Abacus
Overview
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
processes. Today, abaci 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. The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia
Asia
Asia is the world's largest and most populous continent, located primarily in the eastern and northern hemispheres. It covers 8.7% of the Earth's total surface area and with approximately 3.879 billion people, it hosts 60% of the world's current human population...
, Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...
, and elsewhere.
Encyclopedia
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic
processes. Today, abaci 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. The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia
, Africa
, and elsewhere. The user of an abacus is called an abacist.
work borrowed the word from Latin
to describe a sandboard abacus. The Latin word came from Άβακός abakos, the Greek
genitive form
of Άβαξ abax ("calculatingtable"), from Hebrew
ābāq (אבק), "dust". The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use.
ian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.
Some scholars point to a character from the Babylonian cuneiform
which may have been derived from a representation of the abacus. It is the belief of Carruccio (and other Old Babylonian scholars) that Old Babylon
ians "may have used the abacus for the operations of addition
and subtraction
; however, this primitive device proved difficult to use for more complex calculations".
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 lefttoright 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, casting some doubt over the extent to which this instrument was used.
and Sassanian Iran
ian empires, scholars concentrated on exchanging knowledge and inventions by the countries around them – India
, China
, and the Roman Empire
, when it is thought to be expanded over the other countries.
A tablet found on the Greek island Salamis
in 1846 AD 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 bottommost 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.
s 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 tenth century. Due to Pope Sylvester II's reintroduction of the abacus with very useful modifications, it became widely used in Europe once again during the 11th century
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.
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 biquinary coded decimal
system, obviously related to the Roman numerals
. The short grooves on the right may have been used for marking Roman ounces.
The Chinese abacus, known as the suànpán (算盤, lit. "Counting 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 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. If you move them toward the beam, you count their value. If you move away, you don't count their value. The suanpan can be reset to the starting position instantly by a quick jerk along 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. There are currently schools teaching students how to use it.
In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan
(1085–1145 AD) during the Song Dynasty
(960–1297 AD), a suanpan is clearly seen lying 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 abaci may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2, allowing use with a hexadecimal
numeral system. Instead of running on wires as in the Chinese 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
(618907 AD) 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.
. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus. Hindu texts used the term shunya (zero) to indicate the empty column on the abacus.
, imported from China around 1600. The 1/4 abacus, which is suited to decimal calculation, appeared circa 1930, and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China. 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 can arrive at the answer in the same time (or faster) as obtainable with a physical instrument.
around 1400 AD. Koreans call it jupan (주판), supan (수판) or jusan (주산).
culture. This Mesoamerican abacus used a 5digit base20 system.
The word Nepohualtzintzin comes from the Nahuatl and it is formed by the roots; Ne  personal ; pohual or pohualli  the account ; and tzintzin  small similar elements. And its complete meaning was taken as: counting with small similar elements by somebody. Its use was taught in the "Kalmekak" to the "temalpouhkeh", who were students dedicated to take the accounts of skies, from childhood. Unfortunately the Nepohualtzintzin and its teaching were among the victims of the conquering destruction, when a diabolic origin was attributed to them after observing the tremendous properties of representation, precision and speed of calculations.
This arithmetic tool was based on the vigesimal
system (base 20). For the aztec the count by 20s was completely natural. The amount of 4, 5, 13, 20 and other cyclees meant cycles. The Nepohualtzintzin 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 Nepohualtzintzin. 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 Nepohualtzintzin (91) represented the number of days that a season of the year lasts, two Nepohualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepohualtzintzin (273) is the number of days of a baby's gestation, and four Nepohualtzintzin (364) completed a cycle and approximate a year (1 1/4 days short). It is worth mentioning that the Nepohualtzintzin 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, when translated into modern computer arithmetic.
The rediscovery of the Nepohualtzintzin was due to the Mexican engineer David Esparza Hidalgo, 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. . There have also been found very old Nepohualtzintzin attributed to the Olmec
a 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", AustinTexas, 1961 found another base 5, base 4 abacus in the Yucatán 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 used 0, 1, 2 and 3 were used. Note the use of zero at the beginning an end of the two cycles. Sanchez worked with Sylvanus Morley
a noted Mayanist.
The quipu
of the Incas was a system of knotted cords used to record numerical data, like advanced tally stick
s – 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 minimum.
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. 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
. 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 JeanVictor Poncelet
, who served in Napoleon's army and had been a prisoner of war in Russia. The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorism
ic 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.
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 type of abacus shown here is often used to represent numbers without the use of place value. Each bead and each wire has the same value and used in this way it can represent numbers up to 100.
. 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 cubic root.
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. The abacus teaches mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students. Blind students also complete mathematical assignments using a braillewriter 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.
. 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.
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Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
processes. Today, abaci 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. The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia
Asia
Asia is the world's largest and most populous continent, located primarily in the eastern and northern hemispheres. It covers 8.7% of the Earth's total surface area and with approximately 3.879 billion people, it hosts 60% of the world's current human population...
, Africa
Africa
Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...
, and elsewhere. The user of an abacus is called an abacist.
Etymology
The use of the word abacus dates before 1387 AD, when a Middle EnglishMiddle English
Middle English is the stage in the history of the English language during the High and Late Middle Ages, or roughly during the four centuries between the late 11th and the late 15th century....
work borrowed the word from Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
to describe a sandboard abacus. The Latin word came from Άβακός abakos, the Greek
Greek language
Greek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
genitive form
Genitive case
In grammar, genitive is the grammatical case that marks a noun as modifying another noun...
of Άβαξ abax ("calculatingtable"), from Hebrew
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by nonJewish groups, such...
ābāq (אבק), "dust". The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use.
Mesopotamian abacus
The period 2700–2300 BC saw the first appearance of the SumerSumer
Sumer was a civilization and historical region in southern Mesopotamia, modern Iraq during the Chalcolithic and Early Bronze Age....
ian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.
Some scholars point to a character from the Babylonian cuneiform
Cuneiform
Cuneiform can refer to:*Cuneiform script, an ancient writing system originating in Mesopotamia in the 4th millennium BC*Cuneiform , three bones in the human foot*Cuneiform Records, a music record label...
which may have been derived from a representation of the abacus. It is the belief of Carruccio (and other Old Babylonian scholars) that Old Babylon
Babylon
Babylon was an Akkadian citystate of ancient Mesopotamia, the remains of which are found in presentday Al Hillah, Babil Province, Iraq, about 85 kilometers south of Baghdad...
ians "may have used the abacus for the operations of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
and subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
; however, this primitive device proved difficult to use for more complex calculations".
Egyptian abacus
The use of the abacus in Ancient EgyptAncient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
is mentioned by the Greek historian Herodotus
Herodotus
Herodotus was an ancient Greek historian who was born in Halicarnassus, Caria and lived in the 5th century BC . He has been called the "Father of History", and was the first historian known to collect his materials systematically, test their accuracy to a certain extent and arrange them in a...
, who writes that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek lefttoright 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, casting some doubt over the extent to which this instrument was used.
Persian abacus
During the Achaemenid Persian Empire, around 600 BC the Persians first began to use the abacus. Under ParthianParthian Empire
The Parthian Empire , also known as the Arsacid Empire , was a major Iranian political and cultural power in ancient Persia...
and Sassanian Iran
Iran
Iran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as Persia...
ian empires, scholars concentrated on exchanging knowledge and inventions by the countries around them – India
India
India , officially the Republic of India , is a country in South Asia. It is the seventhlargest country by geographical area, the secondmost populous country with over 1.2 billion people, and the most populous democracy in the world...
, China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, and the Roman Empire
Roman Empire
The Roman Empire was the postRepublican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....
, when it is thought to be expanded over the other countries.
Greek abacus
The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. The Greek abacus was a table of wood or marble, preset 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
Salamis Island
Salamis , is the largest Greek island in the Saronic Gulf, about 1 nautical mile offcoast from Piraeus and about 16 km west of Athens. The chief city, Salamina , lies in the westfacing core of the crescent on Salamis Bay, which opens into the Saronic Gulf...
in 1846 AD 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 bottommost 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.
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, jetonJeton
Jetons were token or coinlike medals produced across Europe from the 13th through the 17th centuries. They were produced as counters for use in calculation on a lined board similar to an abacus. They also found use as a money substitute in games, similar to modern casino chips or poker chips...
s 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 tenth century. Due to Pope Sylvester II's reintroduction of the abacus with very useful modifications, it became widely used in Europe once again during the 11th century
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.
One example of archaeological evidence of the Roman abacus
Roman abacus
The Romans developed the Roman hand abacus, a portable, but less capable, base10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors...
, 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 biquinary coded decimal
Biquinary coded decimal
Biquinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, including the Colossus. The term biquinary indicates that the code comprises both a twostate and a fivestate component...
system, obviously related to the Roman numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
. The short grooves on the right may have been used for marking Roman ounces.
Chinese abacus
The earliest known written documentation of the Chinese abacus dates to the 2nd century BC.The Chinese abacus, known as the suànpán (算盤, lit. "Counting 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 for both decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
computation. The beads are usually rounded and made of a hardwood
Hardwood
Hardwood is wood from angiosperm trees . It may also be used for those trees themselves: these are usually broadleaved; in temperate and boreal latitudes they are mostly deciduous, but in tropics and subtropics mostly evergreen.Hardwood contrasts with softwood...
. The beads are counted by moving them up or down towards the beam. If you move them toward the beam, you count their value. If you move away, you don't count their value. The suanpan can be reset to the starting position instantly by a quick jerk along 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
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, division
Division (mathematics)
rightthumb200px20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
and cube root operations at high speed. There are currently schools teaching students how to use it.
In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan
Zhang Zeduan
Zhang Zeduan , alias Zheng Dao, also sometimes translated as Zhang Zerui, was a famous Chinese painter during the twelfth century, during the transitional period from the Northern Song to the Southern Song Dynasty, and was instrumental in the early history of the Chinese art style known as shan...
(1085–1145 AD) during the Song Dynasty
Song Dynasty
The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...
(960–1297 AD), a suanpan is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary
Apothecary
Apothecary is a historical name for a medical professional who formulates and dispenses materia medica to physicians, surgeons and patients — a role now served by a pharmacist and some caregivers....
's (Feibao).
The similarity of the Roman abacus
Roman abacus
The Romans developed the Roman hand abacus, a portable, but less capable, base10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors...
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
Roman Empire
The Roman Empire was the postRepublican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....
and China. However, no direct connection can be demonstrated, and the similarity of the abaci may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2, allowing use with a hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
numeral system. Instead of running on wires as in the Chinese 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
Counting rods
Counting rods are small bars, typically 3–14 cm long, used by mathematicians for calculation in China, Japan, Korea, and Vietnam. They are placed either horizontally or vertically to represent any number and any fraction....
, which operated with a decimal system
Decimal system
Decimal system may refer to:* The decimal number system, used in mathematics for writing numbers and performing arithmetic.* The Dewey Decimal System, a subject classification system used in libraries....
but lacked the concept of zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
as a place holder. The zero was probably introduced to the Chinese in the Tang Dynasty
Tang Dynasty
The Tang Dynasty was an imperial dynasty of China preceded by the Sui Dynasty and followed by the Five Dynasties and Ten Kingdoms Period. It was founded by the Li family, who seized power during the decline and collapse of the Sui Empire...
(618907 AD) when travel in the Indian Ocean
Indian Ocean
The Indian Ocean is the third largest of the world's oceanic divisions, covering approximately 20% of the water on the Earth's surface. It is bounded on the north by the Indian Subcontinent and Arabian Peninsula ; on the west by eastern Africa; on the east by Indochina, the Sunda Islands, and...
and the Middle East
Middle East
The Middle East is a region that encompasses Western Asia and Northern Africa. It is often used as a synonym for Near East, in opposition to Far East...
would have provided direct contact with India
India
India , officially the Republic of India , is a country in South Asia. It is the seventhlargest country by geographical area, the secondmost populous country with over 1.2 billion people, and the most populous democracy in the world...
, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.
Indian abacus
First century sources, such as the Abhidharmakosa describe the knowledge and use of abacus in IndiaIndia
India , officially the Republic of India , is a country in South Asia. It is the seventhlargest country by geographical area, the secondmost populous country with over 1.2 billion people, and the most populous democracy in the world...
. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus. Hindu texts used the term shunya (zero) to indicate the empty column on the abacus.
Japanese abacus
In Japanese, the abacus is called sorobanSoroban
The is an abacus developed in Japan. It is derived from the Chinese suanpan, imported from China via Korea to Japan around 1600. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators....
, imported from China around 1600. The 1/4 abacus, which is suited to decimal calculation, appeared circa 1930, and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China. 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, primarily as an aid to faster mental calculation. Using visual imagery of a soroban can arrive at the answer in the same time (or faster) as obtainable with a physical instrument.
Korean abacus
The Chinese abacus migrated from China to KoreaKorea
Korea ) is an East Asian geographic region that is currently divided into two separate sovereign states — North Korea and South Korea. Located on the Korean Peninsula, Korea is bordered by the People's Republic of China to the northwest, Russia to the northeast, and is separated from Japan to the...
around 1400 AD. Koreans call it jupan (주판), supan (수판) or jusan (주산).
Native American abaci
Some sources mention the use of an abacus called a nepohualtzintzin in ancient MayanMaya civilization
The Maya is a Mesoamerican civilization, noted for the only known fully developed written language of the preColumbian Americas, as well as for its art, architecture, and mathematical and astronomical systems. Initially established during the PreClassic period The Maya is a Mesoamerican...
culture. This Mesoamerican abacus used a 5digit base20 system.
The word Nepohualtzintzin comes from the Nahuatl and it is formed by the roots; Ne  personal ; pohual or pohualli  the account ; and tzintzin  small similar elements. And its complete meaning was taken as: counting with small similar elements by somebody. Its use was taught in the "Kalmekak" to the "temalpouhkeh", who were students dedicated to take the accounts of skies, from childhood. Unfortunately the Nepohualtzintzin and its teaching were among the victims of the conquering destruction, when a diabolic origin was attributed to them after observing the tremendous properties of representation, precision and speed of calculations.
This arithmetic tool was based on the vigesimal
Vigesimal
The vigesimal or base 20 numeral system is based on twenty . Places :...
system (base 20). For the aztec the count by 20s was completely natural. The amount of 4, 5, 13, 20 and other cyclees meant cycles. The Nepohualtzintzin 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 Nepohualtzintzin. 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 Nepohualtzintzin (91) represented the number of days that a season of the year lasts, two Nepohualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepohualtzintzin (273) is the number of days of a baby's gestation, and four Nepohualtzintzin (364) completed a cycle and approximate a year (1 1/4 days short). It is worth mentioning that the Nepohualtzintzin 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, when translated into modern computer arithmetic.
The rediscovery of the Nepohualtzintzin was due to the Mexican engineer David Esparza Hidalgo, 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. . There have also been found very old Nepohualtzintzin attributed to the Olmec
Olmec
The Olmec were the first major PreColumbian civilization in Mexico. They lived in the tropical lowlands of southcentral Mexico, in the modernday states of Veracruz and Tabasco....
a 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", AustinTexas, 1961 found another base 5, base 4 abacus in the Yucatán 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 used 0, 1, 2 and 3 were used. Note the use of zero at the beginning an end of the two cycles. Sanchez worked with Sylvanus Morley
Sylvanus Morley
Sylvanus Griswold Morley was an American archaeologist, epigrapher, and Mayanist scholar who made significant contributions toward the study of the preColumbian Maya civilization in the early twentieth century....
a noted Mayanist.
The quipu
Quipu
Quipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...
of the Incas was a system of knotted cords used to record numerical data, like advanced tally stick
Tally stick
A tally was an ancient memory aid device to record and document numbers, quantities, or even messages. Tally sticks first appear as notches carved on animal bones, in the Upper Paleolithic. A notable example is the Ishango Bone...
s – but not used to perform calculations. Calculations were carried out using a yupana
Yupana
The yupana is a device used by the Incas, presumably as a type of calculator. Though some researchers have hypothesized how this implement might function like an abacus, others are less certain that it was used for this purpose...
(Quechua
Quechua languages
Quechua is a Native South American language family and dialect cluster spoken primarily in the Andes of South America, derived from an original common ancestor language, ProtoQuechua. It is the most widely spoken language family of the indigenous peoples of the Americas, with a total of probably...
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 minimum.
Russian abacus
The Russian abacus, the schoty (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire which has four beads, for quarterruble fractions. This wire is usually near the user). (Older models have another 4bead wire for quarterkopeks, 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 colour 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
Commonwealth of Independent States
The Commonwealth of Independent States is a regional organization whose participating countries are former Soviet Republics, formed during the breakup of the Soviet Union....
, and the usage of it was taught in most schools until the 1990s. Even the 1874 invention of mechanical calculator
Mechanical calculator
A mechanical calculator is a device used to perform the basic operations of arithmetic. Mechanical calculators are comparable in size to small desktop computers and have been rendered obsolete by the advent of the electronic calculator....
, Odhner arithmometer
Odhner Arithmometer
The Odhner Arithmometer was a very successful pinwheel calculator invented in Russia in 1873 by W. T. Odhner, a Swedish immigrant. Its industrial production officially started in 1890 in Odhner's Saint Petersburg workshop...
, had not replaced them in Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semipresidential republic, comprising 83 federal subjects...
and likewise the mass production of Felix arithmometers since 1924 did not significantly reduce their use in the Soviet Union
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
. 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 JeanVictor Poncelet
JeanVictor Poncelet
JeanVictor Poncelet was a French engineer and mathematician who served most notably as the commandant general of the École Polytechnique...
, who served in Napoleon's army and had been a prisoner of war in Russia. The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorism
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...
ic 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.
School abacus
Around the world, abaci have been used in preschools and elementary schools as an aid in teaching the numeral systemNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
and arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
.
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 type of abacus shown here is often used to represent numbers without the use of place value. Each bead and each wire has the same value and used in this way it can represent numbers up to 100.
Uses by the blind
An adapted abacus, invented by Tim Cranmer, called a Cranmer abacus is still commonly used by individuals who are blindBlindness
Blindness is the condition of lacking visual perception due to physiological or neurological factors.Various scales have been developed to describe the extent of vision loss and define blindness...
. 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
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, division
Division (mathematics)
rightthumb200px20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
and cubic root.
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. The abacus teaches mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students. Blind students also complete mathematical assignments using a braillewriter and Nemeth code
Nemeth Braille
The Nemeth Braille Code for Mathematics is a Braille code for encoding mathematical and scientific notation linearly using standard sixdot Braille cells for tactile reading by the visually impaired. The code was developed by Abraham Nemeth....
(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.
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 via ASCIIASCII
The American Standard Code for Information Interchange is a characterencoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...
. 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
 Abacus logicAbacus logicA logical abacus is a mechanical digital computer.Also referred to as a "logical machine", the logical abacus is analogous to the ordinary abacus...
 Abacus systemAbacus systemThe abacus system of mental calculation is a system where users mentally visualize an abacus to do calculations. No physical abacus is used; only the answers are written down....
 Chisanbop
 Napier's bonesNapier's bonesNapier's bones is an abacus created by John Napier for calculation of products and quotients of numbers that was based on Arab mathematics and lattice multiplication used by Matrakci Nasuh in the Umdetul Hisab and Fibonacci writing in the Liber Abaci. Also called Rabdology...
 Sand tableSand tableSand table is a term for using constrained sand for modeling or educational purposes. The original version of a sand table may be the abax used by early Greek students.Abax:...
 Slide RuleSlide ruleThe slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.Slide rules come in a...
 Suanpan
 SorobanSorobanThe is an abacus developed in Japan. It is derived from the Chinese suanpan, imported from China via Korea to Japan around 1600. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators....
Abacus curiosities
 Abacus in Various Number Systems at cuttheknotCuttheknotCuttheknot is a free, advertisementfunded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
 Java applet of Chinese, Japanese and Russian abaci
 An atomicscale abacus
 Examples of Abaci
Sister projects
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 “Abacus,” from A Dictionary of Greek and Roman Antiquities, 3rd ed., 1890.
 “abacus,” from A Dictionary of the English Language by Samuel JohnsonSamuel JohnsonSamuel Johnson , often referred to as Dr. Johnson, was an English author who made lasting contributions to English literature as a poet, essayist, moralist, literary critic, biographer, editor and lexicographer...
.
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