**MSRI Mathematical Circles Library**

Volume: 19;
2016;
124 pp;
Softcover

MSC: Primary 97;

Print ISBN: 978-1-4704-3464-9

Product Code: MCL/19

List Price: $41.00

Individual Member Price: $32.80

**Electronic ISBN: 978-1-4704-3595-0
Product Code: MCL/19.E**

List Price: $41.00

Individual Member Price: $32.80

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#### Supplemental Materials

# Algebraic Inequalities: New Vistas

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*Titu Andreescu; Mark Saul*

A co-publication of the AMS and the Mathematical Sciences Research Institute

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.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

#### Readership

Undergraduate students interested in learning and teaching high school algebra.

#### Reviews & Endorsements

This slim book moves economically, effectively, and elegantly from basic arithmetic to sophisticated inequality topics...Part of what makes the presentation effective, despite requiring no more than high school algebra, are the detailed solutions for the problem sets...This compact and concentrated work will be a useful service to students and teachers in fundamental mathematics.

-- Tom Schulte, MAA Reviews

#### Table of Contents

# Table of Contents

## Algebraic Inequalities: New Vistas

- Cover Cover11
- Title page iii4
- Contents v6
- Acknowledgments vii8
- Introduction ix10
- Chapter 0. Some Introductory Problems 112
- Chapter 1. Squares Are Never Negative 718
- Chapter 2. The Arithmetic-Geometric Mean Inequality, Part I 2132
- Chapter 3. The Arithmetic-Geometric Mean Inequality, Part II 3344
- Chapter 4. The Harmonic Mean 4354
- Chapter 5. Symmetry in Algebra, Part I 5768
- Chapter 6. Symmetry in Algebra, Part II 6980
- Chapter 7. Symmetry in Algebra, Part III 8596
- Chapter 8. The Rearrangement Inequality 91102
- Chapter 9. The Cauchy-Schwarz Inequality 107118
- Back Cover Back Cover1137