Irrational Numbers and their representation by sequences and series /

By:

William J. Stanton

Language: Inglés Series: SeriePublication details: JOHN WILEY Y SONS EUA Edition: 1Description: 118 Contien graficos 18cm de ancho X 23cm de largoISBN:

1007091900

LOC classification:

Contents:

CONTENTS CHAPTER I IRRATIONAL NUMBERS I. Infinite Sets of Objects.. II. Definition of Irrational Numbers.. III. Operations upon Irrational Numbers. IV. Exponents and Logarithms. PAGE 5 14 25 CHAPTER II SEQUENCES I. Representation of Numbers by Sequences. 34 II Regular Sequences. 46 III. Operations upon Sequences.. 48 IV. The Theory of Limits... 57 CHAPTER III SERIES I. Convergency of Series.. 65 81 89 II. Operations upon Series. III. Absolute Corvergence. CHAPTER IV POWER SERIES 1. The Radius of Convergence.. II. "Undetermined Coefficients". 100 108 V CONTENTS CHAPTER V THE EXPONENTIAL, BINOMIAL, AND LOGARITHMIC SERIES PAGE I. The Exponential Series. 112 II. The Binomial Theorem for any Rational Exponent. 116 III. The Binomial Theorem for an Irrational Exponent and the Logarithmic Series.

Summary: PREFACE THIS book is intended to explain the nature of irra-tional numbers, and those parts of Algebra which depend on what is usually called The Theory of Limits. Many of our text-books define irational numbers by means of sequences; but to the author it has seemed more natural to define a number, or at least to con-sider a number as determined, by the place which it occupies among rational numbers, and to assume that a separation of all rational numbers into two classes, those of one class less than those of the other, always determines a number which occupies the point of separation. 'Thus we have the definition of Dede-kind, which is adopted by Weber in his Algebra. With-out attempting to inquire too minutely into the sig-nificance of this definition, we have endeavored to show how the fundamental operations are to be per-formed in the case of irrational numbers and to defire the irrational exponent and the logarithm. Defining the irrational number by the place which it occupies among rational numbers, we proceed to speak of its representation by sequences; and when we have proved that a sequence which represents a number is regular and that a sequence which is regular repre-512.4

Holdings
Item type	Current library	Call number	Copy number	Status	Date due	Barcode
Libro	CI Gustavo A. Madero 2	LCC	3	Available

CONTENTS

CHAPTER I

IRRATIONAL NUMBERS

I. Infinite Sets of Objects..

II. Definition of Irrational Numbers..

III. Operations upon Irrational Numbers.

IV. Exponents and Logarithms.

PAGE

5

14

25

CHAPTER II

SEQUENCES

I. Representation of Numbers by Sequences.

34

II Regular Sequences.

46

III. Operations upon Sequences..

48

IV. The Theory of Limits...

57

CHAPTER III

SERIES

I. Convergency of Series..

65

81

89

II. Operations upon Series.

III. Absolute Corvergence.

CHAPTER IV

POWER SERIES

1. The Radius of Convergence..

II. "Undetermined Coefficients".

100

108

V
CONTENTS

CHAPTER V

THE EXPONENTIAL, BINOMIAL, AND LOGARITHMIC SERIES

PAGE

I. The Exponential Series.

112

II. The Binomial Theorem for any Rational Exponent.

116

III. The Binomial Theorem for an Irrational Exponent and the Logarithmic Series.

PREFACE

THIS book is intended to explain the nature of irra-tional numbers, and those parts of Algebra which depend on what is usually called The Theory of Limits.

Many of our text-books define irational numbers by means of sequences; but to the author it has seemed more natural to define a number, or at least to con-sider a number as determined, by the place which it occupies among rational numbers, and to assume that a separation of all rational numbers into two classes, those of one class less than those of the other, always determines a number which occupies the point of separation. 'Thus we have the definition of Dede-kind, which is adopted by Weber in his Algebra. With-out attempting to inquire too minutely into the sig-nificance of this definition, we have endeavored to show how the fundamental operations are to be per-formed in the case of irrational numbers and to defire the irrational exponent and the logarithm.

Defining the irrational number by the place which it occupies among rational numbers, we proceed to speak of its representation by sequences; and when we have proved that a sequence which represents a number is regular and that a sequence which is regular repre-512.4

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