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.
Arthur T. Benjamin Harvey Mudd College; Jennifer J. Quinn, Occidental College
Foreword 1 Fibonacci Identities 1.1 Combinatorial Interpretation of Fibonacci Numbers 1.2 Identities 1.3 A Fun Application 1.4 Notes 1.5 Exercises 2 Gibonacci and Lucas Identities 2.1 Combinatorial Interpretation of Lucas Numbers 2.2 Lucas Identities 2.3 Combinatorial Interpretation of Gibonacci Numbers 2.4 Gibonacci Identities 2.5 Notes 2.6 Exercises 3 Linear Recurrences 3.1 Combinatorial Interpretations of Linear Recurrences 3.2 Identities for Second-Order Recurrences 3.3 Identities for Third-Order Recurrences 3.4 Identities for kth Order Recurrences 3.5 Get Real! Arbitrary Weights and Initial Conditions 3.6 Notes 3.7 Exercises 4 Continued Fractions 4.1 Combinatorial Interpretation of Continued Fractions 4.2 Identities 4.3 NonsimpleContinued Fractions 4.4 Get Real Again! 4.5 Notes 4.6 Exercises 5 Binomial Identities 5.1 Combinatorial Interpretations of Binomial Coefficients 5.2 Elementary Identities 5.3 More Binomial Coefficient Identities 5.4 Multichoosing 5.5 Odd Numbers in Pascal’s Triangle 5.6 Notes 5.7 Exercises 6 Alternating Sign Binomial Identities 6.1 ParityArguments and Inclusion-Exclusion 6.2 Alternating Binomial Coefficient Identities 6.3 Notes 6.4 Exercises 7 Harmonic and Stirling Number Identities 7.1 HarmonicNumbers and Permutations 7.2 StirlingNumbers of the First Kind 7.3 Combinatorial Interpretation of Harmonic Numbers 7.4 Recounting Harmonic Identities 7.5 StirlingNumbers of the Second Kind 7.6 Notes 7.7 Exercises 8 Number Theory 8.1 Arithmetic Identities 8.2 Algebra and Number Theory 8.3 GCDs Revisited 8.4 Lucas’ Theorem 8.5 Notes 8.6 Exercises 9 Advanced Fibonacci & Lucas Identities 9.1 More Fibonacci and Lucas Identities 9.2 Colorful Identities 9.3 Some “Random” Identities and the Golden Ratio 9.4 Fibonacci and Lucas Polynomials 9.5 NegativeNumbers 9.6 Open Problems and Vajda Data Some Hints and Solutions for Chapter Exercises Appendix of Combinatorial Theorems Appendix of Identities Bibliography Index About the Authors