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Reading, Writing, and Proving: A Closer Look at Mathematics

by and

Reading, Writing, and Proving:  A Closer Look at Mathematics Cover

 

Synopses & Reviews

Publisher Comments:

 This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.

Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Synopsis:

This book, based on Pólya's method of problem solving, aids students in their transition to higher-level mathematics. It begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them.

Synopsis:

This book, which is based on Plya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.

Students will follow Plya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Table of Contents

Preface

  1    The How, When, and Why of Mathematics

    Spotlight: George Polya

    Tips on Doing Homework

  2    Logically Speaking

  3    Introducing the Contrapositive and Converse

  4    Set Notation and Quantifiers

    Tips on Quantification

  5    Proof Techniques

    Tips on Definitions

  6    Sets

    Spotlight: Paradoxes

  7    Operations on Sets

  8    More on Operations on Sets

  9    The Power Set and the Cartesian Product

    Tips on Writing Mathematics

  10    Relations

    Tips on Reading Mathematics

  11    Partitions

    Tips on Putting It All Together

  12    Order in the Reals

    Tips: You Solved it. Now What?

  13    Functions, Domain, and Range

    Spotlight: The Definition of Function

  14    Functions, One-to-one, and Onto

  15    Inverses

  16    Images and Inverse Images

    Spotlight: Minimum or Infimum

  17    Mathematical Induction

  18    Sequences

  19    Convergence of Sequences of Real Numbers

  20    Equivalent Sets

  21    Finite Sets and an Infinite Set

  22    Countable and Uncountable Sets

  23    Metric Spaces

  24    Getting to Know Open and Closed Sets

  25    Modular Arithmetic

  26    Fermat's Little Theorem

    Spotlight: Public and Secret Research

  27    Projects

    Tips on Talking about Mathematics

    27.1    Picture Proofs

    27.2    The Best Number of All

    27.3    Set Constructions

    27.4    Rational and Irrational Numbers

    27.5    Irrationality of $e$ and $\pi $

    27.6    When does $f^{-1} = 1/f$?

    27.7    Pascal's Triangle

    27.8    The Cantor Set

    27.9    The Cauchy-Bunyakovsky-Schwarz  Inequality

    27.10    Algebraic Numbers

    27.11    The RSA Code

    Spotlight: Hilbert's Seventh Problem

  28    Appendix

    28.1    Algebraic Properties of $\@mathbb {R}$

    28.2    Order Properties of $\@mathbb {R}$

    28.3    Polya's List

      References

      Index

Product Details

ISBN:
9780387008349
Author:
Ulrich Daepp and Pamela Gorkin
Publisher:
Springer
Author:
Daepp, Ulrich
Author:
Gorkin, Pamela
Location:
New York
Subject:
Mathematics
Subject:
Technical Writing
Subject:
Logic
Subject:
Mathematical Analysis
Subject:
Number Theory
Subject:
Mathematical Logic and Foundations
Subject:
Mathematics -- Study and teaching (Higher)
Subject:
Analysis
Subject:
Mathematics-Logic and Philosophy
Copyright:
Edition Number:
1
Edition Description:
Book
Series:
Undergraduate texts in mathematics
Series Volume:
24
Publication Date:
August 2003
Binding:
HARDCOVER
Language:
English
Illustrations:
Y
Pages:
408
Dimensions:
235 x 155 mm 789 gr

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Related Subjects

Science and Mathematics » Mathematics » Analysis General
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Foundations and Logic
Science and Mathematics » Mathematics » Logic and Philosophy
Science and Mathematics » Mathematics » Number Theory
Science and Mathematics » Mathematics » Topology

Reading, Writing, and Proving: A Closer Look at Mathematics New Hardcover
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Product details 408 pages Springer - English 9780387008349 Reviews:
"Synopsis" by , This book, based on Pólya's method of problem solving, aids students in their transition to higher-level mathematics. It begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them.
"Synopsis" by , This book, which is based on Plya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.

Students will follow Plya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

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