A brief history of numbers

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.