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Mathematics | Proofs from THE BOOK

Proofs from THE BOOK

Aigner, Martin, Ziegler, Günter M.

3rd ed. 2004, VIII, 239 p.


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  • Excellent, easy-to-read book for everyone with an interest in mathematics
  • Two top researchers have made a big effort and selected a list of mathematical problems which can be solved by elegant, esthetically pleasing proofs
  • Fun for everybody with an interest in mathematics requiring only very little previous knowledge
  • Third edition offers two new chapters (on partition identities, and on card shuffling). Three proofs of Euler's most famous infinite series appear in a separate chapter. There are also a number of other improvements, such as an exciting new way to “enumerate the rationals”
  • Has now been published in seven languages, two more editions are in preparation

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 third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such an exciting new way to "enumerate the rationals."

Content Level » Lower undergraduate

Keywords » Finite - Identity - Partition - analysis - combinatorics - geometry - number theory - proof - proofs

Related subjects » Analysis - Computer Science - Geometry & Topology - Mathematics - Number Theory and Discrete Mathematics

Table of contents 

Number Theory.- 1. Six proofs of the infinity of primes.- 2. Bertrand’s postulate.- 3. Binomial coefficients are (almost) never powers.- 4. Representing numbers as sums of two squares.- 5. Every finite division ring is a field.- 6. Some irrational numbers.- 7. Three times ?2/6.- Geometry.- 8. Hilbert’s third problem: decomposing polyhedra.- 9. Lines in the plane and decompositions of graphs.- 10. The slope problem.- 11. Three applications of Euler’s formula.- 12. Cauchy’s rigidity theorem.- 13. Touching simplices.- 14. Every large point set has an obtuse angle.- 15. Borsuk’s conjecture.- Analysis.- 16. Sets, functions, and the continuum hypothesis.- 17. In praise of inequalities.- 18. A theorem of Pólya on polynomials.- 19. On a lemma of Littlewood and Offord.- 20. Cotangent and the Herglotz trick.- 21. Buffon’s needle problem.- Combinatorics.- 22. Pigeon-hole and double counting.- 23. Three famous theorems on finite sets.- 24. Shuffling cards.- 25. Lattice paths and determinants.- 26. Cayley’s formula for the number of trees.- 27. Completing Latin squares.- 28. The Dinitz problem.- 29. Identities versus bijections.- Graph Theory.- 30. Five-coloring plane graphs.- 31. How to guard a museum.- 32. Turán’s graph theorem.- 33. Communicating without errors.- 34. Of friends and politicians.- 35. Probability makes counting (sometimes) easy.- About the Illustrations.

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