Ancient Near East Mesopotamia The
Sumerian abacus appeared between 2700 and 2300 BC. It held a table of successive columns which delimited the successive orders of magnitude of their
sexagesimal (base 60) number system. Primitive forms of the abacus existed in Sumeria, such as the non-fixed bead abacus, a line of strings fixed to a board, counting strings consisting of beads (similar to
prayer beads and
pace count beads), ropes and knots, and counting boards (similar to
tally sticks). The exact form and function of these devices are scattered around different sources, and their exact relation to the modern abacus is unknown but it is almost certain there is a relationship with masihatu, which functioned similar to an abacus utilizing multiple strings and sets of beads. Some scholars point to a character in
Babylonian cuneiform 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 historian
Herodotus 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, there are no known illustrations of this device.
Persia At around 600 BC, Persians first began to use the abacus, during the
Achaemenid Empire. Under the
Parthian,
Sassanian, and
Iranian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India, China, and the
Roman Empire – which is how the abacus may have been exported to other countries.
Europe Greece The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC.
Demosthenes (384–322 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 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. 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. One example of archaeological evidence of the
Roman abacus, 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.) resembling a
bi-quinary coded decimal system related to the
Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions).
Medieval Europe The Roman system of 'counter casting' was used widely in medieval Europe, and persisted in limited use into the nineteenth century. Wealthy abacists used decorative minted counters, called
jetons. Due to
Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe again during the 11th century It used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster and was more easily moved.
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 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, and its usage was taught in most schools until the 1990s. Even the 1874 invention of
mechanical calculator,
Odhner arithmometer, had not replaced them in Russia. 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
arithmometers since 1924 did not significantly reduce abacus use in the
Soviet Union. 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, who had served in
Napoleon's army and had been a
prisoner of war in Russia. 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.
East Asia China '') (the number represented in the picture is 6,302,715,408) The earliest known written documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, also known as the
suanpan (算盤/算盘, lit. "calculating tray"), comes in various lengths and 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, to represent numbers in a
bi-quinary coded decimal-like system. The beads are usually rounded and made of
hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not. 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, and the beads are oval. The
Song dynasty 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 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. 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 1:4 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 abacus with an aluminium frame and plastic beads has been used. The file is next to the four beads, and pressing the "clearing" button puts the upper bead in the upper position, and the lower bead in the lower position. The abacus is still manufactured in Japan, despite 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, one can complete a calculation as quickly as with a physical instrument.
Korea The Chinese abacus migrated from China to
Korea around 1400 AD. Koreans call the abacus
jupan (주판) or
supan (수판), and the act of using a jupan is
jusan (주산). The four-beads abacus (1:4) was introduced during the
Goryeo Dynasty. The 5:1 abacus was introduced to Korea from China during the Ming Dynasty.
India The
Abhidharmakośabhāṣya of
Vasubandhu (316–396), a Sanskrit work on
Buddhist philosophy, says that the second-century CE philosopher
Vasumitra said that "placing a wick (Sanskrit
vartikā) on the number one (
ekāṅka) means it is a one while placing the wick on the number hundred means it is called a hundred, and on the number one thousand means it is a thousand". It is unclear exactly what this arrangement may have been. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the abacus. Hindu texts used the term
śūnya (zero) to indicate the empty column on the abacus. Since 2020, the organization Indian Abacus has run a national abacus competition in India.
Americas Mesoamerica Some sources mention the use of an abacus called a
nepohualtzintzin in ancient
Aztec culture. This
Mesoamerican abacus used a 5-digit base-20 system. The word Nepōhualtzintzin comes from
Nahuatl, 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, 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
Mayan 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 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.
Inca Empire The
quipu of the
Incas was a system of colored knotted cords used to record numerical data, like advanced
tally sticks – but not used to perform calculations. Calculations were carried out using a
yupana (
Quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 Italian mathematician De Pasquale proposed an explanation. By comparing the form of several yupanas, researchers found that calculations were based using the
Fibonacci sequence 1, 1, 2, 3, 5 and powers of 10, 20, and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at a minimum. ==Types==