[Book Cover]

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 0-13-842360-1


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Summary

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.

Features


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 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.


Table of Contents
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.


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