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Brief History of Numbers [Kietas viršelis]

3.92/5 (14 ratings by Goodreads)
  • Formatas: Hardback, 324 pages, aukštis x plotis x storis: 240x162x25 mm, weight: 626 g
  • Išleidimo metai: 27-Aug-2015
  • Leidėjas: Oxford University Press
  • ISBN-10: 0198702590
  • ISBN-13: 9780198702597
Kitos knygos pagal šią temą:
Brief History of NumbersDidesnis paveikslėlis
  • Formatas: Hardback, 324 pages, aukštis x plotis x storis: 240x162x25 mm, weight: 626 g
  • Išleidimo metai: 27-Aug-2015
  • Leidėjas: Oxford University Press
  • ISBN-10: 0198702590
  • ISBN-13: 9780198702597
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780198702597.html
Raktažodžiai:
The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be?

In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century.

Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.

Recenzijos

Corry has compiled a readable account of the history of mathematics focusing on numbers, although for most of the period in question, arithmetic and geometry are not easily separable. The required level of sophistication of the reader is not great, it is certainly at the level of a first-year undergraduate, or a keen sixth-former who is studying mathematics. Even as an experienced university mathematician, the reviewer learnt many interesting things, and has some misconceptions remedied, on reading Corry's Brief History. * Robin Chapman, LMS Newsletter * This fine book gives what its title promises ... a well-written treatment of the subject. * Underwood Dudley, MAA Reviews * It is a highly recommended and pleasant read, not pedantic, but not casual either ... The book is written with great care ... * Adhemar Bultheel, European Mathematical Society * A Brief History of Numbers is a meticulously researched and carefully crafted look at how mathematicians have explored the concept of number. Corry's prose is clear and engaging, and the mathematical content is uniformly accessible to his audience. ... I highly recommend A Brief History of Numbers to mathematics teachers who wish to know more about how our current edifice of natural, rational, real, complex, and infinite numbers came to be built. * James V. Rauff, Mathematics Teacher *

1 The System of Numbers: An Overview
1(16)
1.1 From natural to real numbers
3(6)
1.2 Imaginary numbers
9(2)
1.3 Polynomials and transcendental numbers
11(4)
1.4 Cardinals and ordinals
15(2)
2 Writing Numbers---Now and Back Then
17(14)
2.1 Writing numbers nowadays: positional and decimal
17(7)
2.2 Writing numbers back then: Egypt, Babylon and Greece
24(7)
3 Numbers and Magnitudes in the Greek Mathematical Tradition
31(32)
3.1 Pythagorean numbers
32(3)
3.2 Ratios and proportions
35(4)
3.3 Incommensurability
39(3)
3.4 Eudoxus' theory of proportions
42(3)
3.5 Greek fractional numbers
45(2)
3.6 Comparisons, not measurements
47(3)
3.7 A unit length
50(13)
Appendix 3.1 The incommensurability of √2. Ancient and modern proofs
52(3)
Appendix 3.2 Eudoxus' theory of proportions in action
55(4)
Appendix 3.3 Euclid and the area of the circle
59(4)
4 Construction Problems and Numerical Problems in the Greek Mathematical Tradition
63(24)
4.1 The arithmetic books of the Elements
64(2)
4.2 Geometric algebra?
66(1)
4.3 Straightedge and compass
67(4)
4.4 Diophantus' numerical problems
71(7)
4.5 Diophantus' reciprocals and fractions
78(2)
4.6 More than three dimensions
80(7)
Appendix 4.1 Diophantus' solution of Problem V.9 in Arithmetica
83(4)
5 Numbers in the Tradition of Medieval Islam
87(38)
5.1 Islamicate science in historical perspective
88(2)
5.2 Al-Khwarizmi and numerical problems with squares
90(4)
5.3 Geometry and certainty
94(3)
5.4 Al-jabr wa'l-muqabala
97(3)
5.5 Al-Khwarizmi, numbers and fractions
100(3)
5.6 Abu Kamil's numbers at the crossroads of two traditions
103(4)
5.7 Numbers, fractions and symbolic methods
107(4)
5.8 Al-Khayyam and numerical problems with cubes
111(5)
5.9 Gersonides and problems with numbers
116(9)
Appendix 5.1 The quadratic equation. Derivation of the algebraic formula
120(1)
Appendix 5.2 The cubic equation. Khayyam's geometric solution
121(4)
6 Numbers in Europe from the Twelfth to the Sixteenth Centuries
125(30)
6.1 Fibonacci and Hindu--Arabic numbers in Europe
128(1)
6.2 Abbacus and coss traditions in Europe
129(9)
6.3 Cardano's Great Art of Algebra
138(8)
6.4 Bombelli and the roots of negative numbers
146(3)
6.5 Euclid's Elements in the Renaissance
149(6)
Appendix 6.1 Casting out nines
150(5)
7 Number and Equations at the Beginning of the Scientific Revolution
155(20)
7.1 Viete and the new art of analysis
157(6)
7.2 Stevin and decimal fractions
163(4)
7.3 Logarithms and the decimal system of numeration
167(8)
Appendix 7.1 Napier's construction of logarithmic tables
171(4)
8 Number and Equations in the Works of Descartes, Newton and their Contemporaries
175(32)
8.1 Descartes' new approach to numbers and equations
176(6)
8.2 Wallis and the primacy of algebra
182(5)
8.3 Barrow and the opposition to the primacy of algebra
187(3)
8.4 Newton's Universal Arithmetick
190(17)
Appendix 8.1 The quadratic equation. Descartes' geometric solution
196(2)
Appendix 8.2 Between geometry and algebra in the seventeenth century: The case of Euclid's Elements
198(9)
9 New Definitions of Complex Numbers in the Early Nineteenth Century
207(16)
9.1 Numbers and ratios: giving up metaphysics
208(1)
9.2 Euler, Gauss and the ubiquity of complex numbers
209(3)
9.3 Geometric interpretations of the complex numbers
212(3)
9.4 Hamilton's formal definition of complex numbers
215(2)
9.5 Beyond complex numbers
217(3)
9.6 Hamilton's discovery of quaternions
220(3)
10 "What Are Numbers and What Should They Be?" Understanding Numbers in the Late Nineteenth Century
223(26)
10.1 What are numbers?
224(1)
10.2 Kummer's ideal numbers
225(3)
10.3 Fields of algebraic numbers
228(3)
10.4 What should numbers be?
231(3)
10.5 Numbers and the foundations of calculus
234(3)
10.6 Continuity and irrational numbers
237(12)
Appendix 10.1 Dedekind's theory of cuts and Eudoxus' theory of proportions
243(2)
Appendix 10.2 IVT and the fundamental theorem of calculus
245(4)
11 Exact Definitions for the Natural Numbers: Dedekind, Peano and Frege
249(16)
11.1 The principle of mathematical induction
250(1)
11.2 Peano's postulates
251(6)
11.3 Dedekind's chains of natural numbers
257(2)
11.4 Frege's definition of cardinal numbers
259(6)
Appendix 11.1 The principle of induction and Peano's postulates
262(3)
12 Numbers, Sets and Infinity. A Conceptual Breakthrough at the Turn of the Twentieth Century
265(26)
12.1 Dedekind, Cantor and the infinite
266(3)
12.2 Infinities of various sizes
269(8)
12.3 Cantor's transfinite ordinals
277(3)
12.4 Troubles in paradise
280(11)
Appendix 12.1 Proof that the set of algebraic numbers is countable
287(4)
13 Epilogue: Numbers in Historical Perspective
291(4)
References and Suggestions for Further Reading 295(8)
Name Index 303(3)
Subject Index 306
Leo Corry is a historian of mathematics with a very broad range of interest, that comprise, among other things, the history of modern algebra, the history of number theory, the history of general relativity, and the Euclidean tradition in the middle ages and the early modern period. He has published extensively in all these fields.

He teaches at Tel Aviv University, where he is the Bert and Barbara Cohn Professor of History and Philosophy of Science. Since 2013 he is director of the Zvi Yavetz Graduate School of History. He was head of the Cohn Institute for History and Philosophy of Science (2003-2009), and editor of the international journal Science in Context (1999-2012).