An abacus is a calculation tool, often constructed as a wooden frame with beads sliding on wires. 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 and elsewhere.
The origins of the abacus are disputed, suggestions including invention in Babylonia and in China, to have taken place between 2400 BC and 300 BC. The first abacus was almost certainly based on a flat stone covered with sand or dust. Lines were drawn in the sand and pebbles used to aid calculations. 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.
The use of the word abacus dates back to 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," it is speculated by some linguists that the Greek word may be derived from a Semitic root, abaq, 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 plural of abacus is abaci.
A tablet found on the island of Salamis (near Greece) in 1846 dates back to the Babylonians of 300 BC making it the oldest counting board discovered so far. It was originally thought to be a gaming board.
Its construction is a slab of white marble measuring 149cm in length, 75cm in width and 4.5cm thick, on which are 5 groups of markings. In the center of the tablet are 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.
The Late Empire Roman abacus shown here in reconstruction contains eight long and eight shorter grooves, the former having up to five beads in each and the latter one.
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.
Computations are made by means of beads which would probably have been slid up and down the grooves to indicate the value of each column.
The suanpan (Simplified Chinese: perhaps; Traditional Chinese: perhaps; Hanyu Pinyin: suànpán) of the Chinese is similar to the Roman abacus in principle, though has a different construction, and it was designed to do both decimal and hexadecimal arithmetics.
The Chinese abacus is typically around 20 cm (8 inches) tall and it comes in various widths depending on the application. 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 hard wood. The beads are counted by moving them up or down towards the beam. The abacus 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.
Chinese abaci 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.
Bead arithmetic is the calculating technique used with various types of abaci, in particular the Chinese abacus.
Japanese abacus (Soroban)
The Japanese eliminated (first) one bead from the upper deck and (later) another bead from the lower deck in each column of the Chinese abacus. The Japanese also eliminated the use of Qiuchu (Chinese division table). The method of Chinese division table was still used when there were 5 lower beads. There came the war of the Multiplication Table versus the Division Table. The school of Multiplication table prevailed in 1920s. The rods (number of digits) increase to usually 21, 23, 27 or even 31, thus allowing calculation for more digits.
Soroban is taught in elementary schools as a part of lessons in mathematics. When teaching the soroban, a song-like instruction is given by the tutor. The soroban is about 8 cm (3 inches) tall. The beads on a soroban are usually shaped as a double cone (bi-cone) to facilitate ease of movement.
Often, primary students may bring along with them two sorobans, one with 1 upper bead and 5 lower beads, the other with 1 upper bead with 4 lower beads, when they learn soroban in school.
The size of beads of soroban is standardized, and they come in two types: the "Japanese" classified soroban for native Japanese, and a separate size for foreigners (since Westerners tend to be larger than most Japanese, and therefore have larger hands and fingers). The soroban that are for foreigners are made with a plastic pipe on both the left and right side of the frame, while ones made for native Japanese were all made with wooden frames. In this way the "thickness" of the soroban (for foreigners) is higher, rendering it easier for the non-Japanese to manipulate.
The Russian abacus, the schoty or sjotty (perhaps), usually has a single slanted deck, with ten beads on each wire (except one wire which has four, and acts as a separator or for fractions). This wire is usually near the user.
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 to 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 common use today in shops and markets throughout the former Soviet Union, although it is no longer taught in most schools.
Around the world, abaci have been used in pre-schools and elementary schools as an aid in teaching arithmetics. In Western countries, a bead frame similar to the Russian abacus but with straight wires has been common. It is still often seen as a plastic or wooden toy.
Uses by the visually impaired
Abaci are still commonly used by individuals who have visual impairments. They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction, square root and cubic root. A piece of soft fabric or rubber is placed behind the beads so that they don't move inadvertently. This keeps the beads in place while the user feels or manipulates them.
Recently, abaci have been replaced to some extent by electronic calculators with speech, but only in those countries where they are easily available and affordable. However, even when they are available, many visually impaired people still prefer to use the abacus. In addition, many blind children are required to learn how to use the abacus before they are permitted the use of a talking calculator or similar device. This can be compared to sighted children being required to learn how to solve mathematical problems on paper before they are allowed the use of a calculator.
Native American abacus
Some sources mention the use of an abacus called a Nepohualtzintzin in ancient Mayan culture. This Mesoamerican abacus uses the 5-digit base-20 Mayan numeral system.
The khipu of the Inkas was a system of knotted cords used to record numerical data - like advanced tally sticks -- but was not used to perform calculations.