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Proofs from THE BOOK Hardcover – November 13, 2003

ISBN-13: 978-3540404606 ISBN-10: 3540404600 Edition: 3rd

8 New from $122.94 12 Used from $32.85
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Hardcover, November 13, 2003
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Proofs from THE BOOK
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Product Details

  • Hardcover: 240 pages
  • Publisher: Springer; 3rd edition (November 13, 2003)
  • Language: English
  • ISBN-10: 3540404600
  • ISBN-13: 978-3540404606
  • Product Dimensions: 9.8 x 7.7 x 0.9 inches
  • Shipping Weight: 1.8 pounds
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (21 customer reviews)
  • Amazon Best Sellers Rank: #1,962,030 in Books (See Top 100 in Books)

Editorial Reviews

Review

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

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

--This text refers to an alternate Hardcover edition.

Customer Reviews

Topics in this book range from Number Theory to Analysis, and also discussing Combinatorics and Graph Theory.
Oscar Vega
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.
Alexander C. Zorach
Most of this book could be read by an undergraduate with only background in calculus and basic discrete math.
Ellipsic

Most Helpful Customer Reviews

131 of 138 people found the following review helpful By T. Smith on 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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111 of 121 people found the following review helpful By Dr. Hoenikker on 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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24 of 26 people found the following review helpful By Alexander C. Zorach on October 22, 2006
Format: Hardcover
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 By Chris Hobbs on October 24, 2002
Format: Hardcover
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 By KARTIK KRISHNAN S. on March 20, 2001
Format: Hardcover
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 By Ellipsic on December 29, 2005
Format: Hardcover
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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