WikiPortallus: Counting rods.html

Counting rods (traditional Chinese: 籌; simplified Chinese: 筹; pinyin: chóu; Japanese: 算木, sangi) are small bars, typically 3-14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction.

The written forms based on them are called rod numerals. They are a true positional numeral system with digits for 1-9 and later also for 0.

History

Counting rods were used by ancient Chinese for more than two thousand years. In 1954, forty-odd counting rods of the Warring States Period were found in Zuǒjiāgōngshān (左家公山) Chǔ Grave No.15 in Changsha, Hunan¹².

The use of counting rods must predate it. Laozi, who lived probably in the 4th century BCE, said "a good calculator doesn't use counting rods."³

After the abacus flourished, counting rods were abandoned except in Japan, where rod numerals developed into symbolic notation for algebra.

Using counting rods

Counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternatingly used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. Sun Tzu wrote that "one is vertical, ten is horizontal."⁴

Red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. The Nine Chapters on the Mathematical Art, which was mainly composed in the 1st century CE, stated "(when subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number."⁵⁶ Later, a go stone was sometimes used to represent 0.

In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids.

Rod numerals

Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit. The vertical bar in the horizontal forms 6-9 is drawn shorter to have the same character height.

A circle (〇) is used for 0. Many historians think it was imported from Indian numerals by Gautama Siddha in 718⁵, but some think it was created from the Chinese text space filler "□"⁷.

In the 13th century, Southern Song mathematicians changed digits for 4, 5, and 9 to reduce strokes⁷. The new horizontal forms eventually transformed into Suzhou numerals. Japanese continued to use the traditional forms.

Rod calculus

The method for using counting rods for mathematical calculation was called rod calculation or rod calculus (筹算‎). Rod calculus can be used for a wide range of calculations, including finding the value of π, finding square roots, cube roots, or higher order roots, and solving a system of linear equations. As a result, the character 籌 is extended to connote the concept of planning in Chinese. For example, the science of using counting rods 運籌學 does not refer to counting rods; it means operational research.

Before the introduction of zero, there was no way to separate 10007 and 107 in written forms except by inserting a bigger space between 1 and 7, and so rod numerals were used only for doing calculations with counting rods. Once zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods. One variation of horizontal rod numerals is still in use for book-keeping in Chinatowns in some parts of the world (see the article on Chinese numerals).

Counting rods in Unicode

Unicode 5.0 includes counting rod numerals in their own block in the Supplementary Multilingual Plane (SMP) from U+1D360 to U+1D37F. The code points for the horizontal digits 1-9 are U+1D360 to U+1D368 and those for the vertical digits 1-9 are U+1D369 to U+1D371. The former are called unit digits and the latter are called tens digits⁸, which is opposite of the convention described above. Zero should be represented by U+3007 (〇, ideographic number zero) and the negative sign should be represented by U+20E5 (combining reverse solidus overlay)⁹. As these were recently added to the character set and since they are included in the SMP, font support may still be limited. Grey areas indicate non-assigned code points.

References

^ 先秦时期竹林资源的利用, http://www.yiyangcity.gov.cn/zhuanti/nanzhu/bamboo_produce/20070716094037.htm, retrieved on 16 December 2007
^ 中国独特的计算工具, http://mkd.lyge.cn/zhanzheng/a04/x3/040.htm, retrieved on 16 December 2007
^ 老子: 善數者不用籌策。
^ http://zh.wikisource.org/en/%E5%AD%AB%E5%AD%90%E7%AE%97%E7%B6%93 孫子算經: 先識其位，一從十橫，百立千僵，千十相望，萬百相當。
^ ^a ^b ^c Wáng, Qīngxiáng (1999), Sangi o koeta otoko (The man who exceeded counting rods), Tokyo: Tōyō Shoten, ISBN 4-88595-226-3
^ http://zh.wikisource.org/en/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93 正負術曰: 同名相除，異名相益，正無入負之，負無入正之。其異名相除，同名相益，正無入正之，負無入負之。
^ ^a ^b Qian, Baocong (1964), Zhongguo Shuxue Shi (The history of Chinese mathematics), Beijing: Kexue Chubanshe
^ The Unicode Standard, Version 5.0 - Electronic edition, Unicode, Inc., 2006, pp. 558, http://www.unicode.org/charts/PDF/U1D360.pdf
^ The Unicode Standard, Version 5.0 - Electronic edition, Unicode, Inc., 2006, pp. 499-500, http://www.unicode.org/versions/Unicode5.0.0/ch15.pdf#G28213

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