Excursions in Modern Mathematics / Edition 9

by Peter Tannenbaum

Overview

For courses in Liberal Arts Mathematics.

Math: Applicable, Accessible, Modern

Excursions in Modern Mathematics introduces readers to the power and beauty of math. By developing an appreciation for the aesthetics and applicability of mathematics, readers who previously felt math was an “unknowable” subject can approach it with a new perspective. Contemporary topics ranging from elections, to networks, to analyzing data show readers that math is an accessible tool that can be applicable and interesting for anyone. Refinement and updating of examples and exercises, plus increased resources, makes the 9th Edition a relevant, accessible, and complete program.

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Product Details

ISBN-13:	9780134468372
Publisher:	Pearson Education
Publication date:	01/01/2017
Edition description:	Older Edition
Pages:	600
Product dimensions:	8.50(w) x 10.90(h) x 1.00(d)

About the Author

Peter Tannenbaum is Professor Emeritus of Mathematics at the California State University, Fresno. He has also held faculty positions at the Universidad Simon Bolivar in Caracas, Venezuela, the University of Arizona in Tucson, Arizona, and as a Fulbright visiting scholar at the Universidad de San Luis in San Luis, Argentina.  

Professor Tannenbaum received Bachelor's degrees in Political Science and Mathematics, and a Ph.D. in Pure Mathematics, all from the University of California, Santa Barbara. His mathematical research interests focus primarily in the interface between combinatorics, finite groups and probability, and he has published papers in combinatorial designs, finite projective geometries, partitions of groups, probability, generating functions and the computation of power indexes. 

Professor Tannenbaum has also worked on various aspects of mathematics education, including curriculum development for undergraduate general education mathematics, the preparation of secondary teachers, and Young Scholar Institutes (summer camps for talented high school students). In recognition of this work he has received many grants and awards, including a Mathematical Association of America Award for “Distinguished College or University Teaching of Mathematics.” 

Professor Tannenbaum was born in Genoa, Italy, and at the age of 4 his family moved to Montevideo, Uruguay, where he spent his formative years. Beyond mathematics, his interests and hobbies include travel, languages (he speaks 5), hiking, riding (bikes and horses), cooking and most recently, painting. He and his wife Sally have 3 children (twin boys and a girl) and currently live in Santa Barbara, California. 

Read an Excerpt

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.

Table of Contents

1 The Mathematics of Elections

1.1 The Basic Elements of an Election

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality-with-Elimination Method

1.5 The Method of Pairwise Comparisons

1.6 Fairness Criteria and Arrow’s Impossibility Theorem

2 The Mathematics of Power

2.1 An Introduction to Weighted Voting

2.2 Banzhaf Power

2.3 Shapley-Shubik Power

2.4 Subsets and Permutations

3 The Mathematics of Sharing

3.1 Fair-Division Games

3.2 The Divider-Chooser Method

3.3 The Lone-Divider Method

3.4 The Lone-Chooser Method

3.5 The Method of Sealed Bids

3.6 The Method of Markers

4 The Mathematics of Apportionment

4.1 Apportionment Problems and Apportionment Methods

4.2 Hamilton’s Method

4.3 Jefferson’s Method

4.4 Adams’s and Webster’s Methods

4.5 The Huntington-Hill Method

4.6 The Quota Rule and Apportionment Paradoxes

5 The Mathematics of Getting Around

5.1 Street-Routing Problems

5.2 An Introduction to Graphs

5.3 Euler’s Theorems and Fleury’s Algorithm

5.4 Eulerizing and Semi-Eulerizing Graphs

6 The Mathematics of Touring

6.1 What Is a Traveling Salesman Problem?

6.2 Hamilton Paths and Circuits

6.3 The Brute-Force Algorithm

6.4 The Nearest-Neighbor and Repetitive Nearest-Neighbor Algorithms

6.5 The Cheapest-Link Algorithm

7 The Mathematics of Networks

7.1 Networks and Trees

7.2 Spanning Trees, MSTs, and MaxSTs

7.3 Kruskal’s Algorithm

8 The Mathematics of Scheduling

8.1 An Introduction to Scheduling

8.2 Directed Graphs

8.3 Priority-List Scheduling

8.4 The Decreasing-Time Algorithm

8.5 Critical Paths and the Critical-Path Algorithm

9 Population Growth Models

9.1 Sequences and Population Sequences

9.2 The Linear Growth Model

9.3 The Exponential Growth Model

9.4 The Logistic Growth Model

10 Financial Mathematics

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Retirement Savings

10.5 Consumer Debt

11 The Mathematics of Symmetry

11.1 Rigid Motions

11.2 Reflections

11.3 Rotations

11.4 Translations

11.5 Glide Reflections

11.6 Symmetries and Symmetry Types

11.7 Patterns

12 Fractal Geometry

12.1 The Koch Snowflake and Self-Similarity

12.2 The Sierpinski Gasket and the Chaos Game

12.3 The Twisted Sierpinski Gasket

12.4 The Mandelbrot Set

13 Fibonacci Numbers and the Golden Ratio

13.1 Fibonacci Numbers

13.2 The Golden Ratio

13.3 Gnomons

13.4 Spiral Growth in Nature

14 Censuses, Surveys, Polls, and Studies

14.1 Enumeration

14.2 Measurement

14.3 Cause and Effect

15 Graphs, Charts, and Numbers

15.1 Graphs and Charts

15.2 Means, Medians, and Percentiles

15.3 Ranges and Standard Deviations

16 Probabilities, Odds, and Expectations

16.1 Sample Spaces and Events

16.2 The Multiplication Rule, Permutations, and Combinations

16.3 Probabilities and Odds

16.4 Expectations

16.5 Measuring Risk

17 The Mathematics of Normality

17.1 Approximately Normal Data Sets

17.2 Normal Curves and Normal Distributions

17.3 Modeling Approximately Normal Distributions

17.4 Normality in Random Events

Answers to Selected Exercises

Index

Photo Credits

Preface

PREFACE

To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.
– Ivars Peterson, The Mathematical Tourist

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. We have made a concerted effort to introduce the reader to an entirely 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 takeMath 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: We have found Intermediate Algebra to be an appropriate and sufficient prerequisite. (In the few instances in which more advanced concepts are unavoidable we have endeavored 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 in this century, 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. It is not necessary that the reader love everything in the book—it is sufficient that he or she find one topic about which they can say, "I really enjoyed learning this stuff!" We believe that anyone coming in with an open mind almost certainly will.

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?
• 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 symmetry type of a 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

We have endeavored to write a book that is 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) we have classified them 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. This category also includes an occasional open-ended problem suitable for a project.

THE FOURTH EDITION

This fourth edition of Excursions in Modern Mathematics retains the topics and organization of the third edition, in a more attractive and hopefully more user friendly package. The exercise sets at the end of each chapter have been significantly reorganized and expanded. The Walking exercises are now classified and listed according to topic, and there is now a much wider variety of exercises to choose from in each topic.

TEACHING EXTRAS AVAILABLE WITH THE FOURTH EDITION

• New York Times Supplement 0-13-019892-7 Prentice Hall and The New York Times jointly sponsor "A Contemporary View," a collection of mathematically significant articles taken from the pages of The New York Times.
• Companion Website (www.prenhall.com/tannenbaum) Features a syllabus manager, online quizzes, Internet projects, graphing calculator help, and dozens of additional resource links.
• Instructor's Solutions Manual 0-13-031483-8 Contains solutions to all the exercises in the text. Also includes extra classroom and student project materials developed at Virginia Commonwealth University.
• Printed Test Bank 0-13-031484-6 Contains over 700 multiple choice questions.
• TestGen-EQ win/mac CD 0-13-018695-3 Test generating software that creates randomized tests and offers an onscreen LAN based testing environment, complete with Instructor Gradebook.
• MathPak 0-13-018698-8 Includes the Companion Website plus the Student Solutions Manual, Excel chapter projects developed by Dale Buske, St. Cloud State, and other extra materials designed to enrich the course.

A FINAL WORD

This book grew out of the conviction that a liberal arts mathematics course should teach students more than just a collection of facts and procedures. The ultimate purpose of this book is to instill in the reader an overall appreciation of mathematics as a discipline and an exposure to the subtlety and variety of its many facets: problems, ideas, methods, and solutions. Last, but not least, we have tried to show that mathematics can be fun.

ACKNOWLEDGMENTS

This book is now in its fourth edition, and there are many people who contributed in significant ways to help it along the way. We are thankful to each and every one of them.

Thanks go to St. Cloud State University mathematics faculty for their invaluable insight. Their dedication and resulting comments have helped shape many of the improvements in this revision.

The exercise sets have grown over time, with valuable contributions at various stages from Vahack Haroutunian, Ronald Wagoner, Carlos Valencia, and L. T. Ullmann.

We extend special thanks to Professor Benoit Mandelbrot of Yale University who read the manuscript for Chapter 12 and made several valuable suggestions.

For this fourth edition, the contributions of our copy editor Kathy SessaFederico and our production editor Barbara Mack were invaluable, and much of the improvements in presentation and readability are due to their work.

Last, but not least, the person most responsible for the success of this book is Sally Yagan. There is an editor behind every book, but few that can match her vision, "can-do" attitude, and leadership.

Editorial Reviews

New edition of a text representing a collection of topics chosen to meet the criteria of applicability to real-life problems, accessibility, inclusion of modern mathematics, and aesthetics. Sixteen chapters discuss the mathematics of social choice, management science, growth and symmetry, and statistics. Exercises address a broad spectrum of levels of difficulty. Annotation c. by Book News, Inc., Portland, Or.

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