Preface

Summary

The worst form of inequality is to try to make unequal things equal.

Aristotle

The fundamental results of mathematics are often inequalities rather than equalities.

Edwin Beckenbach and Richard Bellman

Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often, especially in secondary and collegiate mathematics, the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they do not 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 (recall George Pólya's advice “Draw a figure …”). We will 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.

Visual arguments known as proofs without words are published regularly in Mathematics Magazine, The College Mathematics Journal, and other publications. Some involve inequalities, and appear in [Nelsen, 1993 and 2000]. We have also published a book [Alsina and Nelsen, 2006] on how to create images to help one understand mathematical ideas, proofs, and arguments.