Extrême-Orient, Extrême-Occident 16 - 1994

Large numbers and counting rods

Alexeï Volkov l

This paper treats the question of the "finiteness of numbers" raised in the treatise Shu shu ji yi (early 113rd or mid-VIth century AD), which may have provided an answer to this question through a demonstration of the infinity of the natural numbers. I will argue that the notion of the infinity of numbers also played a role in the debates about the Buddhist concept of reincarnation in early medieval China.

At the same time, this treatise contains descriptions of several counting devices. This fact suggests that the instrumental representation of numbers was somehow related to the notion of number. I will argue that the representation of numbers with the aid of counting rods, widely used at this time, may have allowed Chinese mathematicians to suggest the existence of an infinite number of negative powers of 10 and to perceive the two infinities (of very large and very small numbers) to be fundamentally similar to each other.

1. The finiteness of numbers

The treatise Shu shuji yi 2 (SSJY below) was written by Xu Yue in the late End - early nird century AD (or, perhaps, forged by its commentator Zhen Luan in the Vlth century AD 3). This treatise has been mentioned several times as the source text for certain systems of "large numbers" 4, but the primary question raised in it, that of the "finiteness of numbers", is not so well known 5.

The core part of the treatise runs as follows :

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On the Mount Tai,