The Number Line through Guided Inquiry is designed to give future secondary teachers a deep understanding of the real numbers and functions on the reals. By presenting just that part of the subject that underlies the high school curriculum, this book offers an alternative to a standard real analysis sequence for advanced undergraduate or beginning graduate students. It will give any student a much deeper understanding of the mathematics that they were taught in high school. Written in a guided-inquiry format, this book consists of a carefully scaffolded sequence of definitions, problems, and theorems that guides students through each topic. Readers solve the problems and prove the theorems on their own and present their results to their peers with the instructor as a mentor and a guide. Students will learn not only the mathematics, but also how to help others learn mathematics. They will learn to think creatively and to make compelling arguments to justify their conclusions. They will learn to listen critically to others and give constructive feedback. Ultimately, they will learn to work as a team to answer the bigger questions and build a common understanding of the broader subject.
Author(s): David M. Clark, Xiao Xiao
Series: Ams/Maa Textbooks, 69
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 124
Tags: real numbers; functions on reals;
Title page
Copyright
Contents
Preface
To the Student
Acknowledgement
Chapter 1. Rational Numbers and Missing Numbers
1.1. Integers
1.2. Rational Numbers
1.3. Missing Numbers
1.4. Number Lines
1.5. Many Number Lines
1.6. Historical Note: Other Missing Numbers
Chapter 2. Limit Points and Sequences
2.1. Intervals and Limit Points
2.2. Sequences and Convergence
2.3. Historical Note: Zeno’s Paradox
Chapter 3. Decimal Representations of Numbers
3.1. Infinite Decimal Representations
3.2. Rational Numbers as Infinite Decimals
3.3. Existence and Uniqueness
3.4. Historical Note: The Hindu-Arabic Numerals
Chapter 4. Complete Number Lines
4.1. The Completeness Axiom
4.2. The Square Root of 17
4.3. Historical Note: The Archimedean Dilemma
Chapter 5. Continuity
5.1. Definitions and Examples
5.2. Generating Continuous Functions
5.3. Missing Numbers as Intermediate Values
5.4. Uniform Continuity
5.5. Historical Note: Continuity via Infinitesimals
Chapter 6. Calculus
6.1. Integrals
6.2. Derivatives
6.3. The Fundamental Connection
6.4. Historical Note: Calculus via Infinitesimals
Chapter 7. Log and Exponential Functions
7.1. Rational Exponents
7.2. Natural Logarithm
7.3. Exponents Reveal More Missing Numbers
7.4. Historical Note: The 19th-Century Transition
Chapter 8. The Real Number Line
8.1. The Non-Negative Real Numbers PP
8.2. The Real Number Line R
8.3. All Number Lines
8.4. Historical Note: The Hyperreal Numbers
Chapter 9. The Price of Completeness
9.1. Cantor’s Set Theory
9.2. Lebesgue’s Measure Theory
9.3. Historical Note: More Infinities!
Appendix A. Historical Summary
Appendix B. The Reals via Dedekind Cuts
B.1. Rational and Ghost Downsets
B.2. Construction of the Number Line
B.3. Rational and Irrational Numbers
Appendix C. Guidelines for the Instructor
C.1. Chapter Options
C.2. Teaching through Guided Inquiry
Bibliography
Index
