Numbers : rational and irrational / by Ivan Niven.

Format:

Book

Author/Creator:

Niven, Ivan, 1915-1999.

Contributor:

Mathematical Association of America, issuing body.

Series:

Anneli Lax New Mathematical Library ; 1

Language:

English

Subjects (All):

Numbers, Real.

Numbers, Complex.

Physical Description:

1 online resource (viii, 140 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Washington, DC : Mathematical Association of America, 1961.

Language Note:

English

Summary:

A superb development that starts with the natural numbers and carries the reader through the rationals and their decimal representations to algebraic numbers and then to the real numbers. Along the way, you will see characterizations of the rationals and of certain special (Liouville) transcendental numbers. This material is basic to all of algebra and analysis. Professor Niven's book may be read with profit by interested high school students as well as by college students and others who want to know more about the basic aspects of pure mathematics.

Contents:

Front Cover

Numbers: Rational and Irrational

Copyright Page

CONTENTS

Introduction

Chapter 1. Natural Numbers aud Integers

1.1 Primes

1.2 Unique Factorization

1.3 Integers

1.4 Even and Odd Integers

1.5 Closure Properties

1.6 A Remark on the Nature of Proof

Chapter 2. Rational Numbers

2.1 Definition of Rational Numbers

2.2 Terminating and Non-terminating Decimals

2.3 The Many Ways of Stating and Proving Propositions

2.4 Periodic Decimals

2.5 Terminating Decimals Written as Periodic Decimals

2.6 A Summary

Chapter 3. Real Numbers

3.1 The Geometric Viewpoint

3.2 Decimal Representations

3.3 The Irrationality of √2

3.4 The Irrationality of √3

3.5 The Irrationality of v6 and v2 +√3

3.6 The Words We Use

3.7 An Application to Geometry

3.8 A summary

Chapter 4. Irrational Numbers

4.1 Closure Properties

4.2 Polynomial Equations

4.3 Rational Roots of Polynomial Equations

4.4 Further Examples

4.5 A Summary

Chapter 5. Trigonometric and Logarithmic Numbers

5.1 Irrational Values of Trigonometric Functions

5.2 A Chain Device

5.3 Irrational Values of Common Logarithms

5.4 Transcendental Numbers

5.5 Three Famous Construction Problems

5.6 Further Analysis of 3√2

5.7 A Summary

Chapter 6. The Approximation of Irrationals by Rationals

6.1 Inequalities

6.2 Approximation by Integers

6.3 Approximation by Rationals

6.4 Better Approximations

6.5 Approximations to within1/n2

6.6 Limitations on Approximations

6.7 A Summary

Chapter 7. The Existence of Transcendental Numbers

7.1 Some Algebraic Preliminaries

7.2 An Approximation to α

7.3 The Plan of the Proof

7.4 Properties of Polynomials

7.5 The Transcendence of α

7.6 A Summary

Appendix A Proof That There Are Infinitely Many Prime Numbers.

Appendix B Proof of the Fundamental Theorem of Arithmetic

Appendix C Cantor's Proof of the Existence of Transcendental Numbers

Appendix D Trigonometric Numbers

Answers and Suggestions to Selected Problems

Index.

Notes:

Includes index.

Description based on print version record.

ISBN:

0-88385-919-X

OCLC:

929120331