Counting rods

Counting rods (traditional Chinese: 籌; simplified Chinese: 筹; pinyin: chóu; Japanese: 算木; rōmaji: sangi; Korean: sangaji) are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. Counting rods (traditional Chinese: 籌; simplified Chinese: 筹; pinyin: chóu; Japanese: 算木; rōmaji: sangi; Korean: sangaji) are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals. They are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period (circa 475 BCE) to the 16th century. Chinese arithmeticians used counting rods well over two thousand years ago. In 1954 forty-odd counting rods of the Warring States period (5th century BCE to 221 BCE) were found in Zuǒjiāgōngshān (左家公山) Chu Grave No.15 in Changsha, Hunan. In 1973 archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of the Han dynasty (206 BCE to 220 CE). On one of the wooden scripts was written: '当利二月定算'. This is one of the earliest examples of using counting-rod numerals in writing. In 1976 a bundle of Western Han-era (202 BCE to 9 CE) counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it; Sunzi (c. 544 to c. 496 BCE), a military strategist at the end of Spring and Autumn period of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle; Laozi (died 531 BCE), writing in the Warring States period, said 'a good calculator doesn't use counting rods'. The Book of Han (finished 111 CE) recorded: 'they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces'. At first, calculating rods were round in cross-section, but by the time of the Sui dynasty (581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods for negative numbers. After the abacus flourished, counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for algebra. Counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately 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. It is written in Sunzi Suanjing that 'one is vertical, ten is horizontal'. 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 first century CE, stated '(when using 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'. Later, a go stone was sometimes used to represent zero.