Proofs from THE BOOK

by Aigner, Martin; Ziegler, Gunter M.

• ISBN13: 9783540404606
• ISBN10: 3540404600
• Edition: 3rd
• Format: Hardcover
• Copyright: 2003-12-01
• Publisher: Springer Verlag

Summary

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 "... the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." 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 as an exciting new way to "enumerate the rationals".

Table of Contents

Number Theory	1	(42)
1. Six proofs of the infinity of primes	3	(4)
2. Bertrand's postulate	7	(6)
3. Binomial coefficients are (almost) never powers	13	(4)
4. Representing numbers as sums of two squares	17	(6)
5. Every finite division ring is a field	23	(4)
6. Some irrational numbers	27	(8)
7. Three times π2/6	35	(8)
Geometry	43	(48)
8. Hilbert's third problem: decomposing polyhedra	45	(8)
9. Lines in the plane and decompositions of graphs	53	(6)
10. The slope problem	59	(6)
11. Three applications of Euler's formula	65	(6)
12. Cauchy's rigidity theorem	71	(4)
13. Touching simplices	75	(4)
14. Every large point set has an obtuse angle	79	(6)
15. Borsuk's conjecture	85	(6)
Analysis	91	(46)
16. Sets, functions, and the continuum hypothesis	93	(16)
17. In praise of inequalities	109	(8)
18. A theorem of Pólya on polynomials	117	(6)
19. On a lemma of Littlewood and Offord	123	(4)
20. Cotangent and the Herglotz trick	127	(6)
21. Buffon's needle problem	133	(4)
Combinatorics	137	(60)
22. Pigeon-hole and double counting	139	(12)
29. Three famous theorems on finite set	151	(6)
24. Shuffling cards	157	(10)
25. Lattice path and determinants	167	(6)
26. Cayley's formula for the number of trees	173	(6)
27. Completing Latin squares	179	(6)
28. The Dinitz problem	185	(6)
29. Identities versus bijections	191	(6)
Graph Theory	197	(39)
30. Five-coloring plane graphs	199	(4)
31. How to guard a museum	203	(4)
32. Turán's graph theory	207	(6)
33. Communicating without errors	213	(10)
34. Of friends and politicians	223	(4)
35. Probability makes counting (sometimes) easy	227	(9)
About the Illustrations	236	(1)
Index	237