Proofs from the Book
by Aigner, Martin; Ziegler, Gunter M.; Hofmann, Karl H.Edition: 4th
Format: Hardcover
Pub. Date: 2009-12-15
Publisher(s): Springer Verlag
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Summary
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. Book jacket.
Table of Contents
| Number Theory | p. 1 |
| Six proofs of the infinity of primes | p. 3 |
| Betrand's postulate | p. 7 |
| Binomial coefficients are (almost) never powers | p. 13 |
| Representing numbers as sums of two squares | p. 17 |
| The Law of quadratic reciprocity | p. 23 |
| Every finite division ring is a field | p. 31 |
| Some irrational numbers | p. 35 |
| Three times ¿2/6 | p. 43 |
| Geometry | p. 51 |
| Hilbert's third problem: decomposing polyhedra | p. 53 |
| Lines in the plane and decomposing of graphs | p. 63 |
| The slope problem | p. 69 |
| Three applications of Euler's formula | p. 75 |
| Cauchy's rigidity theorem | p. 81 |
| Touching simplices | p. 85 |
| Every large point set has an obtuse angle | p. 89 |
| Borsuk's conjecture | p. 95 |
| Analysis | p. 101 |
| Sets, functions, and the continuum hypothesis | p. 103 |
| In praise of inequalities | p. 119 |
| The fundamental theorem of algebra | p. 127 |
| One square and an odd number of triangles | p. 131 |
| A theorem of Pólya on polynomials | p. 139 |
| On a lemma of Littlewood and Offord | p. 145 |
| Cotangent and the Herglotz trick | p. 149 |
| Buffon's needle problem | p. 155 |
| Combinatorics | p. 159 |
| Pigeon-hole and double counting | p. 161 |
| Tiling rectangles | p. 173 |
| Three famous theorems on finite sets | p. 179 |
| Shuffling cards | p. 185 |
| Lattice paths and determinants | p. 195 |
| Cayley's formula for the number of trees | p. 201 |
| Identities versus bijections | p. 207 |
| Completing Latin Squares | p. 213 |
| Graph Theory | p. 219 |
| The Dinitz problem | p. 221 |
| Five-coloring plane graphs | p. 227 |
| How to guard a museum | p. 231 |
| Turán's graph theorem | p. 235 |
| Communicating without errors | p. 241 |
| The chromatic number of Kneser graphs | p. 251 |
| Of friends and politicians | p. 257 |
| Probability makes counting (sometimes) easy | p. 261 |
| About the illustrations | p. 270 |
| Index | p. 271 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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