This collection of "excursions" into modern mathematics is written in an informal, very readable style, with features that make the material interesting, clear, and easy-to-learn. It centers on an assortment of real-world examples and applications, demonstrating attractive, useful, and modern coverage of liberal arts mathematics.The book consists of four independent parts, each consisting of four chapters1) Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics.For the study of mathematics.
Table of Contents
(Note: Each chapter concludes with a Conclusion, Biographical Profile, Exercises, References and Further Readings.)
I. THE MATHEMATICS OF SOCIAL CHOICE.
1. The Mathematics of Voting: The Paradoxes of Democracy.
2. Weighted Voting Systems: The Power Game.
3. Fair Division: The Mathematics of Sharing.
4. The Mathematics of Apportionment: Making the Rounds.
II. MANAGEMENT SCIENCE.
5. Euler Circuits: The Circuit Comes to Town.
6. The Traveling-Salesman Problem: Hamilton Joins the Circuit.
7. The Mathematics of Networks: It's All about Being Connected.
8. The Mathematics of Scheduling: Directed Graphs and Critical Paths.
III. GROWTH AND SYMMETRY.
9. Spiral Growth in Nature: Fibonacci Numbers and the Golden Ratio.
10. The Mathematics of Population Growth: There is Strength in Numbers.
11. Symmetry: Mirror, Mirror, off the Wall…
12. Fractal Geometry: Fractally Speaking.
13. Collecting Statistical Data: Censuses, Surveys, and Studies.
14. Descriptive Statistics: Graphing and Summarizing Data.
15. Chances, Probability, and Odds: Measuring Uncertainty.
16. Normal Distributions: Everything is Back to Normal (Almost).
Answers to Selected Problems.
Excursions in Modern Mathematicsis, 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!" OutlineThe 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 li