The objective of this book is to illustrate how the use of visualization can be a powerful tool for better understanding of some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and the authors will convince you that the same is true when working with inequalities. They 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 cannot only show two things unequal, but also help the observer see just how unequal they are.

### Table of Contents

Preface

Introduction

1. Representing positive numbers as lengths of segments

2. Representing positive numbers as areas or volumes

3. Inequalities and the existence of triangles

4. Using incircles and circumcircles

5. Using reflections

6. Using rotations

7. Employing non-isometric transformations

8. Employing graphs of functions

9. Additional topics

Solutions to the Challenges

Notation and symbols

References

Index

About the Authors

### About the Authors

**Claudi Alsina** was born on 30 January 1952 in Barcelona, Spain. He received his BA and PhD in mathematics from the University of Barcelona. His post-doctoral studies were at the University of Massachusetts, Amherst. Claudi, Professor of Mathematics at the Technical University of Catalonia, has developed a wide range of international activities, research papers, publications and hundreds of lectures on mathematics and mathematics education. His latest books include *Associative Functions: Triangular Norms and Copulas* with M.J. Frank and B. Schweizer, WSP, 2006; *Math Made Visual: Creating Images for Understanding Mathematics* (with Roger B. Nelsen), MAA, 2006; *Vitaminas Matematicas* and *El Club de la Hipotenusa*, Ariel, 2008.

**Roger B. Nelsen** was born on 20 December 1942 in Chicago, Illinois. He received his BA in mathematics from DePauw University in 1964 and his PhD in mathematics from Duke University in 1969. Roger was elected to Phi Beta Kappa and Sigma Xi. His previous books include *Proofs Without Words: Exercises in Visual Thinking*, MAA 1993; *An Introduction to Copulas*, Springer, 1999 (2nd ed. 2006); *Proofs Without Words II: More Exercises in Visual Thinking*, MAA, 2000; and *Math Made Visual: Creating Images for Understanding Mathematics* (with Claudi Alsina), MAA, 2006.