An abacus (pl. abaci or abacuses), also called a counting frame, is a hand-operated calculating tool which was used from ancient times, in the ancient Near East, Europe, China, and Russia, until largely replaced by handheld electronic calculators, during the 1980s, with some ongoing attempts to revive their use. An abacus consists of a two-dimensional array of slidable beads (or similar objects). In their earliest designs, the 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.

Bi-quinary coded decimal-like abacus representing 1,352,964,708

Any particular abacus design supports multiple methods to perform calculations, including addition, subtraction, multiplication, division, and square and cube roots. The beads are first arranged to represent a number, then are manipulated to perform a mathematical operation with another number, and their final position can be read as the result (or can be used as the starting number for subsequent operations).

In the ancient world, abacuses were a practical calculating tool. It was widely used in Europe as late as the 17th century, but fell out of use with the rise of decimal notation and algorismic methods. Although calculators and computers are commonly used today instead of abacuses, abacuses remain in everyday use in some countries. The abacus has an advantage of not requiring a writing implement and paper (needed for algorism) or an electric power source. Merchants, traders, and clerks in some parts of Eastern Europe, Russia, China, and Africa 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 that prevents the use of a calculator. The abacus is still used to teach the fundamentals of mathematics to children in many countries such as Japan and China.

History

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.

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.

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. 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, Ancient Rome, and the Western Christian world until the French Revolution.

The Salamis Tablet, found on the Greek island Salamis in 1846 AD, dates to 300 BC, making it the oldest counting board discovered so far. […].

Rome

The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles (Latin: calculi) were used. Marked lines indicated units, fives, tens, etc. as in the Roman numeral system.

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.

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.

China

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

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. 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.

The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, given evidence of a trade relationship between the Roman Empire and China.

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.

Japan

In Japan, the abacus is called soroban (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.

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 it jupan (주판), supan (수판) or 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.

Native America

Representation of an Inca quipu

A yupana as used by the Incas

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 Nahuatl comes from Nahuatl, formed by the roots; Ne – personal -; pōhual or pōhualli Nahuatl – the account -; and tzintzin Nahuatl – small similar elements. Its complete meaning was taken as: counting with small similar elements. Its use was taught in the Calmecac to the temalpouhqueh Nahuatl, who were students dedicated to taking the accounts of skies, from childhood.

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.

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.

Russia

The Russian abacus, the schoty (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 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.

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.

https://en.m.wikipedia.org/wiki/Abacus

Abacus, which represents numbers via a visuospatial format, is a traditional device to facilitate arithmetic operations. Skilled abacus users, who have acquired the ability of abacus-based mental calculation (AMC), can perform fast and accurate calculations by manipulating an imaginary abacus in mind. Due to this extraordinary calculation ability in AMC users, there is an expanding literature investigating the effects of AMC training on cognition and brain systems. This review study aims to provide an updated overview of important findings in this fast-growing research field. Here, findings from previous behavioral and neuroimaging studies about AMC experts as well as children and adults receiving AMC training are reviewed and discussed. Taken together, our review of the existing literature suggests that AMC training has the potential to enhance various cognitive skills including mathematics, working memory and numerical magnitude processing. Besides, the training can result in functional and anatomical neural changes that are largely located within the frontal-parietal and occipital-temporal brain regions. Some of the neural changes can explain the training-induced cognitive enhancements. Still, caution is needed when extend the conclusions to a more general situation. Implications for future research are provided.

https://pmc.ncbi.nlm.nih.gov/articles/PMC7492585/

Presentation of methods for building a mathematical universe in children that gives meaning to addition and subtraction by rooting them in basic concepts of geometry and logic.

Introduction

Numbers are fascinating, and mathematics is often identified with calculation. Strategies for performing calculations have been refined over time. First came abacuses, devices with several rows of movable pieces used for arithmetic calculations. Then came tables of values, which slowly evolved into graph tables or nomograms, i.e., a network of lines or points giving a result by simple reading or by a basic manipulation process. This expertise expanded considerably until the mid-twentieth century in the fields of physics, finance, and architecture, and the epistemological study of the underlying processes gave this discipline the name nomography.

These empirical mechanical or graphical tools, based on clever mathematical processes, required hours of intensive practice during which constant verification of units and consistency of results was essential. They quickly fell into disuse in the 1980s with the rise of computers and the development of digitization and its methods of analysis. Teachers, freed from the responsibility of teaching calculation, were thus able to focus their attention on developing other approaches and concepts.

However, due to the tragic principle of communicating vessels in human intelligence, the downstream expansion of the field of possibilities offered to science has had the effect, upstream, of disrupting the level of calculation among students. The gradual disappearance of certain crafts or their evolution, which goes hand in hand with that of familiar elements of the mechanistic era that promote learning through manipulation and observation (pendulum, balance, etc.), may also have contributed to this decline. It is therefore essential to clearly distinguish between the value of digital methods in engineering and the value of mastering basic arithmetic, which is acquired in the early years.

This presentation, divided into three parts, aims to refocus children’s attention on a few fundamental objects, whose mathematical interest and richness they will discover through long-term observation and manipulation. You will encounter abacuses and nomograms, unusual and aesthetic objects that arouse curiosity and make you want to handle or examine them. The educational benefits include reconciling calculation and geometric vision in order to develop children’s mathematical intuition empirically from an early age. This article also proposes a vertical reflection on the elementary operations induced by these objects, i.e., analyzing the angle of approach to elementary operations that these objects offer and their ability to accompany children from a naive representation to a more abstract model. In this first part, we will present processes that enable children to construct a mathematical universe that gives meaning to addition and subtraction by rooting them in elementary concepts of geometry and logic.

Translation of the introduction to the articles written by Ivan Riou

Des abaques pour reprendre le contrôle des opérations I

Des abaques pour reprendre le contrôle des opérations II

How to Use an Abacus

Counting

Adding and Subtracting

Multiplying

Dividing

https://www.wikihow.com/Use-an-Abacus