Mathematics for the Liberal Arts E-Book

Donald Bindner

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 702

Veröffentlichungsjahr: 2014

Beschreibung

Presents a clear bridge between mathematics and the liberal arts

Mathematics for the Liberal Arts provides a comprehensible and precise introduction to modern mathematics intertwined with the history of mathematical discoveries. The book discusses mathematical ideas in the context of the unfolding story of human thought and highlights the application of mathematics in everyday life.

Divided into two parts, Mathematics for the Liberal Arts first traces the history of mathematics from the ancient world to the Middle Ages, then moves on to the Renaissance and finishes with the development of modern mathematics. In the second part, the book explores major topics of calculus and number theory, including problem-solving techniques and real-world applications. This book emphasizes learning through doing, presents a practical approach, and features:

• A detailed explanation of why mathematical principles are true and how the mathematical processes work
• Numerous figures and diagrams as well as hundreds of worked examples and exercises, aiding readers to further visualize the presented concepts
• Various real-world practical applications of mathematics, including error-correcting codes and the space shuttle program
• Vignette biographies of renowned mathematicians
• Appendices with solutions to selected exercises and suggestions for further reading

Mathematics for the Liberal Arts is an excellent introduction to the history and concepts of mathematics for undergraduate liberal arts students and readers in non-scientific fields wishing to gain a better understanding of mathematics and mathematical problem-solving skills.

Leseprobe

CONTENTS

Preface

Part I Mathematics In History

1 The Ancient Roots of Mathematics

1.1 Introduction

1.2 Ancient Mesopotamia and Egypt

1.3 Early Greek Mathematics: The First Theorists

1.4 The Apex: Third Century Hellenistic Mathematics

1.5 The Slow Decline

2 The Growth of Mathematics to 1600

2.1 China

2.2 India

2.3 Islam

2.4 European Mathematics Awakens

3 Modern Mathematics

3.1 The 17th Century: Scientific Revolution

3.2 The 18th Century: Consolidation

3.3 The 19th Century: Expansion

3.4 The 20th and 21st Centuries: Explosion

3.5 The Future

Part II Two Pillars Of Mathematics

4 Calculus

4.1 What Is Calculus?

4.2 Average and Instantaneous Velocity

4.3 Tangent Line to a Curve

4.4 The Derivative

4.5 Formulas for Derivatives

4.6 The Product Rule and Quotient Rule

4.7 The Chain Rule

4.8 Slopes and Optimization

4.9 Applying Optimization Methods

4.10 Differential Notation and Estimates

4.11 Marginal Revenue, Cost, and Profit

4.12 Exponential Growth

4.13 Periodic Functions and Trigonometry

4.14 The Fundamental Theorem of Calculus

4.15 The Riemann Integral

4.16 Signed Areas and Other Integrals

4.17 Application: Rocket Science

4.18 Infinite Sums

4.19 Exponential Growth and Doubling Times

4.20 Beyond Calculus

5 Number Theory

5.1 What Is Number Theory?

5.2 Divisibility

5.3 Irrational Numbers

5.4 Greatest Common Divisors

5.5 Primes

5.6 Relatively Prime Integers

5.7 Mersenne and Fermat Primes

5.8 The Fundamental Theorem of Arithmetic

5.9 Diophantine Equations

5.10 Linear Diophantine Equations

5.11 Pythagorean Triples

5.12 An Introduction to Modular Arithmetic

5.13 Congruence

5.14 Arithmetic with Congruences

5.15 Division with Congruences; Finite Fields

5.16 Fermat’s Last Theorem

5.17 Unfinished Business

A Answers to Selected Exercises

B Suggested Reading

Index

Cover Credit: @ usetrick/iStockphoto

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Bindner, Donald, author.

Mathematics for the liberal arts / Donald Bindner, Department of Mathematics and Computer Science, Truman State University, Kirksville, MO, Martin J. Erickson, Department of Mathematics and Computer Science, Truman State University, Kirksville, MO, Joe Hemmeter, Farmington, MI.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-35291-5 (hardback)

1. Mathematics—History—Textbooks. I. Erickson, Martin J., 1963–author. II. Hemmeter, Joe, 1950–author. III. Title.

QA21.B56 2013

510.9—dc23

2012023745

To Linda, Christine, and Debbie

PREFACE

This book is an introduction to mathematics history and mathematical concepts for liberal arts students. Students majoring in all fields can understand and appreciate mathematics, and exposure to mathematics can enhance and invigorate students’ thinking.

The book can be used as the basis for introductory courses on mathematical thinking. These courses may have titles such as “Introduction to Mathematical Thinking.” We describe the history of mathematical discoveries in the context of the unfolding story of human thought. We explain why mathematical principles are true and how the mathematics works. The emphasis is on learning about mathematical ideas and applying mathematics to real-world settings. Summaries of historical background and mini-biographies of mathematicians are interspersed throughout the mathematical discussions.

What mathematical knowledge should students have to read this book? An understanding of basic arithmetic, algebra, and geometry is necessary. This material is often taught in high school or beginning college-level courses. Beyond this background, the book is self-contained. Students should be willing to read the text and work through the examples and exercises. In mathematics, the best way (perhaps the only way) to learn is by doing.

Part I, comprising the first three chapters, gives an overview of the history of mathematics. We start with mathematics of the ancient world, move on to the Middle Ages, and then discuss the Renaissance and some of the developments of modem mathematics. Part II gives detailed coverage of two major areas of mathematics: calculus and number theory. These areas loom large in the world of mathematics, and they have many applications. The text is rounded out by appendices giving solutions to selected exercises and recommendations for further reading.

A variety of courses can be constructed from the text, depending on the aims of the instructor and the needs of the students. A one-semester course would likely focus on selected chapters, while a two-semester course sequence could cover all five chapters.

We hope that by working through the book, readers will attain a deeper appreciation of mathematics and a greater facility for using mathematics.

Thanks to the people who gave us valuable feedback about our writing: Linda Bindner, Robert Dobrow, Suren Fernando, David Garth, Amy Hemmeter, Mary Hemmeter, Daniel Jordan, Kenneth Price, Phil Ryan, Frank Sottile, Anthony Vazzana, and Dana Vazzana.

Thanks also to the Wiley staff for their assistance in publishing our book: Liz Belmont, Sari Friedman, Danielle LaCourciere, Jacqueline Palmieri, Susanne Steitz-Filler, and Stephen Quigley.

PART I

MATHEMATICS IN HISTORY

CHAPTER 1

THE ANCIENT ROOTS OF MATHEMATICS

Mathematics—the unshaken Foundation of Sciences, and the plentiful Fountain of Advantage to human affairs.

Isaac Barrow (1630-1677)

1.1 Introduction

Mathematics is a human enterprise, which means that it is part of history. It has been shaped by that history, and in turn has helped to shape it. In this chapter we will trace these connections.

Many societies have contributed to mathematics, but a main historical thread is discernible, one that has led directly to today’s mathematics. That thread began in the ancient Mediterranean world, swelled mightily in ancient Greece, dwindled at the time of the Roman empire, was kept alive and augmented in the Muslim world, re-entered Western Europe in the Renaissance, developed in Europe for several centuries, then spread throughout the world in the 20th century. We will spend most of our time on this thread, in part because so much is known about it, with a few excursions into other cultures.

Eurasia and Africa.1

Fingers, Knots, and Tally Sticks

Experiments have shown that humans, and other animals, are bom with innate mathematical abilities. They regularly distinguish between, say, one tree and two trees. The next logical step is counting, that is, establishing a one-to-one correspondence between sets of objects. This is no doubt also an ancient ability.

Once we can count objects, how do we communicate numbers to others? Most of what follows in this chapter is based on the historical, i.e., written record. But writing is a fairly recent invention. Before the written word, people used a variety of methods to represent numbers. Surely one of the first, and still important, methods was the use of various parts of the body. Some quite elaborate systems have been developed. The Torres Strait islanders, an indigenous Australian people, used fingers, toes, elbows, shoulders, knees, hips, wrists, and sternum to represent different numbers. Many languages preserve the remnants of such systems: the word for “five,” for example, is “hand” in Persian, Russian, and Sanskrit. And it is no coincidence that our number system is based on ten, the number of fingers.

Perhaps the most popular numbering system used notches on sticks or bones, so-called tally sticks, from the French word tailler, to cut. These date back at least 35,000 years, and must rank as one of the most successful technologies ever. As recently as 1826, tally sticks were used in official English tax records.

Another popular counting device was the stone. Our word “calculation” derives from the Latin calculus, which is a small stone. Early versions of the abacus were stones on the ground; “abacus” likely derives from the Hebrew abhaq, dust.

Knotted strings were a popular accounting tool throughout the world. The most notable examples of these were the amazing Incan quipu, which consisted of multiple knotted cords (up to 2000 of them) joined together.

A leading theory of the origin of writing in Mesopotamia, proposed by Denise Schmandt-Besserat, relates to a different method of recording numbers. It starts with the use of small clay tokens, found in archaeological sites, beginning circa 8000 BCE1. These tokens, in various standard shapes, were used for accounting: one shape might represent one sheep, for example, another one goat, or ten sheep. Imagine you are a merchant, and have hired someone to deliver a herd of 27 sheep to a neighboring city. The buyer needs to have some way to verify that the number of sheep that arrive is the same number sent. The solution was to encase tokens representing 27 sheep in a clay “envelope,” a hollow ball. The ball could be broken open at the destination, and the number of sheep verified.

Now imagine that the sheep’s journey has two legs; person A delivers them to person B, who in turn delivers them to the buyer. If B breaks open the ball to verify the count, what is the buyer to do? The solution found was to make impressions on the ball, using the tokens, before they were placed inside. After the clay ball hardened, these impressions could serve as a record as well as the tokens. Eventually, it was realized that the tokens were unnecessary. The “writing” on the ball sufficed.

Agriculture and Civilizations

Some time around 10,000 years ago, humans began developing agriculture, inaugurating the Neolithic, the “new stone age.” The first important crops were grains—large-seeded grasses—including wheat, sorghum, millet, and rice. Gradually, various animals were domesticated, notably cattle, sheep, oxen, pigs, and goats. This whole set of developments dramatically changed the way people lived. Instead of living in relatively small nomadic bands of “hunter-gatherers,” they started settling into villages. This allowed a larger population density.

In some areas of the world, usually in the flood plains of great river valleys, the agricultural settlements developed civilizations. Among these areas were Mesopotamia, the Nile in Egypt, the Yellow River in China, the Indus River in Pakistan, and the Ganges in India. Civilizations were characterized by more central organization, often including irrigation, granaries to store surplus grain, and cities.

The civilizations were based on the existence of agricultural surplus, which freed people to work on other things. This led to the development of many new technologies. Among these were the plow, wheeled vehicles, and, most notably, writing and metallurgy.

Civilizations required different, more sophisticated, types of mathematics. Geometry was needed for surveying land, building canals, dikes, and ditches, and constructing larger buildings like granaries and palaces. Administering the new city-states, apportioning taxes, and paying workers made increasing demands on arithmetic and algebra, as did the expanded commercial activity.

With the rise of civilization came new class structures. Most people were farmers, but some became blacksmiths, leather workers, engineers, architects, merchants, priests, scribes, surveyors, and of course kings. Some of the new, specialized professions (such as surveyors) nurtured their own mathematical techniques, handing them down through the generations. In some societies, small groups inside the new classes turned their collective attention to developing mathematics generally. Society provided practical inspiration for the new mathematics, but some mathematicians pursued knowledge for its own sake.