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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010192603 | QA164.8 B46 2003 | Open Access Book | Book | Searching... |
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Summary
Summary
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.
Reviews 1
Choice Review
The notion of an elegant proof in mathematics is usually only accessible to those who have devoted a great deal of time, if not their lives, to its study. This little book by Benjamin (Harvey Mudd College) and Quinn (Occidental College) provides an eye-opening experience to the uninitiated of just what an elegant proof is. The common combinatorial approach of counting the same thing in two different ways is maintained throughout. Most of the results are stated as an identity, appearing to be simply an equation. This is followed by a question (the essence of the equation) and two answers, each explaining one side of the equation. It is counting at its finest. There is no actual "counting," no computation, simply thinking and reasoning. The initial equation becomes a concise representation of a combinatorial truth. The book is very well written; it can be read from beginning to end or, in most cases, one can pick and choose the topics of immediate interest. This book should be in every undergraduate library and would be a wonderful book for introducing students to combinatorial reasoning. ^BSumming Up: Essential. General readers; lower- and upper-division undergraduates. J. R. Burke Gonzaga University
Table of Contents
| Foreword | p. ix |
| 1 Fibonacci Identities | p. 1 |
| 1.1 Combinatorial Interpretation of Fibonacci Numbers | p. 1 |
| 1.2 Identities | p. 2 |
| 1.3 A Fun Application | p. 11 |
| 1.4 Notes | p. 12 |
| 1.5 Exercises | p. 13 |
| 2 Gibonacci and Lucas Identities | p. 17 |
| 2.1 Combinatorial Interpretation of Lucas Numbers | p. 17 |
| 2.2 Lucas Identities | p. 18 |
| 2.3 Combinatorial Interpretation of Gibonacci Numbers | p. 23 |
| 2.4 Gibonacci Identities | p. 23 |
| 2.5 Notes | p. 32 |
| 2.6 Exercises | p. 32 |
| 3 Linear Recurrences | p. 35 |
| 3.1 Combinatorial Interpretations of Linear Recurrences | p. 36 |
| 3.2 Identities for Second-Order Recurrences | p. 38 |
| 3.3 Identities for Third-Order Recurrences | p. 40 |
| 3.4 Identities for kth Order Recurrences | p. 43 |
| 3.5 Get Real! Arbitrary Weights and Initial Conditions | p. 44 |
| 3.6 Notes | p. 45 |
| 3.7 Exercises | p. 45 |
| 4 Continued Fractions | p. 49 |
| 4.1 Combinatorial Interpretation of Continued Fractions | p. 49 |
| 4.2 Identities | p. 52 |
| 4.3 Nonsimple Continued Fractions | p. 58 |
| 4.4 Get Real Again! | p. 59 |
| 4.5 Notes | p. 59 |
| 4.6 Exercises | p. 60 |
| 5 Binomial Identities | p. 63 |
| 5.1 Combinatorial Interpretations of Binomial Coefficients | p. 63 |
| 5.2 Elementary Identities | p. 64 |
| 5.3 More Binomial Coefficient Identities | p. 68 |
| 5.4 Multichoosing | p. 70 |
| 5.5 Odd Numbers in Pascal's Triangle | p. 75 |
| 5.6 Notes | p. 77 |
| 5.7 Exercises | p. 78 |
| 6 Alternating Sign Binomial Identities | p. 81 |
| 6.1 Parity Arguments and Inclusion-Exclusion | p. 81 |
| 6.2 Alternating Binomial Coefficient Identities | p. 84 |
| 6.3 Notes | p. 89 |
| 6.4 Exercises | p. 89 |
| 7 Harmonic and Stirling Number Identities | p. 91 |
| 7.1 Harmonic Numbers and Permutations | p. 91 |
| 7.2 Stirling Numbers of the First Kind | p. 93 |
| 7.3 Combinatorial Interpretation of Harmonic Numbers | p. 97 |
| 7.4 Recounting Harmonic Identities | p. 98 |
| 7.5 Stirling Numbers of the Second Kind | p. 103 |
| 7.6 Notes | p. 106 |
| 7.7 Exercises | p. 106 |
| 8 Number Theory | p. 109 |
| 8.1 Arithmetic Identities | p. 109 |
| 8.2 Algebra and Number Theory | p. 114 |
| 8.3 GCDs Revisited | p. 118 |
| 8.4 Lucas' Theorem | p. 120 |
| 8.5 Notes | p. 123 |
| 8.6 Exercises | p. 123 |
| 9 Advanced Fibonacci & Lucas Identities | p. 125 |
| 9.1 More Fibonacci and Lucas Identities | p. 125 |
| 9.2 Colorful Identities | p. 130 |
| 9.3 Some "Random" Identities and the Golden Ratio | p. 136 |
| 9.4 Fibonacci and Lucas Polynomials | p. 141 |
| 9.5 Negative Numbers | p. 143 |
| 9.6 Open Problems and Vajda Data | p. 143 |
| Some Hints and Solutions for Chapter Exercises | p. 147 |
| Appendix of Combinatorial Theorems | p. 171 |
| Appendix of Identities | p. 173 |
| Bibliography | p. 187 |
| Index | p. 191 |
| About the Authors | p. 194 |
