Hardcover ISBN:  9780883853429 
Product Code:  DOL/36 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781614442028 
Product Code:  DOL/36.E 
List Price:  $60.00 
MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Hardcover ISBN:  9780883853429 
eBook: ISBN:  9781614442028 
Product Code:  DOL/36.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 
Hardcover ISBN:  9780883853429 
Product Code:  DOL/36 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781614442028 
Product Code:  DOL/36.E 
List Price:  $60.00 
MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Hardcover ISBN:  9780883853429 
eBook ISBN:  9781614442028 
Product Code:  DOL/36.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 

Book DetailsDolciani Mathematical ExpositionsVolume: 36; 2009; 181 pp
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 wellknown 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.

Table of Contents

Chapters

Chapter 1. Representing positive numbers as lengths of segments

Chapter 2. Representing positive numbers as areas or volumes

Chapter 3. Inequalities and the existence of triangles

Chapter 4. Using incircles and circumcircles

Chapter 5. Using reflections

Chapter 6. Using rotations

Chapter 7. Employing nonisometric transformations

Chapter 8. Employing graphs of functions

Chapter 9. Additional topics


Additional Material

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


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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 wellknown 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.

Chapters

Chapter 1. Representing positive numbers as lengths of segments

Chapter 2. Representing positive numbers as areas or volumes

Chapter 3. Inequalities and the existence of triangles

Chapter 4. Using incircles and circumcircles

Chapter 5. Using reflections

Chapter 6. Using rotations

Chapter 7. Employing nonisometric transformations

Chapter 8. Employing graphs of functions

Chapter 9. Additional topics

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