# Titu Andreescu and Mark Saul

American Mathematical Soc., Dec 19, 2016 - Geometry, Algebraic - 124 pages

This book starts with simple arithmetic inequalities and builds to sophisticated inequality results such as the Cauchy-Schwarz and Chebyshev inequalities. Nothing beyond high school algebra is required of the student. The exposition is lean. Most of the learning occurs as the student engages in the problems posed in each chapter. And the learning is not “linear”. The central topic of inequalities is linked to others in mathematics. Often these topics relate to much more than algebraic inequalities.

There are also “secret” pathways through the book. Each chapter has a subtext, a theme which prepares the student for learning other mathematical topics, concepts, or habits of mind. For example, the early chapters on the arithmetic mean/geometric mean inequality show how very simple observations can be leveraged to yield useful and interesting results. Later chapters give examples of how one can generalize a mathematical statement. The chapter on the Cauchy-Schwarz inequality provides an introduction to vectors as mathematical objects. And there are many other secret pathways that the authors hope the reader will discover—and follow.

In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### Contents

 Chapter 0 Some Introductory Problems 1 Chapter 1 Squares Are Never Negative 7 Chapter 2 The ArithmeticGeometric Mean Inequality Part I 21 Chapter 3 The ArithmeticGeometric Mean Inequality Part II 33 Chapter 4 The Harmonic Mean 43 Chapter 5 Symmetry in Algebra Part I 57
 Chapter 6 Symmetry in Algebra Part II 69 Chapter 7 Symmetry in Algebra Part III 85 Chapter 8 The Rearrangement Inequality 91 Chapter 9 The CauchySchwarz Inequality 107 Back Cover 126 Copyright

### About the author (2016)

Titu Andreescu: The University of Texas at Dallas, Richardson, TX,
Mark Saul: Executive Director, Julia Robinson Math Festivals