Proofs from the Book [Hardcover]

Martin Aigner (Author), Gunter Ziegler (Author)

Book Description

ISBN-10: 3540678654 | ISBN-13: 978-3540678656 | Publication Date: December 28, 2000 | Edition: 02

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 Erd/s, 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. According to the great mathematician Paul Erd/s, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, sis, com binatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

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

--This text refers to an out of print or unavailable edition of this title.

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.

Product Details

• Hardcover: 224 pages
• Publisher: Springer; 02 edition (December 28, 2000)
• Language: English
• ISBN-10: 3540678654
• ISBN-13: 978-3540678656
• Product Dimensions: 9.8 x 7.8 x 0.8 inches
• Shipping Weight: 1.9 pounds
• Average Customer Review: 4.5 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #2,174,705 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

71 of 71 people found the following review helpful:

4.0 out of 5 stars original, June 4, 2000

By 

Giuseppe A. Paleologo "gappy" (Riverdale, NY United States) - See all my reviews
(REAL NAME)   

This review is from: Proofs from THE BOOK (Hardcover)

This book was conceived as a tribute to Paul Erdos for his 85th birthday. It is clearly inspired by his aestetics and research interests. The proofs are from number theory, combinatorial geometry, inequalities, combinatorics and graph theory. The statements are very often easy to understand; for example "there always exists a prime number between n and 2n", "every set of more than 2^d points in R^d determines at least one obtuse angle". Theorems and proofs are chosen because of their simplicity and elegance, not their relevance to modern or past mathematics. The book is, graphically and stilistically, a gem.

Overall, this is great reading for mathematicians and mathematically literate readers alike. It's also a bit odd, since the book is neither a reference nor a textbook. The only criticism I have is not directed to the book itself. It would be much appreciated to have similar books, but focused on different topics. For example "probabilistic proofs from the book", or "topological proofs from the book".

92 of 95 people found the following review helpful:

1.0 out of 5 stars Kindle edition is worthless: avoid at all costs, April 23, 2008

By 

T. Smith "tzs" - See all my reviews
(REAL NAME)   

This review is from: Proofs from THE BOOK (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.

82 of 87 people found the following review helpful:

1.0 out of 5 stars Do NOT get the Kindle edition, July 7, 2008

By 

Dr. Hoenikker "drhoenikker" - See all my reviews

Amazon Verified Purchase

This review is from: Proofs from THE BOOK (Kindle Edition)

The Kindle edition of the book is missing or misrepresenting math symbols in so many places it makes it completely unreadable.