Includes bibliographical references (pages 295-301) and index.

The system of numbers : an overview: From natural to real numbers ; Imaginary numbers ; Polynomials and transcendental numbers ; Cardinals and ordinals -- Writing mumbers : now and back then: Writing numbers nowadays : positional and decimal ; Writing numbers back then : Egypt, Babylon and Greece -- Numbers and magnitudes in the Greek mathematical tradition: Pythagorean numbers ; Ratios and proportions ; Incommensurability ; Eudoxus' theory of proportions ; Greek fractional numbers ; Comparisons, not measurements ; A unit length ; Appendix 3.1: The incommensurability of 2 -- ancient and modern proofs ; Appendix 3.2: Eudoxus' theory of proportions in action ; Appendix 3.3: Euclid and the area of the circle -- Construction problems and numerical problems in the Greek mathematical tradition: The arithmetic books of the elements ; Geometric algebra? ; Straightedge and compass ; Diophantus' numerical problems ; Diophantus' reciprocals and fractions ; More than three dimensions ; Appendix 4.1: Diophantus' solution of Problem V.9 in Arithmetica -- Numbers in the tradition of medieval Islam: Islamicate science in historical perspective ; Al-Khwārizmī and numerical problems with squares ; Geometry and certainty ; Al-jabr wa'l-muqābala ; Al-Khwārizmī, numbers and fractions ; Abū Kāmil's numbers at the crossroads of two traditions ; Numbers, fractions and symbolic methods ; Al-Khayyām and numerical problems with cubes ; Gersonides and problems with numbers ; Appendix 5.1: The quadratic equation. Derivation of the algebraic formula ; Appendix 5.2: The cubic equation. Khayyam's geometric solution -- Numbers in Europe from the twelfth to the sixteenth centuries: Fibonacci and Hindu-Arabic numbers in Europe ; Abbacus and coss traditions in Europe ; Cardano's Great art of algebra ; Bombelli and the roots of negative numbers ; Euclid's Elements in the Renaissance ; Appendix 6.1: Casting out nines -- Number and equations at the beginning of the scientific revolution: Viète and the new art of analysis ; Stevin and decimal fractions ; Logarithms and the decimal system of numeration ; Appendix 7.1: Napier's construction of logarithmic tables -- Number and equations in the works of Descartes, Newton and their contemporaries: Descartes' new approach to numbers and equations ; Wallis and the primacy of algebra ; Barrow and the opposition to the primacy of algebra ; Newton's Universal arithmetick ; Appendix 8.1: The quadratic equation. Descartes' geometric solution ; Appendix 8.2: Between geometry and algebra in the seventeenth century : the case of Euclid's Elements -- New definitions of complex numbers in the early nineteenth century: Numbers and ratios : giving up metaphysics ; Euler, Gauss and the ubiquity of complex numbers ; Geometric interpretations of the complex numbers ; Hamilton's formal definition of complex numbers ; Beyond complex numbers ; Hamilton's discovery of quaternions -- "What are numbers and what should they be?" : understanding numbers in the late nineteenth century: What are numbers? ; Kummer's ideal numbers ; Fields of algebraic numbers ; What should numbers be? ; Numbers and the foundations of calculus ; Appendix 10.1: Dedekind's theory of cuts and Eudoxus' theory of proportions ; Appendix 10.2: IVT and the fundamental theorem of calculus -- Exact definitions for the natural numbers : Dedekind, Peano and Frege: The principle of mathematical induction ; Peano's postulates ; Dedekind's chains of cardinal numbers -- Appendix 11.1: The principle of induction and Peano's postulates -- Numbers, sets and infinity : a conceptual breakthrough at the turn of the twentieth century: Dedekind, Cantor and the infinite ; Infinities of various sizes ; Cantor's transfinite ordinals ; Troubles in paradise ; Appendix 12.1: Proof that the set of algebraic numbers is countable -- Epilogue: Numbers in historical perspective.

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. --