Abacus

The abacus (plural abaci or abacuses), additionally alleged a counting frame, is a artful apparatus that was in use in the age-old Abreast East, Europe, China, and Russia, centuries afore the acceptance of the accounting Arabic appearance system.[1] The exact agent of the abacus is still unknown. The abacus about consists of a cardinal of rows of adaptable chaplet or added objects, which represent digits. One of two numbers is set up, and the chaplet are manipulated to apparatus an operation involving a additional cardinal (e.g., addition), or rarely a aboveboard or cubic root.

In age-old use the rows of chaplet could be apart on a collapsed surface, or sliding in grooves. Afterwards the chaplet were fabricated to accelerate on rods of some array congenital into a frame, acceptance faster manipulation. Abacuses are still made, about as a bamboo anatomy with chaplet sliding on wires. In the age-old world, decidedly afore the accession of positional notation, abacuses were a activated artful tool.

There are characteristic avant-garde implementations of the abacus. Some designs, like the bean anatomy consisting of chaplet disconnected into tens, are acclimated mainly to advise arithmetic, although they abide accepted in the post-Soviet states as a tool. Added designs, such as the Japanese soroban, accept been acclimated for activated calculations alike involving numbers of several digits. For any authentic abacus design, there are usually abundant altered methods to accomplish calculations, which may accommodate the four basal operations, and additionally aboveboard and cube roots. Some of these methods assignment with non-natural numbers (numbers such as 1.5 and ​3⁄4).

Although today calculators and computers are usually acclimated instead of abacuses, abacuses still abide in accepted use in some countries. Merchants, traders and clerks in some genitalia of Eastern Europe, Russia, China and Africa use abacuses, and they are still acclimated to advise accession to children.[1] Some bodies who are clumsy to use a calculator because of beheld crime may use an abacus. Etymology

The use of the chat abacus dates from afore 1387 AD, aback a Average English assignment adopted the chat from Latin to alarm a sandboard abacus. The Latin chat came from age-old Greek ἄβαξ (abax) which agency article afterwards base, and improperly, any allotment of ellipsoidal lath or plank.[2][3][4] Alternatively, afterwards advertence to age-old texts on etymology, it has been appropriate that it agency "a aboveboard book broadcast with dust",[5] or "drawing-board covered with dust (for the use of mathematics)"[6] (the exact actualization of the Latin conceivably reflects the animal anatomy of the Greek word, ἄβακoς abakos). While the table broadcast with dust analogue is popular, some disagree, adage that it is not proven.[7][nb 1] Greek ἄβαξ itself is apparently a borrowing of a Northwest Semitic language, conceivably Phoenician, and affiliated with the Hebrew chat ʾābāq (אבק‎), or “dust” (in post-Biblical faculty acceptation "sand acclimated as a autograph surface").[8]

Both abacuses[9] and abaci[9] (soft or adamantine "c") are acclimated as plurals. The user of an abacus is alleged an abacist.[10]
History
Mesopotamian

The aeon 2700–2300 BC saw the aboriginal actualization of the Sumerian abacus, a table of alternating columns which belted the alternating orders of consequence of their sexagesimal cardinal system.[11]

Some advisers point to a appearance in Babylonian cuneiform which may accept been acquired from a representation of the abacus.[12] It is the acceptance of Old Babylonian[13] advisers such as Carruccio that Old Babylonians "may accept acclimated the abacus for the operations of accession and subtraction; however, this archaic accessory accepted difficult to use for added circuitous calculations".[14]
Egyptian

The use of the abacus in Age-old Egypt is mentioned by the Greek historian Herodotus, who writes that the Egyptians manipulated the dust from appropriate to left, adverse in administration to the Greek left-to-right method. Archaeologists accept begin age-old disks of assorted sizes that are anticipation to accept been acclimated as counters. However, bank depictions of this apparatus accept not been discovered.[15]
Persian

During the Achaemenid Empire, about 600 BC the Persians aboriginal began to use the abacus.[16] Beneath the Parthian, Sassanian and Iranian empires, advisers concentrated on exchanging ability and inventions with the countries about them – India, China, and the Roman Empire, aback it is anticipation to accept been exported to added countries.
Greek
An aboriginal photograph of the Salamis Tablet, 1899. The aboriginal is marble and is captivated by the National Museum of Epigraphy, in Athens.

The age-old archaeological affirmation for the use of the Greek abacus dates to the 5th aeon BC.[17] Additionally Demosthenes (384 BC–322 BC) talked of the charge to use dust for calculations too difficult for your head.[18][19] A comedy by Alexis from the 4th aeon BC mentions an abacus and dust for accounting, and both Diogenes and Polybius acknowledgment men that sometimes stood for added and sometimes for less, like the dust on an abacus.[19] The Greek abacus was a table of copse or marble, pre-set with baby counters in copse or metal for algebraic calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Age-old Rome and, until the French Revolution, the Western Christian world.

A book begin on the Greek island Salamis in 1846 AD (the Salamis Tablet), dates aback to 300 BC, authoritative it the oldest counting lath apparent 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 centermost of the book is a set of 5 alongside curve appropriately disconnected by a vertical line, capped with a semicircle at the circle of the bottom-most accumbent band and the distinct vertical line. Below these curve is a avant-garde amplitude with a accumbent able adding it. Below this able is accession accumulation of eleven alongside lines, afresh disconnected into two sections by a band erect to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these curve are apparent with a cantankerous area they bisect with the vertical line.[20] Additionally from this time anatomy the Darius Vase was unearthed in 1851. It was covered with pictures including a "treasurer" captivation a wax book in one duke while manipulating counters on a table with the other.[18]
Chinese
Main article: Suanpan
A Chinese abacus (suanpan) (the cardinal represented in the annual is 6,302,715,408)
Abacus
Traditional Chinese    算盤
Simplified Chinese    算盘
Literal meaning    "calculating tray"
Transcriptions

The age-old accepted accounting affidavit of the Chinese abacus dates to the 2nd aeon BC.[21]

The Chinese abacus, accepted as the suanpan (算盤/算盘, lit. "calculating tray"), is about 20 cm (8 in) alpine and comes in assorted widths depending on the operator. It usually has added than seven rods. There are two chaplet on anniversary rod in the high accouter and bristles chaplet anniversary in the bottom. The chaplet are usually angled and fabricated of a hardwood. The chaplet are counted by affective them up or bottomward arise the beam; chaplet confused against the axle are counted, while those confused abroad from it are not.[22] One of the top chaplet is 5, while one of the basal chaplet is 1. Anniversary rod has a cardinal beneath it, assuming the abode value.The suanpan can be displace to the starting position instantly by a quick movement forth the accumbent arbor to circuit all the chaplet abroad from the accumbent axle at the center.

The ancestor of the Chinese abacus appeared during the Han Dynasty, and the chaplet are oval. The Song Absolutism and beforehand acclimated the 1:4 blazon or four-beads abacus agnate to the avant-garde abacus including the actualization of the chaplet frequently accepted as Japanese-style abacus.[citation needed]

In the aboriginal Ming Dynasty, the abacus began to arise in the anatomy of 1:5 abacus. The high accouter had one bean and the basal had bristles beads.[citation needed]

In the backward Ming Dynasty, the abacus styles appeared in the anatomy of 2:5.[citation needed] The high accouter had two beads, and the basal had bristles beads.

Various adding techniques were devised for Suanpan enabling able calculations. There are currently schools teaching acceptance how to use it.

In the affiliated annal Forth the River During the Qingming Festival corrective by Zhang Zeduan during the Song absolutism (960–1297), a suanpan is acutely arresting beside an annual book and doctor's prescriptions on the adverse of an apothecary's (Feibao).

The affinity of the Roman abacus to the Chinese one suggests that one could accept aggressive the other, as there is some affirmation of a barter accord amid the Roman Authority and China. However, no complete affiliation can be demonstrated, and the affinity of the abacuses may be coincidental, both ultimately arising from counting with bristles fingers per hand. Area the Roman archetypal (like best avant-garde Korean and Japanese) has 4 additional 1 bean per decimal place, the accepted suanpan has 5 additional 2. Incidentally, this allows use with a hexadecimal appearance adjustment (or any abject up to 18) which may accept been acclimated for acceptable Chinese measures of weight. (Instead of active on affairs as in the Chinese, Korean, and Japanese models, the chaplet of Roman archetypal run in grooves, apparently authoritative accession calculations abundant slower.

Another accessible antecedent of the suanpan is Chinese counting rods, which operated with a decimal adjustment but lacked the abstraction of aught as a abode holder. The aught was apparently alien to the Chinese in the Tang absolutism (618–907) aback biking in the Indian Ocean and the Average East would accept provided complete acquaintance with India, acceptance them to access the abstraction of aught and the decimal point from Indian merchants and mathematicians.
Roman
Main article: Roman abacus
Copy of a Roman abacus

The accustomed adjustment of adding in age-old Rome, as in Greece, was by affective counters on a bland table. Originally dust (calculi) were used. Later, and in medieval Europe, jetons were manufactured. Apparent curve adumbrated units, fives, tens etc. as in the Roman appearance system. This adjustment of 'counter casting' affiliated into the backward Roman authority and in medieval Europe, and persisted in bound use into the nineteenth century.[23] Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became broadly acclimated in Europe already afresh during the 11th century[24][25] This abacus acclimated chaplet on wires, clashing the acceptable Roman counting boards, which meant the abacus could be acclimated abundant faster.[26]

Writing in the 1st aeon BC, Horace refers to the wax abacus, a lath covered with a attenuate band of atramentous wax on which columns and abstracts were inscribed application a stylus.[27]

One archetype of archaeological affirmation of the Roman abacus, apparent actuality in reconstruction, dates to the 1st aeon AD. It has eight affiliated grooves complete up to bristles chaplet in anniversary and eight beneath grooves accepting either one or no chaplet in each. The canal apparent I indicates units, X tens, and so on up to millions. The chaplet in the beneath grooves denote fives –five units, bristles tens etc., about in a bi-quinary coded decimal system, accompanying to the Roman numerals. The abbreviate grooves on the appropriate may accept been acclimated for appearance Roman "ounces" (i.e. fractions).
Indian

The Abhidharmakośabhāṣya of Vasubandhu (316-396), a Sanskrit assignment on Buddhist philosophy, says that the second-century CE philosopher Vasumitra said that "placing a wick (Sanskrit vartikā) on the cardinal one (ekāṅka) agency it is a one, while agreement the wick on the cardinal hundred agency it is alleged a hundred, and on the cardinal one thousand agency it is a thousand". It is cryptic absolutely what this adjustment may accept been. About the 5th century, Indian clerks were already award new agency of recording the accommodation of the Abacus.[28] Hindu texts acclimated the appellation śūnya (zero) to announce the abandoned cavalcade on the abacus.[29]
Japanese
Main article: Soroban
Japanese soroban

In Japanese, the abacus is alleged soroban (算盤, そろばん, lit. "Counting tray"), alien from China in the 14th century.[30] It was apparently in use by the alive chic a aeon or added afore the cardinal chic started, as the chic anatomy did not acquiesce for accessories acclimated by the lower chic to be adopted or acclimated by the cardinal class.[31] The 1/4 abacus, which removes the hardly acclimated additional and fifth bean became accepted in the 1940s.

Today's Japanese abacus is a 1:4 type, four-bead abacus was alien from China in the Muromachi era. It adopts the anatomy of the high accouter one bean and the basal four beads. The top bean on the high accouter was according to bristles and the basal one is according to one like the Chinese or Korean abacus, and the decimal cardinal can be expressed, so the abacus is advised as one four abacus. The chaplet are consistently in the actualization of a diamond. The caliber analysis is about acclimated instead of the analysis method; at the aforementioned time, in adjustment to accomplish the multiplication and analysis digits consistently use the analysis multiplication. Later, Japan had a 3:5 abacus alleged 天三算盤, which is now the Ize Rongji accumulating of Shansi Village in Yamagata City. There were additionally 2:5 blazon abacus.

With the four-bead abacus spread, it is additionally accepted to use Japanese abacus about the world. There are additionally bigger Japanese abacus in assorted places. One of the Japanese-made abacus fabricated in China is an aluminum anatomy artificial bean abacus. The book is abutting to the four beads, and the "clearing" button, columnist the allowance button, anon put the high bean to the high position, the lower bean is dialed to the lower position, anon clearing, accessible to use.

The abacus is still bogus in Japan today alike with the proliferation, practicality, and affordability of abridged cyberbanking calculators. The use of the soroban is still accomplished in Japanese primary schools as allotment of mathematics, primarily as an aid to faster brainy calculation. Application beheld adumbration of a soroban, one can access at the acknowledgment in the aforementioned time as, or alike faster than, is accessible with a concrete instrument.[32]
Korean

The Chinese abacus migrated from China to Korea about 1400 AD.[18][33][34] Koreans alarm it jupan (주판), supan (수판) or jusan (주산).[35] The four chaplet abacus( 1:4 ) was alien to Korea Goryeo Absolutism from the China during Song Dynasty, afterwards the bristles chaplet abacus (5:1) abacus was alien to Korean from China during the Ming Dynasty.
Native American
Representation of an Inca quipu
A yupana as acclimated by the Incas.

Some sources acknowledgment the use of an abacus alleged a nepohualtzintzin in age-old Aztec culture.[36] This Mesoamerican abacus acclimated a 5-digit base-20 system.[37] The chat Nepōhualtzintzin [nepoːwaɬˈt͡sint͡sin] comes from Nahuatl and it is formed by the roots; Ne – claimed -; pōhual or pōhualli [ˈpoːwalːi] – the annual -; and tzintzin [ˈt͡sint͡sin] – baby agnate elements. Its complete acceptation was taken as: counting with baby agnate elements by somebody. Its use was accomplished in the Calmecac to the temalpouhqueh [temaɬˈpoʍkeʔ], who were acceptance committed to booty the accounts of skies, from childhood.

The Nepōhualtzintzin was disconnected in two capital genitalia afar by a bar or average cord. In the larboard allotment there were four beads, which in the aboriginal row accept unitary ethics (1, 2, 3, and 4), and in the appropriate ancillary there are three chaplet with ethics of 5, 10, and 15 respectively. In adjustment to apperceive the amount of the agnate chaplet of the high rows, it is abundant to accumulate by 20 (by anniversary row), the amount of the agnate annual in the aboriginal row.

Altogether, there were 13 rows with 7 chaplet in anniversary one, which fabricated up 91 chaplet in anniversary Nepōhualtzintzin. This was a basal cardinal to understand, 7 times 13, a abutting affiliation conceived amid accustomed phenomena, the abyss and the cycles of the heavens. One Nepōhualtzintzin (91) represented the cardinal of canicule that a analysis of the year lasts, two Nepōhualtzitzin (182) is the cardinal of canicule of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the cardinal of canicule of a baby's gestation, and four Nepōhualtzintzin (364) completed a aeon and almost a year (11/4 canicule short). Aback translated into avant-garde computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to the 18 in amphibian point, which affected arch as able-bodied as atomic amounts with complete precision, meant that no annular off was allowed.

The rediscovery of the Nepōhualtzintzin was due to the Mexican architect David Esparza Hidalgo,[38] who in his wanderings throughout Mexico begin assorted engravings and paintings of this apparatus and reconstructed several of them fabricated in gold, jade, encrustations of shell, etc.[39] There accept additionally been begin actual old Nepōhualtzintzin attributed to the Olmec culture, and alike some bracelets of Mayan origin, as able-bodied as a assortment of forms and abstracts in added cultures.

George I. Sanchez, "Arithmetic in Maya", Austin-Texas, 1961 begin accession abject 5, abject 4 abacus in the Yucatán Peninsula that additionally computed agenda data. This was a feel abacus, on one duke 0, 1, 2, 3, and 4 were used; and on the added duke 0, 1, 2 and 3 were used. Note the use of aught at the alpha and end of the two cycles. Sanchez formed with Sylvanus Morley, a acclaimed Mayanist.

The quipu of the Incas was a adjustment of black askance cords acclimated to almanac afterwards data,[40] like avant-garde account sticks – but not acclimated to accomplish calculations. Calculations were agitated out application a yupana (Quechua for "counting tool"; see figure) which was still in use afterwards the acquisition of Peru. The alive assumption of a yupana is unknown, but in 2001 an account of the algebraic base of these instruments was proposed by Italian mathematician Nicolino De Pasquale. By comparing the anatomy of several yupanas, advisers begin that calculations were based application the Fibonacci arrangement 1, 1, 2, 3, 5 and admiral of 10, 20 and 40 as abode ethics for the altered fields in the instrument. Application the Fibonacci arrangement would accumulate the cardinal of grains aural any one acreage at a minimum.[41]
Russian
Russian abacus

The Russian abacus, the schoty (Russian: счёты, plural from Russian: счёт, counting), usually has a distinct slanted deck, with ten chaplet on anniversary wire (except one wire, usually positioned abreast the user, with four chaplet for quarter-ruble fractions). Older models accept accession 4-bead wire for quarter-kopeks, which were minted until 1916. The Russian abacus is about acclimated vertically, with anniversary wire from larboard to appropriate like curve in a book. The affairs are usually angled to appendage advancement in the center, to accumulate the chaplet affianced to either of the two sides. It is austere aback all the chaplet are confused to the right. During manipulation, chaplet are confused to the left. For accessible viewing, the average 2 chaplet on anniversary wire (the 5th and 6th bead) usually are of a altered blush from the added eight beads. Likewise, the larboard bean of the bags wire (and the actor wire, if present) may accept a altered color.

As a simple, bargain and reliable device, the Russian abacus was in use in all shops and markets throughout the above Soviet Union, and the acceptance of it was accomplished in best schools until the 1990s.[42][43] Alike the 1874 apparatus of automated calculator, Odhner arithmometer, had not replaced them in Russia; according to Yakov Perelman, alike in his times, some businessmen attempting to acceptation such accessories into the Russian Authority were accepted to accord up and leave in anguish afterwards actuality apparent the assignment of a accomplished abacus operator.[44] Likewise the accumulation assembly of Felix arithmometers back 1924 did not decidedly abate their use in the Soviet Union.[45] The Russian abacus began to lose acceptance alone afterwards the accumulation assembly of microcalculators had started in the Soviet Union in 1974. Today it is admired as an archaism and replaced by the handheld calculator.

The Russian abacus was brought to France about 1820 by the mathematician Jean-Victor Poncelet, who served in Napoleon's army and had been a captive of war in Russia.[46] The abacus had collapsed out of use in western Europe in the 16th aeon with the acceleration of decimal characters and algorismic methods. To Poncelet's French contemporaries, it was article new. Poncelet acclimated it, not for any activated purpose, but as a teaching and affirmation aid.[47] The Turks and the Armenian bodies additionally acclimated abacuses agnate to the Russian schoty. It was alleged a coulba by the Turks and a choreb by the Armenians.[48]
School abacus
Early-20th-century abacus acclimated in Danish elementary school.
A twenty bean rekenrek

Around the world, abacuses accept been acclimated in pre-schools and elementary schools as an aid in teaching the appearance adjustment and arithmetic.

In Western countries, a bean anatomy agnate to the Russian abacus but with beeline affairs and a vertical anatomy has been accepted (see image). It is still about apparent as a artificial or board toy.

The wire anatomy may be acclimated either with positional characters like added abacuses (thus the 10-wire adaptation may represent numbers up to 9,999,999,999), or anniversary bean may represent one assemblage (so that e.g. 74 can be represented by alive all chaplet on 7 affairs and 4 chaplet on the 8th wire, so numbers up to 100 may be represented). In the bean anatomy shown, the gap amid the 5th and 6th wire, agnate to the blush change amid the 5th and the 6th bean on anniversary wire, suggests the closing use. Teaching multiplication, e.g. 6 times 7 may be represented by alive 7 chaplet on 6 wires.

The red-and-white abacus is acclimated in abreast primary schools for a avant-garde ambit of number-related lessons. The twenty bean version, referred to by its Dutch name rekenrek ("calculating frame"), is about used, sometimes on a cord of beads, sometimes on a adamant framework.[49]
Speed

Eminent physicist Richard Feynman was acclaimed for ability in algebraic calculations. He wrote about an appointment in Brazil with a Japanese abacus expert, who challenged him to acceleration contests amid Feynman's pen and paper, and the abacus. The abacus was abundant faster for addition, somewhat faster for multiplication, but Feynman was faster at division. Aback the abacus was acclimated for a absolutely difficult challenge, cube roots, Feynman won easily, but by a fluke, as the cardinal alleged at accidental was abutting to a cardinal Feynman happened to apperceive was an exact cube, acceptance almost methods to be used.[50]
Neurological analysis

By acquirements how to account with abacus, one can advance one's brainy adding which becomes faster and added authentic in accomplishing ample cardinal calculations. Abacus‐based brainy adding (AMC) was acquired from the abacus which agency accomplishing calculation, including addition, subtraction, multiplication, and division, in apperception with an absurd abacus. It is a high-level cerebral accomplishment that runs through calculations with an able algorithm. Bodies accomplishing abiding AMC training appearance college afterwards anamnesis accommodation and accept added finer affiliated neural pathways.[51][52] They are able to retrieve anamnesis to accord with circuitous processes to calculate.[53] The processing of AMC involves both the visuospatial and visuomotor processing which accomplish the beheld abacus and accomplish the movement of the abstract bead.[54] Back the alone affair bare to be remembered is the final position of beads, it takes beneath anamnesis and beneath ciphering time.[54]
Renaissance abacuses gallery

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Binary abacus
Two bifold abacuses complete by Dr. Robert C. Good, Jr., fabricated from two Chinese abacuses

The bifold abacus is acclimated to explain how computers dispense numbers.[55] The abacus shows how numbers, letters, and signs can be stored in a bifold adjustment on a computer, or via ASCII. The accessory consists of a alternation of chaplet on alongside affairs abiding in three abstracted rows. The chaplet represent a about-face on the computer in either an "on" or "off" position.
Uses by dark people

An acclimatized abacus, invented by Tim Cranmer, alleged a Cranmer abacus is still frequently acclimated by individuals who are blind. A allotment of bendable bolt or elastic is placed abaft the chaplet so that they do not move inadvertently. This keeps the chaplet in abode while the users feel or dispense them. They use an abacus to accomplish the algebraic functions multiplication, division, addition, subtraction, aboveboard base and cube root.[56]

Although dark acceptance accept benefited from talking calculators, the abacus is still actual about accomplished to these acceptance in aboriginal grades, both in accessible schools and accompaniment schools for the blind.[57] Dark acceptance additionally complete algebraic assignments application a braille-writer and Nemeth cipher (a blazon of braille cipher for mathematics) but ample multiplication and affiliated analysis problems can be affiliated and difficult. The abacus gives dark and visually broken acceptance a apparatus to compute algebraic problems that equals the acceleration and algebraic ability appropriate by their sighted aeon application pencil and paper. Many dark bodies acquisition this cardinal apparatus a actual advantageous apparatus throughout life.