86 A History ofMathematics
998877665544332211Fig. 1Rod numbers.Fig. 260390 as a rod number- As in our system, numbers read along the rows, while the various numbers belonging to a sum
(multiplier, multiplicand, product) occupied different rows of the board.
To clarify, here is the rather simple operation of squaring 81, from theSunzi suanjing. This is
no more advanced than theNine Chapters, in fact less so, but in places more explicit. The text
clearly does describesomeprocedure with counting rods. The pictures in rod-numbers are the
reconstruction in Lam and Ang (1992) of how the calculation would have been done, since the
numbers in the text arenotrod-numbers. In the first sentence, for example, ‘nine’ would be
written , and ‘81’ would be. The roman numbers refer to the diagrams below, showing
the progress of the reconstructed calculation.
Nine nines are 81, find the amount when this is multiplied by itself. Answer: 6561.
Method: Set up the two positions [upper and lower] (i). The upper 8 calls the lower 8; eight eights are 64, so put down
6400 in the middle position (ii). The upper 8 calls the lower 1: one eight is 8, so put down 80 in the middle position
(iii). Shift the lower numeral one place [to the right] and putaway the 80 in theupper position (iv). The upper 1 calls
the lower 8; one eight is 8, so put down 80 in the middle position (v). The upper 1 calls the lower 1; one one is 1, so put
down 1 in the middle position (vi). Remove the numerals in the upper and lower positions leaving 6561 in the middle
position (vii). (Lam and Ang 1992, p. 34)The progress of this very simple example is illustrated by the rod-number diagrams (i)–(vii)
(Fig. 3); you should translate these into ‘Arabic’ numbers for yourself. Note that the terms of
the upper number are removed when they are finished with; and that the author takes it for gran-
ted that when you have put down the second 8 (stage v) you use basic rod addition to amalgamate
it with the 648 you have already and get 656.
No one has come up with a better explanation of how the system worked. The firstwritten
records containing rod-numbers used mathematically date from the fifth to tenth centuriesceand
the most coherent ones from much later again. In the meantime, the use of rod-numbers could
have evolved. Martzloff ’s scepticism (it is no more than that) is based on the absence of evidence for
two key assumptions: the use (a) of a ‘board’ to order the calculation, and (b) of blank spaces as a
zero-equivalent at such an early date.
Let us, though, suppose the system granted, as it is widely believed to have been used and
is a reasonable interpretation of the words in theSunzi suanjing. The question of whether this