ChineseMathematics 87
[i] [ii] [iii][v] [vi] [vii][iv]Fig. 3The stages in calculating 81×81 by rod numbers.constitutes an ‘invention’ of the decimal place-value number system, like other priority questions,
then becomes serious, and needs some clarification. The counting rods, in some form, were certainly
being used about a thousand years before our first certain record of ‘Indian’ place-value numbers
(ninth centurycesee next chapter). By that time, as we have seen, the Buddhists had established
a fairly regular route for pilgrims between China and India, and in the seventh century the most
famous of them, Xuan-Tsang, was defending his country’s civilization to an Indian audience:
They have taken the Heavens as their model, and they know how to calculate the movements of the Seven Luminaries;
they have invented all kinds of instruments, fixed the seasons of the year, and discovered the hidden properties of the
six tones and of music. (Needham 1954, p. 210)
Although the shapes of Indian–Arabic numbers are quite unlike those of counting-rod numbers,
Lim and Ang argue that theideaof decimal place-value computation must have been transmitted
from China, probably to India, since the way of perfoming calculations in the earliest textbooks—
in particular the Arabic texts of al-Khw ̄arizm ̄i and al-Uql ̄idis ̄i—is almost identical. Only the form
of the symbols from 1 to 9 was changed, with the zero or dot being devised for the empty space.
Again, this is plausible, although the records which would establish it may never be found.
However, as we have noticed, the counting-rod numbers, until the Tang dynasty (about 600ce),
retained the status not of a way of writing numbers but of a way ofworkingwith them. The
early texts say what you must do, but nowhere do they even draw a picture of the counting-rod
number, much less insert one into the text. The counting-rod numbers, in their early life, were
‘fleeting’ (to borrow Lam and Ang’s term) and dynamic, there to be erased and transformed as
the example shows.^9 Later, (perhaps from the tenth centuryce—we canot be sure) they were
written in mathematical textbooks to illustrate procedure; and in the thirteenth century a zero
symbol was introduced^10 but even so they did not become the current way of writing numbers.
If the idea of using a decimal number representation did diffuse to India, then it underwent an
important change, since the system became something which (a) was written down and (b) became
theprincipal representation of a number in everyday transactions. There may then have been
diffusion, for example, of the zero symbol, in the other direction, from India or the Middle East to
China. Assuming that the ‘Indian’ numbers did derive from the rods, they changed in becoming the
- Interestingly, it seems that the early use of Indian numbers in the Islamic world involved erasure—they were traced on the
‘dust-board’ to which al-Uql ̄idis ̄i refers (see next chapter). - By Qin Jiushao, apparently (see later).