Proofs from THE BOOK

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

Texte du rabat

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.

Contenu

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 Pólya 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.- Turán's graph theorem.- Communicating without errors.- The chromatic number of Kneser graphs.- Of friends and politicians.- Probability makes counting (sometimes) easy.