Making up Numbers : a history of invention in mathematics / Ekkehard Kopp.

Format:

Book

Author/Creator:

Kopp, P. E., 1944- author.

Contributor:

Open Book Publishers, Publisher.

Language:

English

Subjects (All):

Mathematics--History.

Mathematics.

Inventions--Mathematical models.

Inventions.

Physical Description:

1 online resource (ix, 267 pages) : illustrations (some colour).

Place of Publication:

Open Book Publishers 2020

Cambridge, UK : OpenBook Publishers, [2020]

Language Note:

English

Summary:

"Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject."

Contents:

Intro; Preface; Prologue: Naming Numbers; 1. Naming large numbers; 2. Very large numbers; 3. Archimedes' Sand-Reckoner; 4. A long history; Chapter 1. Arithmetic in Antiquity; Summary; 1. Babylon: sexagesimals, quadratic equations; 2. Pythagoras: all is number; 3. Incommensurables; 4. Diophantus of Alexandria; Chapter 2. Writing and Solving Equations; Summary; 1. The Hindu-Arabic number system; 2. Reception in mediaeval Europe; 3. Solving equations: cubics and beyond; Chapter 3. Construction and Calculation; Summary; 1. Constructions in Greek geometry; 2. `Famous problems' of antiquity; 3. Decimals and logarithms; Chapter 4. Coordinates and Complex Numbers; Summary; 1. Descartes' analytic geometry; 2. Paving the way; 3. Imaginary roots and complex numbers; 4. The fundamental theorem of algebra; Chapter 5. Struggles with the Infinite; Summary; 1. Zeno and Aristotle; 2. Archimedes' `Method'; 3. Infinitesimals in the calculus; 4. Critique of the calculus; Chapter 6. From Calculus to Analysis; Summary; 1. D'Alembert and Lagrange; 2. Cauchy's `Cours d'Analyse'; 3. Continuous functions; 4. Derivative and integral Chapter 7. Number Systems; Summary; 1. Sets of numbers; 2. Natural numbers; 3. Integers and rationals; 4. Dedekind cuts; 5. Cantor's construction of the reals; 6. Decimal expansions; 7. Algebraic and constructible numbers; 8. Transcendental numbers; Chapter 8. Axioms for number systems; Summary; 1. The axiomatic method; 2. The Peano axioms; 3. Axioms for the real number system; 4. Appendix: arithmetic and order in C; Chapter 9. Counting beyond the finite; Summary; 1. Cantor's continuum; 2. Cantor's transfinite numbers; 3. Comparison of cardinals; Chapter 10. Solid Foundations?; Summary; 1. Avoiding paradoxes: the ZF axioms; 2. The axiom of choice; 3. Tribal conflict; 4. Gödel's incompleteness theorems; 5. A logician's revenge?; Epilogue; Bibliography; Name Index; Index.

Notes:

Includes bibliographical references and indexes.

Description based on print version record.

ISBN:

1-80064-097-8

9781800640979

1800640978