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A Bridge to Advanced Mathematics: From Natural to Complex Numbers
 
Sebastian M. Cioabă University of Delaware, Newark, DE
Werner Linde Friedrich-Schiller University Jena, Jena, Germany
Softcover ISBN:  978-1-4704-7148-4
Product Code:  AMSTEXT/58
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7213-9
Product Code:  AMSTEXT/58.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7148-4
eBook: ISBN:  978-1-4704-7213-9
Product Code:  AMSTEXT/58.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
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A Bridge to Advanced Mathematics: From Natural to Complex Numbers
Sebastian M. Cioabă University of Delaware, Newark, DE
Werner Linde Friedrich-Schiller University Jena, Jena, Germany
Softcover ISBN:  978-1-4704-7148-4
Product Code:  AMSTEXT/58
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7213-9
Product Code:  AMSTEXT/58.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7148-4
eBook ISBN:  978-1-4704-7213-9
Product Code:  AMSTEXT/58.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: 582023; 525 pp
    MSC: Primary 00; 05

    Most introduction to proofs textbooks focus on the structure of rigorous mathematical language and only use mathematical topics incidentally as illustrations and exercises. In contrast, this book gives students practice in proof writing while simultaneously providing a rigorous introduction to number systems and their properties. Understanding the properties of these systems is necessary throughout higher mathematics. The book is an ideal introduction to mathematical reasoning and proof techniques, building on familiar content to ensure comprehension of more advanced topics in abstract algebra and real analysis with over 700 exercises as well as many examples throughout. Readers will learn and practice writing proofs related to new abstract concepts while learning new mathematical content. The first task is analogous to practicing soccer while the second is akin to playing soccer in a real match. The authors believe that all students should practice and play mathematics.

    The book is written for students who already have some familiarity with formal proof writing but would like to have some extra preparation before taking higher mathematics courses like abstract algebra and real analysis.

    Ancillaries:

    Readership

    Undergraduate students interested in an introduction to proofs in context, topics based introduction to proofs, and a bridge course to algebra and analysis.

  • Table of Contents
     
     
    • Cover
    • Title page
    • Copyright
    • Contents
    • Preface
    • 1. The Content of the Book
    • 2. How to Use This Book?
    • Chapter 1. Natural Numbers N
    • 1.1. Basic Properties
    • 1.2. The Principle of Induction
    • 1.3. Arithmetic and Geometric Progressions
    • 1.4. The Least Element Principle
    • 1.5. There are 10 Kinds of People in the World
    • 1.6. Divisibility
    • 1.7. Counting and Binomial Formula
    • 1.8. More Exercises
    • Chapter 2. Integer Numbers Z
    • 2.1. Basic Properties
    • 2.2. Integer Division
    • 2.3. Euclidean Algorithm Revisited
    • 2.4. Congruences and Modular Arithmetic
    • 2.5. Modular Equations
    • 2.6. The Chinese Remainder Theorem
    • 2.7. Fermat and Euler Theorems
    • 2.8. More Exercises
    • Chapter 3. Rational Numbers Q
    • 3.1. Basic Properties
    • 3.2. Not Everything Is Rational
    • 3.3. Fractions and Decimal Representations
    • 3.4. Finite Continued Fractions
    • 3.5. Farey Sequences and Pick’s Formula
    • 3.6. Ford Circles and Stern–Brocot Trees
    • 3.7. Egyptian Fractions
    • 3.8. More Exercises
    • Chapter 4. Real Numbers R
    • 4.1. Basic Properties
    • 4.2. The Real Numbers Form a Field
    • 4.3. Order and Absolute Value
    • 4.4. Completeness
    • 4.5. Supremum and Infimum of a Set
    • 4.6. Roots and Powers
    • 4.7. Expansion of Real Numbers
    • 4.8. More Exercises
    • Chapter 5. Sequences of Real Numbers
    • 5.1. Basic Properties
    • 5.2. Convergent and Divergent Sequences
    • 5.3. The Monotone Convergence Theorem and Its Applications
    • 5.4. Subsequences
    • 5.5. Cauchy Sequences
    • 5.6. Infinite Series
    • 5.7. Infinite Continued Fractions
    • 5.8. More Exercises
    • Chapter 6. Complex Numbers C
    • 6.1. Basic Properties
    • 6.2. The Conjugate and the Absolute Value
    • 6.3. Polar Representation of Complex Numbers
    • 6.4. Roots of Complex Numbers
    • 6.5. Geometric Applications
    • 6.6. Sequences of Complex Numbers
    • 6.7. Infinite Series of Complex Numbers
    • 6.8. More Exercises
    • Epilogue
    • Appendix. Sets, Functions, and Relations
    • A.1. Logic
    • A.2. Sets
    • A.3. Functions
    • A.4. Cardinality of Sets
    • A.5. Relations
    • A.6. Proofs
    • A.7. Peano’s Axioms and the Construction of Integers
    • A.8. More Exercises
    • Bibliography
    • Index
    • Back Cover
  • Reviews
     
     
    • It aims to ease the transition from primarily calculus-based mathematics courses to more conceptually advanced proof-based courses. As such, it is intended for early undergraduate students who wish to become familiar with the language, fundamental knowledge and methods of abstract mathematics. Although this is a niche market with a lot of competition, the authors — Sebastian M. Cioaba and Werner Linde — have nevertheless come up with a relevant and unique proposal.

      Frederic Morneau-Guérin (Université TELUQ), MAA Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 582023; 525 pp
MSC: Primary 00; 05

Most introduction to proofs textbooks focus on the structure of rigorous mathematical language and only use mathematical topics incidentally as illustrations and exercises. In contrast, this book gives students practice in proof writing while simultaneously providing a rigorous introduction to number systems and their properties. Understanding the properties of these systems is necessary throughout higher mathematics. The book is an ideal introduction to mathematical reasoning and proof techniques, building on familiar content to ensure comprehension of more advanced topics in abstract algebra and real analysis with over 700 exercises as well as many examples throughout. Readers will learn and practice writing proofs related to new abstract concepts while learning new mathematical content. The first task is analogous to practicing soccer while the second is akin to playing soccer in a real match. The authors believe that all students should practice and play mathematics.

The book is written for students who already have some familiarity with formal proof writing but would like to have some extra preparation before taking higher mathematics courses like abstract algebra and real analysis.

Ancillaries:

Readership

Undergraduate students interested in an introduction to proofs in context, topics based introduction to proofs, and a bridge course to algebra and analysis.

  • Cover
  • Title page
  • Copyright
  • Contents
  • Preface
  • 1. The Content of the Book
  • 2. How to Use This Book?
  • Chapter 1. Natural Numbers N
  • 1.1. Basic Properties
  • 1.2. The Principle of Induction
  • 1.3. Arithmetic and Geometric Progressions
  • 1.4. The Least Element Principle
  • 1.5. There are 10 Kinds of People in the World
  • 1.6. Divisibility
  • 1.7. Counting and Binomial Formula
  • 1.8. More Exercises
  • Chapter 2. Integer Numbers Z
  • 2.1. Basic Properties
  • 2.2. Integer Division
  • 2.3. Euclidean Algorithm Revisited
  • 2.4. Congruences and Modular Arithmetic
  • 2.5. Modular Equations
  • 2.6. The Chinese Remainder Theorem
  • 2.7. Fermat and Euler Theorems
  • 2.8. More Exercises
  • Chapter 3. Rational Numbers Q
  • 3.1. Basic Properties
  • 3.2. Not Everything Is Rational
  • 3.3. Fractions and Decimal Representations
  • 3.4. Finite Continued Fractions
  • 3.5. Farey Sequences and Pick’s Formula
  • 3.6. Ford Circles and Stern–Brocot Trees
  • 3.7. Egyptian Fractions
  • 3.8. More Exercises
  • Chapter 4. Real Numbers R
  • 4.1. Basic Properties
  • 4.2. The Real Numbers Form a Field
  • 4.3. Order and Absolute Value
  • 4.4. Completeness
  • 4.5. Supremum and Infimum of a Set
  • 4.6. Roots and Powers
  • 4.7. Expansion of Real Numbers
  • 4.8. More Exercises
  • Chapter 5. Sequences of Real Numbers
  • 5.1. Basic Properties
  • 5.2. Convergent and Divergent Sequences
  • 5.3. The Monotone Convergence Theorem and Its Applications
  • 5.4. Subsequences
  • 5.5. Cauchy Sequences
  • 5.6. Infinite Series
  • 5.7. Infinite Continued Fractions
  • 5.8. More Exercises
  • Chapter 6. Complex Numbers C
  • 6.1. Basic Properties
  • 6.2. The Conjugate and the Absolute Value
  • 6.3. Polar Representation of Complex Numbers
  • 6.4. Roots of Complex Numbers
  • 6.5. Geometric Applications
  • 6.6. Sequences of Complex Numbers
  • 6.7. Infinite Series of Complex Numbers
  • 6.8. More Exercises
  • Epilogue
  • Appendix. Sets, Functions, and Relations
  • A.1. Logic
  • A.2. Sets
  • A.3. Functions
  • A.4. Cardinality of Sets
  • A.5. Relations
  • A.6. Proofs
  • A.7. Peano’s Axioms and the Construction of Integers
  • A.8. More Exercises
  • Bibliography
  • Index
  • Back Cover
  • It aims to ease the transition from primarily calculus-based mathematics courses to more conceptually advanced proof-based courses. As such, it is intended for early undergraduate students who wish to become familiar with the language, fundamental knowledge and methods of abstract mathematics. Although this is a niche market with a lot of competition, the authors — Sebastian M. Cioaba and Werner Linde — have nevertheless come up with a relevant and unique proposal.

    Frederic Morneau-Guérin (Université TELUQ), MAA Reviews
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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