Summing it up : from one plus one to modern number theory / Robert Gross, Avner Ash.

Format:

Book

Author/Creator:

Ash, Avner, author.

Gross, Robert, author.

Language:

English

Subjects (All):

Number theory.

Mathematics--Popular works.

Mathematics.

Physical Description:

1 online resource (249 p.)

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

English

System Details:

Mode of access: World Wide Web.

Summary:

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.

Contents:

Frontmatter

CONTENTS

PREFACE

ACKNOWLEDGMENTS

INTRODUCTION: WHAT THIS BOOK IS ABOUT

PART ONE. FINITE SUMS

CHAPTER 1. PROEM

CHAPTER 2. SUMS OF TWO SQUARES

CHAPTER 3. SUMS OF THREE AND FOUR SQUARES

CHAPTER 4. SUMS OF HIGHER POWERS: WARING'S PROBLEM

CHAPTER 5. SIMPLE SUMS

CHAPTER 6. SUMS OF POWERS, USING LOTS OF ALGEBRA

PART TWO. INFINITE SUMS

CHAPTER 7. INFINITE SERIES

CHAPTER 8. CAST OF CHARACTERS

CHAPTER 9. ZETA AND BERNOULLI

CHAPTER 10. COUNT THE WAYS

PART III. MODULAR FORMS AND THEIR APPLICATIONS

CHAPTER 11. THE UPPER HALF-PLANE

CHAPTER 12. MODULAR FORMS

CHAPTER 13. HOW MANY MODULAR FORMS ARE THERE?

CHAPTER 14. CONGRUENCE GROUPS

CHAPTER 15. PARTITIONS AND SUMS OF SQUARES REVISITED

CHAPTER 16. MORE THEORY OF MODULAR FORMS

CHAPTER 17. MORE THINGS TO DO WITH MODULAR FORMS: APPLICATIONS

BIBLIOGRAPHY

INDEX

Notes:

Description based upon print version of record

Includes bibliographical references and index

Description based on online resource; title from PDF title page (publisher's Web site, viewed 23. Mai 2019)

ISBN:

9781400880539

140088053X

OCLC:

946158222