Anmerkungen:
Includes bibliographical references (pages 259-260) and index
Beschreibung:
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 -- Blank Page -- Blank Page
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