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.
By:
Avner Ash,
Robert Gross
Imprint: Princeton University Pres
Country of Publication: United States
Dimensions:
Height: 235mm,
Width: 152mm,
Weight: 312g
ISBN: 9780691178516
ISBN 10: 0691178518
Pages: 248
Publication Date: 09 April 2018
Audience:
College/higher education
,
Primary
Format: Paperback
Publisher's Status: Active
Frontmatter, pg. iCONTENTS, pg. viiPREFACE, pg. xiACKNOWLEDGMENTS, pg. xvINTRODUCTION: WHAT THIS BOOK IS ABOUT, pg. 1CHAPTER 1. PROEM, pg. 11CHAPTER 2. SUMS OF TWO SQUARES, pg. 22CHAPTER 3. SUMS OF THREE AND FOUR SQUARES, pg. 32CHAPTER 4. SUMS OF HIGHER POWERS: WARING'S PROBLEM, pg. 37CHAPTER 5. SIMPLE SUMS, pg. 42CHAPTER 6. SUMS OF POWERS, USING LOTS OF ALGEBRA, pg. 50CHAPTER 7. INFINITE SERIES, pg. 73CHAPTER 8. CAST OF CHARACTERS, pg. 96CHAPTER 9. ZETA AND BERNOULLI, pg. 103CHAPTER 10. COUNT THE WAYS, pg. 110CHAPTER 11. THE UPPER HALF-PLANE, pg. 127CHAPTER 12. MODULAR FORMS, pg. 147CHAPTER 13. HOW MANY MODULAR FORMS ARE THERE?, pg. 160CHAPTER 14. CONGRUENCE GROUPS, pg. 179CHAPTER 15. PARTITIONS AND SUMS OF SQUARES REVISITED, pg. 186CHAPTER 16. MORE THEORY OF MODULAR FORMS, pg. 201CHAPTER 17. MORE THINGS TO DO WITH MODULAR FORMS: APPLICATIONS, pg. 213BIBLIOGRAPHY, pg. 225INDEX, pg. 227
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).
Reviews for Summing It Up: From One Plus One to Modern Number Theory
"""Offers a clear and beautiful progression from addition to modern number theory.""--Math-Blog ""The authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book.""--Adhemar Bultheel, European Mathematical Society ""Ash and Gross do a masterful job of leading students from finite sums to modular forms and to the forefront of modern number theory... This is an excellent piece of mathematical writing.""--Choice ""[A]n accessible and fun introduction to modular forms... [Summing It Up] is engaging and conversational, without losing accuracy or essential rigor.""--Dominic Lanphier, American Mathematical Monthly"
