Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient

East Asia East Asia is the eastern region of Asia, which is defined in both Geography, geographical and culture, ethno-cultural terms. The modern State (polity), states of East Asia include China, Japan, Mongolia, North Korea, South Korea, and Taiwan. ...

. 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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which th ...

with digits for 1–9 and a blank for 0, from the Warring states period (circa 475 BCE) to the 16th century.

History

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𝍥". 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, from Mawangdui M3, Changsha, Hunan Province. 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 Sun Tzu ( ; zh, t=孫子, s=孙子, first= t, p=Sūnzǐ) was a Chinese military general, strategist, philosopher, and writer who lived during the Eastern Zhou period of 771 to 256 BCE. Sun Tzu is traditionally credited as the author of ''The ...

( 544 to 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 The ''Book of Han'' or ''History of the Former Han'' (Qián Hàn Shū,《前汉书》) is a history of China finished in 111AD, covering the Western, or Former Han dynasty from the first emperor in 206 BCE to the fall of Wang Mang in 23 CE. ...

'' (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 The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...

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

Using counting rods

Seki Kowa Katsuyo Sampo Bernoulli numbers

Counting rods represent digits by the number of rods, and the

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...

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 ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...

'' that "one is vertical, ten is horizontal". Red rods represent

positive number In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...

s 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 ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...

, 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. 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.

Place value

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 mathematician

Jia Xian Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty. Biography According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...

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 The ''Yongle Encyclopedia'' () or ''Yongle Dadian'' () is a largely-lost Chinese ''leishu'' encyclopedia commissioned by the Yongle Emperor of the Ming dynasty in 1403 and completed by 1408. It comprised 22,937 manuscript rolls or chapters, in 1 ...

. 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: 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). Examples:

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 are drawn shorter to have the same character height. A circle (〇) is used for 0. Many historians think it was imported from

Indian numerals Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...

Gautama Siddha Gautama Siddha, (fl. 8th century) astronomer, astrologer and compiler of Indian descent, known for leading the compilation of the '' Treatise on Astrology of the Kaiyuan Era'' during the Tang Dynasty. He was born in Chang'an, and his family was ori ...

in 718, 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''. In the 13th century, Southern Song mathematicians changed digits for 4, 5, and 9 to reduce strokes. The new horizontal forms eventually transformed into

Suzhou numerals The Suzhou numerals, also known as ' (), is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numerals are also known as ' (), ' (), ' (), ' () and ' (). History The Suzhou numeral system is the only survivin ...

. Japanese continued to use the traditional forms.

Examples: In Japan,

Seki Takakazu , Selin, Helaine. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890 also known as ,Selin, was a Japanese mathematician and author of the Edo period. Seki laid foundations for the subs ...

developed the rod numerals into symbolic notation for algebra and drastically improved

Japanese mathematics denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...

. After his period, the

using Chinese numeral characters was developed, and the rod numerals were used only for the

plus and minus signs The plus and minus signs, and , are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, represents the operation of addition, which results in a sum, while represents subtraction, resul ...

Fractions

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).

Rod calculus

The method for using counting rods for mathematical calculation was called ''rod calculation'' or

rod calculus Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Ro ...

(筹算). Rod calculus can be used for a wide range of calculations, including finding the value of , finding square roots, cube roots, or higher order roots, and solving a system of linear equations. Before the introduction of written zero, there was no way to distinguish 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 written zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods, after its replacement by

. One variation of horizontal rod numerals, the

is still in use for book-keeping and in herbal medicine prescription in Chinatowns in some parts of the world.

Unicode

Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...

5.0 includes counting rod numerals in their own block in the

Supplementary Multilingual Plane In the Unicode standard, a plane is a continuous group of 65,536 (216) code points. There are 17 planes, identified by the numbers 0 to 16, which corresponds with the possible values 00–1016 of the first two positions in six position hexadecimal ...

(SMP) from U+1D360 to U+1D37F. The

code point In character encoding terminology, a code point, codepoint or code position is a numerical value that maps to a specific character. Code points usually represent a single grapheme—usually a letter, digit, punctuation mark, or whitespace—but ...

s 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 digits'', 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. 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). As these were recently added to the character set and since they are included in the SMP, font support may still be limited.

References

External links

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 * * {{DEFAULTSORT:Counting Rods Chinese inventions Chinese mathematics Japanese mathematics Korean mathematics Mathematical tools Numerals Science and technology in China