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4.1 POSITIONAL NUMBER SYSTEMS 183 1685), 18-22, 301. The fact that any integer greater than 1 could serve as radix was apparently first stated in print by Blaise Pascal in De numeris multiplicibus, which was written about 1658 [see Pascal s QXuvres Compl&tes (Paris: hditions de Seuil, 1963), 84-891. Pascal wrote, Denaria enim ex institute hominum, non ex necessitate nature ut vulgus arbitratur, et sane satis inepte, posita est ; i.e., The decimal system has been established, somewhat foolishly to be sure, according to man s custom, not from a natural necessity as most people would think. He stated that the duodecimal (radix twelve) system would be a welcome change, and he gave a rule for testing a duodecimal number for divisibility by nine. Erhard Weigel tried to drum up enthusiasm for the quaternary (radix four) system in a series of publications beginning in 1673. A detailed discussion of radix-twelve arithmetic was given by Joshua Jordaine, Duodecimal Arithmetick (London, 1687). Although decimal notation was almost exclusively used for arithmetic during that era, other systems of weights and measures were rarely if ever based on multiples of 10, and many business transactions required a good deal of skill in adding quantities such as pounds, shillings, and pence. For centuries merchants had therefore learned to compute sums and differences of quantities expressed in peculiar units of currency, weights, and measures; and this was actually arithmetic in a nondecimal number system. The common units of liquid measure in England, dating from the 13th century or earlier, are particularly noteworthy: 2 gills = 1 chopin 2 demibushels = 1 bushel or firkin 2 chopins = 1 pint 2 firkins = 1 kilderkin 2 pints = 1 quart 2 kilderkins = 1 barrel 2 quarts = 1 pottle 2 pottles = 1 gallon 2 barrels = 1 hogshead 2 gallons = 1 peck 2 hogsheads = 1 pipe 2 pecks = 1 demibushel 2 pipes = 1 tun Quantities of liquid expressed in gallons, pottles, quarts, pints, etc. were essen- tially written in binary notation. Perhaps the true inventors of binary arithmetic were English wine merchants! The first known appearance of pure binary notation was about 1605 in some unpublished manuscripts of Thomas Harriot (1560-1621). Harriot was a creative man, who first became famous by coming to America as a representative of Sir Walter Raleigh. He invented (among other things) a notation like that now used for less than and greater than relations; but for some reason he chose not to publish many of his discoveries. Excerpts from his notes on binary arithmetic have been reproduced by John W. Shirley, Amer. J. Physics 19 (1951), 452-454. The first published discussion of the binary system was given in a comparatively little-known work by a Spanish bishop, Juan Caramuel Lobkowitz, Mathesis biceps 1 (Campan&, 1670), 45-48; Caramuel discussed the representation of numbers in radices 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, and 60 at some length, but gave no examples of arithmetic operations in nondecimal systems (except for the trivial operation of adding unity).

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,

4.1 POSITIONAL NUMBER SYSTEMS 181 Our decimal notation, which differs from the more ancient forms primarily because of its fixed radix point, together with its symbol for zero to mark an empty position, was developed first in India within the Hindu culture. The exact date when this notation first appeared is quite uncertain; about 600 A.D. seems to be a good guess. Hindu science was rather highly developed at that time, particularly in astronomy. The earliest known Hindu manuscripts that show this notation have numbers written backwards (with the most significant digit at the right), but soon it became standard to put the most significant digit at the left. About 750 A.D., the Hindu principles of decimal arithmetic were brought to Persia, as several important works were translated into Arabic; a picturesque account of this development is given in a Hebrew document, which has been translated into English in AA4M 15 (1918), 99-108. Not long after this, al- Khwarizmi wrote his Arabic textbook on the subject. (As noted in Chapter 1, our word algorithm comes from al-Khwbrizmi s name.) His work was trans- lated into Latin and was a strong influence on Leonardo Pisano (Fibonacci), whose book on arithmetic (1202 A.D.) played a major role in the spreading of Hindu-Arabic numerals into Europe. It is interesting to note that the left-to-right order of writing numbers was unchanged during these two transitions, although Arabic is written from right to left while Hindu and Latin scholars generally wrote from left to right. A detailed account of the subsequent propagation of decimal numeration and arithmetic into all parts of Europe during the period from 1200 to 1600 A.D. has been given by David Eugene Smith in his History of Mathematics 1 (Boston: Ginn and Co., 1923), Chapters 6 and 8. Decimal notation was applied at first only to integer numbers, not to frac- tions. Arabic astronomers, who required fractions in their star charts and other tables, continued to use the notation of Ptolemy (the famous Greek astronomer), a notation based on sexagesimal fractions. This system still survives today, in our trigonometric units of degrees, minutes, and seconds, and also in our units of time, as a remnant of the original Babylonian sexagesimal notation. Early European mathematicians also used sexagesimal fractions when dealing with noninteger numbers; for example, Fibonacci gave the value 1 22 7 42 33N 4v 40 as an approximation to the root of the equation x3 + 2s2 + lOa: = 20. (The correct answer is lo 22 7 42 33N 4 38M 30W1 50* 151x 43x . . . .) The use of decimal notation also for tenths, hundredths, etc., in a similar way seems to be a comparatively minor change; but, of course, it is hard to break with tradition, and sexagesimal fractions have an advantage over decimal fractions in that numbers such as 4 can be expressed exactly, in a simple way. Chinese mathematicians-who never used sexagesimals-were apparently the first people to work with the equivalent of decimal fractions, although their numeral system (lacking zero) was not originally a positional number system in the strict sense. Chinese units of weights and measures were decimal, so that Tsu Chhung-Chih (who died c. 500 A.D.) was able to express an approximation

180 ARITHMETIC 4.1 During the twentieth century, historians of mathematics have made exten- sive studies of early cuneiform tablets found by archeologists in the Middle East. These studies show that the Babylonian people actually had two distinct systems of number representation: Numbers used in everyday business transactions were written in a notation based on grouping by tens, hundreds, etc.; this notation was inherited from earlier Mesopotamian civilizations, and large numbers were seldom required. When more difficult mathematical problems were considered, however, Babylonian mathematicians made extensive use of a sexagesimal (radix sixty) positional notation that was highly developed at least as early as 1750 B.C. This notation was unique in that it was actually a Aoatingpoint form of representation with exponents omitted; the proper scale factor or power of sixty was to be sup- plied by the context, so that, for example, the numbers 2, 120, 7200, and & were all written in an identical manner. The notation was especially convenient for multiplication and division, using auxiliary tables, since radix-point alignment had no effect on the answer. As examples of this Babylonian notation, consider the following excerpts from early tables: The square of 30 is 15 (which may also be read, The square of 4 is a ); the reciprocal of 81 = (1 21)ao is (44 26 4O)eo; and the square of the latter is (32 55 18 31 6 4O)so. The Babylonians had a sym- bol for zero, but because of their floating point philosophy, it was used only within numbers, not at the right end to denote a scale factor. For the interesting story of early Babylonian mathematics, see 0. Neugebauer, The Exact Sciences in Antiquity (Princeton, N. J.: Princeton University Press, 1952), and B. L. van der Waerden, Science Awakening, tr. by A. Dresden (Groningen: P. Noordhoff, 1954); see also D. E. Knuth, CACM 15 (1972), 671-677; 19 (1976), 108. Fixed point positional notation was apparently first conceived by the Maya Indians in central America 2000 years ago; their radix-20 system was highly developed, especially in connection with astronomical records and calendar dates. But the Spanish conquerors destroyed nearly all of the Maya books on history and science, so we have comparatively little knowledge about how sophisticated the native Americans had become at arithmetic; special-purpose multiplication tables have been found, but no examples of division are known [cf. J. Eric S. Thompson, Contributions to Amer. Anthropology and History 7 (Carnegie Inst. of Washington, 1942), 37-621. Several centuries before Christ, the Greek people employed an early form of the abacus to do their arithmetical calculations, using sand and/or pebbles on a board that had rows or columns corresponding in a natural way to our decimal system. It is perhaps surprising to us that the same positional notation was never adapted to written forms of numbers, since we are so accustomed to reckoning with the decimal system using pencil and paper; but the greater ease of calculating by abacus (since handwriting was not a common skill, and since abacus calculation makes it unnecessary to memorize addition and multiplication tables) probably made the Greeks feel it would be silly even to suggest that computing could be done better on scratch paper. At the same time Greek astronomers did make use of a sexagesimal positional notation for fractions, which they had learned from the Babylonians.