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Frederick W. Stevenson , University of Arizona

Published December 1999 by Engineering/Science/Mathematics

Copyright 2000, 365 pp. Cloth ISBN 0-13-040261-3

Intended primarily for a course for future high school teachers. Can also serve as an introduction to mathematical thought, a short course in number theory, an honors course at the high school level, or an introduction to mathematical education research.

As much a book about numbers as a number theory text, Exploring the Real Numbers answers the need for future teachers to understand the real number system. Experienced educator Frederick Stevenson brings students up to date with the study of the nature of real numbers and provides a sense of the historical journey that has led to our current knowledge of the subject. Many interesting topics that arise during the study of the real numbers are presented and students are given the opportunity to study topics further on their own.

• Unique exploratory approach-An entire chapter (5) composed of twenty research projects provides students with the opportunity to discover new and significant results stretching their knowledge beyond the text.

  • Addresses the trend of students being encouraged to pursue open-ended topics instead of always working to a "right answer."

• Flexible presentation:

  • If teaching all four chapters seems too ambitious, it would be easy to settle on the first three chapters and topics of the instructor's choosing from Chapter 4. The projects from Chapter 5 can be integrated into earlier chapters at will.

• Presents 4 different aspects of irrational numbers in Chapter 4 -Algebraic, geometric, trigonometric, and analytic. The last section, 4.4, deals with transcendental numbers.

• Includes 350 exercises that keep the reader current with the text.

• More than 100 carefully worked examples make the material accessible to laymen as well as students.
  1. The Natural Numbers. The Basics. The Fundamental Theorem of Arithmetic. Searching for Primes. Number Fascinations.

2. The Integers. Diophantine Equations. Congruence Arithmetic. Pell and Pythagoras. Factoring Large Numbers.
3. The Rational Numbers. Rational Numbers as Decimals. Decimals as Rational Numbers. Continued Fractions. Solving Equations on the Rational Plane
4. The Real Numbers. Algebraic Representations. Geometric Representations. Analytic Representations. Searching for Transcendental Numbers.
5. Mathematical Projects. Rings of Factors. Sums of Consecutive Numbers. Measuring Abundance. Inside the Fibonacci Numbers. Pictures at an Iteration. Eenie Meenie Miney Mo. Factoring with the Pollard ...r Method. Charting the Integral Universe. Triangles on the Integral Lattice. The Gaussian Integers. Writing Fractions the Egyptian Way. Building Polygons with Dots. The Decimal Universe of Fractions, I. The Decimal Universe of Fractions, II. The Making of a Star. Making Your Own Real Numbers. Building 1 the Egyptian Way. Continued Fraction Expansions of x1/2 N. A Special Kind of Triangle. Polygon Numbers. Continued Fraction Expansions.