The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since

Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

; according to Yakov Perelman. Some businessmen attempting to import calculators into the Russian Empire were known to leave in despair after watching a skilled abacus operator. Likewise, the mass production of Felix

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

Min Multimedia

*

Abacus in Various Number Systems

at

Java applet of Chinese, Japanese and Russian abaci

Indian Abacus

{{Authority control Mathematical tools Chinese mathematics Egyptian mathematics Greek mathematics Indian mathematics Japanese mathematics Korean mathematics Roman mathematics

ancient times
Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history cov ...

. It was used in the ancient Near East
The ancient Near East was the home of early civilizations within a region roughly corresponding to the modern Middle East: Mesopotamia (modern Iraq, southeast Turkey, southwest Iran and northeastern Syria), ancient Egypt, ancient Iran (Elam, M ...

, Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entire ...

, China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones an ...

, and Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

, centuries before the adoption of the Hindu-Arabic numeral system. The exact origin of the abacus has not yet emerged. It consists of rows of movable beads, or similar objects, strung on a wire. They represent digits. One of the two numbers is set up, and the beads are manipulated to perform an operation such as addition, or even a square or cubic root.
In their earliest designs, the rows of beads could be loose on a flat surface or sliding in grooves. Later the beads were made to slide on rods and built into a frame, allowing faster manipulation. Abacuses are still made, often as a bamboo
Bamboos are a diverse group of evergreen perennial flowering plants making up the subfamily Bambusoideae of the grass family Poaceae. Giant bamboos are the largest members of the grass family. The origin of the word "bamboo" is uncertain, b ...

frame with beads sliding on wires. In the ancient world, particularly before the introduction of positional notation
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 the ...

, abacuses were a practical calculating tool. The abacus is still used to teach the fundamentals of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

to some children, for example, in Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

.
Designs such as the Japanese soroban
The 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 elect ...

have been used for practical calculations of up to multi-digit numbers. Any particular abacus design supports multiple methods to perform calculations, including addition, subtraction, multiplication, division, and square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

and cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...

s. Some of these methods work with non-natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...

numbers (numbers such as and ).
Although calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...

s and computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These p ...

s are commonly used today instead of abacuses, abacuses remain in everyday use in some countries. Merchants, traders, and clerks in some parts of Eastern Europe
Eastern Europe is a subregion of the European continent. As a largely ambiguous term, it has a wide range of geopolitical, geographical, ethnic, cultural, and socio-economic connotations. The vast majority of the region is covered by Russia, whi ...

, Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eigh ...

, China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones an ...

, and Africa
Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area ...

use abacuses. The abacus remains in common use as a scoring system in non- electronic table games. Others may use an abacus due to visual impairment
Visual impairment, also known as vision impairment, is a medical definition primarily measured based on an individual's better eye visual acuity; in the absence of treatment such as correctable eyewear, assistive devices, and medical treatment ...

that prevents the use of a calculator.
Etymology

The word ''abacus'' dates to at least AD 1387 when aMiddle English
Middle English (abbreviated to ME) is a form of the English language that was spoken after the Norman conquest of 1066, until the late 15th century. The English language underwent distinct variations and developments following the Old English ...

work borrowed the word from Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of ...

that described a sandboard abacus. The Latin word is derived from ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic ...

(''abax'') which means something without a base, and colloquially, any piece of rectangular material. Alternatively, without reference to ancient texts on etymology, it has been suggested that it means "a square tablet strewn with dust", or "drawing-board covered with dust (for the use of mathematics)" (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, (''abakos'')). While the table strewn with dust definition is popular, some argue evidence is insufficient for that conclusion. Greek probably borrowed from a Northwest Semitic language
Northwest Semitic is a division of the Semitic languages comprising the indigenous languages of the Levant. It emerged from Proto-Semitic in the Early Bronze Age. It is first attested in proper names identified as Amorite in the Middle Bronze A ...

like Phoenician, evidenced by a cognate with the Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserve ...

word ''ʾābāq'' (), or “dust” (in the post-Biblical sense "sand used as a writing surface").
Both ''abacuses'' and ''abaci'' (soft or hard "c") are used as plurals. The user of an abacus is called an ''abacist''.
History

Mesopotamia

TheSumer
Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles o ...

ian abacus appeared between 2700–2300 BC. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

(base 60) number system.
Some scholars point to a character in Babylonian cuneiform
Cuneiform is a logo- syllabic script that was used to write several languages of the Ancient Middle East. The script was in active use from the early Bronze Age until the beginning of the Common Era. It is named for the characteristic wedge-s ...

that may have been derived from a representation of the abacus. It is the belief of Old Babylonian scholars, such as Ettore Carruccio, that Old Babylonians "seem to have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations".
Egypt

Greek historianHerodotus
Herodotus ( ; grc, , }; BC) was an ancient Greek historian and geographer from the Greek city of Halicarnassus, part of the Persian Empire (now Bodrum, Turkey) and a later citizen of Thurii in modern Calabria (Italy). He is known for hav ...

mentioned the abacus in Ancient Egypt. He wrote 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 are yet to be discovered.
Persia

At around 600 BC, Persians first began to use the abacus, during theAchaemenid Empire
The Achaemenid Empire or Achaemenian Empire (; peo, wikt:𐎧𐏁𐏂𐎶, 𐎧𐏁𐏂, , ), also called the First Persian Empire, was an History of Iran#Classical antiquity, ancient Iranian empire founded by Cyrus the Great in 550 BC. Bas ...

. Under the Parthian, Sassanian, and Iran
Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkm ...

ian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India
India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on th ...

, China, and the Roman Empire
The Roman Empire ( la, Imperium Romanum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn Rhōmaíōn) was the post- Republican period of ancient Rome. As a polity, it included large territorial holdings around the Medit ...

- which is how the abacus may have been exported to other countries.
Greece

The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC.Demosthenes
Demosthenes (; el, Δημοσθένης, translit=Dēmosthénēs; ; 384 – 12 October 322 BC) was a Greek statesman and orator in ancient Athens. His orations constitute a significant expression of contemporary Athenian intellectual prow ...

(384 BC–322 BC) complained that the need to use pebbles for calculations was too difficult. A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius
Polybius (; grc-gre, Πολύβιος, ; ) was a Greek historian of the Hellenistic period. He is noted for his work , which covered the period of 264–146 BC and the Punic Wars in detail.
Polybius is important for his analysis of the mixe ...

use the abacus as a metaphor for human behavior, stating "that men that sometimes stood for more and sometimes for less" like the pebbles on an abacus. 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 was used in Achaemenid Persia, the Etruscan civilization
The Etruscan civilization () was developed by a people of Etruria in ancient Italy with a common language and culture who formed a federation of city-states. After conquering adjacent lands, its territory covered, at its greatest extent, rough ...

, Ancient Rome, and the Western Christian world until the French Revolution
The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are consider ...

.
A tablet found on the Greek island Salamis in 1846 AD (the Salamis Tablet) dates to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble in length, wide, and thick, on which are 5 groups of markings. In the tablet's center 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. 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.
Rome

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 theRoman numeral
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

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. Due to Pope Sylvester II
Pope Sylvester II ( – 12 May 1003), originally known as Gerbert of Aurillac, was a French-born scholar and teacher who served as the bishop of Rome and ruled the Papal States from 999 to his death. He endorsed and promoted study of Arab and Gr ...

's reintroduction of the abacus with modifications, it became widely used in Europe again during the 11th century This abacus used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster and was more easily moved.
Writing in the 1st century BC, Horace
Quintus Horatius Flaccus (; 8 December 65 – 27 November 8 BC), known in the English-speaking world as Horace (), was the leading Roman lyric poet during the time of Augustus (also known as Octavian). The rhetorician Quintilian regarded his ...

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.
One example of archaeological evidence of the Roman abacus
The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the Greeks and Babylonians.
Origin
The Roman abacus was the first portable calculating device for ...

, shown nearby 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
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...

. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions).
China

The earliest known written documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, also known as the ''suanpan
The suanpan (), 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 ...

'' (算盤/算盘, lit. "calculating tray"), is typically 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 one. The beads are usually rounded and made of hardwood
Hardwood is wood from dicot trees. These are usually found in broad-leaved temperate and tropical forests. In temperate and boreal latitudes they are mostly deciduous, but in tropics and subtropics mostly evergreen. Hardwood (which comes fro ...

. 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. One of the top beads is 5, while one of the bottom beads is 1. Each rod has a number under it, showing the place value. 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 appeared during the Han Dynasty
The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...

, and the beads are oval. The Song Dynasty
The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...

and earlier used the 1:4 type or four-beads abacus similar to the modern abacus including the shape of the beads commonly known as Japanese-style abacus.
In the early Ming Dynasty
The Ming dynasty (), officially the Great Ming, was an imperial dynasty of China, ruling from 1368 to 1644 following the collapse of the Mongol-led Yuan dynasty. The Ming dynasty was the last orthodox dynasty of China ruled by the Han peo ...

, the abacus began to appear in a 1:5 ratio. The upper deck had one bead and the bottom had five beads. In the late Ming Dynasty, the abacus styles appeared in a 2:5 ratio. The upper deck had two beads, and the bottom had five.
Various calculation techniques were devised for ''Suanpan'' enabling efficient calculations. Some schools teach students how to use it.
In the long scroll ''Along the River During the Qingming Festival
''Along the River During the Qingming Festival'' (''Qingming Shanghe Tu'') is a handscroll painting by the Song dynasty painter Zhang Zeduan (1085–1145) and copied many times in the following centuries. It captures the daily life of people a ...

'' painted by Zhang Zeduan during the Song dynasty
The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...

(960–1297), a ''suanpan'' is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary
''Apothecary'' () is a mostly archaic term for a medical professional who formulates and dispenses '' materia medica'' (medicine) to physicians, surgeons, and patients. The modern chemist (British English) or pharmacist (British and North Ameri ...

's (Feibao).
The similarity of the Roman abacus
The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the Greeks and Babylonians.
Origin
The Roman abacus was the first portable calculating device for ...

to the Chinese one suggests that one could have inspired the other, given evidence of a trade relationship between the Roman Empire
The Roman Empire ( la, Imperium Romanum ; grc-gre, Βασιλεία τῶν Ῥωμαίων, Basileía tôn Rhōmaíōn) was the post- Republican period of ancient Rome. As a polity, it included large territorial holdings around the Medit ...

and China. However, no direct connection has been 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
Japanese may refer to:
* Something from or related to Japan, an island country in East Asia
* Japanese language, spoken mainly in Japan
* Japanese people, the ethnic group that identifies with Japan through ancestry or culture
** Japanese diaspo ...

) has 4 plus 1 bead per decimal place, the standard ''suanpan'' has 5 plus 2. Incidentally, this allows use with a hexadecimal
In mathematics and computing, the hexadecimal (also base-16 or simply hex) numeral system is a positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system representing numbers using 10 symbols, hex ...

numeral system (or any base up to 18) which may have been used for traditional Chinese measures of weight. (Instead of running on wires as in the Chinese, Korean, and Japanese models, the Roman model used grooves, presumably making arithmetic calculations much slower.)
Another possible source of the ''suanpan'' is Chinese counting rods
Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number.
The written ...

, which operated with a decimal system but lacked the concept of zero as a placeholder. The zero was probably introduced to the Chinese in the Tang dynasty
The Tang dynasty (, ; zh, t= ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907 AD, with an interregnum between 690 and 705. It was preceded by the Sui dynasty and followed by the Five Dynasties and Ten Kingd ...

(618–907) when travel in the Indian Ocean
The Indian Ocean is the third-largest of the world's five oceanic divisions, covering or ~19.8% of the water on Earth's surface. It is bounded by Asia to the north, Africa to the west and Australia to the east. To the south it is bounded by th ...

and the Middle East
The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabia (including the Arabian Peninsula and Bahrain), Asia Minor (Asian part of Turkey except Hatay Province), East Thrace (Europe ...

would have provided direct contact with India
India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on th ...

, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.
India

The '' Abhidharmakośabhāṣya'' ofVasubandhu
Vasubandhu (; Tibetan: དབྱིག་གཉེན་ ; fl. 4th to 5th century CE) was an influential Buddhist monk and scholar from ''Puruṣapura'' in ancient India, modern day Peshawar, Pakistan. He was a philosopher who wrote commentar ...

(316-396), a Sanskrit work on Buddhist philosophy
Buddhist philosophy refers to the philosophical investigations and systems of inquiry that developed among various schools of Buddhism in India following the parinirvana of The Buddha and later spread throughout Asia. The Buddhist path comb ...

, 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. Hindu texts used the term ''śūnya'' (zero) to indicate the empty column on the abacus.
Japan

In Japan, the abacus is called ''soroban
The 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 elect ...

'' (, lit. "counting tray"). It was imported from China in the 14th century. It was probably in use by the working class a century or more before the ruling class adopted it, as the class structure obstructed such changes. The 1:4 abacus, which removes the seldom-used second and fifth bead became popular in the 1940s.
Today's Japanese abacus is a 1:4 type, four-bead abacus, 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 similar to the Chinese or Korean abacus, and the decimal number can be expressed, so the abacus is designed as a one:four device. 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 in the Ize Rongji collection of Shansi Village in Yamagata City. Japan also used a 2:5 type abacus.
The four-bead abacus spread, and became common around the world. Improvements to the Japanese abacus arose in various places. In China an aluminium frame plastic bead abacus was used. The file is next to the four beads, and pressing the "clearing" button put the upper bead in the upper position, and the lower bead in the lower position.
The abacus is still manufactured in Japan even with the proliferation, practicality, and affordability of pocket electronic calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...

s. The use of the soroban is still taught in Japanese primary school
A primary school (in Ireland, the United Kingdom, Australia, Trinidad and Tobago, Jamaica, and South Africa), junior school (in Australia), elementary school or grade school (in North America and the Philippines) is a school for primary e ...

s as part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, primarily as an aid to faster mental calculation. Using visual imagery can complete a calculation as quickly as a physical instrument.
Korea

The Chinese abacus migrated from China to Korea around 1400 AD. Koreans call it ''jupan'' (주판), ''supan'' (수판) or ''jusan'' (주산). The four-beads abacus (1:4) was introduced during theGoryeo Dynasty
Goryeo (; ) was a Korean kingdom founded in 918, during a time of national division called the Later Three Kingdoms period, that unified and ruled the Korean Peninsula until 1392. Goryeo achieved what has been called a "true national unificat ...

. The 5:1 abacus was introduced to Korea
Korea ( ko, 한국, or , ) is a peninsular region in East Asia. Since 1945, it has been divided at or near the 38th parallel, with North Korea (Democratic People's Republic of Korea) comprising its northern half and South Korea (Republic o ...

from China during the Ming Dynasty.
Native America

Some sources mention the use of an abacus called a ''nepohualtzintzin'' in ancientAztec
The Aztecs () were a Mesoamerican culture that flourished in central Mexico in the post-classic period from 1300 to 1521. The Aztec people included different ethnic groups of central Mexico, particularly those groups who spoke the Nahuatl ...

culture. This Mesoamerica
Mesoamerica is a historical region and cultural area in southern North America and most of Central America. It extends from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica. ...

n abacus used a 5-digit base-20 system. The word Nepōhualtzintzin comes from Nahuatl
Nahuatl (; ), Aztec, or Mexicano is a language or, by some definitions, a group of languages of the Uto-Aztecan language family. Varieties of Nahuatl are spoken by about Nahua peoples, most of whom live mainly in Central Mexico and have smalle ...

, formed by the roots; ''Ne'' – personal -; ''pōhual'' or ''pōhualli'' – the account -; and ''tzintzin'' – small similar elements. Its complete meaning was taken as: counting with small similar elements. Its use was taught in the Calmecac to the ''temalpouhqueh'' , who were students dedicated to taking the accounts of skies, from childhood.
The Nepōhualtzintzin was divided into two main parts separated by a bar or intermediate cord. In the left part were four beads. Beads in the first row have unitary values (1, 2, 3, and 4), and on the right side, three beads had 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 count in the first row.
The device featured 13 rows with 7 beads, 91 in total. This was a basic number for this culture. It had a close relation to 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 approximated one year. When translated into modern computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to 18 in floating point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...

, which precisely calculated large and small amounts, although round off was not allowed.
The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo, who in his travels throughout Mexico found diverse engravings and paintings of this instrument and reconstructed several of them in gold, jade, encrustations of shell, etc. Very old Nepōhualtzintzin are attributed to the Olmec culture, and some bracelets of Maya
Maya may refer to:
Civilizations
* Maya peoples, of southern Mexico and northern Central America
** Maya civilization, the historical civilization of the Maya peoples
** Maya language, the languages of the Maya peoples
* Maya (Ethiopia), a populat ...

n origin, as well as a diversity of forms and materials in other cultures.
Sanchez wrote in ''Arithmetic in Maya'' that another base 5, base 4 abacus had been found in the Yucatán Peninsula
The Yucatán Peninsula (, also , ; es, Península de Yucatán ) is a large peninsula in southeastern Mexico and adjacent portions of Belize and Guatemala. The peninsula extends towards the northeast, separating the Gulf of Mexico to the north ...

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.
The quipu
''Quipu'' (also spelled ''khipu'') are recording devices fashioned from strings historically used by a number of cultures in the region of Andean South America.
A ''quipu'' usually consisted of cotton or camelid fiber strings. The Inca people ...

of the Incas was a system of colored knotted cords used to record numerical data, like advanced tally stick
A tally stick (or simply tally) was an ancient memory aid device used to record and document numbers, quantities and messages. Tally sticks first appear as animal bones carved with notches during the Upper Palaeolithic; a notable example is the ...

s – but not used to perform calculations. Calculations were carried out using a yupana (Quechua
Quechua may refer to:
*Quechua people, several indigenous ethnic groups in South America, especially in Peru
*Quechuan languages, a Native South American language family spoken primarily in the Andes, derived from a common ancestral language
**So ...

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 Italian mathematician De Pasquale proposed an explanation. By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

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.
Russia

The Russian abacus, the ''schoty'' (, plural from , counting), usually has a single slanted deck, with ten beads on each wire (except one wire with four beads for quarter-ruble
The ruble (American English) or rouble ( Commonwealth English) (; rus, рубль, p=rublʲ) is the currency unit of Belarus and Russia. Historically, it was the currency of the Russian Empire and of the Soviet Union.
, currencies named '' ...

fractions). 4-bead wire was introduced for quarter- kopeks, which were minted until 1916. The Russian abacus is used vertically, with each wire running horizontally. The wires are usually bowed upward in the center, to keep the beads pinned to either side. 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. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color.
The Russian abacus was in use in shops and markets throughout the former Soviet Union
The post-Soviet states, also known as the former Soviet Union (FSU), the former Soviet Republics and in Russia as the near abroad (russian: links=no, ближнее зарубежье, blizhneye zarubezhye), are the 15 sovereign states that wer ...

, and its usage was taught in most schools until the 1990s. Even the 1874 invention of mechanical calculator
A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or (historically) a simulation such as an analog computer or a slide rule. Most mechanical calculators we ...

, Odhner arithmometer, had not replaced them in arithmometer
The arithmometer (french: arithmomètre) was the first digital mechanical calculator strong enough and reliable enough to be used daily in an office environment. This calculator could add and subtract two numbers directly and could perform lon ...

s since 1924 did not significantly reduce abacus use in the Soviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...

. The Russian abacus began to lose popularity only after the mass production of domestic microcalculators in 1974.
The Russian abacus was brought to France around 1820 by mathematician Jean-Victor Poncelet
Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work '' ...

, who had served in Napoleon
Napoleon Bonaparte ; it, Napoleone Bonaparte, ; co, Napulione Buonaparte. (born Napoleone Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French military commander and political leader who ...

's army and had been a prisoner of war
A prisoner of war (POW) is a person who is held captive by a belligerent power during or immediately after an armed conflict. The earliest recorded usage of the phrase "prisoner of war" dates back to 1610.
Belligerents hold prisoners of w ...

in Russia. 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. The Turks and the Armenian people used abacuses similar to the Russian schoty. It was named a ''coulba'' by the Turks and a ''choreb'' by the Armenians.
School abacus

Around the world, abacuses have been used in pre-schools and elementary schools as an aid in teaching thenumeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
The same sequence of symbo ...

and arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...

.
In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame is common (see image).
The wireframe 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 (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. Teaching multiplication, e.g. 6 times 7, may be represented by shifting 7 beads on 6 wires.
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
Dutch commonly refers to:
* Something of, from, or related to the Netherlands
* Dutch people ()
* Dutch language ()
Dutch may also refer to:
Places
* Dutch, West Virginia, a community in the United States
* Pennsylvania Dutch Country
People ...

name ''rekenrek'' ("calculating frame"), is often used, either on a string of beads or on a rigid framework.
Feynman vs the abacus

PhysicistRichard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...

was noted for facility in mathematical calculations. He wrote about an encounter in Brazil with a Japanese abacus expert, who challenged him to speed contests between Feynman's pen and paper, and the abacus. The abacus was much faster for addition, somewhat faster for multiplication, but Feynman was faster at division. When the abacus was used for a really difficult challenge, i.e. cube roots, Feynman won easily. However, the number chosen at random was close to a number Feynman happened to know was an exact cube, allowing him to use approximate methods.
Neurological analysis

Learning how to calculate with the abacus may improve capacity for mental calculation. Abacus-based mental calculation (AMC), which was derived from the abacus, is the act of performing calculations, including addition, subtraction, multiplication, and division, in the mind by manipulating an imagined abacus. It is a high-level cognitive skill that runs calculations with an effective algorithm. People doing long-term AMC training show higher numerical memory capacity and experience more effectively connected neural pathways. They are able to retrieve memory to deal with complex processes. AMC involves both visuospatial and visuomotor processing that generate the visual abacus and move the imaginary beads. Since it only requires that the final position of beads be remembered, it takes less memory and less computation time.Renaissance abacuses

Binary abacus

The binary abacus is used to explain how computers manipulate numbers. The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or viaASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...

. 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.
Visually impaired users

An adapted abacus, invented by Tim Cranmer, and called a Cranmer abacus is commonly used by visually impaired users. A piece of soft fabric or rubber is placed behind the beads, keeping them in place while the users manipulate them. The device is then used to perform the mathematical functions of multiplication, division, addition, subtraction, square root, and cube root. Although blind students have benefited from talking calculators, the abacus is often taught to these students in early grades. Blind students can also complete mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication andlong division
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps ...

problems are tedious. The abacus gives these 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 useful tool throughout life.
See also

* Chinese Zhusuan * Chisanbop * Logical abacus * Mental abacus * Napier's bones * Sand table * Slide rule *Soroban
The 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 elect ...

* Suanpan
The suanpan (), 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 ...

Notes

Footnotes

References

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *Reading

* * * * * *External links

*Tutorials

*Min Multimedia

*

Abacus curiosities

*Abacus in Various Number Systems

at

cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Ma ...

Java applet of Chinese, Japanese and Russian abaci

Indian Abacus

{{Authority control Mathematical tools Chinese mathematics Egyptian mathematics Greek mathematics Indian mathematics Japanese mathematics Korean mathematics Roman mathematics