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When Less is More: Visualizing Basic Inequalities (Dolciani Mathematical Expositions) [Hardcover]

by Claudi Alsina, Roger Nelsen
5.0 out of 5 stars  See all reviews (2 customer reviews)

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Book Description

April 1, 2009 0883853426 978-0883853429 1

Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don't possess the richness and variety that one finds with inequalities.

The objective of this book is to illustrate how use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and we would like to convince you that the same is true when working with inequalities. We show how to produce figures in a systematic way for the illustration of inequalities; and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument can not only show two things unequal, but also help the observer see just how unequal they are.

The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.

Frequently Bought Together

When Less is More: Visualizing Basic Inequalities (Dolciani Mathematical Expositions) + Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions) + Proofs Without Words II: More Exercises in Visual Thinking (Classroom Resource Materials) (v. 2)
Price for all three: $151.85

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Editorial Reviews


The book will be valuable for secondary and collegiate teachers as a source of inspiration for preparing lessons. It is also recommended for self study to students having adquate knowledge of geometry and a sufficiently rigorous mind. --Zentralblatt fur Mathematik

The Dolciani Mathematical Expositions series has once again provided an excellent text in the publication When Less is More: Visualizing Basic Inequalities. Alsina and Nelsen offer readers visual arguments for many historical inequalities, allowing a deeper understanding of those inequalities. Perhaps even more significant than the deeper meaning of these particular inequalities is the fact that the authors present a methodology for producing such visual arguments. Each of the nine chapters features a different approach and concludes with "challenge" problems (with solutions) to support the understanding of the chapter's method. The text is highly recommended for instructors as a course supplement and could prove an excellent undergraduate text for a course in inequalities. --Matthew J. Haines (Minneapolis, MN) Mathematical Reviews

Book Description

An introduction to basic inequalities in mathematics (including numbers, functions and geometry) motivated and illustrated by drawings that make it possible to 'see' the results in an elegant manner. Applications of the inequalities, historical details, and exercises for the reader (with solutions) are also included.

Product Details

  • Series: Dolciani Mathematical Expositions
  • Hardcover: 164 pages
  • Publisher: The Mathematical Association of America; 1 edition (April 1, 2009)
  • Language: English
  • ISBN-10: 0883853426
  • ISBN-13: 978-0883853429
  • Product Dimensions: 0.8 x 6.2 x 9.2 inches
  • Shipping Weight: 14.1 ounces (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,806,433 in Books (See Top 100 in Books)

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3 of 3 people found the following review helpful
The vast majority of problems that math students solve are based on equalities, a situation that is somewhat artificial when related to the real world. Many engineering problems are based on an inequality, in other words find the minimum value that will work and then add a safety factor. There is also the additional fact that many inequalities are proved by reference to known inequalities, so if you don't have the basics, it is difficult to go very far. Finally, all humans, even abstract mathematicians, understand things better when there is a way to visualize the situation.
As the name implies, the authors make heavy use of figures in their arguments used to prove inequalities of many different types. While there is the natural restriction that many types of inequalities do not lend themselves very well to figurative representations, the authors certainly do as much as possible. Sets of problems called challenges appear at the end of each chapter and solutions to the challenges are included in an appendix.
As the authors state on the back cover, this book is an attempt to provide a resource for the teaching of geometric inequalities. Since illustrations are used, there is also an emphasis on a systematic way in which the images of the inequalities can be constructed. This book would be an excellent text for a course in inequalities or as a supplement for any instructor that wants to place more emphasis on introducing and proving inequalities.
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1 of 1 people found the following review helpful
5.0 out of 5 stars Awesomely inspiring. April 19, 2012
Format:Hardcover|Verified Purchase
Very beautiful stuff. Right now I'm making colorful tutorials based on the images inside this book. It really does give you a geometric and intuitive understand of many inequalities. It's very helpful to use this in conjunction with Michael J. Steele's Cauchy-Schwarz Masterclass.
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