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Summing it Up: From One Plus One to Modern Number Theory

9780691178516
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We use addition on a daily basis—yet how many of us stop to truly consider the enormous and remarkable ramifications of this mathematical activity? Summing It Up uses addition as a springboard to present a fascinating and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition's most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research.

Ash and Gross tailor their succinct and engaging investigations for math enthusiasts of all backgrounds. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series—long sums that can only be defined by the concept of limit, as in the example of 1+1/2+1/4+. . .=? With the help of some group theory and geometry, the third section ties together the first two parts of the book through a discussion of modular forms—the analytic functions on the upper half-plane of the complex numbers that have growth and transformation properties. Ash and Gross show how modular forms are indispensable in modern number theory, for example in the proof of Fermat's Last Theorem.

Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.

Category:
Science
Cover:
Paperback
Edition Number:
1
ISBN:
9780691178516
Pages:
248
Author:
Ash Avner
Publisher:
PRINCETON UNIVERSITY PRESS
Release Year:
2018

Avner Ash is professor of mathematics at Boston College.

Robert Gross is associate professor of mathematics at Boston College.

They are the coauthors of Elliptic Tales: Curves, Counting, and Number Theory and Fearless Symmetry: Exposing the Hidden Patterns of Numbers (both Princeton).

  • Frontmatter, pg. i
  • CONTENTS, pg. vii
  • PREFACE, pg. xi
  • ACKNOWLEDGMENTS, pg. xv
  • INTRODUCTION: WHAT THIS BOOK IS ABOUT, pg. 1
  • CHAPTER 1. PROEM, pg. 11
  • CHAPTER 2. SUMS OF TWO SQUARES, pg. 22
  • CHAPTER 3. SUMS OF THREE AND FOUR SQUARES, pg. 32
  • CHAPTER 4. SUMS OF HIGHER POWERS: WARING’S PROBLEM, pg. 37
  • CHAPTER 5. SIMPLE SUMS, pg. 42
  • CHAPTER 6. SUMS OF POWERS, USING LOTS OF ALGEBRA, pg. 50
  • CHAPTER 7. INFINITE SERIES, pg. 73
  • CHAPTER 8. CAST OF CHARACTERS, pg. 96
  • CHAPTER 9. ZETA AND BERNOULLI, pg. 103
  • CHAPTER 10. COUNT THE WAYS, pg. 110
  • CHAPTER 11. THE UPPER HALF-PLANE, pg. 127
  • CHAPTER 12. MODULAR FORMS, pg. 147
  • CHAPTER 13. HOW MANY MODULAR FORMS ARE THERE?, pg. 160
  • CHAPTER 14. CONGRUENCE GROUPS, pg. 179
  • CHAPTER 15. PARTITIONS AND SUMS OF SQUARES REVISITED, pg. 186
  • CHAPTER 16. MORE THEORY OF MODULAR FORMS, pg. 201
  • CHAPTER 17. MORE THINGS TO DO WITH MODULAR FORMS: APPLICATIONS, pg. 213
  • BIBLIOGRAPHY, pg. 225
  • INDEX, pg. 227
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