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Proofs That Really Count The Art of Combinatorial Proof

ISBN-10: 0883853337
ISBN-13: 9780883853337
Edition: 2003
List price: $45.95 Buy it from $22.01
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Description: 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  More...

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Book details

List price: $45.95
Copyright year: 2003
Publisher: Mathematical Association of America
Publication date: 11/13/2003
Binding: Hardcover
Pages: 206
Size: 7.25" wide x 10.00" long x 0.75" tall
Weight: 1.496
Language: English

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 emphasizes numbers that are often not 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.

Jennifer Quinn received her PhD in Combinatorics from the University of Wisconsin. She currently teaches at Occidental College. With Arthur Benjamin she the editor of "Math Horizons," a student magazine published by the Mathematical Association of America.

Foreword
Fibonacci Identities
Combinatorial Interpretation of Fibonacci Numbers
Identities
A Fun Application
Notes
Exercises
Gibonacci and Lucas Identities
Combinatorial Interpretation of Lucas Numbers
Lucas Identities
Combinatorial Interpretation of Gibonacci Numbers
Gibonacci Identities
Notes
Exercises
Linear Recurrences
Combinatorial Interpretations of Linear Recurrences
Identities for Second-Order Recurrences
Identities for Third-Order Recurrences
Identities for kth Order Recurrences
Get Real! Arbitrary Weights and Initial Conditions
Notes
Exercises
Continued Fractions
Combinatorial Interpretation of Continued Fractions
Identities
Nonsimple Continued Fractions
Get Real Again!
Notes
Exercises
Binomial Identities
Combinatorial Interpretations of Binomial Coefficients
Elementary Identities
More Binomial Coefficient Identities
Multichoosing
Odd Numbers in Pascal's Triangle
Notes
Exercises
Alternating Sign Binomial Identities
Parity Arguments and Inclusion-Exclusion
Alternating Binomial Coefficient Identities
Notes
Exercises
Harmonic and Stirling Number Identities
Harmonic Numbers and Permutations
Stirling Numbers of the First Kind
Combinatorial Interpretation of Harmonic Numbers
Recounting Harmonic Identities
Stirling Numbers of the Second Kind
Notes
Exercises
Number Theory
Arithmetic Identities
Algebra and Number Theory
GCDs Revisited
Lucas' Theorem
Notes
Exercises
Advanced Fibonacci & Lucas Identities
More Fibonacci and Lucas Identities
Colorful Identities
Some "Random" Identities and the Golden Ratio
Fibonacci and Lucas Polynomials
Negative Numbers
Open Problems and Vajda Data
Some Hints and Solutions for Chapter Exercises
Appendix of Combinatorial Theorems
Appendix of Identities
Bibliography
Index
About the Authors

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