An abacus (plurals abacuses or abaci), also called a counting frame, is a calculating tool for
performing arithmetical processes, often constructed as a wooden frame with beads sliding on wires. The user, called an
abacist, slides counters by hand on rods or in grooves.[1] It was in use centuries before the adoption of the written Hindu-Arabic numeral system and is still widely used by merchants and clerks in
China, Japan, Africa and
elsewhere.
Origins
The first abacus was almost certainly based on a flat stone covered with sand or dust. Words and letters were drawn in the
sand; eventually numbers were added[2] and pebbles used to
aid calculations. The Babylonians used this dust abacus as early as 2400 BC.[3] The origin of the
counter abacus with strings is obscure, but India, Mesopotamia or Egypt are seen as probable points of origin.[4] China played an essential part
in the development and evolution of the abacus.
From this, a variety of abaci were developed; the most popular were based on the bi-quinary system, using a combination of two bases (base-2 and base-5) to represent decimal
numbers. But the earliest abaci used first in Mesopotamia and later by scribes in Egypt and Greece used sexagesimal numbers represented with factors of 5, 2, 3, and 2 for each digit.
The use of the word abacus dates from before 1387, when a Middle English work
borrowed the word from Latin to describe a sandboard abacus. The Latin word came from
abakos, the Greek genitive form of
abax ("calculating-table"). Because abax also had the sense of "table sprinkled with sand or dust, used for drawing
geometric figures", some linguists speculate that the Greek word may be derived from a Semitic root, ābāq (pronounced "a-vak"), the
Hebrew word for "dust". Though details of the transmission are obscure, it may also be
derived from the Phoenician word abak, meaning "sand". The preferred plural
of abacus is a subject of disagreement, but both abacuses[5] and abaci[6] are in
use.
Babylonian abacus
Babylonians may have used the abacus for mathematical operations of addition and subtraction. However, this primitive device
proved difficult to use for more complex calculations.[7]
Some scholars point to a character from the Babylonian cuniform which may have been derived from a representation of the
abacus.[8]
Egyptian abacus
The use of the abacus in ancient Egypt is mentioned by the Greek historian
Herodotus, who writes that the manner of its usage by the Egyptians was opposite in direction
when compared with the Greek method. Archaeologists have found ancient disks of various sizes that are thought to have been used
as counters. However, wall depictions of this instruments have not been discovered, casting some doubt over the extent of use of
this instrument.[9]
Greek abacus
A tablet found on the Greek island Salamis in 1846 dates back to 300 BC, making it the
oldest counting board discovered so far. It is a slab of white marble 149 cm long, 75 cm wide, and 4.5 cm 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
semi-circle 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 semi-circle 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
Reconstructed 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, 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.[10]
In addition to the more common method using loose counters, several specimens have been found of a Roman abacus, shown here in reconstruction. 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.
Indian abacus
1st century sources, such as the Abhidharmakosa describe the knowledge and use of
abacus in India.[11]
Around the 5th century, Indian clerks were already finding new ways of recording the
contents of the Abacus.[12] Hindu texts used the term
shunya(means Zero) to indicate the empty column on the abacus.[13]
Chinese abacus
Suanpan (the number represented in the picture is 6,302,715,408)
-
The earliest mention of a suanpan is found in a First Century book of the Eastern Han
Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue.[14] However, the exact design of this suanpan is not known.
Usually, a suanpan is about 20 cm tall and it 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. 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.
In the famous long scroll Riverside Scenes at Qingming Festival painted by
Zhang Zeduan (1085-1145) during
the Song Dynasty (960-1297), 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 less challenging
arithmetic algorithms, and also 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 groves, 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)
when travel in the Indian Ocean and the Middle East
would have provided direct contact with India and Islam
allowing them to acquire the concept of zero and the decimal point from Indian and
Islamic merchants and mathematicians.
Japanese abacus
-
A soroban (算盤, そろばん, lit. "Counting tray") is a Japanese-modified version of the Chinese abacus.
It is devised from the suanpan, imported from China to Japan through the Korean
peninsula in the 15th century. Like the suanpan, the soroban is still used in Japan
today, even with the proliferation, practicality, and affordability of pocket electronic calculators.
Korea also has its own called the supan (수판), which is basically the soroban before it took its modern form in the
1930s. The modern soroban also goes by this Korean name.[15]
Native American abaci
Representation of an
Inca quipu
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 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 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 quarter-ruble fractions). This wire is usually near the user. (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, in order 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 have a colour different from the other 8 beads.
Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color.
The Russian abacus is still in use today in shops and markets throughout the former Soviet Union, although it is no longer taught in most schools.
School abacus
School abacus used in Danish elementary school. Early 20th century.
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.
The most significant educational advantage of using an abacus, rather than loose beads or counters, when practicing counting
and simple addition is that it gives the student an awareness of the groupings of 10 which are the foundation of our number
system. Although adults take this base 10 structure for granted, it is actually difficult to learn. Many 6-year-olds can count to
100 by rote with only a slight awareness of the patterns involved.
Uses by the blind
An adapted abacus, 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 math skills that can never be replaced
with talking calculators and is an important learning tool for blind students. Blind students also complete math assignments
using a braille-writer and Nemeth code (a type of braille code for math) but large
multiplication and long division problems can be long and difficult. The abacus gives blind and visually impaired students a tool
to compute math 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.
Notes
- ^ "abacist", "abacus", in Merriam-Webster's Third New International
Dictionary Unabridged, 2000, Version 2.5.
- ^ "abacus." Encyclopædia Britannica. 3 February
2007
- ^ Reilly, page 825
- ^ Smith, page 159
- ^ Merriam-Webster's 2003
- ^ Oxford English Dictionary, 1989
- ^ Carruccio, page 14
- ^ Crump, page 188
- ^ Smith, page 160
- ^ Pullan, page 18
- ^ Stearns, page 44
- ^ Körner, page 232
- ^ Mollin, page 3
- ^ Peng Yoke Ho, page 71
- ^ This term is used extensively in the Korean website http://www.supan.net, which not only offers lessons on using the
abacus but also sells soroban, particularly from Japanese dealer Tomoe.
References
- Reilly, Edwin D.; William Leonard Langer (2004). Concise Encyclopedia of
Computer Science. John Wiley and Sons. ISBN ISBN 0470090952.
- Körner, Thomas William; William Leonard Langer (1996). The Pleasures of
Counting. Houghton Mifflin Books. ISBN ISBN 0521568234.
- Mollin, Richard Anthony (1998). Fundamental Number Theory with
Applications. CRC Press. ISBN ISBN 0849339871.
- Smith, David Eugene. History of Mathematics (Volume 2). Courier Dover
Publications. ISBN ISBN 0486204308.
- Crump, Thomas (1992). The Japanese Numbers Game: The Use and Understanding of
Numbers in Modern Japan. Routledge. ISBN ISBN 0415056098.
- Carruccio, Ettore (2006). Mathematics And Logic in History And in
Contemporary Thought. Aldine Transaction. ISBN ISBN 0202308502.
- Stearns, Peter N.; William Leonard Langer (2001). The Encyclopedia of World
History: Ancient, Medieval, and Modern, Chronologically Arranged. Houghton Mifflin Books. ISBN ISBN
0395652375.
- Peng Yoke Ho (2000). Li, Qi and Shu: An Introduction to Science and
Civilization in China. Courier Dover Publications. ISBN ISBN 0486414450.
- (2003) Merriam-Webster's Collegiate Dictionary, 11th edtion,
Merriam-Webster, Inc. ISBN 0877798095.
- "abacus". Oxford English Dictionary. Oxford University Press. 2nd ed.
1989.
- Pullan, J. M. (1968). The History of the
Abacus. London: Books That Matter. ISBN 0-09-089410-3.
See also
Further reading
- Menninger, Karl W. (1969). Number Words and
Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 0-262-13040-8.
- Kojima, Takashi (1954). The Japanese Abacus: its
Use and Theory. Tokyo: Charles E. Tuttle. ISBN 0-8048-0278-5.
External links
Tutorials
Abacus curiosities
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