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Softcover ISBN:  9781470471484 
Product Code:  AMSTEXT/58 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Sale Price:  $57.85 
eBook ISBN:  9781470472139 
Product Code:  AMSTEXT/58.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Sale Price:  $55.25 
Softcover ISBN:  9781470471484 
eBook ISBN:  9781470472139 
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 
Sale Price:  $113.10 $85.48 

Book DetailsPure and Applied Undergraduate TextsVolume: 58; 2023; 525 ppMSC: 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:
ReadershipUndergraduate 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


Additional Material

Reviews

It aims to ease the transition from primarily calculusbased mathematics courses to more conceptually advanced proofbased 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 MorneauGuérin (Université TELUQ), MAA Reviews


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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:
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 calculusbased mathematics courses to more conceptually advanced proofbased 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 MorneauGuérin (Université TELUQ), MAA Reviews