Preface

Summary

Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement (appropriately by a philosopher) is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art. Secondly, the fundamental results of mathematics are often inequalities rather than equalities.

In the pages that follow, we have presented three aspects of the theory of inequalities. First, in Chapters 1, 2, and 3, we have the axiomatic aspect. Secondly, in Chapter 4, we use the products of the preceding chapters to derive the basic inequalities of analysis, results that are used over and over by the practicing mathematician. In Chapter 5, we show how to use these results to derive a number of interesting and important maximum and minimum properties of the elementary symmetric figures of geometry: the square, cube, equilateral triangle, and so on. Finally, in Chapter 6, some properties of distance are studied and some unusual distance functions are exhibited.

There is thus something for many tastes, material that may be read consecutively or separately. Some readers will want to understand the axiomatic approach that is basic to higher mathematics.

They will enjoy the first three chapters. In addition, in Chapter 3 there are many illuminating graphs associated with inequalities.