Numbers: Their Tales, Types, and Treasures.

Chapter 3: Numbers in History

3.5.THE CHINESE PLACE-VALUE NOTATION

About two thousand years ago, during the Han dynasty, the Chinese developed another numeral system, based on the representation of numbers on a counting board. A counting board was an early calculator—a sort of checkerboard with square fields in which counting rods (little sticks made of bamboo and sometimes even of ivory) were arranged to symbolize numbers. It was easy to rearrange the rods in a field in order to represent different numbers, and this had to be done frequently during a calculation. Later, the counting rods found their way into writing, in two different, closely related forms: one in a vertical layout and one in a horizontal layout. The numerals in the vertical layout are shown in figure 3.9, the numbers in the horizontal layout are shown in figure 3.10.

Figure 3.9: Rod numbers in the vertical layout.

Figure 3.10: Rod numbers in the horizontal layout.

These rod-number symbols were either written or represented with counting rods placed in the square fields of a counting board. Here, the rightmost column symbolized the number of units, the next column to the left contained the tens, and then the hundreds went into the next column, and so on.

In principle, the number 2345 could now be represented, for example, by using rod numbers in the vertical layout, putting II III IIII IIIII into adjacent squares, but you can see the problem that this could cause, if by chance one of the sticks slid over into the next square. This would create the risk of confusion, changing the configuration, for example, to II III IIIII IIII, or 2354. The solution was simple and elegant—namely to arrange the sticks alternatingly in vertical and horizontal layout, as shown in figure 3.11. Typically, one would start with the vertical layout for the units, followed by the horizontal placement for the tens, and so on.

Figure 3.11: Rod number notation.

The rod numerals are, thus, written in a place-value system. The value of a “digit” depends on the column in which it is written. As long as the digits were placed within square fields, there was no need for a special symbol for zero because the square corresponding to a missing digit was simply left empty, as in figure 3.12

Figure 3.12: Empty space instead of “zero.”

The written notation often omitted the squares around the digits, and they were moved closer together. The missing symbol for zero usually was not a problem because two adjacent symbols in horizontal (or vertical) layout would indicate a “missing digit” in between. The symbol for zero was introduced to China in the eighth century through the influence of Indian scholars. Still, the alternating vertical-horizontal layout was kept as in , which means 106929. Rod numbers in this style were in use for many centuries, not only in China, but also in Japan, Korea, and Vietnam. Figure 5.10 in section 5.9 shows a Chinese example of the use of rod numerals from the thirteenth century CE.