Writing numbers

The procedure of the Celestial Source testifies not only of a discursive computation, but also of a process based on manipulation of peculiar tool: counting rods on a counting support. The presence of tools for computation next to the text is testified by the usage of vocabulary indicating manipulation. Here, I will present the way of writing numbers. Two systems for writing numbers are used: a system based on the representation of counting rods (rod numerals) and a system based on natural language.

The procedure of Celestial Source and the Sections of Pieces [of Areas] both use the decimal system of writing numbers using the actual characters 一, 二, 三, 四, 五, 六, 七, 八, 九 for 1, 2, 3, 4, 5, 6, 7, 8, 9 with the character indicating the position: 十, 百, 千, 萬 for 10, 100, 1000, 10000; and 分, 釐 for tenths and cents. In the procedure of the Celestial Source, this system is used to transcript dimensions of areas or distances, therefore those are only positive quantities. The negative quantities appear only in coefficient of polynomials transcript with rod numerals. As the procedure of Section of Pieces [of Areas] relies on another type of procedure, the rod numerals transcription is never used, and consequently negative quantities are also transcript in natural language. For example, in the art of celestial source: 一千二百一十二萬七千五百 is 12127500. Quantities in the Yigu yanduan never exceed10⁷. In my translation, as this numeration is borrowed from natural language, I translated these quantities with their names in letters.

70 Sometime a zero is used in this numeration system like a place value zero: 二萬二千 二百 O 二步八分八釐 is 22202.88. In this example from the pb.3, the markers of units and hundreds should be sufficient to show that there are no tens. If one writes 二萬二千二百二 步八分八釐, the quantity is the same. For two cases concerning fractions, one read: 零一十 四分, 0.14, where 0 is named by the character ling, 零. These cases are seldom and most of the time the zero is not used at all. For decimal quantities usually the zero is not written, like 0.47 is 四分七釐.

Another decimal system, this time place value, is used only in the art of the celestial source to represent results from computation with polynomials, where the different coefficients are represented in columns. This place-value system is inspired from representation of numbers with counting rods. The counting rods, called chou, 籌¹⁴⁵, were probably small bamboo rods arranged into different configurations to represent numbers and perform calculations. Up to now no reliable evidence has been found to determine when the counting rods started to be used, on which kind of support (table, board?), and how they were like in the Song-Yuan dynasty¹⁴⁶. Historians already reconstructed how they were used for calculation and the dating of this system¹⁴⁷. In my translation these numbers are transcript with Indo-Arabic numbers. Here follows a table representing the numbers, vertical and horizontal strokes are written alternatively to indicate values:

Line 1: 1, 2, 3, 4, 5, 6, 7, 8, 9 for unit, cents and ten thousands Line2: 1, 2, 3, 4, 5, 6, 7, 8, 9 for tens and thousands

Line 3: 12317

145 [Li Yan, Du Shiran, 1987], p. 6.

146 [Li Yan, Du Shiran, 1987], p.8. The counting rods were small bamboo rods. In august 1971 more than 30 rods of 140 mm were excavated dating back to the time of the emperor Xuan (73-49 BC) from the western Han dynasty in Shanxi. In the 1975, in Hubei, a bundle of rods were unearthed dated back from the reign of emperor Wen (179-157 BC). In 1978, a quantity of earthenware with the signs and marks of rods dating from the time the warring states period (475-221 BC) was found in Henan. In the Lü Li Zhi chapter of the Sui shu (隋書.

Memoir on the calendar, chapter of History of the Sui dynasty, 7^th century) there is also reported: “to calculate one uses bamboo, two fen wide, three inch long”. That is 70 mm according to [Li Yan, Du Shiran, 1987], p.7.

The counting rods were gradually shortened. But as no later artifacts were excavated since, we do not know how the rods were like in the Song-Yuan dynasties. See also [Needham, Joseph, 1955], p.365. [Martzloff Jean-Claude, 1987], p. 194.

147[Volkov Alexei, 2001] Alexei Volkov wrote that the system of counting rods takes its place at the 3^rd century, the latest.

See also [Guo Shuchun, 郭書春,1991], pp. 26-27; [Li Yan, Du Shiran, 1987], pp. 6-24 [Chemla Karine, 1982], p. 4.3. [Chemla Karine, 1996] and [Chemla Karine, Guo Shuchun, 2004], pp. 15-20.

71 Numbers are set in one column, one above the other. In the Ceyuan haijing, when a number is too long markers of positions, 步, 十, 百, 千, 萬 are sometime written under the

digits. For example¹⁴⁸: This polynomial contains the character 億, yi, “billion” under

the fourth line. . This polynomial contains the character 百, bai, “hundreds” under the second line.

This system is never used in the Yigu yanduan, even for big quantities. Only the character 步 is sometimes marked down to indicate the place of the unit when the reading

of a number could be ambiguous. Like in the following example (1): ; this number is 392. This marker, 步, appears in two cases: either when the column contains only one row of number, or when one has to write a decimal quantity in one of the rows. Like in the next

two examples (2) ; the number in the first row is 2070.25; and (3) in the last line is read: 1.47. But this marker is not systematically used for each of the decimal quantities.

When expressions are clearly written without any ambiguity concerning the place of the

unit, the character 步 is absent. For example (4): ; the number in the last row is 0.96.

Indeed, if a quantity starts with a zero, it means that one has a decimal quantity and the character 步 does not need to be written.

148 Ceyuan haijing, Zhibuzu zhai congshu edition, ch.5. p. 15 for both two examples (p. 798 of Guo Shuchun’s collectanea). It seems, at first sight, that this process of marking positions is more frequently used in the Ming dynasty edition than in Li Rui’s edition.

72 These examples above (2) (3) (4) also show how the place value 0 is used to mark

empty position. This last example shows zero at terminal position (5): ¹⁴⁹.

In the Yigu yanduan, the diagonal stroke added by Li Rui in the counting rod system is the only kind of sign used for negative quantities. We noticed that numbers written in natural language do not need to be marked by any sign for the reason that all these numbers are measures of lengths of segment or of areas. These quantities are always positive.

Concerning other mention of negative quantity, one of the coefficient of polynomials written in counting rods in pb.11b, is qualified by the character “negative” (負, fu). The problem 11.b has for peculiarity the polynomials are described discursively. Li Ye enumerates the different rows of the polynomial and signals that one of the coefficients is negative: 三百三十九步 O 八釐負, that is -339.08. It is the only occurrence of this character in the procedure of the Celestial Source in the Yigu yanduan, and the presence of this character denotes the insistence of Li Ye to differentiate positive coefficient from negative ones.