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.
Flexible presentation:
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.