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Proofs that Really Count: The Art of Combinatorial Proof

Proofs that Really Count: The Art of Combinatorial Proof

By Arthur T. Benjamin and Jennifer J. Quinn

Print ISBN: 978-0-88385-333-7
Electronic ISBN: 978-1-61444-208-0
208 pp., Hardbound, 2003
List Price: $58.00
Member Price: $43.50
Series: Dolciani Mathematical Expositions

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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 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.

Table of Contents

Table of Contents
Foreword
1. Fibonacci Identities
2. Gibonacci and Lucas Identities
3. Linear Recurrences
4. Continued Fractions
5. Binomial Identities
6. Alternating Sign Binomial Identities
7. Harmonic and Stirling Number Identities
8. Number Theory
9. Advanced Fibonacci & Lucas Identities
Some Hints and Solutions for Chapter Exercises
Appendix of Combinatorial Theorems
Appendix of Identities
Bibliography
Index
About the Authors

About the Authors

Arthur T. Benjamin was born in Cleveland, Ohio. He received his BS in applied mathematics from Carnegie Mellon. In 1989, he received his PhD in mathematical sciences from Johns Hopkins University, and was awarded the Nicholson Prize from the Operations Research Society of America. Currently, Art is Professor of Mathematics at Harvey Mudd College, and is a Fellow of the Institute for Combinatorics and its Applications. In 2000, he was awarded the Haimo Prize from MAA for Distinguished Teaching. Art Benjamin is also a professional magician and lightning calculator, frequently performing at The Magic Castle in Hollywood and for uncountably many schools and organizations around the world. Art lives in Claremont, California with his wife Deena and daughters Laurel and Ariel.

Jennifer J. Quinn was born in Princeton, NJ though she spent the bulk of her formative years growing up in Rhode Island. She received her BA magna cum laude from Williams College with majors in mathematics and biology and was elected to the Phi Beta Kappa Society. Working as an actuary in Chicago after graduation convinced her to return to academic life. In 1987, she earned her MS in pure mathematics from the University of Illinois at Chicago and in 1993, earned her PhD in combinatorics from the University of Wisconsin, Madison. She is currently Professor of Mathematics at the University of Washington Tacoma.

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