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.
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Transition to Advanced Mathematics-Mathematics
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
Contains much material that is already familiar to students,
allowing them to focus on writing and reading proofs immediately.
Engages students with interesting expositions using
history, etymology, humor, and many practical examples and exercises.
- Starts with topics they have seen (induction, binomial
theorem, polygonal areas, etc.) and progresses to more sophisticatedbut
also familiarsubjects (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.).
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
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 onas
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 arithmetic-geometric 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.
2. THINGS PYTHAGOREAN.
3. CIRCLES AND PI.
The Pythagorean Theorem.
The Method of Diophantus.
The Area of a Disk.
The Circumference of a Disk.
De Moivre's Theorem.
Cubics and Quartics.
Glossary of Logic.