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

by and

 Comment on this title and you could win free books!Synopses & Reviews Table of Contents ISBN13: 9780387008349 ISBN10: 0387008349 Condition: All Product Details

### Synopses & Reviews

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.

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

7    Operations on Sets

8    More on Operations on Sets

9    The Power Set and the Cartesian Product

Tips on Writing Mathematics

10    Relations

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

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:
Publisher:
Springer
Author:
Author:
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
Edition Number:
1
Edition Description:
Book
Series:
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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