# The History of Chinese Mathematics

Humphries, Mark, Scott, Paul, Australian Mathematics Teacher

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The Chinese counting system

There were two schemes of notation in use in ancient China.

Rod numeral system

Under this system, the digits for 1 to 9 appeared as

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and the first nine multiples of 10 appeared as

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These symbols are alternated for successive powers of 10:

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Using this notation numbers as large as desired could be represented. For example, 635 814 would appear as

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The problem with this notation is the potential for ambiguity representing numbers with zeros in them. For example,

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could read as 18, or 1008, or 100008. To rectify this, a place in which a zero was required was left blank, however this did not fully solve the problem as it was not necessarily obvious how many zeros a certain gap represented. In around the 13th century, a round symbol O appeared for zero, and so 4096 was written as

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Traditional system

Under this scheme, the multiplicative principle is used. There were distinct symbols for the digits from 1 to 9, and additional symbols for the powers of ten:

The digits from 1 to 9 would be multiplied by the power of ten which follows it, and so

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647 would be written as

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6 x 100 + 4 x 10 + 7

Note that this eliminates the ambiguity described above, as if a certain power of 10 has no value, then it is simply omitted.

For example, 2030 would be written as

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Computation of pi

The Chinese put a lot of emphasis on computing [pi], the ratio of a circle's circumference to its diameter. The computation of [pi] was an area of mathematics where the Chinese were far ahead of the Western world. From the Arithmetic in Nine Sections, the area of a circle is approximated as 3/4 x the square of the diameter, or one-twelfth of the square of the circumference, which is consistent with a value of 3 for [pi].

Much progress was made in computing [pi] during the Post-Han period, which lasted from AD 220 to around AD 600. In the third Century, a general named Wang Fu established the rational approximation 144/45 for [pi], which gives a value of around 3.1511. Slightly after Wang Fu, Liu Hui established the relation 3.1410 < [pi] < 3.1427.

During the 5th Century, Tsu Ch'ung-chi and his son did even better, finding that 3.1415926 < [pi] < 3.1415927 and arriving at the rational approximation [pi] = 355/113, which yields [pi] correct to 6 decimal places. This level of precision was not surpassed until 1420, when [pi] was found correct to 16 places by Jamshid Al-Kashr of Smirkand. Western mathematics did not surpass the approximation of Tsu until around 1600. Tsu's achievements were considered so remarkable that a landmark on the Moon now bears his name.

The approximation of [pi] was done by calucating the perimeter of a polygon of a certain number of sides, inscribed inside a circle of a known diameter. As the number of sides in the polygon is increased, the closer the polygon approximates the circle, and so the closer the approximation for [pi] will be.

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Magic squares

Although they are of little mathematical use, magic squares are an important aspect of ancient Chinese mathematics. Legend has it that magic squares were discovered in around 2200 BC by the Chinese Emperor Yu, who first saw it decorated upon the back of a divine tortoise along the banks of the Yellow River. It is a square array of numerals indicated by knots in strings, as shown below:

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Note that there are black knots for the even numbers (representing yin) and white knots for the odd numbers (representing yang). …

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