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Book Cover
E-book
Author Ash, Avner, 1949-

Title Summing it up : from one plus one to modern number theory / Avner Ash and Robert Gross
Published Princeton : Princeton University Press, [2016]
©20

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Description 1 online resource
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
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
Analysis Absolute value
Addition
Analytic continuation
Analytic function
Automorphic form
Axiom
Bernoulli number
Big O notation
Binomial coefficient
Binomial theorem
Book
Calculation
Chain rule
Coefficient
Complex analysis
Complex number
Complex plane
Computation
Congruence subgroup
Conjecture
Constant function
Constant term
Convergent series
Coprime integers
Counting
Cusp form
Determinant
Diagram (category theory)
Dirichlet series
Division by zero
Divisor
Elementary proof
Elliptic curve
Equation
Euclidean geometry
Existential quantification
Exponential function
Factorization
Fourier series
Function composition
Fundamental domain
Gaussian integer
Generating function
Geometric series
Geometry
Group theory
Hecke operator
Hexagonal number
Hyperbolic geometry
Integer factorization
Integer
Line segment
Linear combination
Logarithm
Mathematical induction
Mathematician
Mathematics
Matrix group
Modular form
Modular group
Natural number
Non-Euclidean geometry
Number theory
Parity (mathematics)
Pentagonal number
Periodic function
Polynomial
Power series
Prime factor
Prime number theorem
Prime number
Pythagorean theorem
Quadratic residue
Quantity
Radius of convergence
Rational number
Real number
Remainder
Riemann surface
Root of unity
Scientific notation
Semicircle
Series (mathematics)
Sign (mathematics)
Square number
Square root
Subgroup
Subset
Sum of squares
Summation
Taylor series
Theorem
Theory
Transfinite number
Triangular number
Two-dimensional space
Unique factorization domain
Upper half-plane
Variable (mathematics)
Vector space
Bibliography Includes bibliographical references and index
Notes In English
Print version record
Subject Mathematik.
Number theory.
Mathematics -- Popular works
MATHEMATICS -- Algebra -- Intermediate.
MATHEMATICS -- General.
Mathematics
Number theory
Addition
Zahl
Zahlentheorie
Philosophie
Genre/Form Electronic books
Popular works
Obras de divulgación.
Form Electronic book
Author Gross, Robert, 1959-
LC no. 2015037578
ISBN 9781400880539
140088053X