Algebraic Inequalities: New Vistas (MSRI Mathematical Circles Library)

Description

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. Cover Title page Contents Acknowledgments Introduction Chapter 0. Some Introductory Problems How Do Inequalities Behave? Solutions Chapter 1. Squares Are Never Negative Problems Solutions Chapter 2. The Arithmetic-Geometric Mean Inequality, Part I Problems Solutions Chapter 3. The Arithmetic-Geometric Mean Inequality, Part II Problems Solutions Chapter 4. The Harmonic Mean Introduction It’s more than just \boldmath{𝐷=𝑅𝑇} Notes and Summary Problems The Harmonic Mean of Several Quantities Problems The Harmonic Mean in Geometry Problems Solutions Chapter 5. Symmetry in Algebra, Part I Problems Symmetry in Inequalities Problems Solutions Chapter 6. Symmetry in Algebra, Part II Problems Solutions Chapter 7. Symmetry in Algebra, Part III Problems Solutions Chapter 8. The Rearrangement Inequality The General Rearrangement Inequality Permutations Problems Chebyshev’s Inequality Problems Solutions Chapter 9. The Cauchy-Schwarz Inequality Problems Higher Dimensions Problems More Generally… An Important Problem The Importance of the Problem Problems Solutions Back Cover