Abacus

 ( Mathematical Tools ) 

The Salamis Tablet, found on the Greek island Salamis Island in 1846 AD, dates to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble in length, wide, and thick, on which are 5 groups of markings. In the tablet's center 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. Also from this time frame, the Darius Vase was unearthed in 1851. It was covered with pictures, including a "treasurer" holding a wax tablet in one hand while manipulating counters on a table with the other.

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 nearby 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.) resembling a bi-quinary coded decimal system related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces" (i.e. fractions).

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.

The Chinese abacus, also known as the suanpan (算盤/算盘, lit. "calculating tray"), comes in various lengths and 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 one, to represent numbers in a bi-quinary coded decimal-like system. The beads are usually rounded and made of hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not. One of the top beads is 5, while one of the bottom beads is 1. Each rod has a number under it, showing the place value. The suanpan can be reset to the starting position instantly by a quick movement along the horizontal axis to spin all the beads away from the horizontal beam at the center.

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.

In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty (960–1297), a suanpan is clearly visible 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, given evidence of a trade relationship between the Roman Empire and China. However, no direct connection has been demonstrated, and the similarity of the abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Korean and Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. Incidentally, this ancient Chinese calculation system 市用制 ( Shì yòng zhì) allows use with a hexadecimal numeral system (or any Radix up to 18) which is used for traditional Chinese measures of weight ( jīn. (Instead of running on wires as in the Chinese, Korean, and Japanese models, the Roman model used grooves, presumably making arithmetic calculations much slower).

Another possible source of the suanpan is Chinese counting rods, which operated with a decimal but lacked the concept of zero as a placeholder. 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, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.

Today's Japanese abacus is a 1:4 type, four-bead abacus, introduced from China in the Muromachi period. It adopts the form of the upper deck one bead and the bottom four beads. The top bead on the upper deck was equal to five and the bottom one is similar to the Chinese or Korean abacus, and the decimal number can be expressed, so the abacus is designed as a 1:4 device. The beads are always in the shape of a diamond. The quotient division is generally used instead of the division method; at the same time, in order to make the multiplication and division digits consistently use the division multiplication. Later, Japan had a 3:5 abacus called 天三算盤, which is now in the Ize Rongji collection of Shansi Village in Yamagata City. Japan also used a 2:5 type abacus.

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 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.

The Nepōhualtzintzin was divided into two main parts separated by a bar or intermediate cord. In the left part were four beads. Beads in the first row have unitary values (1, 2, 3, and 4), and on the right side, three beads had 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 count in the first row.

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 Maya peoples 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 was a system of colored knotted cords used to record numerical data, like advanced – 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.

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. Arithmetic for Entertainment, Yakov Perelman, page 51. Likewise, the mass production of Felix 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 Turkic peoples and the Armenians people used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians.

In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame is common (see image).

Each bead represents one unit (e.g. 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented). In the bead frame shown, the gap between the 5th and 6th wire, corresponding to the color change between the 5th and the 6th bead on each wire, suggests the latter use. Teaching multiplication, e.g. 6 times 7, may be represented by shifting 7 beads on 6 wires.

The red-and-white abacus is used in contemporary primary schools for a wide range of number-related lessons. The twenty bead version, referred to by its Dutch language name rekenrek ("calculating frame"), is often used, either on a string of beads or on a rigid framework.