The Development of Hindu-Arabic and Traditional Chinese Arithmetic
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Chinese Science 13 (1996): 35-54 The Development of Hindu-Arabic and Traditional Chinese Arithmetic Lam Lay Yong [Lam Lay Yong is Professor of Mathematics at the National University of Singapore. She is an Effective Member of the International Academy of the History of Science. Her latest research paper is "Zhang Qiujian suanjing (The Mathematical Classic of Zhang Qiujian): An Overview. '1 * * * The Basics rithmetic is a branch of mathematics with which all of us are familiar. AIrrespective of the countries we come from, we are taught the Hindu Arabic numerals and their place value notation at a very early age. Thereafter we learn how to add, subtract, multiply, and divide with the numerals. We are also taught how to write the common fractions and how to add, subtract, multiply, and divide with them. It was from these basic techniques that arithmetic developed. Arithmetic is an important subject in our schools as its applications have become essential in our everyday life; from the man-in-the-street to the scientist, government officer, and businessman-all of us need to know arithmetic. That arithmetic is useful is clear, but the history of the beginnings of the subject is still obscure. In recent decades, there have been some studies on Arabic texts related to early arithmetical procedures, and these have helped to shed some light. When we compare this knowledge with what we know about the arithmetic of ancient China, what is immediately and impressively apparent are certain strikingly similar features. The aim of this paper is to draw attention to these similarities and to show how vital they were in the development of arithmetic. The Hindu-Arabic numeral system on which arithmetic is built possesses the intrinsic property of a place value notation which has ten as base. The digits of a numeral are arranged from left to right in descending order of ranks. Initially, 35 Downloaded from Brill.com10/04/2021 06:55:27PM via free access 36 Chinese Science 13 (1996) the system employed only nine signs which represented the numbers from one to nine. For example, the number sixty-five thousand three hundred and ninety two was written in the form 65392 so that the signs representing six, five, three, nine, and two were written in a horizontal line from left to right. The place occupied by each digit indicated the rank of that digit; in this example, the ranks of the digits 6, 5, 3, 9, and 2 were ten thousands, thousands, hundreds, tens, and units, respectively. A number such as sixty-five thousand three hundred and two was indicated by the appropriate signs for 6, 5, 3, and 2 which occupied their due places, while the place which represented the tens rank was left vacant. This blank space was called sunya in India and sifr in the Arab world, both words meaning empty. 1 A tenth sign, which was usually in the form of the zero symbol (occasionally it was a dot), was introduced later to occupy this vacant place. The rules for manipulating the Hindu-Arabic numerals and the processes of calculation were known in the West through Latin translations of Arabic manuscripts from the twelfth century. The procedure of computation was called "algorism." This word is generally accepted as associated with the name of Muhammad ibn Musa al-Khwarizmi of the ninth century. The Latin text of Cambridge University Library Ms. Ii.vi.5, dating from the thirteenth century, is generally accepted as a copy of an earlier Latin translation of an Arabic manuscript based on the arithmetic text by al-Khwarizmi. The incomplete manuscript has only sixteen pages2 of which the first four and a half pages are devoted to a description of the Hindu-Arabic numerals with detailed explanations of the place value notation. Except for the last part on fractions and two short paragraphs on halving and doubling and casting out nines, the rest of the manuscript is concerned with the operations of addition, subtraction, multiplication, and division. An English translation of the part describing the division of 46468 by 324 is reproduced below so as to give the reader some idea of how the process of division was described and performed.3 And know that division is similar to multiplication, but this is done inversely, because in division we subtract and there we add, i.e., in multiplication is its 1 Karl Menninger, Number Words and Number Symbols: A Cultural History of Numbers, translated by Paul Broneer from the revised 1958 German edition (Cambridge, Mass.: MIT Press, 1969), pp. 400-401; Tobias Dantzig, Number: The Language of Science (London: George Allen & Unwin, 1930, 3rd ed. 1947), pp. 29-30; Louis C. Karpinski, The History ofArithmetic (Chicago: Rand McNally, 1925), p. 41. 2 The manuscript occupies folios I 04r-l I Iv, which were previously numbered as I 02- 109 in A Catalogue of the Manuscripts Preserved in the Library of the University of Cambridge, vol. 3 (Cambridge: Cambridge Univ. Press, 1858), pp. 500-501. 3 John N. Crossley and Alan S. Henry, "Thus Spake Al-Khwarizmi: A Translation of the Text of Cambridge University Library Ms. Ii. vi.5," Historia Mathematica vol. 17, no. 2 (May 1990), pp. 118-19. Downloaded from Brill.com10/04/2021 06:55:27PM via free access Lam Lay Yong: Hindu-Arabic and Traditional Chinese Arithmetic 37 model. But when we wanted to divide forty-six thousand and four hundred and sixty-eight by three hundred and XXIIII, we first put eight on the right side, then we put six toward the left which is sixty, then IIII which is four hundred, then six which is VI thousand, then IIII which is forty-thousand. Of these places the last toward the left will be III!, and the first of them toward the right eight; after this write under them the number by which you are dividing, and write the last place of the number by which you are dividing, which is the form of three and is three hundred, under the last place of the upper number which is IIII, provided that it is less than that which is above it: and if it were more than it, we shall move it by one place and put it under the six. After this we shall put in that place, that comes after the three, the form of two which is XX, beneath the six; then we shall put in that place, which is beneath the 1111, likewise !III and this will be their form: /46468) \324 After this to begin with let us write one in the column of the first place of the number by which you are dividing, above the upper number that you are dividing which is four. And if we had put it beneath the IIII, it would be equally appro priate. Let us multiply it (i.e., one) by three, and we shall subtract it (i.e., 3) from that which is above it, and there will remain one. Then let us multiply it by two and subtract it (i.e., 2) from that which is above it, which is VI, and there will remain IIII. After this let us multiply it again by IIII and subtract it (i.e., 4) from that which is above it, which is 1111 and nothing will remain; and we shall put a circle in its place. Next move the beginning of the number by which you are dividing, i.e., IIII, beneath VI and there will be two beneath the circle and III beneath IIII. Then write in the column of the lower number something in the column [sic, should be "row"] of the one, i.e., IIII, and multiply it by three and there will be XII; and subtract it (i.e., 12) from that which is above three which is XIIII and there will remain II; after this multiply also !III itself by the two that follows the three and there will be VIII, and subtract it (i.e., 8) from that which is above it, that is XX and there will remain XII, i.e., two above the II and one above the three. Again multiply IIII by IIII which comes next on the right, and there will be XVI; and subtract it (i.e., 16) from that which is above it which is CXXVI and there will remain above the !III a circle and above the two, one, and above the three, one. Again move the number by which you are dividing, i.e., III!, beneath VIII, and there will be two beneath the circle and three beneath the one; next write three in the column of IIII above the upper number that you are dividing in the row of IIII and one (i-.e., 143), multiply it (i.e., 3) by three, and there will be IX; and subtract it (i.e., 9) from that which is above three which is XI and there will remain two above the three. Multiply also the three by the two that follows the three and there will be VI, and subtract it (i.e., 6) from that which is above the three, which is XX; there will remain XIIII. Once more multiply the aforesaid three by III! that follows the two and there will be XII, and Downloaded from Brill.com10/04/2021 06:55:27PM via free access 38 Chinese Science 13 (1996) subtract it from that which is above it, which is CXXVIII ... ;4 and there will remain six above the IIII, and three above the two, and one above the three.