2 CHINA 135FIGURE 2. The Shang numerals.call § of a hekat of grain. Since this amount of grain goes into one jug, it follows
that the pesu of that beer is what we call the reciprocal of that number, namely
2 3. The author gives this result immediately, apparently assuming that by now
the reader will know how to "calculate with 4 8 until 1 is reached." The Rule of
Three procedure is invoked in Problem 73, which asks how many loaves of 15-pesw
bread are required to provide the same amount of grain as 100 loaves of 10-pesu
bread. The answer is found by dividing 100 by 10, then multiplying by 15, which
is precisely the Rule of Three.
2. China
In contrast to the Egyptians, who computed with ink on papyrus, the ancient Chi-
nese, starting in the time of the Shang dynasty, used rods representing numerals to
carry out computations. Chinese documents from the second century BCE men-
tion the use of counting rods, and a set of such rods from the first century BCE
was discovered in 1970. The rods can be arranged to form the Shang numerals
(Fig. 2) and thereby represent decimal digits. They were used in conjunction with
a counting board, which is a board ruled into squares so that each column (or row,
depending on the direction of writing) represents a particular item. In pure com-
putations, the successive rows in the board indexed powers of 10. These rods could
be stacked to represent any digit from 1 to 9. Since they were placed on a board in
rows and columns, the empty places are logically equivalent to a use of 0, but not
psychologically equivalent. The use of a circle for zero in China is not found before
the thirteenth century. On the other hand, according to Lam and Ang [1987, p.
102), the concept of negative numbers (fu), represented by black rods instead of
the usual red ones for positive numbers (cheng), was also present as early as the
fourth century BCE.
It is difficult to distinguish between, say, 22 (|| ||) and 4 (||||) if the rods are
placed too close together. To avoid that difficulty, the Chinese rotated the rods in
alternate rows through a right angle, in effect using a positional system based on
100 rather than 10. Since this book is being published in a language that is read
from left to right, then from top to bottom, we shall alternate columns rather than
rows. In our exposition of the system the number 22 becomes = 11 and 4 remains
||||. The Shang numerals are shown in Fig. 2, the top row being used to represent
digits multiplied by an even power of 10 and the bottom row digits multiplied by
an odd power of 10.
Addition and subtraction with rods representing Shang numerals are obvious
operations. Multiplication and division require somewhat more work, and those
procedures are explained in the Sun Zi Suan Jing.
Except that multiplication was carried out starting with the largest denomi-
nations rather than the smallest, the procedure for multiplying digits and carrying
resembles all other systems for multiplying. Using numerals in place of the rods,