A Brief Journey in Discrete Mathematics

1. Front Matter

   Pages i-xiii

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2. Introduction

   • Randolph Nelson

   Pages 1-2

3. Let Me Count the Ways

   • Randolph Nelson

   Pages 3-26

4. Syntax Precedes Semantics

   • Randolph Nelson

   Pages 27-38

5. Fearful Symmetry

   • Randolph Nelson

   Pages 39-61

6. All That Glitters Is Not Gold

   • Randolph Nelson

   Pages 63-77

7. Heads I Win, Tails You Lose

   • Randolph Nelson

   Pages 79-92

8. Sums of the Powers of Successive Integers

   • Randolph Nelson

   Pages 93-108

9. As Simple as 2 + 2 = 1

   • Randolph Nelson

   Pages 109-118

10. Hidden in Plain Sight

    • Randolph Nelson

    Pages 119-132

11. Running Off the Page

    • Randolph Nelson

    Pages 133-159

12. Back Matter

    Pages 161-185

    PDF

The goal of this book is to showcase the beauty of mathematics as revealed in nine topics of discrete mathematics. In each chapter, properties are explored through a series of straightforward questions that terminate with results that lie at the doorstep of a field of study. Each step along the way is elementary and requires only algebraic manipulation. This frames the wonder of mathematics and highlights the complex world that lies behind a series of simple, mathematical, deductions.

Topics addressed include combinatorics, unifying properties of symmetric functions, the Golden ratio as it leads to k-bonacci numbers, non-intuitive and surprising results found in a simple coin tossing game, the playful, trick question aspect of modular systems, exploration of basic properties of prime numbers and derivations of bewildering results that arise from approximating irrational numbers as continued fraction expansions.  The Appendix contains the basic tools of mathematics that are used in the text along with a numerous list of identities that are derived in the body of the book.

The mathematics in the book is derived from first principles.  On only one occasion does it rely on a result not derived within the text. Since the book does not require calculus or advanced techniques, it should be accessible to advanced high school students and undergraduates in math or computer science.  Senior mathematicians might be unfamiliar with some of the topics addressed in its pages or find interest in the book's unified approach to discrete math.

• permutations
• binomial identities
• polynomial solutions
• Stirling numbers
• combinatorial interpretation
• Stirling transformation
• golden ratio
• Fibonacci numbers
• k-bonacci numbers
• closed form solution
• Catalan numbers
• Cauchy's theorem
• Wilson's theorem
• Fermat's Little Theorem
• prime counting function
• Lagrange's theorem
• induction

• (Home address), Beverly, USA

  Randolph Nelson

Randolph Nelson received his PhD from UCLA exploring topics in probability and stochastic processes that had applications to computer performance modeling.  For more than a decade he was a research scientist at the Thomas J. Watson research center managing fellow workers in the system performance group.  Currently, Dr. Nelson is a managing general partner at OTA LLC, a private financial firm engaged in algorithmic trading in US equity markets. Dr. Nelson is also the author of another book published by Springer  Probability, Stochastic Processes and Queueing Theory: The Mathematics of Computer Performance Modeling.

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