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About this book

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.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
Hamlet’s advice to actors, quoted above, speaks to the essence of mathematics: the action and word are neither more nor less than what is required, the virtue of pure thought and its ageless body transcends time and, finally, that mathematics is the mirror that reflects the face of nature.
Randolph Nelson

Chapter 2. Let Me Count the Ways

Abstract
Elizabeth Browning probably didn’t realize that she was really talking about mathematics when she penned her 43rd sonnet, How Do I Love Thee? This chapter provides a more comprehensive answer to this question than Browning was able to present in the remaining stanzas where she enumerates the ways she loves the veiled object of her sonnet. With the power of mathematics, equations are derived that provide a thorough enumeration, leaving no stone untouched. This is done through the simple expedient of selecting a set of items from a set. It is surprising, as when one falls in love, how fast innocent simplicity explodes into a tangled web of complexity. Perhaps this is what makes love stories, and mathematics, so enduringly interesting.
Randolph Nelson

Chapter 3. Syntax Precedes Semantics

Abstract
How you say something is often as important as what you say. A simple “Good Morning” can confuse even a wizard like Gandalf and this can be no more apparent than in writing mathematics where ambiguity is not tolerated. This explains one reason why LATE X has made such a major impact on mathematics even though it only deals with the syntax of mathematical writing and not its content. The TE X project started by Donald Knuth (1938–) gave mathematicians the tools they needed to be able to write beautifully typeset papers and books that brought to light the semantics of math in a crystal clear format. In this way, syntax precedes semantics.
Randolph Nelson

Chapter 4. Fearful Symmetry

Abstract
Symmetry might be fearful in a Tiger as William Blake alludes to in his poem, The Tyger, but in mathematics it is wholly a thing of beauty. Symmetry can often be used as a tool to cut a simple, elegant, path through a labyrinth of mathematical obstacles. Abstractly, a mathematical object displays the property of symmetry if it is invariant to parametric change.
Randolph Nelson

Chapter 5. All That Glitters Is Not Gold

Abstract
Despite the temptations of gold alluded to in Shakespeare’s verse above from The Merchant of Venice, the pursuit of mathematical gold leads not to gilded tombs but to the paradise of the Elysian fields of ancient Greece. Our journey in this chapter takes us back to the days of Phidias (480–430 BC), a Greek sculptor and mathematician who is said to have helped with the design of the Parthenon. The approach in this chapter uses a simple artifice—the ratio of two line segments.
Randolph Nelson

Chapter 6. Heads I Win, Tails You Lose

Abstract
Consider a game where two players toss a coin. If the coin lands heads up, player 1 wins a dollar. Otherwise player 2 wins a dollar. If the coin is fair, then each player has the same chance of winning. A few things about the game are obvious from the outset. Since neither player has an edge over the other, there is little chance that one of them will win a lot of money. Thus, the game should hover around break even most of the time. Additionally, each player should be ahead of the other about half of the time. Another feature of the game concerns its duration if there is an agreed stopping event. For example, suppose the game stops the first time heads is ahead of tails. Then, clearly, the game should end fairly quickly. These observations are all straightforward which suggests that coin tossing does not have much to offer in terms of mathematical results. To show this, and move on to a more interesting topic, let us quickly dispense with the mathematical analysis that establishes these obvious, intuitive, observations.
Randolph Nelson

Chapter 7. Sums of the Powers of Successive Integers

Abstract
What happens when you sum successive powers of integers? To investigate this, define
$$\displaystyle S_{k,n} = 1 + 2^k + 3^k + \cdots + n^k = \sum _{i=1}^n i^k, \ \ \ \ k=0, 1, \ldots $$
An easy program generates the following table of numeric values for small k and n.
Randolph Nelson

Chapter 8. As Simple as 2 + 2 = 1

Abstract
Orwell was not speaking about mathematics in the quote above from his book 1984. Rather, he was commenting on how totalitarian governments attempt to define, and impose, their own notion of reality on the public. Speaking mathematically, it is as clear as the back of your hand that 2 + 2 = 1 and 1 + 2 = 0. That is, if you belong to a three fingered species. We have grown so used to the ten fingers on our hands, that we forget that there is nothing special about base 10. Since the invention of the number 0 by Indian mathematicians of the fifth century, this means that all of our numbers are composed of the digits 0 through 9. To three fingered species this means that their number system uses the digits 0 through 2 so that 3 wraps around to 0 and 4 to 1. Thus 2 + 2 = 1 and 1 + 2 = 0 in base 3. Orwell’s above statement is thus valid for all bases 5 and larger unless, of course as he alludes, the totalitarian regime in power says otherwise.
Randolph Nelson

Chapter 9. Hidden in Plain Sight

Abstract
Take any number and keep finding factors of that number that cannot be factored themselves. For example, 84 = 2 ⋅ 2 ⋅ 3 ⋅ 7, 455 = 5 ⋅ 7 ⋅ 13, or 897 = 3 ⋅ 13 ⋅ 23. These examples show that a number can be written as the product of prime numbers. This is called a prime factorization. A separate argument, that we will shortly get to, shows that this factorization is unique. This result has far reaching consequences and is called the Fundamental Theorem of Arithmetic. This theorem shows that primes are the DNA of the number system. Essentially all of the results of number theory are theorems of the primes, the topic of this chapter.
Randolph Nelson

Chapter 10. Running Off the Page

Abstract
The analysis in this chapter illustrates Temple’s observation regarding the necessity for creative imagination in mathematics. A simple expression is all that is needed to develop the theory of continued fractions which leads to a deep theorem of Lagrange and also leads to an optimal way to approximate real numbers as rational fractions.
Randolph Nelson

Backmatter

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