136 6. CALCULATIONwe can illustrate the multiplication 324 · 29 as follows:
324 324 324 24 24
—>6 —>8 7 —>8 7 —>9 1 —•
29 29 29 29 292 4 4 4 4
—>9 2 8 —>9 2 8 —>9 3 6 —>9 3 9 6 —>9 3 9 6
29 29 29 29
The sequence of operations is very easy to understand from this illustration.
First, the larger number is written on top (on the right in Chinese, of course, since
the writing is vertical). The smaller number is written on the bottom (actually on
the left), with its units digit opposite the largest digit of the larger number. Then,
working always from larger denominations to smaller, we multiply the digits one at
a time and enter the products between the two numbers. Once a digit of the larger
number has been multiplied by all the digits of the smaller one, it is "erased" (the
rods are picked up), and the rods representing the smaller number are moved one
place to the right (actually downward). At that point, the process repeats until all
the digits have been multiplied. When that happens, the last digit of the larger
number and all the digits of the smaller number are picked up, leaving only the
product.
Long division was carried out in a similar way. The partial quotients were kept
in the top row, and the remainder at each stage occupied the center row (with
the same caveat as above, that rows are actually columns in Chinese writing). For
example, to get the quotient 438 -f- 7, one proceeds as follows.
6 6 6 2
4 3 8 —>4 3 8 —>4 3 8 —>1 8—»1 8—• 4
7 7 7 7 7 7
The first step here is merely a statement of the problem. The procedure be-
gins with the second step, where the divisor (7) is moved to the extreme left, then
moved rightward until a division is possible. Thereafter, one does simple divisions,
replacing the dividend by the remainder at each stage. The original dividend can
be thought of as the remainder of a fictitious "zeroth" division. Except for the "era-
sures" when the rods are picked up, the process looks very much like the algorithm
taught to school children in the United States. The final display allows the answer
to be read off: 438 -j- 7 = 621. It would be only a short step to replace this last
common fraction by a decimal; all one would have to do is continue the algorithm
as if there were zeros on the right of the dividend. However, no such procedure is
described in the Sun Zi Suan Jing. Instead, the answer is expressed as an integer
plus a proper fraction.
2.1. Fractions and roots. The early Chinese way of handling fractions is much
closer to our own ideas than that of the Egyptians. The Sun Zi Suan Jing gives
a procedure for reducing fractions that is equivalent to the familiar Euclidean al-
gorithm for finding the greatest common divisor of two integers. The rule is to
subtract the smaller number from the larger until the difference is smaller than the
originally smaller number. Then begin subtracting the difference from the smaller
number. Continue this procedure until two equal numbers are obtained. That
number can then be divided out of both numerator and denominator.