eBook ISBN:  9780883859551 
Product Code:  NML/41.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 
eBook ISBN:  9780883859551 
Product Code:  NML/41.E 
List Price:  $50.00 
MAA Member Price:  $37.50 
AMS Member Price:  $37.50 

Book DetailsAnneli Lax New Mathematical LibraryVolume: 41; 2000; 176 pp
The Geometry of Numbers presents a selfcontained introduction to the geometry of numbers, beginning with easily understood questions about latticepoints on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems.

Table of Contents

Chapters

Chapter 1. Lattice Points and Straight Lines

Chapter 2. Counting Lattice Points

Chapter 3. Lattice Points and the Area of Polygons

Chapter 4. Lattice Points in Circles

Chapter 5. Minkowski’s Fundamental Theorem

Chapter 6. Applications of Minkowski’s Theorems

Chapter 7. Linear Transformations and Integral Lattices

Chapter 8. Geometric Interpretations of Quadratic Forms

Chapter 9. A New Principle in the Geometry of Numbers

Chapter 10. A Minkowski Theorem (Optional)

Appendix I. Gaussian Integers, by Peter D. Lax

Appendix II. The Closest Packing of Convex Bodies

Appendix III. Brief Biographies

Solutions and Hints


Reviews

The interplay between geometric methods and apparently nongeometric questions is intriguing, and many of the proofs are very elegant. Much of the geometry of numbers can be explained at a relatively elementary level, but there are few books on the subject aimed at beginning students. This book is likely the most accessible treatment of this material ever written. It should play a valuable role in exposing bright high school students, or college math majors, to the geometry of numbers.
Henry Cohn, MAA Reviews


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The Geometry of Numbers presents a selfcontained introduction to the geometry of numbers, beginning with easily understood questions about latticepoints on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems.

Chapters

Chapter 1. Lattice Points and Straight Lines

Chapter 2. Counting Lattice Points

Chapter 3. Lattice Points and the Area of Polygons

Chapter 4. Lattice Points in Circles

Chapter 5. Minkowski’s Fundamental Theorem

Chapter 6. Applications of Minkowski’s Theorems

Chapter 7. Linear Transformations and Integral Lattices

Chapter 8. Geometric Interpretations of Quadratic Forms

Chapter 9. A New Principle in the Geometry of Numbers

Chapter 10. A Minkowski Theorem (Optional)

Appendix I. Gaussian Integers, by Peter D. Lax

Appendix II. The Closest Packing of Convex Bodies

Appendix III. Brief Biographies

Solutions and Hints

The interplay between geometric methods and apparently nongeometric questions is intriguing, and many of the proofs are very elegant. Much of the geometry of numbers can be explained at a relatively elementary level, but there are few books on the subject aimed at beginning students. This book is likely the most accessible treatment of this material ever written. It should play a valuable role in exposing bright high school students, or college math majors, to the geometry of numbers.
Henry Cohn, MAA Reviews