4.5 out of 5 stars  See all reviews (21 customer reviews)

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Book Description

October 13, 2009 3642008550 978-3642008559 4th ed. 2010. Corr. 3rd printing 2013
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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Proofs from THE BOOK + The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (MAA Problem Books)
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Editorial Reviews


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

Product Details

  • Hardcover: 274 pages
  • Publisher: Springer; 4th ed. 2010. Corr. 3rd printing 2013 edition (October 13, 2009)
  • Language: English
  • ISBN-10: 3642008550
  • ISBN-13: 978-3642008559
  • Product Dimensions: 9.7 x 7.7 x 0.9 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (21 customer reviews)
  • Amazon Best Sellers Rank: #108,926 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
128 of 134 people found the following review helpful
1.0 out of 5 stars Kindle edition is worthless: avoid at all costs April 23, 2008
Format:Kindle Edition
NOTE: This review is JUST for the Kindle edition.

The Kindle edition is completely worthless, because it is missing many symbols. It appears to have been done using OCR, and it was confused by mathematical symbols. For example, there are some places where I THINK it was supposed to be the greek letter phi, but it comes out as a left parenthesis and a right parenthesis. At least with that you can figure out what it was supposed to be. There is much worse--places where symbols are completely gone. E.g., there is a place where you just get a capital sigma with a subscript giving a summation limit, a blank space, a less than sign, and another blank space. So, the proof is saying the some of *something* is less than *something else*.

This is a shame, because the book itself, from what I can see, is EXCELLENT.
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109 of 118 people found the following review helpful
1.0 out of 5 stars Do NOT get the Kindle edition July 7, 2008
Format:Kindle Edition|Verified Purchase
The Kindle edition of the book is missing or misrepresenting math symbols in so many places it makes it completely unreadable.
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23 of 25 people found the following review helpful
5.0 out of 5 stars Excellent, but bizarre and atypical book on mathematics October 22, 2006
I agree with what most of the other authors have said. This is a wonderful book, if only a bit strange. The book is very clearly written and easy to follow, although many of the proofs require a great deal of contemplation in order to fully understand.

This is one of those books that a serious mathematician will probably enjoy picking up and reading from time to time. It is neither a reference nor a textbook, but more a source of mathematical inspiration, a collection of particularly elegant mathematical results. My favourite aspect of this book is the way it focuses on proofs and results that draw connections between different areas of mathematics. The interconnectedness of mathematics is too often ignored by researchers nowadays, who have become specialized to the point that their work is often inaccessible. This book is a healthy step in the opposite direction.

My only complaint about this book (and it's not really about this book at all) is that there are not more books like this. The book is heavily slanted towards number theory, combinatorics, and graph theory. The chapter on analysis is beautiful, but atypical of analysis as a whole. I think that it would be wonderful for people to write similar books in other topics, in particular, analysis, algebra, or probability or topology like the other reviewer mentioned.

As a final note I would like to give credit to the publisher for doing a good job on this one. The typesetting is excellent, the layout is clear with all the illustrations, the paper is of high quality, and the binding is oustanding. It has held up well to moderate use and still looks brand new. I only wish some of my more heavily-used reference texts were bound like this.
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26 of 29 people found the following review helpful
5.0 out of 5 stars A breath of pure air October 24, 2002
I stumbled across this book and am amazed that I had not heard about it before. Since buying it, I have kept it by my bedside and have now read the whole book four or five times, picking up more of the subtleties at each reading.
The proofs are almost all magnificent (although I wonder how Buffon and his needles got in there) and even the well-known and time-honoured ones have a new twist or new extension.
The level of mathematics required to follow the proofs is reasonably low (high-school 'A' levels in the British system, no idea about other countries) although the book gives a deeper explanation in some areas (e.g. trans-finite arithmetic) than in others (e.g. number theory). I wonder if this unevenness reflects the interests of the authors.
But these are tiny nit-pickings. This is a wonderful and inspiring book and reading it should be made compulsory by the government in all high-school mathematics classes.
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17 of 19 people found the following review helpful
5.0 out of 5 stars A fitting tribute to the great Paul Erdos March 20, 2001
Paul Erdos once remarked that you need not believe in God, but you certainly have to believe in the book in which God maintains the "perfect" mathematical proofs. Martin Aigner and Gunter Ziegler have certainly done a great job with this book, a fitting tribute to the great Erdos himself.
I had purchased a copy of the 1st edition of this book and was plesantly surprised that the authors had come up with a 2nd edition, with a few more "perfect" proofs.
My personal favorites are "The Shannon capacity of a graph". where the Lovasz theta number would eventually lead to semidefinite programming, Erdos' probabilistic method where probability makes counting sometimes easy, computing the number of trees in a graph, how many guards it takes to guard a museum, and the section on Turan's theorem.
This book deserves to be on the bookshelves of both amateur and professional mathematicians.
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13 of 14 people found the following review helpful
5.0 out of 5 stars Proofs from the Book August 23, 2006
Format:Hardcover|Verified Purchase
The famous eccentric mathematician had a theory that God (who he liked to call the SF-Supreme Fascist) had a book with every mathematical theorem and that the book had the most beautiful and elegant proofs for the theorems.

This book has that philosophy and presents great proofs for many beautiful theorems.

It is a masterpiece. A book that shows why mathematicians love what they do.
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13 of 14 people found the following review helpful
Most of this book could be read by an undergraduate with only background in calculus and basic discrete math. The proofs are mostly self-contained, and there are helpful appendices to each chapter when prerequisite material is needed. I think this would be a good book for undergraduates or enthusiastic high school students to read for fun, or for lecturers to draw interesting examples from.

The proofs, while "elementary," are sometimes quite involved, however, and do require some maturity to be able to appreciate the "big picture" of each proof, and it's tempting to "island-hop" (i.e., just check that each if... then... works out in the proof, without looking at the whole thing put together). This would be especially nice material for an undergraduate math club or for math enthusiasts to read at liesure. The problems are also highly "integrated," in that they make use of ideas from several different types of math, and are not usually straightforward, so they are good examples of creative solutions to challenging problems.
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Most Recent Customer Reviews
5.0 out of 5 stars A journey into the "double life" of mathematics and the art of...
The late mathematician and philosopher Gian-Carlo Rota referring to the age old (and perhaps unanswerable) question - "are mathematical ideas invented or discovered? Read more
Published 4 months ago by Shubhendu Trivedi
3.0 out of 5 stars Even Erdos would have wanted a Glossery
" ...a modest amount of technique from undergraduate mathematics...." The authors say is all that is needed to follow the proofs in this book. No, modest will not do it. Read more
Published 5 months ago by Richard H. Pratt
5.0 out of 5 stars Amazing...
Each proof is a brilliant piece of mathematics, and the exposition is very clear and accessible for non-specialists. This book shows mathematics at its best.
Published 9 months ago by a student
5.0 out of 5 stars A wonderful book
This book is a gem that holds the most beautiful proofs of well-known problems of Mathematics. A feast for any lover of Math.
Published 11 months ago by ricardo mansilla
5.0 out of 5 stars Most beautiful
This book has a splendid collection of arguably the most beautiful theorems in mathematics. The authors also present the most elegant proofs for these theorems. Read more
Published 22 months ago by ARollingStone
5.0 out of 5 stars great books for the people who think math is fun
Not a professional mathematician, I feel this book is challenging enough to be interesting, but not beyond what I can follow. Read more
Published on September 19, 2011 by joe
5.0 out of 5 stars excellent book
I think that Proofs from the Book, is a title that every mathematicians should have on his desk.
Wonderful and very elegant proofs of some interesting Theorems in several area... Read more
Published on January 4, 2011 by eliogabalo
5.0 out of 5 stars Elegance Proofs from the Book
My French Professors in France always demanded rigourous proof from us, but never emphasized elegance is equally important, if not more. Read more
Published on May 23, 2009 by Wu Bing
5.0 out of 5 stars A different kind of book
This is a different kind of math book. It is not a textbook. Neither is it a monograph on a special math topic. Read more
Published on July 27, 2008 by Chee Lim Cheung
5.0 out of 5 stars this is THE BOOK!
One of the most important mathematicians in the last quarter of a century was, arguably, Paul Erdos. Read more
Published on September 13, 2007 by Oscar Vega
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