##### Decimal Numbers

One of the first things that leaps out at you when you examine the *Nine Chapters* is that although it’s an ancient work, it uses a “modern” decimal number system.

Decimal numeration arose, along with the earliest writing of Chinese characters, during the transition from slave-ownership to feudalism. In China this transition occurred during the Spring and Autumn (770-476 BCE) and the Warring States (475-221 BCE) periods, a time when the most advanced societies in Europe, those of Greece and Rome, were still based on slavery. Later, in medieval times, decimal place-value numeration diffused outward from Asia, spreading from India through the Middle East and North Africa to Europe.

The Chinese counting board is a good example of how a technological invention can influence how science develops, and even how people think. The counting board, in use by 400 BCE, was made of polished wood and had rulings that formed a grid of square cells (see illustration on this page). Since Chinese characters are written in columns, proceeding from right to left, it was natural to adapt such columns to represent numbers according to their units, tens, hundreds, etc., from right to left. Each of the digits 1 through 9 was represented by its own Chinese character made with tally-like strokes. These digit characters were formed easily with counting rods (*chousuan*), which were small bamboo sticks having a square or triangular cross-section, and varnished with lacquer. Rods with a red dot were used for positive numbers, and those with a black dot for negative numbers. To “write” a number on the counting board, its digit characters could be placed, one per cell, on one row of the grid. (A blank cell stood for what we would call the zero digit.) To do an arithmetic problem, two or more numbers could be placed on neighboring rows, and the results calculated on successive rows of the board, much as we might do today in working a lengthy addition or multiplication on paper.

You can see how a merchant or a scholar who started doing arithmetic on such a board would quickly acquire the mental habit of breaking quantities into their components column-by-column. This became ingrained into Chinese culture. The abacus, which wasn’t widely used in East Asia until medieval times, retained the decimal place-value system in its columns of beads.

For more permanent records, numerals and other characters could also be written directly on plain strips of bamboo or on bone, tortoise shell, silk fabric, or paper, the latter invented in China in the First Century CE. (Paper, silk fabric, and lacquer were all Chinese inventions.)

No old drawings of counting boards survive from China, but counting boards like the one for which a paper facsimile is shown above were known to have been used in Edo Period Japan (1603-1867) and furthermore were known to have been introduced into that country from China. The *sanghi* rods placed on the board above show the numbers 23, 144, 48, and 1. (This image shows about a quarter of the board. The full image appears in *Convergence*'s Mathematical Treasures: Japanese Mathematics in the Edo Period and is presented courtesy of the National Diet Library in Japan.)

##### System of Units

The decimal number system also influenced Chinese units of measurement.

In 221 BCE the ruler Shi Huangdi, based in the Wei Valley, had quickly conquered several nearby warring states and unified the region, ushering in China’s first empire, the Qin Dynasty (221-201 BCE). The Great Wall was completed under the Qin Dynasty. To help ensure the administrative efficiency and cultural unity needed in the empire, the government imposed a standard system of characters for writing as well as a standard system of units for measurement. Thus, China’s uniform system of units was due to the early centralization of its political rule.

###### Units of Length:

The units adopted in the Qin and later dynasties included those for length, based on the* zhang*, approximately 7½ feet:

1 *zhang* = 10 *chi* = 100 *cun*;

for longer distances, the *li* or “Chinese mile” was used, equal to 180* zhang*.

###### Units of area:

Units of area were applied particularly for measuring farm plots, where a man’s “pace” or *bu* (60 *cun*) was a handy reference. The units of area were as follows:

1 *qing* = 100 *mu* = 24,000 square *bu*;

the *qing* was equivalent to about 11.4 of our acres.

###### Units of volume:

1 *hu* = 10 *dou* = 100 *sheng*;

the *hu* was equivalent to about 5.3 of our gallons.

###### Units of weight:

1 *jin* = 16 *liang* = 384 *zhu*;

the *jin* was equivalent to a little over 7 of our ounces.