Counting rods (筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancientEast Asia. They are placed either horizontally or vertically to represent anyinteger orrational number.

The written forms based on them are calledrod numerals. They are a truepositional numeral system withdigits for 1–9 and a blank for 0, from theWarring states period (circa 475 BCE)^[1] to the 16th century.

Chinese arithmeticians used counting rods well over two thousand years ago.

In 1954, forty-odd counting rods of theWarring States period (5th century BCE to 221 BCE) were found inZuǒjiāgōngshān (左家公山)Chu Grave No.15 inChangsha,Hunan.^{[2][3][failed verification]}

In 1973, archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of theHan dynasty (206 BCE to 220 CE). On one of the wooden scripts was written: "当利二月定算𝍥".^{[citation needed]} This is one of the earliest examples of using counting-rod numerals in writing.

A square lacquer box, dating from c. 168 BCE, containing a square chess board with the TLV patterns, chessmen, counting rods, and other items, was excavated in 1972, fromMawangdui M3, Changsha, Hunan Province.^[4][5]

In 1976, a bundle ofWestern Han-era (202 BCE to 9 CE) counting rods made of bones was unearthed fromQianyang County inShaanxi.^[6][7] The use of counting rods must predate it;Sunzi (c. 544 toc. 496 BCE), a military strategist at the end ofSpring and Autumn period of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle;^[8]Laozi (died 531 BCE), writing in the Warring States period, said "a good calculator doesn't use counting rods".^[9] TheBook 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".^[10]

At first, calculating rods were round in cross-section, but by the time of theSui dynasty (581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods fornegative numbers.^{[citation needed]}

After theabacus flourished^[when?], counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation foralgebra.

Counting rods represent digits by the number of rods, and theperpendicular 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 inSunzi Suanjing that "one is vertical, ten is horizontal".^[11]

Red rods representpositive numbers and black rods representnegative numbers.^[12] 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".^[13][14] Later, ago stone was sometimes used to represent zero.

This alternation of vertical and horizontal rod numeral form is very important to understanding written transcription of rod numerals on manuscripts correctly. For instance, in Licheng suanjin, 81 was transcribed as, and 108 was transcribed as; it is clear that the latter clearly had a blank zero on the "counting board" (i.e., floor or mat), even though on the written transcription, there was no blank. In the same manuscript, 405 was transcribed as , with a blank space in between for obvious reasons, and could in no way be interpreted as "45". In other words, transcribed rod numerals may not be positional, but on the counting board, they are positional.  is an exact image of the counting rod number 405 on a table top or floor.

The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position (i.e., to the tens place) gives 9[] or 90. Shifting left again to the third position (to the hundreds place) gives 9[][] or 900. Each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10. This applies to single-digit numbers or multiple-digit numbers.

Song dynasty mathematicianJia 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 ofYongle Encyclopedia. He arranged 七萬一千八百二十四 as

七一八二四

萬千百十步

He treated the Chinese order numbers as place value markers, and 七一八二四 became place value decimal number. He then wrote the rod numerals according to their place value:

七	一	八	二	四
萬	千	百	十	步

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. 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).^[15]

Zero and positive integers

	0	1	2	3	4	5	6	7	8	9
Vertical
Horizontal

Zero and negative numbers

	0	−1	−2	−3	−4	−5	−6	−7	−8	−9
Vertical
Horizontal

Examples:

231
5089
−407
−6720

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 are drawn shorter to have the same character height.

A circle (〇) is used for 0. Many historians think it was imported fromIndian numerals byGautama Siddha in 718,^[13] but some think it was created from the Chinese text space filler "□", and others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for "nothing".^[16]

In the 13th century,Southern Song mathematicians changed digits for 4, 5, and 9 to reduce strokes.^[16] The new horizontal forms eventually transformed intoSuzhou numerals. Japanese continued to use the traditional forms.

Positive numbers (traditional)

	0	1	2	3	4	5	6	7	8	9
Vertical
Horizontal

Negative numbers (traditional)

	0	−1	−2	−3	−4	−5	−6	−7	−8	−9
Vertical
Horizontal

Positive numbers (Southern Song)

	0	1	2	3	4	5	6	7	8	9
Vertical
Horizontal

Examples:

	Traditional	Southern Song
231
5089
−407
−6720

In Japan,Seki Takakazu developed the rod numerals into symbolic notation for algebra and drastically improvedJapanese mathematics.^[13] After his period, thepositional numeral system usingChinese numeral characters was developed, and the rod numerals were used only for theplus and minus signs.

Western	Seki	After Seki
x +y + 246	甲乙	甲乙二四六
5x − 6y	甲乙	五甲六乙
7xy	甲乙	七甲乙
8x /y	N/A	乙八甲[dubious –discuss]

A fraction was expressed with rod numerals as two rod numerals one on top of another (without any other symbol, like the modern horizontal bar).^{[citation needed]}

The method for using counting rods for mathematical calculation was calledrod calculation orrod calculus (筹算). Rod calculus can be used for a wide range of calculations, including finding the value ofπ, findingsquare roots,cube roots, orhigher order roots, and solving asystem of linear equations.

Before the introduction of a written zero, a space was used to indicate no units, and the rotation of the character in the subsequent unit column, by 90°, adopted, to help reduce the ambiguity in record values calculated on the rods.^[17] For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, though multiple zero units could lead to ambiguity, eg. 1007 (𝍩 𝍧) , and 10007 (𝍠 𝍧). Once written zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods, after its replacement byabacus. One variation of horizontal rod numerals, theSuzhou numerals is still in use for book-keeping and in herbal medicine prescription inChinatowns in some parts of the world.

Unicode 5.0 includes counting rod numerals in their own block in theSupplementary Multilingual Plane (SMP) from U+1D360 to U+1D37F. Thecode 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 calledunit digits and the latter are calledtens digits,^[18][19] which is opposite of the convention described above. 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.^[20] 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).^[21] As these were recently added to the character set and since they are included in the SMP, font support may still be limited.

For a look of the ancient counting rods, and further explanation, you can visit the sites

• https://web.archive.org/web/20010217175749/http://www.math.sfu.ca/histmath/China/Beginning/Rod.html
• http://mathforum.org/library/drmath/view/52557.html
• Counting rods in China (in Chinese) (Translate to English: Google, Bing, Yandex)
• Counting rods and go stones of a Japanese mathematician around 1872 (in Japanese) (Translate to English: Google, Bing, Yandex)

• Chinese inventions
• Chinese mathematics
• Japanese mathematics
• Korean mathematics
• Mathematical tools
• Numerals
• Science and technology in China

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