
Journey into Mathematics, A: An Introduction to Proofs, 1/e
Joseph Rotman, University of Illinois  Urbana
Published August, 1997 by Prentice Hall Engineering/Science/Mathematics
Copyright 1998, 237 pp.
Cloth
ISBN 0138423601

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Transition to Advanced MathematicsMathematics

Prompting students to do mathematics, not merely read about
it, this interesting and uniquely enjoyable text prepares students
for reading and writing proofs by having them do just that at the
outset. Complete proofs are given from the start and coverage begins
with elementary mathematics to allow students to focus on the writing
and reading of proofs without the distraction of absorbing new ideas
simultaneously.
Contains much material that is already familiar to students,
allowing them to focus on writing and reading proofs immediately.
 Starts with topics they have seen (induction, binomial
theorem, polygonal areas, etc.) and progresses to more sophisticated—but
also familiar—subjects (using the Diophantine parametrization
of the circle by rational functions to solve trigonometric identities;
parametrizing conic sections by rational functions to integrate certain
complicated functions, cubic and quartic formulas, etc.).
Engages students with interesting expositions using
history, etymology, humor, and many practical examples and exercises.
The modern notion of convergence of sequences is given only
after one proves the area and circumference formulas for circles,
using the easier classical Greek notion of limit. The story of the
attempted legislation of …p by the Indiana House in 1897 is told
in full.
Presents the material as a 'single story' to aid in
the subject's progressive growth and development, with the answers
to one question raising new questions, and new answers, and so on—as
students continue their query, they can then begin to understand how
the whole picture fits together.
Offers a light introduction to complex numbers, and eases
students into the topic gently.
 Initially provides easy examples to induction, then gives
the arithmeticgeometric mean inequality; discusses the quadratic
formula first, then presents a variant so that the formula can estimate
roots when the leading coefficient of the polynomial is very small.
1. SETTING OUT.
Induction.
Binomial Coefficients.
2. THINGS PYTHAGOREAN.
Area.
The Pythagorean Theorem.
Pythagorean Triples.
The Method of Diophantus.
Trigonometry.
Integration.
3. CIRCLES AND PI.
Approximations.
The Area of a Disk.
The Circumference of a Disk.
Sequences.
4. POLYNOMIALS.
Quadratics.
Complex Numbers.
De Moivre's Theorem.
Cubics and Quartics.
Irrationalities.
Glossary of Logic.
