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  1. Home
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  3. The Geometry of Numbers
  • This book is no longer available to purchase from Cambridge Core
  • Cited by 28
Publisher:
Mathematical Association of America
Online publication date:
January 2012
Print publication year:
2000
Online ISBN:
9780883859551
DOI:
https://doi.org/10.5948/UPO9780883859551
Subjects:
Mathematics (general), Mathematics
Series:
Anneli Lax New Mathematical Library
Information

Book description

The Geometry of Numbers presents a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice-points 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.

Reviews

The Geometry of Numbers is a very well-written expository book. It is well-paced and enjoyable to read. The material is interesting, well-chosen, and presented at a level appropriate for high school students, undergraduates and teachers at all levels.For anyone who has, or might have an interest in the geometry of numbers, this is a great book.

John G. McLoughlin Source: Crux Mathematicorum

I found the book to be fascinating, in that shifting to the geometric approach makes so many of the hard problems much easier. One of the most difficult things to teach students who are just beginning to be exposed to proofs is that there are many ways to verify results. They just seem to get stuck on one approach and cannot see that there may be another, much simpler strategy. The examples in this book demonstrate cases where a problem appears hard, but in fact is fairly easy if geometric representations are used. This would make it very valuable as a tool to help build confidence and breadth of knowledge in students.

Charles Ashbacher Source: Journal of Recreational Mathematics

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