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 Erdös, 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.
Rezensionen / Stimmen
"...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." - Notices of the AMS
Sprache
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Verlagsgruppe
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Dateigröße
ISBN-13
978-3-662-22343-7 (9783662223437)
DOI
10.1007/978-3-662-22343-7
Schweitzer Klassifikation
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.- Geometry.- 7. Hilbert's third problem: decomposing polyhedra.- 8. Lines in the plane and decompositions of graphs.- 9. The slope problem.- 10. Three applications of Euler's formula.- 11. Cauchy's rigidity theorem.- 12. The problem of the thirteen spheres.- 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.- Combinatorics.- 20. Pigeon-hole and double counting.- 21. Three famous theorems on finite sets.- 22. Cayley's formula for the number of trees.- 23. Completing Latin squares.- 23. The Dinitz problem.- Graph Theory.- 25. Five-coloring plane graphs.- 26. How to guard a museum.- 27. Turán's graph theorem.- 28. Communicating without errors.- 29. Of friends and politicians.- 30. Probability makes counting (sometimes) easy.- About the Illustrations.
