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New edition of a text representing a collection of topics chosen to meet the criteria of applicability to reallife 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.Product Details
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The "excursions" in this book represent a collection of topics chosen to meet a few simple criteria.
The material in the book is divided into four independent parts. Each of these parts in turn contains four chapters dealing with interrelated topics.
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:
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:
Table of Contents
PART 1. SOCIAL CHOICE
1. The Mathematics of Elections: The Paradoxes of Democracy
1.1 The Basic Elements of an Election
1.2 The Plurality Method
1.3 The Borda Count Method
1.4 The PluralitywithElimination Method
1.5 The Method of Pairwise Comparisons
1.6 Fairness Criteria and Arrow’s Impossibility Theorem
Conclusion
Key Concepts
Exercises
Projects and Papers
2. The Mathematics of Power: Weighted Voting
2.1 An Introduction to Weighted Voting
2.2 Banzhaf Power
2.3 ShapleyShubik Power
2.4 Subsets and Permutations
Conclusion
Key Concepts
Exercises
Projects and Papers
3. The Mathematics of Sharing: FairDivision Games
3.1 FairDivision Games
3.2 The DividerChooser Method
3.3 The LoneDivider Method
3.4 The LoneChooser Method
3.5 The Method of Sealed Bids
3.6 The Method of Markers
Conclusion
Key Concepts
Exercises
Projects and Papers
4. The Mathematics of Apportionment: Making the Rounds
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 HuntingtonHill Method
4.6 The Quota Rule and Apportionment Paradoxes
Conclusion
Key Concepts
Exercises
Projects and Papers
PART 2. MANAGEMENT SCIENCE
5. The Mathematics of Getting Around: Euler Paths and Circuits
5.1 StreetRouting Problems
5.2 An Introduction to Graphs
5.3 Euler’s Theorems and Fleury’s Algorithm
5.4 Eulerizing and SemiEulerizing Graphs
Conclusion
Key Concepts
Exercises
Projects and Papers
6. The Mathematics of Touring: Traveling Salesman Problems
6.1 What Is a Traveling Salesman Problem?
6.2 Hamilton Paths and Circuits
6.3 The BruteForce Algorithm
6.4 The NearestNeighbor and Repetitive NearestNeighbor Algorithms
6.5 The CheapestLink Algorithm
Conclusion
Key Concepts
Exercises
Projects and Papers
The Mathematics of Networks
7. The Cost of Being Connected
7.1 Networks and Trees
7.2 Spanning Trees, MST’s, and MaxST’s
7.3 Kruskal’s Algorithm
Conclusion
Key Concepts
Exercises
Projects and Papers
8. The Mathematics of Scheduling: Chasing the Critical Path
8.1 An Introduction to Scheduling
8.4 Directed Graphs
8.3 PriorityList Scheduling
8.4 The DecreasingTime Algorithm
8.5 Critical Paths and the CriticalPath Algorithm
Conclusion
Key Concepts
Exercises
Projects and Papers
PART 3. GROWTH
9. Population Growth Models: There Is Strength in Numbers
9.1 Sequences and Population Sequences
9.2 The Linear Growth Model
9.3 The Exponential Growth Model
9.4 The Logistic Growth Model
Conclusion
Key Concepts
Exercises
Projects and Papers
10. Financial Mathematics: Money Matters
10.1 Percentages
10.2 Simple Interest
10.3 Compound Interest
10.4 Consumer Debt
Conclusion
Key Concepts
Exercises
Projects and Papers
PART 4. SHAPE AND FORM
11. The Mathematics of Symmetry: Beyond Reflection
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
Conclusion
Key Concepts
Exercises
Projects and Papers
12. Fractal Geometry: The Kinky Nature of Nature
12.1 The Koch Snowflake and SelfSimilarity
12.2 The Sierpinski Gasket and the Chaos Game
12.3 The Twisted Sierpinski Gasket
13.4 The Mandelbrot Set
Conclusion
Key Concepts
Exercises
Projects and Papers
13. Fibonacci Numbers and the Golden Ratio: Tales of Rabbits and Gnomons
13.1 Fibonacci Numbers
13.2 The Golden Ratio
13.3 Gnomons
13.4 Spiral Growth in Nature
Conclusion
Key Concepts
Exercises
Projects and Papers
PART 5. STATISTICS
14. Censuses, Surveys, Polls, and Studies: The Joys of Collecting Data
14.1 Enumeration
14.2 Measurement
14.3 Cause and Effect
Conclusion
Key Concepts
Exercises
Projects and Papers
15. Graphs, Charts, and Numbers: The Data Show and Tell
15.1 Graphs and Charts
15.2 Means, Medians, and Percentiles
15.3 Ranges and Standard Deviations
Conclusion
Key Concepts
Exercises
Projects and Papers
16. Probabilities, Odds, and Expectations: Measuring Uncertainty and Risk
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
Conclusion
Key Concepts
Exercises
Projects and Papers
17. The Mathematics of Normality: The Call of the Bell
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
Conclusion
Key Concepts
Exercises
Projects and Papers
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 collegelevel liberal arts mathematics courseits 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.
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.
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 difficultyfrom the routine drill to the ultimate challenge. For convenience (but with some trepidation) we have classified them into three levels of difficulty:
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
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, "cando" attitude, and leadership.