History of the Abacus

The abacus is a calculation tool used to perform arithmetic operations, typically constructed on a wooden, metal, or plastic frame with beads sliding on rods.

The user slides the beads by hand along rods or within grooves.

The abacus was in use centuries before the adoption of the written Hindu–Arabic numeral system and is still widely used by merchants in China, Japan, and elsewhere.

The abacus is a simple, inexpensive, yet powerful calculating tool. Although electronic calculators have replaced it for practical purposes, abacus use continues to be taught in technologically advanced countries like Japan.

Similar to popular board games such as chess, shogi, and Go, local, regional, and national abacus competitions are held, demonstrating that even though the abacus has fallen out of use for many applications, it continues to generate enthusiasm.

Boulier or Abacus?

What exactly is the difference between these two terms?

An abacus is defined as a calculation tool: it can be graphical (like the Smith chart) or could have been, before our era, a diagram drawn in the dust.

The definition of the abacus is broader and encompasses the different types of boulier (counting frames), which are by definition abacuses in a specific physical form using beads on rods.

Origin of the Abacus

Invention of the Abacus

The first abacus was most likely based on a flat stone covered with sand or dust. Words and letters were drawn in the sand, then numbers were added, and small stones were used to facilitate calculations.

The Babylonians used this dust abacus as early as 2400 BC. The origin of the string abacus is obscure, but India, Mesopotamia, or Egypt are considered likely points of origin.

China also played a key role in the development and evolution of the abacus.

From there, a variety of abacuses was developed; the most popular were based on the bi-quinary system, using a combination of two bases (base 1 and base 5) to represent decimal numbers. But the earliest abacuses, first used in Mesopotamia and then by Egyptian and Greek scribes, used a hexadecimal base.

Origin of the Name

The word boulier is unique to French; in other Latin languages or those with some Latin roots, a word derived from the term abacus is used.

The use of the word abacus dates from before the 14th century and is borrowed from the Latin word for describing a sand abacus.

The Latin word comes from abakos, the Greek genitive form of abax ("calculating table"). Since abax also had the meaning of "table sprinkled with sand or dust, used for drawing geometric figures," some linguists speculate that the Greek word could be derived from a Semitic root, ābāq (pronounced "a-vak"), the Hebrew word for "dust."

Although the details of the transmission are obscure, it might also be derived from the Phoenician word abak, meaning "sand." The preferred plural of abacus is a matter of disagreement, but both abacuses and abaci are used.

Babylonian Abacus

The Babylonians may have used the abacus for addition and subtraction. However, this primitive device proved difficult to use for more complex calculations. Some scholars note a Babylonian cuneiform character that may have been derived from a representation of the abacus.

Egyptian Abacus

The use of the abacus in ancient Egypt is mentioned by the Greek historian Herodotus, who writes that the way Egyptians used it was the opposite of the Greek method. Archaeologists have found ancient discs of different sizes that may have been used as counters. However, no mural depictions of these instruments have been discovered, casting doubt on the extent of this instrument's use.

Greek Abacus

A tablet found on the Greek island of Salamis in 1846 dates to 300 BC, making it the oldest counting board discovered to date. It is a white marble slab 149 cm long, 75 cm wide, and 4.5 cm thick, bearing 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, topped by a semicircle at the intersection of the lowest horizontal line and the single vertical line. Below these lines is a wide space divided by a horizontal crack. 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 the vertical line.

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, tokens were manufactured. Marked lines indicated units, fives, tens, etc., as in the Roman numeral system. This system of "counter-casting" continued through the late Roman Empire and into medieval Europe and persisted in limited use until the 19th century.

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 each and eight shorter grooves having one or no bead each.

The groove marked I indicates units, X indicates tens, and so on up to millions. The beads in the shorter grooves indicate five units, five tens, etc., essentially in a bi-quinary coded decimal system, obviously related to Roman numerals. The short grooves on the right may have been used for marking Roman ounces.

Indian Abacus

First-century sources, such as the Abhidharmakośa, describe knowledge and use of the abacus in India. By the fifth century, Indian clerics were already finding new ways to record the contents of the abacus. Hindu texts used the term śūnya (meaning zero) to indicate the empty column on the abacus.

Chinese Abacus

The earliest mention of a suanpan is found in a book from the first century of the Eastern Han dynasty.

Typically, a suanpan is about 20 cm tall and comes in different 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 lower deck for decimal and hexadecimal calculation. The beads are usually rounded and made of hardwood. Beads are counted by moving them up or down toward the beam. The suanpan can be reset to its starting position instantly by a quick shake along the horizontal axis to move all beads away from the horizontal center beam.

Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, highly efficient techniques have been developed for rapidly performing multiplication, division, addition, subtraction, square root, and cube root operations.

In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan (1085–1145) during the Song dynasty (960–1297), a suanpan can be clearly seen lying beside an account book and medical prescriptions on the counter of an apothecary (Feibao).

The resemblance between the Roman abacus and the Chinese abacus suggests that one might have inspired the other, as there is evidence of a trade relationship between the Roman Empire and China. However, no direct link can be demonstrated, and the similarity of the abacuses could be merely a coincidence, both ultimately resulting from counting with five fingers per hand. Whereas the Roman model (like most modern Japanese ones) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2 for less complex arithmetic algorithms in a hexadecimal numeral system. Instead of running on wires as in the Chinese and Japanese models, the beads of the Roman model run in grooves, which probably made 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 placeholder. Zero was probably introduced to the Chinese during the Tang dynasty (618–907), when travel in the Indian Ocean and the Middle East would have allowed them to come into direct contact with India and Islam and acquire the concept of zero and the decimal point from Indian and Islamic merchants and mathematicians.

The Japanese Abacus

The abacus migrated from China to Korea around the year 1400 and later to Japan, around 1600. The Korean version of the abacus is called jupan (주판) or supan (수판) or jusan (주산).

In Japan, the Chinese suanpan was called soroban (算盤, そろばん). Like the suanpan, the soroban is still used in Japan today, even with the widespread use of electronic calculators.

The Russian Abacus

The Russian abacus, the schoty (счёты), typically has a single slanted deck, with ten beads on each wire (except one wire that has four beads, for quarter-kopeks). This wire is usually near the user. (Older models have another four-bead wire for quarter-kopeks, which were minted until 1916.) The Russian abacus is often used vertically, with wires running left to right in a book-like fashion. The wires are usually bowed to bulge upward in the center, to keep the beads pinned to either side. The abacus is cleared when all beads are moved to the right. During manipulation, beads are moved to the left. To aid in visualization, the two middle beads on each wire (the 5th and 6th bead) are usually a different color from the other eight beads. Similarly, the leftmost bead of the thousands wire (and the millions wire, if present) may be a different color.

The Russian abacus is still used today in shops and markets throughout the former Soviet Union, although it is no longer taught in most schools.

The School Abacus

Worldwide, abacuses have been used in nursery schools and elementary schools as an aid to 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).

The type of abacus presented here is often used to represent numbers without using 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 important pedagogical advantage of using an abacus, rather than loose beads or counters, for practicing counting and simple addition is that it makes the student aware of groupings of 10, which are the foundation of our number system. Although adults take this base-10 structure for granted, it is actually not necessarily the easiest to learn.