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Real Analysis: A Constructive Approach Through Interval Arithmetic
 
Mark Bridger Northeastern University, Boston, MA
Real Analysis
Hardcover ISBN:  978-1-4704-5144-8
Product Code:  AMSTEXT/38
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5289-6
Product Code:  AMSTEXT/38.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-5144-8
eBook: ISBN:  978-1-4704-5289-6
Product Code:  AMSTEXT/38.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
Real Analysis
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Real Analysis: A Constructive Approach Through Interval Arithmetic
Mark Bridger Northeastern University, Boston, MA
Hardcover ISBN:  978-1-4704-5144-8
Product Code:  AMSTEXT/38
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-5289-6
Product Code:  AMSTEXT/38.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-5144-8
eBook ISBN:  978-1-4704-5289-6
Product Code:  AMSTEXT/38.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 382019; 302 pp
    MSC: Primary 03; 97; 26;

    Real Analysis: A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this book: All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering.

    The first part of the book contains all the usual material in a standard one-semester course in analysis of functions of a single real variable: continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material—including an introduction to complex variables and Fourier series—to fill out a full-year course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects—many with hints—at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout.

    Readership

    Undergraduate and graduate students interested in real analysis, constructivism, and logic.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Contents
    • Preface
    • Acknowledgments
    • Introduction
    • Chapter 0. Preliminaries
    • 0.1. The Natural Numbers
    • 0.2. The Rationals
    • Chapter 1. The Real Numbers and Completeness
    • 1.0. Introduction
    • 1.1. Interval Arithmetic
    • 1.2. Families of Intersecting Intervals
    • 1.3. Fine Families
    • 1.4. Definition of the Reals
    • 1.5. Real Number Arithmetic
    • 1.6. Rational Approximations
    • 1.7. Real Intervals and Completeness
    • 1.8. Limits and Limiting Families
    • Appendix: The Goldbach Number and Trichotomy
    • Chapter 2. An Inverse Function Theorem and Its Applications
    • 2.0. Introduction
    • 2.1. Functions and Inverses
    • 2.2. An Inverse Function Theorem
    • 2.3. The Exponential Function
    • 2.4. Natural Logs and the Euler Number e
    • Chapter 3. Limits, Sequences, and Series
    • 3.1. Sequences and Convergence
    • 3.2. Limits of Functions
    • 3.3. Series of Numbers
    • Appendix I: Some Properties of Exp and Log
    • Appendix II: Rearrangements of Series
    • Chapter 4. Uniform Continuity
    • 4.1. Definitions and Elementary Properties
    • 4.2. Limits and Extensions
    • Appendix I: Are There Non-Continuous Functions?
    • Appendix II: Continuity of Double-Sided Inverses
    • Appendix III: The Goldbach Function
    • Chapter 5. The Riemann Integral
    • 5.1. Definition and Existence
    • 5.2. Elementary Properties
    • 5.3. Extensions and Improper Integrals
    • Chapter 6. Differentiation
    • 6.1. Definitions and Basic Properties
    • 6.2. The Arithmetic of Differentiability
    • 6.3. Two Important Theorems
    • 6.4. Derivative Tools
    • 6.5. Integral Tools
    • Chapter 7. Sequences and Series of Functions
    • 7.1. Sequences of Functions
    • 7.2. Integrals and Derivatives of Sequences
    • 7.3. Power Series
    • 7.4. Taylor Series
    • 7.5. The Periodic Functions
    • Appendix: Raabe’s Test and Binomial Series
    • Chapter 8. The Complex Numbers and Fourier Series
    • 8.0. Introduction
    • 8.1. The Complex Numbers ℂ
    • 8.2. Complex Functions and Vectors
    • 8.3. Fourier Series Theory
    • References
    • Index
  • Reviews
     
     
    • Here is another new textbook for undergraduate analysis, but it's far from a traditional approach. This one is dedicated to being entirely constructive. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.

      Bill Satzer, MAA Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 382019; 302 pp
MSC: Primary 03; 97; 26;

Real Analysis: A Constructive Approach Through Interval Arithmetic presents a careful treatment of calculus and its theoretical underpinnings from the constructivist point of view. This leads to an important and unique feature of this book: All existence proofs are direct, so showing that the numbers or functions in question exist means exactly that they can be explicitly calculated. For example, at the very beginning, the real numbers are shown to exist because they are constructed from the rationals using interval arithmetic. This approach, with its clear analogy to scientific measurement with tolerances, is taken throughout the book and makes the subject especially relevant and appealing to students with an interest in computing, applied mathematics, the sciences, and engineering.

The first part of the book contains all the usual material in a standard one-semester course in analysis of functions of a single real variable: continuity (uniform, not pointwise), derivatives, integrals, and convergence. The second part contains enough more technical material—including an introduction to complex variables and Fourier series—to fill out a full-year course. Throughout the book the emphasis on rigorous and direct proofs is supported by an abundance of examples, exercises, and projects—many with hints—at the end of every section. The exposition is informal but exceptionally clear and well motivated throughout.

Readership

Undergraduate and graduate students interested in real analysis, constructivism, and logic.

  • Cover
  • Title page
  • Contents
  • Preface
  • Acknowledgments
  • Introduction
  • Chapter 0. Preliminaries
  • 0.1. The Natural Numbers
  • 0.2. The Rationals
  • Chapter 1. The Real Numbers and Completeness
  • 1.0. Introduction
  • 1.1. Interval Arithmetic
  • 1.2. Families of Intersecting Intervals
  • 1.3. Fine Families
  • 1.4. Definition of the Reals
  • 1.5. Real Number Arithmetic
  • 1.6. Rational Approximations
  • 1.7. Real Intervals and Completeness
  • 1.8. Limits and Limiting Families
  • Appendix: The Goldbach Number and Trichotomy
  • Chapter 2. An Inverse Function Theorem and Its Applications
  • 2.0. Introduction
  • 2.1. Functions and Inverses
  • 2.2. An Inverse Function Theorem
  • 2.3. The Exponential Function
  • 2.4. Natural Logs and the Euler Number e
  • Chapter 3. Limits, Sequences, and Series
  • 3.1. Sequences and Convergence
  • 3.2. Limits of Functions
  • 3.3. Series of Numbers
  • Appendix I: Some Properties of Exp and Log
  • Appendix II: Rearrangements of Series
  • Chapter 4. Uniform Continuity
  • 4.1. Definitions and Elementary Properties
  • 4.2. Limits and Extensions
  • Appendix I: Are There Non-Continuous Functions?
  • Appendix II: Continuity of Double-Sided Inverses
  • Appendix III: The Goldbach Function
  • Chapter 5. The Riemann Integral
  • 5.1. Definition and Existence
  • 5.2. Elementary Properties
  • 5.3. Extensions and Improper Integrals
  • Chapter 6. Differentiation
  • 6.1. Definitions and Basic Properties
  • 6.2. The Arithmetic of Differentiability
  • 6.3. Two Important Theorems
  • 6.4. Derivative Tools
  • 6.5. Integral Tools
  • Chapter 7. Sequences and Series of Functions
  • 7.1. Sequences of Functions
  • 7.2. Integrals and Derivatives of Sequences
  • 7.3. Power Series
  • 7.4. Taylor Series
  • 7.5. The Periodic Functions
  • Appendix: Raabe’s Test and Binomial Series
  • Chapter 8. The Complex Numbers and Fourier Series
  • 8.0. Introduction
  • 8.1. The Complex Numbers ℂ
  • 8.2. Complex Functions and Vectors
  • 8.3. Fourier Series Theory
  • References
  • Index
  • Here is another new textbook for undergraduate analysis, but it's far from a traditional approach. This one is dedicated to being entirely constructive. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.

    Bill Satzer, MAA Reviews
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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