SYLLABUS

I. MATH 440, Construction of the Real Number System 4 cr. hrs.

II. Spring, 1998 INSTRUCTOR: Herman Matthews

Office: 207 Farr-Chinnock, Ext. 6226

Office Hours: Office hours are posted on the door of my office, together with my complete schedule.

III. COURSE PREREQUISITES: MATH 232-232 or equivalent, or permission of instructor.

IV. COURSE DESCRIPTION/COURSE GOALS: A construction of the real number system from axioms for the natural numbers. The concept of isomorphic mappings plays a central role as each subsystem is extended to the next system. The reals are introduced through Cauchy sequences in the rationals or through Dedekind cuts, as the text may require, and either approach is used to develop various wordings for the completeness property. Some time may be spent on the complex number system or on cardinal numbers as class progress allows. Major goals: 1) To aid the student’s ability to think creatively and to reason mathematically. 2) To develop an appreciation of the complexity and beauty of the real number system and its place in analysis. 3) To prepare the student for courses at the graduate level that may make extensive use of the completeness property of the reals.

V. RELATIONHIP OF THIS COURSE TO CONTENT AREA KNOWLEDGE AND SKILLS: This course uniquely seeks to develop the ability to analyze and to synthesize mathematical ideas of great importance in analysis. It will aid the student’s ability to communicate mathematics, both orally and in writing. Both inductive and deductive thinking must be used in the process of solving problems, so the student’s under-standing of these modes of thought will be enhanced.

VI. TEXT: Roberts, J.B., The Real Number System in an Algebraic Setting,

W.H. Freeman and Company, San Francisco, 1962 NOTE: Copies made by permission

VII. COURSE OBJECTIVES: 1) To understand the axiomatic approach by using a set of axioms first for the natural numbers and proving further results of importance. 2) To see how the natural numbers are “nested” in the positive rationals by using the notion of isomorphism. 3) To develop successively the analogous facts for positive rationals in the positive reals and the positive reals in the reals. 4) If time allows, the complex number system is treated similarly. 5) To develop a clear concept of the notion of proof and its relation to definitions, axioms, and both inductive and deductive reasoning.

VIII. OUTLINE OF COURSE CONTENT/UNITS OF INSTRUCTION:

1. Introduction: sets, relations, functions (mappings), isomorphic mappings, and applications to systems or extensions of systems.

2. The natural numbers: axioms, consequences (such as linear order properties, mathematical induction, the Archimedean principle, properties of natural exponents), prime numbers, the long division algorithm, and subtraction properties.

3. Positive rationals: equivalence classes of Z ´ Z, definitions of addition, subtractions, and order, proofs of such properties as commutative, associative, and distributive laws, and solutions of equations.

4. Introduction to cardinal numbers (if time allows): equivalence of sets, finite cardinals, infinite sets, the cardinal number À₀ (“aleph nought”).

5. Preparation for construction of reals: denseness, special inequality notation, limits of sequences, bounds and Cauchy sequences, and the equivalence of Cauchy sequences.

6. The nonnegative reals: equivalence classes of Cauchy sequences to define R⁺, operations in R⁺, linear order, R⁺ as an extension of the positive rationals, convergence of Cauchy sequences to real numbers.

7. The real numbers: In this unit, which is the last step in the construction process, students will be expected to supply proofs of all theorems. Topics discussed: fields, order, and completeness, Dedekind cuts, base ten representation of integers, and the decimal notation. Emphasis is given to expository writing.

8. Optional topics as time allows: complex numbers, Peano axioms, and additional properties of cardinal numbers.

IX. REQUIRED READING: Textbook

X. SUGGESTED READINGS/BIBLIOGRAPHY: None

XI. METHODS OF INSTRUCTION AND LEARNING: Minimum lecture, but maximum problem-solving activity will be required. The relationship between analysis of theorems and writing of proofs (synthesis) will be explored throughout the study. Classroom discussion for ideas will be used to find possible ways to prove some theorems and to get (hopefully) some “bright ideas.” A short expository paper will be required for the final step in the construction process, in which proofs of theorems must be supplied by the students.

XII. COURSE REQUIREMENTS/METHODS OF ASSESSMENT/EVALUATION/DOCUMENTATION: All students are expected to attend class regularly and to keep up homework and study. Three or four major tests and a final exam will constitute the testing, but grades will be determined by input of the class relative to the weighting of oral presentations, discussion, and a written paper. The goal here is to let the class determine the grading process under the instructor's guidance. The final grading will conform to the scheme

A 90.0 and above

B 80.0 – 89.99

C 70.0 – 79.99

D 60.0 – 69.99

F below 60.0.

The final exam will be given only on the official date designated by the University, excepting the requirements of early grades for seniors or emergency cases. Students who for some valid reason must take an incomplete must contract for this grade and complete the work within a designated time frame. Such accommodation must be provided for before the date of the final. See the Catalog for more information.

XIII. CLINICAL/LABORATORY/FIELD EXPERIENCES: None required.

XIV. DATE OF REVISION: January 17, 1998