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The World of Whole Numbers: Number Theory for the Novice

The World of Whole Numbers: Number Theory for the Novice

Author(s): Agnes M Rash

Edition: 3

Copyright: 2023

Pages: 252

Edition: 3

Copyright: 2023

Pages: 282

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The World of Whole Numbers is an introduction to the field of Number Theory for students in non-math and non-science majors who have studied at least two years of high school algebra. Rather than giving brief introductions to a wide variety of topics, this book provides an in-depth introduction to the field of Number Theory. The topics covered include material in an introductory Number Theory course for mathematics majors, but the presentation is carefully tailored to meet the needs of elementary education, liberal arts, and other non-mathematical majors with a minimal mathematical background. The text covers logic and proofs, so that students may understand how to prove mathematical statements and to give counterexamples when statements are false.  The book contains an abundance of worked examples and exercises to both clearly illustrate concepts and evaluate the students’ mastery of the material. The text is enhanced with historic notes, biographical sketches and links to videos that reiterate what is presented in the text.

Testimonial
Preface to the Instructor
What’s New in the Third Edition
Preface to the Student
Acknowledgments
About the Author

Chapter 0 Introduction

0.1 Communication in Mathematics
0.2 Problem-Solving Strategies
0.3 Number Systems
0.4 Summary
0.5 Tidbits
0.6 Test Study Guide Notes and Important Concepts
0.7 Review Exercises are online in Ancillary Materials
0.8 Activities are online in Ancillary Materials

Chapter 1 Conjectures, Proofs and Counterexamples

1.1 What Is Number Theory?
1.2 Formal Definitions
1.3 Unsolved Problems and Unanswered Questions in Number Theory
1.4 Inductive and Deductive Reasoning
1.5 Statements and Connectives
1.6 Properties of the Integers
1.7 Rules of Logic and Direct Proofs
1.7.1 Symbolic Logic—Rules of Logic
1.7.2 General Properties of a Direct Proof
1.8 Interlude (Optional)
1.9 Indirect Proofs and Proofs by Contradiction
1.9.1 Indirect Proofs
1.9.2 Proof by Contradiction
1.9.3 A Conditional Statement and its Contrapositive are Logically Equivalent
1.10 Counterexamples—Proving a Statement is False
1.11 Divisors and the Greatest Common Divisor
1.12 Divisibility Rules
1.13 Summary
1.14 Tidbits
1.15 Test Study Guide Notes and Important Concepts
1.16 Review Exercises are online in Ancillary Materials
1.17 Activities are online in Ancillary Materials

Chapter 2 Pythagorean Triples

2.1 Right Triangles and the Pythagorean Theorem
2.2 Primitive Pythagorean Triples
2.3 The Proof of the Primitive Pythagorean Triples Formula
2.4 Summary
2.5 Tidbits
2.6 Test Study Guide Notes and Important Concepts
2.7 Review Exercises are online in Ancillary Materials
2.8 Activities are online in Ancillary Materials

Chapter 3 Prime Numbers and Unique Factorization

3.1 Why Are Prime Numbers Important?
3.2 The Unique Factorization Theorem
3.3 Proof of the Unique Factorization Theorem
3.4 The Search for Primes
3.5 Summary
3.6 Tidbits
3.7 Test Study Guide Notes and Important Concepts
3.8 Review Exercises are online in Ancillary Materials
3.9 Activities are online in Ancillary Materials

Chapter 4 The Euclidean Algorithm

4.1 The Division Algorithm
4.2 The Euclidean Algorithm
4.3 Solving Linear Equations in Two Variables and the Euclidean Algorithm Backwards
4.4 More about Solutions to ax + by = gcd(a, b)
4.5 What if ax + by ≠ gcd(a, b)?
4.6 Return to Primitive Pythagorean Triples (Optional)
4.7 Summary
4.8 Tidbits
4.9 Test Study Guide Notes and Important Concepts
4.10 Review Exercises are online in Ancillary Materials
4.11 Activities are online in Ancillary Materials

Chapter 5 Congruences

5.1 Introduction to Congruence
5.2 Congruences Versus Linear Equations
5.3 Solving Linear Congruences
5.4 An Application of Congruences—Identification Numbers and Check Digits
5.4.1 Universal Product Codes
5.4.2 ISBN Numbers
5.4.3 Bank Checks
5.5 The Chinese Remainder Theorem
5.6 Summary
5.7 Tidbit
5.8 Test Study Guide Notes and Important Concepts
5.9 Review Exercises are online in Ancillary Materials
5.10 Activities are online in Ancillary Materials

Chapter 6 Special Congruences and Numerical Functions

6.1 Introduction
6.2 Wilson’s Theorem
6.3 Fermat’s Little Theorem
6.4 Euler’s Function
6.5 Euler’s Theorem
6.6 More Numerical Functions
6.7 Summary
6.8 Tidbit
6.9 Test Study Guide Notes and Important Concepts
6.10 Review Exercises are online in Ancillary Materials
6.11 Activities are online in Ancillary Materials

Chapter 7 Cryptography

7.1 Introduction
7.2 Private Key Cryptography
7.2.1 Substitution Cipher
7.2.2 Caesar Cipher
7.2.3 Vigenère Cipher
7.3 Encryption by Exponentiation
7.4 Public Key Cryptography—the RSA Cryptosystem
7.4.1 RSA Encryption
7.4.2 RSA Decryption
7.4.3 Why RSA Encryption Works
7.5 Summary
7.6 Tidbit
7.7 Test Study Guide Notes and Important Concepts
7.8 Review Exercises are online in Ancillary Materials
7.9 Activities are online in Ancillary Materials
 
Bibliography
Epilogue
Index
Agnes M Rash

Inspired teaching of General Education courses is a challenging task due to a student audience that often is not interested in the subject matter. Similarly, high school mathematics teachers lack enrichment courses other than calculus. The World of Whole Numbers - Number Theory for the Novice is the book that evolved from the extensive practice and research of Dr. Agnes M. Rash. Its latest edition is a wonderful treatment of the Number Theory, with a broad range of innovative problems and mathematical proofs for nonmathematics majors or advanced high schoolers.

The book begins with a presentation of mathematical problem-solving strategies and methods of communicating mathematical ideas. It is a distinguishing feature for a typical mathematics textbook that allows students to learn how to study mathematics.

As an introductory Number Theory text, the book contains various types of learning tasks including puzzles, projects, real-life applications, and mathematical proofs. Each chapter contains numerous solved examples, proven statements, and practice exercises of varying degrees of difficulty. Also, each chapter concludes with a list of new terms and symbols, followed by review exercises and activities.

Humanizing mathematics is one of the methods for motivating students’ learning of the subject. Dr. Rash has included numerous historical facts and engaging stories, with pictures to illustrate the evolution of human thought and power of its knowledge.

The World of Whole Numbers - Number Theory for the Novice is an opportunity to foster important development of mathematical and analytical thinking intended for a novice audience. The book will introduce the student to some of the genesis of mathematical algorithms including Euclidean, the Sieve of Eratosthenes, generation of primes and many others.

The instructor teaching from this book will find it an invaluable course book for capstone problem-solving or first-year seminar courses, enrichment courses, and a springboard for a number of interesting number theory projects, term papers, or honors studies.

Dr. Tetyana Berezovski
Saint Joseph’s University
Mathematics Department
2019

The World of Whole Numbers is an introduction to the field of Number Theory for students in non-math and non-science majors who have studied at least two years of high school algebra. Rather than giving brief introductions to a wide variety of topics, this book provides an in-depth introduction to the field of Number Theory. The topics covered include material in an introductory Number Theory course for mathematics majors, but the presentation is carefully tailored to meet the needs of elementary education, liberal arts, and other non-mathematical majors with a minimal mathematical background. The text covers logic and proofs, so that students may understand how to prove mathematical statements and to give counterexamples when statements are false.  The book contains an abundance of worked examples and exercises to both clearly illustrate concepts and evaluate the students’ mastery of the material. The text is enhanced with historic notes, biographical sketches and links to videos that reiterate what is presented in the text.

Testimonial
Preface to the Instructor
What’s New in the Third Edition
Preface to the Student
Acknowledgments
About the Author

Chapter 0 Introduction

0.1 Communication in Mathematics
0.2 Problem-Solving Strategies
0.3 Number Systems
0.4 Summary
0.5 Tidbits
0.6 Test Study Guide Notes and Important Concepts
0.7 Review Exercises are online in Ancillary Materials
0.8 Activities are online in Ancillary Materials

Chapter 1 Conjectures, Proofs and Counterexamples

1.1 What Is Number Theory?
1.2 Formal Definitions
1.3 Unsolved Problems and Unanswered Questions in Number Theory
1.4 Inductive and Deductive Reasoning
1.5 Statements and Connectives
1.6 Properties of the Integers
1.7 Rules of Logic and Direct Proofs
1.7.1 Symbolic Logic—Rules of Logic
1.7.2 General Properties of a Direct Proof
1.8 Interlude (Optional)
1.9 Indirect Proofs and Proofs by Contradiction
1.9.1 Indirect Proofs
1.9.2 Proof by Contradiction
1.9.3 A Conditional Statement and its Contrapositive are Logically Equivalent
1.10 Counterexamples—Proving a Statement is False
1.11 Divisors and the Greatest Common Divisor
1.12 Divisibility Rules
1.13 Summary
1.14 Tidbits
1.15 Test Study Guide Notes and Important Concepts
1.16 Review Exercises are online in Ancillary Materials
1.17 Activities are online in Ancillary Materials

Chapter 2 Pythagorean Triples

2.1 Right Triangles and the Pythagorean Theorem
2.2 Primitive Pythagorean Triples
2.3 The Proof of the Primitive Pythagorean Triples Formula
2.4 Summary
2.5 Tidbits
2.6 Test Study Guide Notes and Important Concepts
2.7 Review Exercises are online in Ancillary Materials
2.8 Activities are online in Ancillary Materials

Chapter 3 Prime Numbers and Unique Factorization

3.1 Why Are Prime Numbers Important?
3.2 The Unique Factorization Theorem
3.3 Proof of the Unique Factorization Theorem
3.4 The Search for Primes
3.5 Summary
3.6 Tidbits
3.7 Test Study Guide Notes and Important Concepts
3.8 Review Exercises are online in Ancillary Materials
3.9 Activities are online in Ancillary Materials

Chapter 4 The Euclidean Algorithm

4.1 The Division Algorithm
4.2 The Euclidean Algorithm
4.3 Solving Linear Equations in Two Variables and the Euclidean Algorithm Backwards
4.4 More about Solutions to ax + by = gcd(a, b)
4.5 What if ax + by ≠ gcd(a, b)?
4.6 Return to Primitive Pythagorean Triples (Optional)
4.7 Summary
4.8 Tidbits
4.9 Test Study Guide Notes and Important Concepts
4.10 Review Exercises are online in Ancillary Materials
4.11 Activities are online in Ancillary Materials

Chapter 5 Congruences

5.1 Introduction to Congruence
5.2 Congruences Versus Linear Equations
5.3 Solving Linear Congruences
5.4 An Application of Congruences—Identification Numbers and Check Digits
5.4.1 Universal Product Codes
5.4.2 ISBN Numbers
5.4.3 Bank Checks
5.5 The Chinese Remainder Theorem
5.6 Summary
5.7 Tidbit
5.8 Test Study Guide Notes and Important Concepts
5.9 Review Exercises are online in Ancillary Materials
5.10 Activities are online in Ancillary Materials

Chapter 6 Special Congruences and Numerical Functions

6.1 Introduction
6.2 Wilson’s Theorem
6.3 Fermat’s Little Theorem
6.4 Euler’s Function
6.5 Euler’s Theorem
6.6 More Numerical Functions
6.7 Summary
6.8 Tidbit
6.9 Test Study Guide Notes and Important Concepts
6.10 Review Exercises are online in Ancillary Materials
6.11 Activities are online in Ancillary Materials

Chapter 7 Cryptography

7.1 Introduction
7.2 Private Key Cryptography
7.2.1 Substitution Cipher
7.2.2 Caesar Cipher
7.2.3 Vigenère Cipher
7.3 Encryption by Exponentiation
7.4 Public Key Cryptography—the RSA Cryptosystem
7.4.1 RSA Encryption
7.4.2 RSA Decryption
7.4.3 Why RSA Encryption Works
7.5 Summary
7.6 Tidbit
7.7 Test Study Guide Notes and Important Concepts
7.8 Review Exercises are online in Ancillary Materials
7.9 Activities are online in Ancillary Materials
 
Bibliography
Epilogue
Index

Agnes M Rash
Agnes M Rash

Inspired teaching of General Education courses is a challenging task due to a student audience that often is not interested in the subject matter. Similarly, high school mathematics teachers lack enrichment courses other than calculus. The World of Whole Numbers - Number Theory for the Novice is the book that evolved from the extensive practice and research of Dr. Agnes M. Rash. Its latest edition is a wonderful treatment of the Number Theory, with a broad range of innovative problems and mathematical proofs for nonmathematics majors or advanced high schoolers.

The book begins with a presentation of mathematical problem-solving strategies and methods of communicating mathematical ideas. It is a distinguishing feature for a typical mathematics textbook that allows students to learn how to study mathematics.

As an introductory Number Theory text, the book contains various types of learning tasks including puzzles, projects, real-life applications, and mathematical proofs. Each chapter contains numerous solved examples, proven statements, and practice exercises of varying degrees of difficulty. Also, each chapter concludes with a list of new terms and symbols, followed by review exercises and activities.

Humanizing mathematics is one of the methods for motivating students’ learning of the subject. Dr. Rash has included numerous historical facts and engaging stories, with pictures to illustrate the evolution of human thought and power of its knowledge.

The World of Whole Numbers - Number Theory for the Novice is an opportunity to foster important development of mathematical and analytical thinking intended for a novice audience. The book will introduce the student to some of the genesis of mathematical algorithms including Euclidean, the Sieve of Eratosthenes, generation of primes and many others.

The instructor teaching from this book will find it an invaluable course book for capstone problem-solving or first-year seminar courses, enrichment courses, and a springboard for a number of interesting number theory projects, term papers, or honors studies.

Dr. Tetyana Berezovski
Saint Joseph’s University
Mathematics Department
2019