An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes. Today, abaci are often constructed as a wooden 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 China, Japan, Africa, India and elsewhere.
The user of an abacus is called an abacist; he or she slides the beads of the abacus by hand.
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. Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus.
A tablet found on the Salamis in 1846 AD dates back to 300 BC. 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.
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.
The earliest known written documentation of the Chinese abacus dates to the 14th century AD.
The top of the abacus is called the heaven and the bottom is called the earth.
The Chinese abacus known as the suànpán is typically 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. Modern abacuses have one bead on the top deck and four beads on the bottom deck. 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.
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.
In Japanese, the abacus is called Soroban (算盤, そろばん, lit. "Counting tray") in Japan, imported from China around 1600. The 1/4 abacus appeared circa 1930, and it is preferred and 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 a part of math.
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 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 is 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. But now we know with certainty that it is a concrete example of the great scientific and technological development that the majority of the native cultures already had in those times.
This arithmetic tool is based on the vigesimal system (base 20). For the aztec the count by 20s was completely natural, since the use of "huaraches" (native sandals) allowed them to also use the toes for their calculations. In this way, the amount of 20 meant to them a complete human being. The Nepohualtzintzin is divided in two main parts separated by a bar or intermediate cord. In the left part there are 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 are 13 rows with 7 beads in each one, which makes up 91 beads in each Nepohualtzintzin. This is a basic number to understand the close relation conceived between the exact accounts and the natural phenomena. This is so that one Nepohualtzintzin (91) represents 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 complete a cycle and form a year. It is worth to mention that in the Nepohualtzintzin, amounts in the rank from 10 to the 18 can be calculated, with floating point, which allows calculating stellar as well as infinitesimal amounts with absolute precision.
The rediscovering of the Nepohualtzintzin is due to the teacher David Esparza Hidalgo, who in his wandering by all Mexico has found diverse engravings and paintings of this instrument and has reconstructed several of them made in gold, jade, incrustations of shell, etc. There have been also found very old Nepohualtzintzin attributed to the Olmeca culture, and even some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures. This gives us an idea of the so early epochs in which our ancestors already had the sufficient knowledge to devise and to handle a device of such complexity, and the notion of the extension of its use in their daily activities.
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.
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 are of a different colour than the other eight beads. 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 all shops and markets throughout the former Soviet Union, and the usage of it was taught in most schools until the 1990s. Today it is regarded as an archaism and replaced by microcalculator. The use of calculators has been taught since the 1990s.
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.
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.