When Less is More: Visualizing Basic Inequalities (Dolciani Mathematical Expositions)

The vast majority of problems that math students solve are based on equalities, a situation that is somewhat artificial when related to the real world. Many engineering problems are based on an inequality, in other words find the minimum value that will work and then add a safety factor. There is also the additional fact that many inequalities are proved by reference to known inequalities, so if you don't have the basics, it is difficult to go very far. Finally, all humans, even abstract mathematicians, understand things better when there is a way to visualize the situation. As the name implies, the authors make heavy use of figures in their arguments used to prove inequalities of many different types. While there is the natural restriction that many types of inequalities do not lend themselves very well to figurative representations, the authors certainly do as much as possible. Sets of problems called challenges appear at the end of each chapter and solutions to the challenges are included in an appendix. As the authors state on the back cover, this book is an attempt to provide a resource for the teaching of geometric inequalities. Since illustrations are used, there is also an emphasis on a systematic way in which the images of the inequalities can be constructed. This book would be an excellent text for a course in inequalities or as a supplement for any instructor that wants to place more emphasis on introducing and proving inequalities.