The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system.[1] The exact origin of the abacus is still unknown. Today, abacuses are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal.

Abacuses come in different designs. Some designs, like the bead frame consisting of beads divided into tens, are used mainly to teach arithmetic, although they remain popular in the post-Soviet states as a tool. Other designs, such as the Japanese soroban, have been used for practical calculations even involving several digits. For any particular abacus design, there usually are numerous different methods to perform a certain type of calculation, which may include basic operations like addition and multiplication, or even more complex ones, such as calculating square roots. Some of these methods may work with non-natural numbers (numbers such as 1.5 and 34).

Although today many use calculators and computers instead of abacuses to calculate, abacuses still remain in common use in some countries. Merchants, traders and clerks in some parts of Eastern Europe, Russia, China and Africa use abacuses, and they are still used to teach arithmetic to children.[1] Some people who are unable to use a calculator because of visual impairment may use an abacus.

A Chinese abacus, Suanpan
Houghton Typ 520.03.736 - Margarita philosophica
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (inaccurately represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.[1]


The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus. The Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank.[2][3][4] Alternatively, without reference to ancient texts on etymology, it has been suggested that it means "a square tablet strewn with dust",[5] or "drawing-board covered with dust (for the use of mathematics)"[6] (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς abakos). Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all and in fact state that it is not proven.[7][nb 1] Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq (אבק), "dust" (or in post-Biblical sense meaning "sand used as a writing surface").[8]

The preferred plural of abacus is a subject of disagreement, with both abacuses[9] and abaci[9] (hard "c") in use. The user of an abacus is called an abacist.[10]



The period 2700–2300 BC saw the first appearance of the Sumerian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[11]

Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus.[12] It is the belief of Old Babylonian[13] scholars such as Carruccio that Old Babylonians "may have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations".[14]


The use of the abacus in Ancient Egypt is mentioned by the Greek historian Herodotus, who writes that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek left-to-right method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument have not been discovered.[15]


During the Achaemenid Empire, around 600 BC the Persians first began to use the abacus.[16] Under the Parthian, Sassanian and Iranian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India, China, and the Roman Empire, when it is thought to have been exported to other countries.


Salaminische Tafel Salamis Tablet nach Wilhelm Kubitschek Numismatische Zeitschrift Bd 31 Wien 1899 p. 394 ff
An early photograph of the Salamis Tablet, 1899. The original is marble and is held by the National Museum of Epigraphy, in Athens.

The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC.[17] Also Demosthenes (384 BC–322 BC) talked of the need to use pebbles for calculations too difficult for your head.[18][19] A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius mention men that sometimes stood for more and sometimes for less, like the pebbles on an abacus.[19] The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world.

A tablet found on the Greek island Salamis in 1846 AD (the Salamis Tablet), dates back to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm (59 in) long, 75 cm (30 in) wide, and 4.5 cm (2 in) thick, on which are 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, capped with a semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line.[20] Also from this time frame the Darius Vase was unearthed in 1851. It was covered with pictures including a "treasurer" holding a wax tablet in one hand while manipulating counters on a table with the other.[18]


Abacus 6
A Chinese abacus (suanpan) (the number represented in the picture is 6,302,715,408)
Literal meaning"calculating tray"

The earliest known written documentation of the Chinese abacus dates to the 2nd century BC.[21]

The Chinese abacus, known as the suanpan (算盤, lit. "calculating tray"), is typically 20 cm (8 in) tall and comes in various widths depending on the operator. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads each in the bottom. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not.[22] The suanpan can be reset to the starting position instantly by a quick movement along the horizontal axis to spin all the beads away from the horizontal beam at the center.

The prototype of the Chinese abacus is the appeared during the Han Dynasty, and the beads are oval. In the Song Dynasty or before used the 4:1 type or four beads abacus similar to the modern abacus or commony known as Japanese style abacus, "you can make a number by hand," and "beads are counted", which can be expressed as a decimal number. Therefore, the abacus is designed as a four-bead abacus.

In the early Ming Dynasty, the abacus began to appear in the form of 1:5 abacus. The upper deck had one bead and the bottom had five beads. "you can make a number by hand," and "the number of beads will be counted". Binary or any of the following numbers, so the abacus is designed as a five-bead abacus.

In the late Ming Dynasty, the abacus styles that appeared in the form of 2:5. The upper deck had two beads, and the bottom had five beads. "You can make a number by hand," and "Beads are counted." It can be expressed in hexadecimal or any of the following numbers, and because the calculation method at that time is a Chinese catty equal to sixteen tael(一斤十六兩)which means hexadecimal, the abacus is designed as a two-five bead.

Suanpan can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed. There are currently schools teaching students how to use it.

In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty (960–1297), a suanpan is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao).

The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, as there is some evidence of a trade relationship between the Roman Empire and China. However, no direct connection can be demonstrated, and the similarity of the abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Korean and Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. (Incidentally, this allows use with a hexadecimal numeral system, which was used for traditional Chinese measures of weight.) Instead of running on wires as in the Chinese, Korean, and Japanese models, the beads of Roman model run in grooves, presumably making arithmetic calculations much slower.

Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a place holder. The zero was probably introduced to the Chinese in the Tang dynasty (618–907) when travel in the Indian Ocean and the Middle East would have provided direct contact with India, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.


Copy of a Roman abacus

The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles (calculi) were used. Later, and in medieval Europe, jetons were manufactured. Marked lines indicated units, fives, tens etc. as in the Roman numeral system. This system of 'counter casting' continued into the late Roman empire and in medieval Europe, and persisted in limited use into the nineteenth century.[23] Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe once again during the 11th century[24][25] This abacus used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster.[26]

Writing in the 1st century BC, Horace refers to the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus.[27]

One example of archaeological evidence of the Roman abacus, shown here in reconstruction, dates to the 1st century AD. It has eight long grooves containing up to five beads in each and eight shorter grooves having either one or no beads in each. The groove marked I indicates units, X tens, and so on up to millions. The beads in the shorter grooves denote fives –five units, five tens etc., essentially in a bi-quinary coded decimal system, related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions).


The decimal number system invented in India replaced the abacus in Western Europe.[28]

The Abhidharmakośabhāṣya of Vasubandhu (316-396), a Sanskrit work on Buddhist philosophy, says that the second-century CE philosopher Vasumitra said that "placing a wick (Sanskrit vartikā) on the number one (ekāṅka) means it is a one, while placing the wick on the number hundred means it is called a hundred, and on the number one thousand means it is a thousand". It is unclear exactly what this arrangement may have been. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus.[29] Hindu texts used the term śūnya (zero) to indicate the empty column on the abacus.[30]


Japanese soroban

In Japanese, the abacus is called soroban (算盤, そろばん, lit. "Counting tray"), imported from China in the 14th century.[31] It was probably in use by the working class a century or more before the ruling class started, as the class structure did not allow for devices used by the lower class to be adopted or used by the ruling class.[32] The 1/4 abacus, which is suited to decimal calculation popular appeared circa 1930, and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China.

Today's Japanese abacus is a 1:4 type, four-bead abacus was introduced from China in the Muromachi era. It adopts the form of the upper deck one bead and the bottom four beads. The top bead on the upper deck was equal to five and the bottom one is equal to one like the Chinese or Korean abacus, and the decimal number can be expressed, so the abacus is designed as one four abacus. The beads are always in the shape of a diamond. The quotient division is generally used instead of the division method; at the same time, in order to make the multiplication and division digits consistently use the division multiplication. Later, Japan had a 3:5 abacus called天三算盤, which is now the Ize Rongji collection of Shansi Village in Yamagata City. There were also had 2:5 beads abacus. With the four-bead abacus spread, it is also common to use Japanese abacus around the world. There are also improved Japanese abacus in various places. One of the Japanese-made abacus made in China is an aluminum frame plastic bead abacus. The file is next to the four beads, and the "clearing" button, press the clearing button, immediately put the upper bead to the upper position, the lower bead is dialed to the lower position, immediately clearing, easy to use.

The abacus is still manufactured in Japan today even with the proliferation, practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation. Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.[33]


The Chinese abacus migrated from China to Korea around 1400 AD.[18][34][35] Koreans call it jupan (주판), supan (수판) or jusan (주산).[36] The four beads abacus( 1:4 ) was introduced to Korea Goryeo Dynaty from the China during Song Dynasty, later the five beads abacus (5:1) abacus was introduced to Korean from China during the Ming Dynasty.

Native American

Representation of an Inca quipu
Yupana 1
A yupana as used by the Incas.

Some sources mention the use of an abacus called a nepohualtzintzin in ancient Aztec culture.[37] This Mesoamerican abacus used a 5-digit base-20 system.[38] The word Nepōhualtzintzin [nepoːwaɬˈt͡sint͡sin] comes from Nahuatl and it is formed by the roots; Ne – personal -; pōhual or pōhualli [ˈpoːwalːi] – the account -; and tzintzin [ˈt͡sint͡sin] – small similar elements. Its complete meaning was taken as: counting with small similar elements by somebody. Its use was taught in the Calmecac to the temalpouhqueh [temaɬˈpoʍkeʔ], who were students dedicated to take the accounts of skies, from childhood.

The Nepōhualtzintzin was divided in two main parts separated by a bar or intermediate cord. In the left part there were four beads, which in the first row have unitary values (1, 2, 3, and 4), and in the right side there are three beads with values of 5, 10, and 15 respectively. In order to know the value of the respective beads of the upper rows, it is enough to multiply by 20 (by each row), the value of the corresponding account in the first row.

Altogether, there were 13 rows with 7 beads in each one, which made up 91 beads in each Nepōhualtzintzin. This was a basic number to understand, 7 times 13, a close relation conceived between natural phenomena, the underworld and the cycles of the heavens. One Nepōhualtzintzin (91) represented the number of days that a season of the year lasts, two Nepōhualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the number of days of a baby's gestation, and four Nepōhualtzintzin (364) completed a cycle and approximate a year (11/4 days short). When translated into modern computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to the 18 in floating point, which calculated stellar as well as infinitesimal amounts with absolute precision, meant that no round off was allowed.

The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo,[39] who in his wanderings throughout Mexico found diverse engravings and paintings of this instrument and reconstructed several of them made in gold, jade, encrustations of shell, etc.[40] There have also been found very old Nepōhualtzintzin attributed to the Olmec culture, and even some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures.

George I. Sanchez, "Arithmetic in Maya", Austin-Texas, 1961 found another base 5, base 4 abacus in the Yucatán Peninsula that also computed calendar data. This was a finger abacus, on one hand 0, 1, 2, 3, and 4 were used; and on the other hand 0, 1, 2 and 3 were used. Note the use of zero at the beginning and end of the two cycles. Sanchez worked with Sylvanus Morley, a noted Mayanist.

The quipu of the Incas was a system of colored knotted cords used to record numerical data,[41] like advanced tally sticks – but not used to perform calculations. Calculations were carried out using a yupana (Quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 an explanation of the mathematical basis of these instruments was proposed by Italian mathematician Nicolino De Pasquale. By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20 and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at a minimum.[42]


Schoty abacus
Russian abacus

The Russian abacus, the schoty (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire, usually positioned near the user, with four beads for quarter-ruble fractions). Older models have another 4-bead wire for quarter-kopeks, which were minted until 1916. The Russian abacus is often used vertically, with wires from left to right in the manner of a book. The wires are usually bowed to bulge upward in the center, to keep the beads pinned to either of the two sides. It is cleared when all the beads are moved to the right. During manipulation, beads are moved to the left. For easy viewing, the middle 2 beads on each wire (the 5th and 6th bead) usually are of a different color from the other eight beads. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color.

As a simple, cheap and reliable device, the Russian abacus was in use in all shops and markets throughout the former Soviet Union, and the usage of it was taught in most schools until the 1990s.[43][44] Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia and likewise the mass production of Felix arithmometers since 1924 did not significantly reduce their use in the Soviet Union.[45] The Russian abacus began to lose popularity only after the mass production of microcalculators had started in the Soviet Union in 1974. Today it is regarded as an archaism and replaced by the handheld calculator.

The Russian abacus was brought to France around 1820 by the mathematician Jean-Victor Poncelet, who served in Napoleon's army and had been a prisoner of war in Russia.[46] The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorismic methods. To Poncelet's French contemporaries, it was something new. Poncelet used it, not for any applied purpose, but as a teaching and demonstration aid.[47] The Turks and the Armenian people also used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians.[48]

School abacus

Early-20th-century abacus used in Danish elementary school.
A twenty bead rekenrek

Around the world, abacuses have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic.

In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame has been common (see image). It is still often seen as a plastic or wooden toy.

The wire frame may be used either with positional notation like other abacuses (thus the 10-wire version may represent numbers up to 9,999,999,999), or each bead may represent one unit (so that e.g. 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented). In the bead frame shown, the gap between the 5th and 6th wire, corresponding to the color change between the 5th and the 6th bead on each wire, suggests the latter use.

The red-and-white abacus is used in contemporary primary schools for a wide range of number-related lessons. The twenty bead version, referred to by its Dutch name rekenrek ("calculating frame"), is often used, sometimes on a string of beads, sometimes on a rigid framework.[49]

Neurological analysis

By learning how to calculate with abacus, one can improve one's mental calculation which becomes faster and more accurate in doing large number calculations. Abacus‐based mental calculation (AMC) was derived from the abacus which means doing calculation, including addition, subtraction, multiplication, and division, in mind with an imagined abacus. It is a high-level cognitive skill that run through calculations with an effective algorithm. People doing long-term AMC training show higher numerical memory capacity and have more effectively connected neural pathways.[50][51] They are able to retrieve memory to deal with complex processes to calculate.[52] The processing of AMC involves both the visuospatial and visuomotor processing which generate the visual abacus and perform the movement of the imaginary bead.[53] Since the only thing needed to be remembered is the finial position of beads, it takes less memory and less computation time.[53]

Renaissance abacuses gallery

Gregor Reisch, Margarita Philosophica, 1508 (1230x1615)
Rechnung auff der Linihen und Federn
Köbel Böschenteyn 1514
Rechnung auff der linihen 1525 Adam Ries
1543 Robert Recorde
Peter Apian 1544
Adam riesen
Rekenaar 1553

Uses by the blind

An adapted abacus, invented by Tim Cranmer, called a Cranmer abacus is still commonly used by individuals who are blind. A piece of soft fabric or rubber is placed behind the beads so that they do not move inadvertently. This keeps the beads in place while the users feel or manipulate them. They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction, square root and cube root.[54]

Although blind students have benefited from talking calculators, the abacus is still very often taught to these students in early grades, both in public schools and state schools for the blind.[55] Blind students also complete mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication and long division problems can be long and difficult. The abacus gives blind and visually impaired students a tool to compute mathematical problems that equals the speed and mathematical knowledge required by their sighted peers using pencil and paper. Many blind people find this number machine a very useful tool throughout life.[54]

Binary abacus

Bbinary Abacus 002
Two binary abacuses constructed by Dr. Robert C. Good, Jr., made from two Chinese abaci

The binary abacus is used to explain how computers manipulate numbers.[56] The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or via ASCII. The device consists of a series of beads on parallel wires arranged in three separate rows. The beads represent a switch on the computer in either an "on" or "off" position.

See also


  1. ^ Both C. J. Gadd, a keeper of the Egyptian and Assyrian Antiquities at the British Museum, and Jacob Levy, a Jewish Historian who wrote Neuhebräisches und chaldäisches wörterbuch über die Talmudim und Midraschim [Neuhebräisches and Chaldean dictionary on the Talmuds and Midrashi] disagree with the "dust table" theory.[7]


  1. ^ a b c Boyer & Merzbach 1991, pp. 252–253
  2. ^ de Stefani 1909, p. 2
  3. ^ Gaisford 1962, p. 2
  4. ^ Lasserre & Livadaras 1976, p. 4
  5. ^ Klein 1966, p. 1
  6. ^ Onions, Friedrichsen & Burchfield 1967, p. 2
  7. ^ a b Pullan 1968, p. 17
  8. ^ Huehnergard 2011, p. 2
  9. ^ a b Brown 1993, p. 2
  10. ^ Gove 1976, p. 1
  11. ^ Ifrah 2001, p. 11
  12. ^ Crump 1992, p. 188
  13. ^ Melville 2001
  14. ^ Carruccio 2006, p. 14
  15. ^ Smith 1958, pp. 157–160
  16. ^ Carr 2014
  17. ^ Ifrah 2001, p. 15
  18. ^ a b c Williams 1997, p. 55
  19. ^ a b Pullan 1968, p. 16
  20. ^ Williams 1997, pp. 55–56
  21. ^ Ifrah 2001, p. 17
  22. ^ Fernandes 2003
  23. ^ Pullan 1968, p. 18
  24. ^ Brown 2010, pp. 81–82
  25. ^ Brown 2011
  26. ^ Huff 1993, p. 50
  27. ^ Ifrah 2001, p. 18
  28. ^ Rowlett, Russ (2004-07-04), Roman and "Arabic" Numerals, University of North Carolina at Chapel Hill, retrieved 2009-06-22
  29. ^ Körner 1996, p. 232
  30. ^ Mollin 1998, p. 3
  31. ^ Gullberg 1997, p. 169
  32. ^ Williams 1997, p. 65
  33. ^ Murray 1982
  34. ^ Anon 2002
  35. ^ Jami 1998, p. 4
  36. ^ Anon 2013
  37. ^ Sanyal 2008
  38. ^ Anon 2004
  39. ^ Hidalgo 1977, p. 94
  40. ^ Hidalgo 1977, pp. 94–101
  41. ^ Albree 2000, p. 42
  42. ^ Aimi & De Pasquale 2005
  43. ^ Burnett & Ryan 1998, p. 7
  44. ^ Hudgins 2004, p. 219
  45. ^ Leushina 1991, p. 427
  46. ^ Trogeman & Ernst 2001, p. 24
  47. ^ Flegg 1983, p. 72
  48. ^ Williams 1997, p. 64
  49. ^ West 2011, p. 49
  50. ^ Hu, Yuzheng; Geng, Fengji; Tao, Lixia; Hu, Nantu; Du, Fenglei; Fu, Kuang; Chen, Feiyan (2010-12-14). "Enhanced white matter tracts integrity in children with abacus training". Human Brain Mapping. 32 (1): 10–21. doi:10.1002/hbm.20996. ISSN 1065-9471. PMID 20235096.
  51. ^ Wu, Tung-Hsin; Chen, Chia-Lin; Huang, Yung-Hui; Liu, Ren-Shyan; Hsieh, Jen-Chuen; Lee, Jason J. S. (2008-11-05). "Effects of long-term practice and task complexity on brain activities when performing abacus-based mental calculations: a PET study". European Journal of Nuclear Medicine and Molecular Imaging. 36 (3): 436–445. doi:10.1007/s00259-008-0949-0. ISSN 1619-7070.
  52. ^ "Brain activation during abacus-based mental calculation with fMRI: a comparison between abacus experts and normal subjects - IEEE Conference Publication". ieeexplore.ieee.org. Retrieved 2018-07-27.
  53. ^ a b Chen, C.L.; Wu, T.H.; Cheng, M.C.; Huang, Y.H.; Sheu, C.Y.; Hsieh, J.C.; Lee, J.S. (2006-12-20). "Prospective demonstration of brain plasticity after intensive abacus-based mental calculation training: An fMRI study". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 569 (2): 567–571. doi:10.1016/j.nima.2006.08.101. ISSN 0168-9002.
  54. ^ a b Terlau & Gissoni 2006
  55. ^ Presley & D'Andrea 2009
  56. ^ Good Jr. 1985, p. 34


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  • Gaisford, Thomas, ed. (1962) [1848]. Etymologicon Magnum seu verius Lexicon Saepissime vocabulorum origines indagans ex pluribus lexicis scholiastis et grammaticis anonymi cuiusdam opera concinnatum [The Great Etymologicon: Which Contains the Origins of the Lexicon of Words from a Large Number or Rather with a Great Amount of Research Lexicis Scholiastis and Connected Together by the Works of Anonymous Grammarians] (in Latin). Amsterdam, The Netherlands: Adolf M. Hakkert.
  • Good Jr., Robert C. (Fall 1985). "The Binary Abacus: A Useful Tool for Explaining Computer Operations". Journal of Computers in Mathematics and Science Teaching. 5 (1): 34–37.
  • Gove, Philip Babcock, ed. (1976). "abacist". Websters Third New International Dictionary (17th ed.). Springfield, MA: G. & C. Merriam Company. ISBN 978-0-87779-101-0.
  • Gullberg, Jan (1997). Mathematics: From the Birth of Numbers. Illustrated by Pär Gullberg. New York, NY: W. W. Norton & Company. ISBN 978-0-393-04002-9.
  • Hidalgo, David Esparza (1977). Nepohualtzintzin: Computador Prehispánico en Vigencia [The Nepohualtzintzin: An Effective Pre-Hispanic Computer] (in Spanish). Tlacoquemécatl, Mexico: Editorial Diana.
  • Hudgins, Sharon (2004). The Other Side of Russia: A Slice of Life in Siberia and the Russian Far East. Eugenia & Hugh M. Stewart '26 Series on Eastern Europe. Texas A&M University Press. ISBN 978-1-58544-404-5.
  • Huehnergard, John, ed. (2011). "Appendix of Semitic Roots, under the root ʾbq.". American Heritage Dictionary of the English Language (5th ed.). Houghton Mifflin Harcourt Trade. ISBN 978-0-547-04101-8.
  • Huff, Toby E. (1993). The Rise of Early Modern Science: Islam, China and the West (1st ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0-521-43496-6.
  • Ifrah, Georges (2001). The Universal History of Computing: From the Abacus to the Quantum Computer. New York, NY: John Wiley & Sons, Inc. ISBN 978-0-471-39671-0.
  • Jami, Catherine (1998). "Abacus (Eastern)". In Bud, Robert; Warner, Deborah Jean. Instruments of Science: An Historical Encyclopedia. New York, NY: Garland Publishing, Inc. ISBN 978-0-8153-1561-2.
  • Klein, Ernest, ed. (1966). "abacus". A Comprehensive Etymological Dictionary of the English Language. I: A-K. Amsterdam: Elsevier Publishing Company.
  • Körner, Thomas William (1996). The Pleasures of Counting. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-56823-4.
  • Lasserre, Franciscus; Livadaras, Nicolaus, eds. (1976). Etymologicum Magnum Genuinum: Symeonis Etymologicum: Una Cum Magna Grammatica (in Greek and Latin). Primum: α — άμωσϒέπωϛ. Rome, Italy: Edizioni dell'Ateneo. LCCN 77467964.
  • Leushina, A. M. (1991). The development of elementary mathematical concepts in preschool children. National Council of Teachers of Mathematics. ISBN 978-0-87353-299-0.
  • Melville, Duncan J. (May 30, 2001). "Chronology of Mesopotamian Mathematics". St. Lawrence University. It.stlawu.edu. Archived from the original on June 19, 2014. Retrieved Jun 19, 2014.
  • Mish, Frederick C., ed. (2003). "abacus". Merriam-Webster's Collegiate Dictionary (11th ed.). Merriam-Webster, Inc. ISBN 978-0-87779-809-5.
  • Mollin, Richard Anthony (September 1998). Fundamental Number Theory with Applications. Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press. ISBN 978-0-8493-3987-5.
  • Murray, Geoffrey (July 20, 1982). "Ancient calculator is a hit with Japan's newest generation". The Christian Science Monitor. CSMonitor.com. Archived from the original on July 31, 2014. Retrieved July 31, 2014.
  • Onions, C. T.; Friedrichsen, G. W. S.; Burchfield, R. W., eds. (1967). "abacus". The Oxford Dictionary of English Etymology. Oxford, UK: Oxford at the Clarendon Press.
  • Presley, Ike; D'Andrea, Frances Mary (2009). Assistive Technology for Students who are Blind Or Visually Impaired: A Guide to Assessment. American Foundation for the Blind. p. 61. ISBN 978-0-89128-890-9.
  • Pullan, J. M. (1968). The History of the Abacus. New York, NY: Frederick A. Praeger, Inc., Publishers. ISBN 978-0-09-089410-9. LCCN 72075113.
  • Reilly, Edwin D., ed. (2004). Concise Encyclopedia of Computer Science. New York, NY: John Wiley and Sons, Inc. ISBN 978-0-470-09095-4.
  • Sanyal, Amitava (July 6, 2008). "Learning by Beads". Hindustan Times.
  • Smith, David Eugene (1958). History of Mathematics. Dover Books on Mathematics. 2: Special Topics of Elementary Mathematics. Courier Dover Publications. ISBN 978-0-486-20430-7.
  • Stearns, Peter N.; Langer, William Leonard, eds. (2001). "The Encyclopedia of World History: Ancient, Medieval, and Modern, Chronologically Arranged". The Encyclopedia of World History (6th ed.). New York, NY: Houghton Mifflin Harcourt. ISBN 978-0-395-65237-4.
  • Terlau, Terrie; Gissoni, Fred (July 20, 2006). "Abacus: Position Paper". APH.org. Archived from the original on August 1, 2014. Retrieved July 31, 2014.
  • Trogeman, Georg; Ernst, Wolfgang (2001). Trogeman, Georg; Nitussov, Alexander Y.; Ernst, Wolfgang, eds. Computing in Russia: The History of Computer Devices and Information Technology Revealed. Braunschweig/Wiesbaden: Vieweg+Teubner Verlag. ISBN 978-3-528-05757-2.
  • West, Jessica F. (2011). Number sense routines : building numerical literacy every day in grades K-3. Portland, Me.: Stenhouse Publishers. ISBN 978-1-57110-790-9.
  • Williams, Michael R. (1997). Baltes, Cheryl, ed. A History of Computing technology (2nd ed.). Los Alamitos, CA: IEEE Computer Society Press. ISBN 978-0-8186-7739-7. LCCN 96045232.
  • Yoke, Ho Peng (2000). Li, Qi and Shu: An Introduction to Science and Civilization in China. Dover Science Books. Courier Dover Publications. ISBN 978-0-486-41445-4.

Further reading

  • Fernandes, Luis (2013). "The Abacus: A Brief History". ee.ryerson.ca. Archived from the original on July 31, 2014. Retrieved July 31, 2014.
  • Menninger, Karl W. (1969), Number Words and Number Symbols: A Cultural History of Numbers, MIT Press, ISBN 978-0-262-13040-0
  • Kojima, Takashi (1954), The Japanese Abacus: its Use and Theory, Tokyo: Charles E. Tuttle Co., Inc., ISBN 978-0-8048-0278-9
  • Kojima, Takashi (1963), Advanced Abacus: Japanese Theory and Practice, Tokyo: Charles E. Tuttle Co., Inc., ISBN 978-0-8048-0003-7
  • Stephenson, Stephen Kent (July 7, 2010), Ancient Computers, IEEE Global History Network, arXiv:1206.4349, Bibcode:2012arXiv1206.4349S, retrieved 2011-07-02
  • Stephenson, Stephen Kent (2013), Ancient Computers, Part I - Rediscovery, Edition 2, ISBN 978-1-4909-6437-9

External links


Abacus curiosities

Abacus (architecture)

In architecture, an abacus (from the Greek abax, slab; or French abaque, tailloir; plural abacuses or abaci) is a flat slab forming the uppermost member or division of the capital of a column, above the bell. Its chief function is to provide a large supporting surface, tending to be wider than the capital, as an abutment to receive the weight of the arch or the architrave above. The diminutive of abacus, abaculus, is used to describe small mosaic tiles, also called abaciscus or tessera, used to create ornamental floors with detailed patterns of chequers or squares in a tessellated pavement.

Abacus Institute of Engineering and Management

The Abacus Institute of Engineering and Management or AIEM is an undergraduate engineering college in West Bengal, India. It was established in 2008 under a joint venture of JIS and Techno India Group.The college is affiliated with Maulana Abul Kalam Azad University of Technology and all the programmes are approved by the All India Council for Technical Education.The campus is located at Natungram, Magra, Hooghly.

Bead sort

Bead sort, also called gravity sort, is a natural sorting algorithm, developed by Joshua J. Arulanandham, Cristian S. Calude and Michael J. Dinneen in 2002, and published in The Bulletin of the European Association for Theoretical Computer Science. Both digital and analog hardware implementations of bead sort can achieve a sorting time of O(n); however, the implementation of this algorithm tends to be significantly slower in software and can only be used to sort lists of positive integers. Also, it would seem that even in the best case, the algorithm requires O(n2) space.


A calculation is a deliberate process that transforms one or more inputs into one or more results, with variable change. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to the vague heuristics of calculating a strategy in a competition, or calculating the chance of a successful relationship between two people.

For example, multiplying 7 by 6 is a simple algorithmic calculation. Estimating the fair price for financial instruments using the Black–Scholes model is a more complex algorithmic calculation.

Statistical estimations of the likely election results from opinion polls also involve algorithmic calculations, but produces ranges of possibilities rather than exact answers.

To calculate means to determine mathematically in the case of a number or amount, or in the case of an abstract problem to deduce the answer using logic, reason or common sense. The English word derives from the Latin calculus, which originally meant a small stone in the gall-bladder (from Latin calx). It also meant a pebble used for calculating, or a small stone used as a counter in an abacus (Latin abacus, Greek abax). The abacus was an instrument used by Greeks and Romans for arithmetic calculations, preceding the slide-rule and the electronic calculator, and consisted of perforated pebbles sliding on iron bars.

Capital (architecture)

In architecture the capital (from the Latin caput, or "head") or chapiter forms the topmost member of a column (or a pilaster). It mediates between the column and the load thrusting down upon it, broadening the area of the column's supporting surface. The capital, projecting on each side as it rises to support the abacus, joins the usually square abacus and the usually circular shaft of the column. The capital may be convex, as in the Doric order; concave, as in the inverted bell of the Corinthian order; or scrolling out, as in the Ionic order. These form the three principal types on which all capitals in the classical tradition are based. The Composite order (illustration, right), established in the 16th century on a hint from the Arch of Titus, adds Ionic volutes to Corinthian acanthus leaves.

From the highly visible position it occupies in all colonnaded monumental buildings, the capital is often selected for ornamentation; and is often the clearest indicator of the architectural order. The treatment of its detail may be an indication of the building's date.

Century Media Records

Century Media Records is a heavy metal record label with offices in the United States, Germany and London. In August 2015, Century Media was acquired by Sony Music for US $17 million.


Chisanbop or chisenbop (from Korean chi (ji) finger + sanpŏp (sanbeop) calculation 지산법/指算法) is an abacus-like finger counting method used to perform basic mathematical operations. According to The Complete Book of Chisanbop by Hang Young Pai, chisanbop was created in the 1940s in Korea by Sung Jin Pai and revised by his son Hang Young Pai. He then brought the system to the United States c. 1977. With this method it is possible to display all numbers from 0 to 99 with two hands.

Fossil Wrist PDA

The Fossil Wrist PDA is a smartwatch that runs Palm OS. The newer incarnation, which does not include Palm OS, is called the Fossil WristNet watch.

Global distribution system

A global distribution system (GDS) is a computerised network system owned or operated by a company that enables transactions between travel industry service providers, mainly airlines, hotels, car rental companies, and travel agencies. The GDS mainly uses real-time inventory (for e.g. number of hotel rooms available, number of flight seats available, or number of cars available) to service providers. Travel agencies traditionally relied on GDS for services, products and rates in order to provide travel-related services to the end consumers. Thus, a GDS can link services, rates and bookings consolidating products and services across all three travel sectors: i.e., airline reservations, hotel reservations, car rentals.

GDS is different from a computer reservations system, which is a reservation system used by the service providers (also known as vendors). Primary customers of GDS are travel agents (both online and office-based) to make reservation on various reservation systems run by the vendors. GDS holds no inventory; the inventory is held on the vendor's reservation system itself. A GDS system will have real-time link to the vendor's database. For example, when a travel agency requests a reservation on the service of a particular airline company, the GDS system routes the request to the appropriate airline's computer reservations system.

Goldman Sachs

The Goldman Sachs Group, Inc., is an American multinational investment bank and financial services company headquartered in New York City. It offers services in investment management, securities, asset management, prime brokerage, and securities underwriting.

The bank is one of the largest investment banking enterprises in the world, and is a primary dealer in the United States Treasury security market and more generally, a prominent market maker. The bank also owns Goldman Sachs Bank USA, a direct bank. Goldman Sachs was founded in 1869 and is headquartered at 200 West Street in Lower Manhattan with additional offices in other international financial centers.As a result of its involvement in securitization during the subprime mortgage crisis, Goldman Sachs suffered during the 2007-2008 financial crisis, and received a $10 billion investment from the United States Department of the Treasury as part of the Troubled Asset Relief Program, a financial bailout created by the Emergency Economic Stabilization Act of 2008. The investment was made in November 2008 and was repaid in June 2009.

Former employees of Goldman Sachs have moved on to government positions. Notable examples includes former U.S. Secretaries of the Treasury Robert Rubin and Henry Paulson; current United States Secretary of the Treasury Steven Mnuchin; former chief economic advisor Gary Cohn; European Central Bank President Mario Draghi; former Bank of Canada Governor and current Governor of the Bank of England Mark Carney and the former Prime Minister of Australia Malcolm Turnbull. In addition, former Goldman employees have headed the New York Stock Exchange, the World Bank, and competing banks such as Citigroup and Merrill Lynch.

Hachette Book Group

Hachette Book Group (HBG) is a publishing company owned by Hachette Livre, the largest publishing company in France, and the third largest trade and educational publisher in the world. Hachette Livre is a wholly owned subsidiary of Lagardère Group. HBG was formed when Hachette Livre purchased the Time Warner Book Group from Time Warner on March 31, 2006. Its headquarters are located at 1290 Avenue of the Americas, Midtown Manhattan, New York City. Hachette is considered one of the big-five publishing companies, along with Holtzbrinck/Macmillan, Penguin Random House, HarperCollins, and Simon & Schuster. In one year, HBG publishes approximately 1400+ adult books (including 50-100 digital-only titles), 300 books for young readers, and 450 audio book titles (including both physical and downloadable-only titles). In 2016, the company had 214 books on the New York Times bestseller list, 44 of which reached #1.

Liber Abaci

Liber Abaci (also spelled as Liber Abbaci) ("The Book of Calculation") is a 1202 historic book on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.

Liber Abaci was among the first Western books to describe the Hindu–Arabic numeral system and to use symbols traditionally described as "Arabic numerals". By addressing the applications of both commercial tradesmen and mathematicians, it contributed to convincing the public of the superiority of the system, and the use of these glyphs.Although the book's title has also been translated as "The Book of the Abacus", Sigler (2002) writes that this is an error: the intent of the book is to describe methods of doing calculations without aid of an abacus, and as Ore (1948) confirms, for centuries after its publication the algorismists (followers of the style of calculation demonstrated in Liber Abaci) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals).

The second version of Liber Abaci was dedicated to Michael Scot in 1227 CE. No copies of the first version are known to exist.

Little, Brown and Company

Little, Brown and Company is an American publisher founded in 1837 by Charles Coffin Little and his partner, James Brown, and for close to two centuries has published fiction and nonfiction by American authors. Early lists featured Little Women by Louisa May Alcott, Emily Dickinson's poetry, and Bartlett's Familiar Quotations. As of 2016, Little, Brown & Company is a division of the Hachette Book Group.

Register machine

In mathematical logic and theoretical computer science a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent.

Roman abacus

The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those used by the Greeks and Babylonians. It was the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals.

As Karl Menninger says on page 315 of his book, "For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, gives us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi. The Latin word calx means 'pebble' or 'gravel stone'; calculi are thus little stones (used as counters)."

Both the Roman abacus and the Chinese suanpan have been used since ancient times. With one bead above and four below the bar, the systematic configuration of the Roman abacus is coincident to the modern Japanese soroban, although the soroban is historically derived from the suanpan.


The soroban (算盤, そろばん, counting tray) is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators.


The suanpan (simplified Chinese: 算盘; traditional Chinese: 算盤; pinyin: suànpán), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue. However, the exact design of this suanpan is not known.

Usually, a suanpan is about 20 cm (8 in) tall and it comes in various widths depending on the application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. This configuration is used for both decimal and hexadecimal computation. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. The suanpan can be reset to the starting position instantly by a quick jerk around the horizontal axis to spin all the beads away from the horizontal beam at the center.

Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed.

The modern suanpan has 4+1 beads, colored beads to indicate position and a clear-all button. When the clear-all button is pressed, two mechanical levers push the top row beads to the top position and the bottom row beads to the bottom position, thus clearing all numbers to zero. This replaces clearing the beads by hand, or quickly rotating the suanpan around its horizontal center line to clear the beads by centrifugal force.

The Potato Factory

The Potato Factory is a 1995 fictionalised historical novel by Bryce Courtenay, which was made into a television miniseries in Australia in 2000. The book is the first in a three-part series, followed by Tommo & Hawk and Solomon's Song. The Potato Factory has been the subject of some controversy regarding its historical accuracy and its portrayal of Jewish characters.

The book is based on Ikey Solomon, the so-called "Prince of Fences", and the basis of the Fagin character in the Charles Dickens novel Oliver Twist. Courtenay states in the book's introduction that it is a fictional historical novel based on extensive research, but portrays fictionalised versions of the characters. Author Judith Sackville-O'Donnell, who wrote another book on Ikey Solomon, claimed that the book was inaccurate and anti-Semitic.The book's other main character is a completely fictional woman named Mary Abacus. Abacus goes from serving girl, to prostitute, to high-class madam, to prisoner transported to Tasmania, to successful businesswoman. She gets her name for her outstanding ability to use an abacus.

The story starts in London in the early 19th century. Mary and Ikey start working together as business partners. It follows them as they are separately sent to Tasmania, a penal colony at the time.

The book was made into a four-part miniseries that aired in Australia in 2000.

Standard Mandarin
Hanyu Pinyinsuànpán
Yue: Cantonese
Yale Romanizationsyun-pùhn
Southern Min
Hokkien POJsǹg-pôaⁿ

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