Java web server - 182 ARITHMETIC 4.1 to 7r in the following
182 ARITHMETIC 4.1 to 7r in the following form: 3 chang, 1 chhih, 4 tshun, 1 fen, 5 li, 9 hao, 2 miao, 7 hu. Here chang, . . . , hu are units of length; 1 hu (the diameter of a silk thread) equals l/10 miao, etc. The use of such decimal-like fractions was fairly widespread in China after about 1250 A.D. The first known appearance of decimal fractions in a true positional system occurs in a lOth-century arithmetic text written in Damascus by an obscure mathematician named al-Uqlidisf ( the Euclidean ). He used the symbol for a decimal point, for example in connection with a problem about compound interest, the computation of 135 times (l.l)n for 1 2 n 5 5. [See A. S. Saidan, Tee Arithmetic of al-Uqlidisi (Dordrecht: D. Reidel, 1975), 110, 114, 343, 355, 481-485.1 But he did not develop the idea very fully, and his trick was soon forgotten; five centuries passed before decimal fractions were reinvented by a Persian mathematician, al-Kashi, who died c. 1436. Al-Kashi was a highly skillful calculator, who gave the value of 27r as follows, correct to 16 decimal places: integer fractions 0 6 2831853071795865 This was by far the best approximation to 7r known until Ludolph van Ceulen laboriously calculated 35 decimal places during the period 1596-1610. The earliest known example of decimal fractions in Europe occurs in a 15th- century text where, for example, 153.5 is multiplied by 16.25 to get 2494.375; this was referred to as a Turkish method. In 1525, Christof Rudolff of Germany discovered decimal fractions for himself; but like al-Uqlidisi, his work seems to have had little influence. Francois Vi&e suggested the idea again in 1579. Finally, an arithmetic text by Simon Stevin of Belgium, who independently hit on the idea of decimal fractions in 1585, became popular. Stevin s work, and the discovery of logarithms soon afterwards, made decimal fractions commonplace in Europe during the 17th century. [See D. E. Smith, History of Mathematics 2 (Boston: Ginn and Co., 1925), 228-247, and C. B. Boyer, History of Mathematics (New York: Wiley, 1968), for further remarks and references.] The binary system of notation has its own interesting history. Many primi- tive tribes in existence today are known to use a binary or pair system of counting (making groups of two instead of five or ten), but they do not count in a true radix-2 system, since they do not treat powers of 2 in a special manner. See The Diffusion of Counting Practices by Abraham Seidenberg, Univ. Calif. Pub]. in Math. 3 (1960), 215-300, for interesting details about primitive number systems. Another primitive example of an essentially binary system is the conventional musical notation for expressing rhythms and durations of time. Nondecimal number systems were discussed in Europe during the seven- teenth century. For many years astronomers had occasionally used sexagesimal arithmetic both for the integer and the fractional parts of numbers, primarily when performing multiplication [see John Wallis, Deatise of Algebra (Oxford,