Proofs that really count

The art of combinatorial proof

Arthur T. Benjamin, Jennifer J. Quinn

194 pages, parution le 15/01/2003

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Résumé

This is a delightful book. In this wonderful volume, the authors have put together their combinatorial insights in a fascinating package that is appealing and accessible to a very broad audience stretching from clever high school students to university faculty. Benjamin and Quinn begin with the combinatorics of Fibonacci numbers and move on to a variety of interesting related problems. Their book will attract and inspire students young and old. George E. Andrews, Perm State University

This is a magical introduction to the simplicity, power, and beauty ofbijective proofs, proofs that accomplish their task by counting the same set of objects in two different ways. The examples range widely, and the proofs are presented with clarity and originality. Many undergraduates-and not a few more senior mathematicians-will be seduced by this book into searching for their own proofs that count. David M. Bressoud, Macalester College

This book blends the talents of Martin Gardner and Houdini; it gives magical 'aha' proofs that are real mathematics but accessible to everyone. It's as good a way as I've seen to show why we love mathematics. Persi Diaconis, Stanford University

It is a masterpiece. What a great way to introduce students to proofs! Combinatorics is a perfect medium to introduce that forbidding concept of 'formal proof,' and Benjamin and Quinn did a masterful job of making proofs both accessible, and so much fun. Doron Zeilberger, Rutgers University

Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The arguments primarily take one of two forms:

A counting question is posed and answered in two different ways. Since both answers solve the same question they must be equal.
Two different sets are described, counted, and a correspondence found between them. One-to-one correspondences guarantee sets of the same size. Almost one-to-one correspondences take error terms into account. Even many-to-one correspondences are utilized.

The book explores more than 200 identities throughout the text and exercises, frequently emphasizing numbers not often thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels from high school math students to professional mathematicians.

L'auteur - Arthur T. Benjamin

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Sommaire

Fibonacci identities
Gibonacci and Lucas identities
Linear recurrences
Continued fractions
Binominal identities
Alternating sign binominal identities
Number theory
Advanced Fibonacci & Lucas identities

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Caractéristiques techniques

	PAPIER
Éditeur(s)	The Mathematical Association of America (MAA)
Auteur(s)	Arthur T. Benjamin, Jennifer J. Quinn
Parution	15/01/2003
Nb. de pages	194
Format	18,2 x 26
Couverture	Relié
Poids	510g
Intérieur	Noir et Blanc
EAN13	9780883853337
ISBN13	978-0-88385-333-7