2007;
338 pp;
Hardcover

MSC: Primary 00; 11; 52;

Print ISBN: 978-0-8218-4403-8

Product Code: MBK/48

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

**Electronic ISBN: 978-1-4704-1198-5
Product Code: MBK/48.E**

List Price: $53.00

AMS Member Price: $42.40

MAA Member Price: $47.70

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#### Supplemental Materials

# Roots to Research: A Vertical Development of Mathematical Problems

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*Judith D. Sally; Paul J. Sally, Jr.*

Certain contemporary mathematical problems are of particular interest to
teachers and students because their origin lies in mathematics covered in
the elementary school curriculum and their development can be traced
through high school, college, and university level mathematics. This book
is intended to provide a source for the mathematics (from beginning to
advanced) needed to understand the emergence and evolution of five of
these problems: The Four Numbers Problem, Rational Right Triangles,
Lattice Point Geometry, Rational Approximation, and Dissection.

Each chapter begins with the elementary geometry and number theory at the
source of the problem, and proceeds (with the exception of the first
problem) to a discussion of important results in current research. The
introduction to each chapter summarizes the contents of its various
sections, as well as the background required.

The book is intended for students and teachers of mathematics from high
school through graduate school. It should also be of interest to working
mathematicians who are curious about mathematical results in fields other
than their own. It can be used by teachers at all of the above mentioned
levels for the enhancement of standard curriculum materials or
extra-curricular projects.

#### Readership

High school students, undergraduate and graduate students, and teachers of all levels interested in mathematics.

#### Reviews & Endorsements

I am a huge fan of this book! ... "Roots to Research" is very accessible, supported throughout with insightful examples and exercises that motivate both the ideas and the formal notation. I recommend this book to future and current math teachers, math majors, and working mathematicians who are interested in reading about cool math. ... the Sallys have done the mathematical community a service by writing a book that illustrates an approach that more of us should take when teaching upper-level undergraduate and graduate math courses.

-- MAA Monthly

Many references are given but the book is largely self-contained. The authors have done a remarkable job of giving a seamless presentation of material at very different levels of difficulty. Teachers and students will appreciate this book both for the information it contains and as a model of expository writing.

-- Mathematical Reviews

The book gives a very good introduction in how to solve mathematical problems and it is well suited as a basis for a beginner's seminar at universities.

-- Zentralblatt MATH

# Table of Contents

## Roots to Research: A Vertical Development of Mathematical Problems

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents vii8 free
- Chapter 1. The Four Numbers Problem 116 free
- 1. Introduction 116
- 2. The Four Numbers Game Rule 217
- 3. Symmetry and the Four Numbers Game 520
- 4. Does Every Four Numbers Game Have Finite Length? 1126
- 5. Games With Length Independent of the Size of the Numbers 1530
- 6. Long Games 1732
- 7. The Tribonacci Games 2237
- 8. The Length of the Four Real Numbers Game 2641
- 9. The Probability that a Four Numbers Game Ends in n Steps 3247
- 10. The k-Numbers Game 4156
- Bibliography 4560

- Chapter 2. Rational Right Triangles and the Congruent Number Problem 4762
- 1. Introduction 4762
- 2. Right Triangles 4863
- 3. Pythagorean Triples 6378
- 4. Sums of Squares 6984
- 5. Rational Right Triangles 8499
- 6. Congruent Numbers 90105
- 7. Equivalent Definitions of Congruent Number 94109
- 8. 1, 2, and 3 Are Not Congruent Numbers 96111
- 9. Rational Right Triangles and Certain Cubic Curves 101116
- 10. Elliptic Curves 104119
- 11. The Abelian Group of Rational Points on an Elliptic Curve 109124
- 12. E[sub(n)](Q) and Congruent Numbers 112127
- Bibliography 121136

- Chapter 3. Lattice Point Geometry 123138
- 1. Introduction 123138
- 2. Geometric Shapes as Lattice Polygons 126141
- 3. Embedding Regular Polygons in a Lattice 130145
- 4. Basic Algebraic and Geometric Tools 137152
- 5. Pick's Theorem 145160
- 6. Applications of Pick's Theorem 157172
- 7. Lattice Points In and On a Circle 162177
- 8. Integer Points in Bounded Convex Regions in R[sup(2)] 166181
- 9. Minkowski's Theorem in R[sup(2)] 169184
- 10. Embedding Regular Plane Polygons as Lattice Polygons in R[sup(k)] 172187
- 11. Lattice Hypercubes 176191
- 12. Minkowski's Theorem in R[sup(k)] 183198
- 13. Ehrhart's Theorem 185200
- Bibliography 193208

- Chapter 4. Rational Approximation 195210
- 1. Introduction 195210
- 2. Introduction to Approximation Theory 197212
- 3. Properties of Rational Numbers Close to a Real Number 201216
- 4. An Interesting Example, Part I 204219
- 5. Dirichlet's Theorem 205220
- 6. An Interesting Example, Part II 208223
- 7. Hurwitz's Theorem 210225
- 8. Liouville's Theorem 215230
- 9. The Thue-Siegel-Roth Theorem 221236
- 10. The Approximation Exponent 230245
- 11. An Interesting Example, Part III 232247
- 12. An Application to Diophantine Equations 233248
- 13. What About Transcendental Numbers? 236251
- Bibliography 241256

- Chapter 5. Dissection 243258
- 1. Introduction 243258
- 2. Dissection and Area 245260
- 3. Basic Properties of Dissection 251266
- 4. Polygons of Equal Area 257272
- 5. Dissection in Three Dimensions 259274
- 6. The Angles of a Polyhedron 266281
- 7. The Dehn Invariant 269284
- 8. A Solution of Hilbert's Third Problem 277292
- 9. Congruence by Finite Decomposition and Equidecomposability 280295
- 10. Hausdorff's Paradox 283298
- 11. The Banach-Tarski Paradox 290305
- 12. Equidissectabihty and Equidecomposability 295310
- 13. Squaring the Circle 297312
- 14. Borsuk's Problem 298313

- Bibliography 311326
- Appendix A. Volume 315330
- Appendix B. Convexity 325340
- Index 335350
- Back Cover Back Cover1354