How We Got from There to Here: A Story of Real Analysis

(2 reviews)

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Robert Rogers, State University of New York
Eugene Boman, The Pennsylvania State University

Pub Date: 2014

ISBN 13: 978-0-9897226-6-7

Publisher: Open SUNY

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Reviewed by Clark Gaylord, Adjunct, Virginia Tech, on 6/11/2015.

From Number to Cantor's Theorem, this book brings you on a journey of the development of mathematical analysis. Several important stops along the way … read more

 

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Reviewed by Patrick Shipman, Associate Professor, Colorado State University, on 1/8/2016.

Besides standard material for an analysis book, this text runs on themes motivated by discussions of the history of analysis. Along with that come … read more

 

Table of Contents

  • Chapter 1: Numbers, Real (R) and Rational (Q)
  • Chapter 2: Calculus in the 17th and 18th Centuries
  • Chapter 3: Questions Concerning Power Series
  • Chapter 4: Convergence of Sequences and Series
  • Chapter 5: Convergence of the Taylor Series: A “Tayl” of Three Remainders
  • Chapter 6: Continuity: What It Isn’t and What It Is
  • Chapter 7: Intermediate and Extreme Values
  • Chapter 8: Back to Power Series
  • Chapter 9: Back to the Real Numbers

About the Book

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context.

This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.

About the Contributors

Author(s)

Robert Rogers received his BS in Mathematics with Certification in Secondary Education from Buffalo State College in 1979. He earned his MS in Mathematics from Syracuse University in 1980 and his Ph.D. in Mathematics from the University of Buffalo in 1987, specializing in Functional Analysis/Operator Theory. He has been on the faculty of the State University of New York at Fredonia since 1987 where he is currently Professor of Mathematics. He is a recipient of the SUNY Fredonia President’s Award for Excellence in Teaching and the MAA Seaway Section’s Clarence F. Stephens’ Award for Distinguished Teaching. He is also a recipient of the MAA Seaway Section’s Distinguished Service Award. He is currently the editor of the New York State Mathematics Teachers’ Journal.

Eugene Boman received his BA from Reed College in 1984, his MA in 1986 and his Ph.D. in 1993, both of the latter were from the University of Connecticut. He has been teaching math at The Pennsylvania State University since 1996, first at the DuBois campus (1996-2006) and then at the Harrisburg campus. In 2008 he won the Carl B. Allendorfer Award for excellence in expository mathematical writing from the editors of Mathematics Magazine, for the article “Mom! There’s an Astroid in My Closet” (Mathematics Magazine, Vol. 80 (2007), pp. 247-273).