Donald Bindner - Mathematics for the Liberal Arts

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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. MathematicsHistoryTextbooks. I. Erickson, Martin J., 1963author. II. Hemmeter, Joe, 1950author. III. Title.

QA21.B56 2013

510.9dc23

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.

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

Isaac Barrow (1630-1677)

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 todays 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.

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.