Counting rods

Counting rods (筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient East Asia.

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)[1] to the 16th century.

[2][3][failed verification] 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).

[4][5] 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.

[9] 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".

[12] Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter.

Song dynasty mathematician Jia Xian used hand-written Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from a facsimile from a page of Yongle Encyclopedia.

An 18th-century Japanese mathematics book has a checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛" (thousand, hundred, ten, unit, tenth, hundredth, thousandth).

Examples: In Japan, Seki Takakazu developed the rod numerals into symbolic notation for algebra and drastically improved Japanese mathematics.

[17] For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, though multiple zero units could lead to ambiguity, eg.

Unicode 5.0 includes counting rod numerals in their own block in the Supplementary Multilingual Plane (SMP) from U+1D360 to U+1D37F.

The Unicode Standard states that the orientation of the Unicode characters follows Song dynasty convention, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines.

[21] As these were recently added to the character set and since they are included in the SMP, font support may still be limited.