**Dolciani Mathematical Expositions**

Volume: 36;
2009;
181 pp;
Hardcover

**Print ISBN: 978-0-88385-342-9
Product Code: DOL/36**

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-61444-202-8
Product Code: DOL/36.E**

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

# When Less is More: Visualizing Basic Inequalities

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*Claudi Alsina; Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Reviews & Endorsements

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

# Table of Contents

## When Less is More: Visualizing Basic Inequalities

- cover cover11
- copyright page ii3
- title page iii4
- Contents ix10
- Preface xiii14
- Introduction xv16
- CHAPTER 1 Representing positive numbers as lengths of segments 122
- CHAPTER 2 Representing positive numbers as areas or volumes 1940
- 2.1 Three examples 2041
- 2.2 Chebyshev’s inequality 2142
- 2.3 The AM-GM inequality for three numbers 2445
- 2.4 Guha’s inequality 2950
- 2.5 The AM-GM inequality for n numbers 2950
- 2.6 The HM-AM-GM-RMS inequality for nnumbers 3152
- 2.7 The mediant property and Simpson’s paradox 3253
- 2.8 Chebyshev’s inequality revisited 3455
- 2.9 Schur’s inequality 3758
- 2.10 Challenges 3859

- CHAPTER 3 Inequalities and the existence of triangles 4364
- CHAPTER 4 Using incircles and circumcircles 5576