Irrational numbers:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Buffalo]
Mathematical Assoc. of America
2006
|
| Ausgabe: | 6. print. |
| Schriftenreihe: | Carus mathematical monographs
11 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 164 S. |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804137488667639808 |
|---|---|
| adam_text | Titel: Irrational numbers
Autor: Niven, Ivan
Jahr: 2006
CONTENTS
CHAPTER PAGE
I. Rationale and Irrationals
1. The preponderance of irrationals 1
2. Countability 4
3. Dense sets 5
4. Decimal expansions 6
II. Simple Irrationalities
1. Introduction 15
2. The trigonometric functions and r 16
3. The hyperbolic, exponential, and logarithmic functions. 22
III. Certain Algebraic Numbers
1. Introduction 28
2. Further background material 30
3. The factorization of x - 1 33
4. Certain trigonometric values 36
5. Extension to the tangent 38
IV. The Approximation of Irrationals by Rationals
1. The problem 42
2. A generalization 44
3. Linearly dependent sets 48
V. Continued Fractions
1. The Euclidean algorithm 51
2. Uniqueness 53
3. Infinite continued fractions 54
4. Infinite continued fraction expansions 59
5. The convergent^ as approximations 61
6. Periodic continued fractions 63
VI. Further Diophantine Approximations
1. A basic result 68
2. Best possible approximations 70
xi
xii CONTENTS
CHAPTER PAGE
3. Uniform distributions 71
4. A proof by Fourier analysis 75
VII. Algebraic and Transcendental Numbers
1. Closure properties of algebraic numbers 83
2. A property of algebraic integers 85
3. Transcendental numbers 87
4. The order of approximation 88
VIII. Normal Numbers
1. Definition of a normal number 94
2. The measure of the set of normal numbers 98
3. Equivalent definitions 104
4. A normal number exhibited 112
IX. The Generalized Lindemann Theorem
1. Statement of the theorem 117
2. Preliminaries 118
3. Proof of the theorem 124
4. Applications of the theorem 131
5. Squaring the circle 132
X. The Gelfond-Schneider Theorem
1. Hilbert s seventh problem 134
2. Background material 135
3. Two lemmas 137
4. Proof of the Gelfond-Schneider theorem 142
List or Notation 151
Glossary 153
Reference Books 157
Index op Topics 159
Index of Names 163
|
| adam_txt |
Titel: Irrational numbers
Autor: Niven, Ivan
Jahr: 2006
CONTENTS
CHAPTER PAGE
I. Rationale and Irrationals
1. The preponderance of irrationals 1
2. Countability 4
3. Dense sets 5
4. Decimal expansions 6
II. Simple Irrationalities
1. Introduction 15
2. The trigonometric functions and r 16
3. The hyperbolic, exponential, and logarithmic functions. 22
III. Certain Algebraic Numbers
1. Introduction 28
2. Further background material 30
3. The factorization of x" - 1 33
4. Certain trigonometric values 36
5. Extension to the tangent 38
IV. The Approximation of Irrationals by Rationals
1. The problem 42
2. A generalization 44
3. Linearly dependent sets 48
V. Continued Fractions
1. The Euclidean algorithm 51
2. Uniqueness 53
3. Infinite continued fractions 54
4. Infinite continued fraction expansions 59
5. The convergent^ as approximations 61
6. Periodic continued fractions 63
VI. Further Diophantine Approximations
1. A basic result 68
2. Best possible approximations 70
xi
xii CONTENTS
CHAPTER PAGE
3. Uniform distributions 71
4. A proof by Fourier analysis 75
VII. Algebraic and Transcendental Numbers
1. Closure properties of algebraic numbers 83
2. A property of algebraic integers 85
3. Transcendental numbers 87
4. The order of approximation 88
VIII. Normal Numbers
1. Definition of a normal number 94
2. The measure of the set of normal numbers 98
3. Equivalent definitions 104
4. A normal number exhibited 112
IX. The Generalized Lindemann Theorem
1. Statement of the theorem 117
2. Preliminaries 118
3. Proof of the theorem 124
4. Applications of the theorem 131
5. Squaring the circle 132
X. The Gelfond-Schneider Theorem
1. Hilbert's seventh problem 134
2. Background material 135
3. Two lemmas 137
4. Proof of the Gelfond-Schneider theorem 142
List or Notation 151
Glossary 153
Reference Books 157
Index op Topics 159
Index of Names 163 |
| any_adam_object | 1 |
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| dewey-full | 512.786 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 6. print. |
| format | Book |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:10:58Z |
| indexdate | 2024-07-09T21:13:06Z |
| institution | BVB |
| language | English |
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| physical | XII, 164 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Mathematical Assoc. of America |
| record_format | marc |
| series | Carus mathematical monographs |
| series2 | Carus mathematical monographs |
| spelling | Niven, Ivan 1915-1999 Verfasser (DE-588)115101217 aut Irrational numbers Ivan Niven 6. print. [Buffalo] Mathematical Assoc. of America 2006 XII, 164 S. txt rdacontent n rdamedia nc rdacarrier Carus mathematical monographs 11 Irrational numbers Transzendente Zahl (DE-588)4225034-1 gnd rswk-swf Irrationale Zahl (DE-588)4162426-9 gnd rswk-swf Irrationale Zahl (DE-588)4162426-9 s Transzendente Zahl (DE-588)4225034-1 s DE-604 Carus mathematical monographs 11 (DE-604)BV001887236 11 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016394347&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Niven, Ivan 1915-1999 Irrational numbers Carus mathematical monographs Irrational numbers Transzendente Zahl (DE-588)4225034-1 gnd Irrationale Zahl (DE-588)4162426-9 gnd |
| subject_GND | (DE-588)4225034-1 (DE-588)4162426-9 |
| title | Irrational numbers |
| title_auth | Irrational numbers |
| title_exact_search | Irrational numbers |
| title_exact_search_txtP | Irrational numbers |
| title_full | Irrational numbers Ivan Niven |
| title_fullStr | Irrational numbers Ivan Niven |
| title_full_unstemmed | Irrational numbers Ivan Niven |
| title_short | Irrational numbers |
| title_sort | irrational numbers |
| topic | Irrational numbers Transzendente Zahl (DE-588)4225034-1 gnd Irrationale Zahl (DE-588)4162426-9 gnd |
| topic_facet | Irrational numbers Transzendente Zahl Irrationale Zahl |
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