The abacus (plural abaci or abacuses), also caleed a counting frame, is a calculating frame, is a calculating tool used primarily in a parts of asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but orginally 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.
Etymology
The use of the word abacus dates before 1387 AD, when a
Middle English work borrowed the word from Latin to describe a sandboard
abacus. The Latin word came from Greek ἄβαξ abax "board strewn with sand
or dust used for drawing geometric figures or calculating" (the exact
shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς
abakos). Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic,
perhaps Phoenician, word akin to Hebrew ʾābāq (אבק),
"dust" (since dust strewn on wooden boards to draw figures in). The
preferred plural of abacus is a subject of disagreement, with both abacuses and
abaci in use.
Examples
Mesopotamian
The period 2700–2300 BC saw the first appearance of the
Sumerian 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 Old Babylonian scholars such as Carruccio that Old Babylonians
"may have used the abacus for the operations of addition and subtraction;
however, this primitive device proved difficult to use for more complex calculations".
Egyptian
The use of the abacus in Ancient Egypt 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 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 have not been discovered, casting some doubt over the extent to
which this instrument was used.[original research?]
Persian
During the Achaemenid Persian Empire, around 600 BC the
Persians first began to use the abacus. Under Parthian and Sassanian Iranian
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.
Greek
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, pre-set 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 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 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.
Roman
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 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 nineteenth 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 bi-quinary coded decimal system,
obviously related to the Roman numerals. The short grooves on the right may
have been used for marking Roman ounces.
Chinese
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 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.[17] 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 (618-907 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.
Indian
First century sources, such as the Abhidharmakosa describe
the knowledge and use of abacus in India. 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
In Japanese, the abacus is called soroban (算盤, そろばん,
lit. "Counting tray"), 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, one can arrive at the answer in the same time as,
or even faster than, is possible with a physical instrument.Korean
The Chinese abacus migrated from China to Korea around 1400
AD.[24] Koreans call it jupan (주판), supan (수판) or jusan (주산).
Native American
Some sources mention the use of an abacus called a
nepohualtzintzin in ancient Mayan culture. This Mesoamerican abacus used a
5-digit base-20 system. The word Nepōhualtzintzin [nepoːwaɬ't͡sint͡sin] comes
from the Nahuatl and it is formed by the roots; Ne - personal -; pōhual or
pōhualli ['poːwalːi] - the account -; and tzintzin ['t͡sint͡sin] - small
similar elements. And its complete meaning was taken as: counting with small
similar elements by somebody. Its use was taught in the Calmecac [kal'mekak] to
the temalpouhqueh [temaɬ'poʍkeʔ], who were students dedicated to take the
accounts of skies, from childhood. Unfortunately the Nepōhualtzintzin 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.[citation needed]
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.[clarification needed] The
Nepōhualtzintzin 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 Nepōhualtzintzin. 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 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
approximate a year (1¼ days short). It is worth mentioning that the
Nepōhualtzintzin 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 Nepōhualtzintzin 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.[citation
needed]. There have also been found very old Nepōhualtzintzin attributed to the
Olmeca 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",
Austin-Texas, 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 and 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 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 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
The Russian abacus, the schoty (счёты), usually has a single
slanted deck, with ten beads on each wire (except one wire, usually positioned
near the user, with four beads for quarter-ruble fractions). Older models have
another 4-bead wire for quarter-kopeks, 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, 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 Jean-Victor 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 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.
School abacus
Around the world, abaci have been used in pre-schools and
elementary schools as an aid in teaching the numeral system 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.
Early 19th century abacus used in Danish
elementary school.
Uses by the blind
An adapted abacus, invented by Tim Cranmer, called a Cranmer
abacus is still commonly used by individuals who are blind. 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 braille-writer 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.
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 ASCII. 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.
Two binary abaci constructed by
Dr. Robert C. Good, Jr., made from two
Chinese abaci
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