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Proofs from THE BOOK

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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

Author(s): Martin Aigner, Günter M. Ziegler (auth.)
Edition: 4th ed.
Publisher: Springer Berlin Heidelberg
Year: 2010

Language: English
Pages: 282
Tags: Mathematics, general; Number Theory; Geometry; Combinatorics; Analysis; Computer Science, general

Front Matter....Pages I-VIII
Front Matter....Pages 1-1
Six proofs of the infinity of primes....Pages 3-6
Bertrand’s postulate....Pages 7-12
Binomial coefficients are (almost) never powers....Pages 13-16
Representing numbers as sums of two squares....Pages 17-22
The law of quadratic reciprocity....Pages 23-30
Every finite division ring is a field....Pages 31-34
Some irrational numbers....Pages 35-41
Three times Π²/6....Pages 43-50
Front Matter....Pages 51-51
Hilbert’s third problem: decomposing polyhedra....Pages 53-61
Lines in the plane and decompositions of graphs....Pages 63-67
The slope problem....Pages 69-73
Three applications of Euler’s formula....Pages 75-80
Cauchy’s rigidity theorem....Pages 81-84
Touching simplices....Pages 85-88
Every large point set has an obtuse angle....Pages 89-94
Borsuk’s conjecture....Pages 95-100
Front Matter....Pages 101-101
Sets, functions, and the continuum hypothesis....Pages 103-118
In praise of inequalities....Pages 119-125
The fundamental theorem of algebra....Pages 127-129
One square and an odd number of triangles....Pages 131-138
Front Matter....Pages 101-101
A theorem of Pólya on polynomials....Pages 139-144
On a lemma of Littlewood and Offord....Pages 145-148
Cotangent and the Herglotz trick....Pages 149-154
Buffon’s needle problem....Pages 155-158
Front Matter....Pages 159-159
Pigeon-hole and double counting....Pages 161-171
Tiling rectangles....Pages 173-177
Three famous theorems on finite sets....Pages 179-183
Shuffling cards....Pages 185-194
Lattice paths and determinants....Pages 195-200
Cayley’s formula for the number of trees....Pages 201-206
Identities versus bijections....Pages 207-212
Completing Latin squares....Pages 213-218
Front Matter....Pages 219-219
The Dinitz problem....Pages 221-226
Five-coloring plane graphs....Pages 227-230
How to guard a museum....Pages 231-234
Turán’s graph theorem....Pages 235-239
Communicating without errors....Pages 241-250
The chromatic number of Kneser graphs....Pages 251-255
Of friends and politicians....Pages 257-259
Probability makes counting (sometimes) easy....Pages 261-268
Back Matter....Pages 269-274