88 A History ofMathematics
favoured medium for writing numbers, and in this sense we could think of a two-stage ‘invention’
of the place-value system. The reader who takes a little time to try out the method of counting rods
might reflect that thewrittendecimal place-value system which we have is not necessarily the best
for all purposes.
On one other point priority is certainly established: the Chinese from an early period were quite
happy with negative numbers, as Westerners were not. Liu is explicit on this; at the point where
theNine Chaptersgive a detailed and helpful ‘Sign Rule’—‘like signs subtract, opposite signs add’—
he supplies a note on procedure:
Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative
[respectively]. Red counting rods are positive, black counting rods are negative. (Shen et al. 1999, p. 404)Martzloff speculates that this ease in dealing with signs may have arisen not simply from the
manipulation of debts, but from the duality underlying Chinese natural philosophy:
For example, astronomers imagined coupling the planet Jupiter with an anti-Jupiter, whose motion was deduced from
the former by inversion; diviners practised a double-sided divination with symmetrically arranged graphics; not to
mention also, of course,yinyangdualism. (Martzloff 1995, p. 200)
If Indians or Westerners ‘borrowed’ the idea of negative numbers at some much later date, they
made more heavy weather of it.Exercise 3. Make your own set of counting rods and try to perform a simple multiplication on the lines
of the one above.6. Matrices
So far we have only looked at the elementary parts of theNine Chapters. This gives a wrong picture
of early Chinese mathematics, which contained some sophisticated procedures—always framed in
terms of straightforward problems with general explanation. Two in particular stand out:- The extraction of roots, a combination of counting-rod and geometrical arguments, which
would lead to more general algebra. - The solution of systems of linear equations by an equivalent of what we call matrices.
Here we shall consider the second. Once again, to translate it into modern terms (‘we are using
matrices’) is clearly a misrepresentation of the procedure of a Han dynasty mathematician using
counting rods; and yet, the comparison of the methods is an interesting one, since we can see
what elements there are in common. The subject is covered in the eighth chapter, ‘Rectangular
Arrays’, orfangcheng; and the title in itself says something about the material. A large number of
the problems concern different grades of paddy, and Liu comments, ‘it is difficult to comprehend in
mere words, so we simply use paddy to clarify’.
This is a fascinating remark, if we think of the question of abstraction. It almost seems as though
Liu is undercutting the apparent concreteness of theNine Chaptersby claiming that he, at least,
could use an abstract language (‘mere words’). This would not, of course, be algebraic symbols, but
given the nature of Chinese mathematics they could be rather similar, as we shall see. Perhaps we
should think of the characters for ‘low-grade paddy’ (or medium, or high) as a more complicated
version of the symbolsx,y, andz.