A brief history of numbers:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2015
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| Edition: | 1. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 309 S. - graph. Darst. 24 cm |
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Record in the Search Index
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| adam_text | Titel: A brief history of numbers
Autor: Corry, Leo
Jahr: 2015
CONTENTS
1. The System ofNumbers: An Overview 1
1.1 From natural to real numbers 3
1.2 Imaginary numbers 9
1.3 Polynomials and transcendental numbers 11
1.4 Cardinais and ordinals 15
2. Writing Numbers—Now and Back Then 17
2.1 Writing numbers nowadays: positional and decimal 17
2.2 Writing numbers back then: Egypt, Babylon and Greece 24
3. Numbers and Magnitudes in the Greek Mathematical Tradition 31
3.1 Pythagorean numbers 32
3.2 Ratios and proportions 35
3.3 Incommensurability 39
3.4 Eudoxus theory of proportions 42
3.5 Greek fractional numbers 45
3.6 Comparisons, not measurements 47
3.7 A unit length 50
Appendix 3.1 The incommensurability of fl. Ancient and
modern proofs 52
Appendix 3.2 Eudoxus theory of proportions in action 55
Appendix 3.3 Euclid and the area of the circle 59
4. Construction Problems and Numerical Problems
in the Greek Mathematical Tradition 63
4.1 The arithmetic books of the Elements 64
4.2 Geometrie algebra? 66
4.3 Straightedge and compass 67
4.4 Diophantus numerical problems 71
4.5 Diophantus reeiproeals and fractions 78
4.6 More than three dimensions 80
Appendix 4.1 Diophantus Solution of Problem V.9 in Arithmetica 83
CONTENTS I xi
5. Numbers in the Tradition ofMedieval Islam 87
5.1 Islamicate science in historical perspective 88
5.2 Al-Khwärizmi and numerical problems with Squares 90
5.3 Geometry and certainty 94
5.4 Al-jabr wal-muqäbala 97
5.5 Al-Khwärizmi, numbers and fractions 100
5.6 Abü Kämil s numbers at the crossroads of two traditions 103
5.7 Numbers, fractions and symbolic methods 107
5.8 Al-Khayyäm and numerical problems with cubes 111
5.9 Gersonides and problems with numbers 116
Appendix 5.1 The quadratic equation. Derivation of the algebraic formula 120
Appendix 5.2 The cubic equation. Khayyam s geometric Solution 121
6. Numbers in Europe from the Twelfth to the Sixteenth Centuries 125
6.1 Fibonacci and Hindu-Arabic numbers in Europe 128
6.2 Abbacus and coss traditions in Europe 129
6.3 Cardano s Great Art of Algebra 138
6.4 Bombelli and the roots of negative numbers 146
6.5 Euclid s Elements in the Renaissance 149
Appendix 6.1 Casting out nines 150
7. Number and Equations at the Beginning of the Scientific Revolution 155
7.1 Viete and the new art of analysis 157
7.2 Stevin and decimal fractions 163
7.3 Logarithms and the decimal System of numeration 167
Appendix 7.1 Napier s construction of logarithmic tables 171
8. Number and Equations in the Works of Descartes, Newton
and their Contemporaries 175
8.1 Descartes new approach to numbers and equations 176
8.2 Wallis and the primacy of algebra 182
8.3 Barrow and the Opposition to the primacy of algebra 187
8.4 Newton s Universal Arithmetick 190
Appendix 8.1 The quadratic equation. Descartes geometric Solution 196
Appendix 8.2 Between geometry and algebra in the seventeenth
Century: The case of Euclid s Elements 198
9. New Definitions of Complex Numbers in the Early
Nineteenth Century 207
9.1 Numbers and ratios: giving up metaphysics 208
9.2 Euler, Gauss and the ubiquity of complex numbers 209
9.3 Geometric interpretations of the complex numbers 212
9.4 Hamilton s formal definition of complex numbers 215
9.5 Beyond complex numbers 217
9.6 Hamilton s discovery of quaternions 220
xii | CONTENTS
10. What Are Numbers and What Should They Be?
Understanding Numbers in the Late Nineteenth Century 223
10.1 What are numbers? 224
10.2 Kummers ideal numbers 225
10.3 Fields of algebraic numbers 228
10.4 What should numbers be? 231
10.5 Numbers and the foundations of calculus 234
10.6 Continuity and irrational numbers 237
Appendix 10.1 Dedekinds theory of cuts and Eudoxus theory of
proportions 243
Appendix 10.2 IVT and the fundamental theorem of calculus 245
11. Exact Definitions for the Natural Numbers: Dedekind,
Peano and Frege 249
11.1 The principle of mathematical induction 250
11.2 Peano s postulates 251
11.3 Dedekinds chains of natural numbers 257
11.4 Freges definition of cardinal numbers 259
Appendix 11.1 The principle of induction and Peano s postulates 262
12. Numbers, Sets and Infinity. A Conceptual Breakthrough at
the Turn of the Twentieth Century 265
12.1 Dedekind, Cantor and the infinite 266
12.2 Infinities of various sizes 269
12.3 Cantor s transfinite ordinals 277
12.4 Troubles in paradise 280
Appendix 12.1 Proof that the set of algebraic numbers is countable 287
13. Epilogue: Numbers in Historical Perspective 291
References and Suggestions for Further Reading 295
Name Index 303
Subject Index 306
CONTENTS xiii
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| spelling | Corry, Leo 1956- Verfasser (DE-588)172947472 aut A brief history of numbers Leo Corry 1. ed. Oxford Oxford Univ. Press 2015 XIII, 309 S. - graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Geschichte gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahl (DE-588)4067271-2 gnd rswk-swf Zahl (DE-588)4067271-2 s Zahlentheorie (DE-588)4067277-3 s Geschichte z b DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028206011&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Corry, Leo 1956- A brief history of numbers Zahlentheorie (DE-588)4067277-3 gnd Zahl (DE-588)4067271-2 gnd |
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| title | A brief history of numbers |
| title_auth | A brief history of numbers |
| title_exact_search | A brief history of numbers |
| title_full | A brief history of numbers Leo Corry |
| title_fullStr | A brief history of numbers Leo Corry |
| title_full_unstemmed | A brief history of numbers Leo Corry |
| title_short | A brief history of numbers |
| title_sort | a brief history of numbers |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Zahl (DE-588)4067271-2 gnd |
| topic_facet | Zahlentheorie Zahl |
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