Exploring the Real Numbers, 1/e
Frederick W. Stevenson, University of Arizona
Coming December, 1999 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 300 pp.
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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 approachAn 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.
Includes 350 exercises that keep the reader current
with the text.
- Addresses the trend of students being encouraged to pursue
open-ended topics instead of always working to a right answer.
- 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.
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.
Decimal Expansions. Continued Fractions. Keeping the Denominators
Small. Diophantine 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