Excursions in Modern Mathematics - Hardcover

Key Benefit: The book centers around an assortment of real-world examples and applications, demonstrating a more attractive, useful, and modern coverage of liberal arts mathematics. Key Topics: This collection of “excursions” into modern mathematics consists of four independent parts, each consisting of four chapters—1) Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. It is written in an informal, very readable style, with pedagogical features that make material both interesting and clear.

Excursions in Modern Mathematics is, as we hope the title might suggest, a collection of "trips" into that vast and alien frontier that many people perceive mathematics to be. While the purpose of this book is quite conventional—it is intended to serve as a textbook for a college-level liberal arts mathematics course—its contents are not. By design, the topics in this book are chosen with the purpose of showing the reader a different view of mathematics from the one presented in a traditional general education mathematics curriculum. The notion that general education mathematics must be dull, unrelated to the real world, highly technical, and deal mostly with concepts that are historically ancient is totally unfounded.

The "excursions" in this book represent a collection of topics chosen to meet a few simple criteria.

• Applicability. The connection between the mathematics presented here and down-to-earth, concrete real-life problems is direct and immediate. The often heard question, "What is this stuff good for?" is a legitimate one and deserves to be met head on. The often heard answer, "Well, you need to learn the material in Math 101 so that you can understand Math 102 which you will need to know if you plan to take Math 201 which will teach you the real applications," is less than persuasive and in many cases reinforces students' convictions that mathematics is remote, labyrinthine, and ultimately useless to them.
• Accessibility. Interesting mathematics need not always be highly technical and built on layers upon layers of concepts. As a general rule, the choice of topics in this book is such that a heavy mathematical infrastructure is not needed—by and large, Intermediate Algebra is an appropriate and sufficient prerequisite. (In the few instances in which more advanced concepts are unavoidable, an effort has been made to provide enough background to make the material self-contained.) A word of caution—this does not mean that the material is easy! In mathematics, as in many other walks of life, simple and straightforward is not synonymous with easy and superficial.
• Age. Much of the mathematics in this book has been discovered within the last 100 years; some as recently as 20 years ago. Modern mathematical discoveries do not have to be only within the grasp of experts.
• Aesthetics. The notion that there is such a thing as beauty in mathematics is surprising to most casual observers. There is an important aesthetic component in mathematics and, just as in art and music (which mathematics very much resembles), it often surfaces in the simplest ideas. A fundamental objective of this book is to develop an appreciation for the aesthetic elements of mathematics. Hopefully, every open-minded reader will find some topics about which they can say, "I really enjoyed learning this stuff!"

Outline

The material in the book is divided into four independent parts. Each of these parts in turn contains four chapters dealing with interrelated topics.

• Part 1 (Chapters 1 through 4). The Mathematics of Social Choice. This part deals with mathematical applications in social science. How do groups make decisions? How are elections decided? What is power? How can power be measured? What is fairness? How are competing claims on property resolved in a fair and equitable way? How are seats apportioned in the House of Representatives?
• Part 2 (Chapters 5 through 8). Management Science. This part deals with methods for solving problems involving the organization and management of complex activities-that is, activities involving either a large number of steps and/or a large number of variables (routing the delivery of packages, landing a spaceship on Mars, organizing a banquet, scheduling classrooms at a big university, etc.). Efficiency is the name of the game in all these problems. Some limited or precious resource (time, money, raw materials) must be managed in such a way that waste is minimized. We deal with problems of this type (consciously or unconsciously) every day of our lives.
• Part 3 (Chapters 9 through 12). Growth and Symmetry. This part deals with nontraditional geometric ideas. How do sunflowers and seashells grow? How do animal populations grow? What are the symmetries of a snowflake? What is the true pattern behind that wallpaper pattern? What is the geometry of a mountain range? What kind of symmetry lies hidden in our circulatory system?
• Part 4 (Chapters 13 through 16). Statistics. In one way or another, statistics affects all of our lives. Government policy, insurance rates, our health, our diet, and public opinion are all governed by statistical laws. This part deals with some of the most basic aspects of statistics. How should statistical data be collected? How is data summarized so that it is intelligible? How should statistical data be interpreted? How can we measure the inherent uncertainty built into statistical data? How can we draw meaningful conclusions from statistical information? How can we use statistical knowledge to predict patterns in future events?

Exercises and Projects

An important goal for this book is that it be flexible enough to appeal to a wide range of readers in a variety of settings. The exercises, in particular, have been designed to convey the depth of the subject matter by addressing a broad spectrum of levels of difficulty—from the routine drill to the ultimate challenge. For convenience (but with some trepidation) the exercises are classified into three levels of difficulty:

• Walking. These exercises are meant to test a basic understanding of the main concepts, and they are intended to be within the capabilities of students at all levels.
• Jogging. These are exercises that can no longer be considered as routine—either because they use basic concepts at a higher level of complexity, or they require slightly higher order critical thinking skills, or both.
• Running. This is an umbrella category for problems that range from slightly unusual or slightly above average in difficulty to problems that can be a real challenge to even the most talented of students.

Traditional exercises sometimes are not sufficient to convey the depth and richness of a topic. A new feature in this edition is the addition of a Projects and Papers section following the exercise sets at the end of each chapter. One of the nice things about the "excursions" in this book is that they often are just a starting point for further exploration and investigation. This section offers some potential topics and ideas for some of these explorations, often accompanied with suggested readings and leads for getting started. In most cases, the projects are well suited for group work, be it a handful of students or an entire small class.

What Is New in This Edition?

The two most visible additions to this edition are the Projects and Papers section discussed above and a biographical profile at the end of each chapter (in the chapter on Apportionment, a historical section detailing the checkered story of apportionment in the U.S. House of Representatives was added instead). Each biographical profile features a scientist (they are not always mathematicians) who made a significant contribution to the subject covered in the chapter. In keeping with the spirit of modernity, most are contemporary and in many cases still alive.

Other changes in this edition worth mentioning are:

• In Chapter 2 the European Union is introduced as another important example of a weighted voting system. Both the Banzhaf and the Shapley-Shubik power distribution of the member nations in the EU are given. The power distribution of the Electoral College has been updated to reflect the 2000 Census data.
• In Chapter 4 I expanded the discussion of how to use trial and error to find divisors for Jefferson's and Adams's methods. The new explanations are illustrated with flowcharts. An historical section on apportionment (which was an appendix in earlier editions) has been expanded and moved to the end of the chapter. I added a new appendix (Appendix 2), showing the apportionments in the U.S. House of Representatives for each state under each of the methods discussed in the chapter.
• In Chapter 5 I added a brief discussion on algorithms in general.
• In Chapter 10 several new examples have been added to give more realistic illustrations of the use of exponential growth models.
• In Chapter 11 several new tables have been added to better clarify the classification of symmetry types. I also included a flowchart for the classification of wallpaper patterns.
• In Chapter 14 anew section on computing pth percentiles in general has been added. The computations of the median and the quartiles now follow as special cases of the general case.