Chinese abacus
Encyclopedia
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...
of Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
origin first described in a 190 CE book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue. 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 application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. This configuration is used for both decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
and hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
computation. The beads are usually rounded and made of a hardwood
Hardwood
Hardwood is wood from angiosperm trees . It may also be used for those trees themselves: these are usually broad-leaved; in temperate and boreal latitudes they are mostly deciduous, but in tropics and subtropics mostly evergreen.Hardwood contrasts with softwood...
. 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
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
and cube root operations at high speed.
The modern suanpan has 4+1 beads, colored beads to indicate position and a clear-all button. When the clear-all button is pressed, two mechanical levers push the top row beads to the top position and the bottom row beads to the bottom position, thus clearing all numbers to zero. This replaces clearing the beads by hand, or quickly rotating the suanpan around its horizontal center line to clear the beads by centrifugal force.
History
Zhang Zeduan
Zhang Zeduan , alias Zheng Dao, also sometimes translated as Zhang Zerui, was a famous Chinese painter during the twelfth century, during the transitional period from the Northern Song to the Southern Song Dynasty, and was instrumental in the early history of the Chinese art style known as shan...
(1085–1145) during the Song Dynasty
Song Dynasty
The Song Dynasty was a ruling dynasty in China between 960 and 1279; it succeeded the Five Dynasties and Ten Kingdoms Period, and was followed by the Yuan Dynasty. It was the first government in world history to issue banknotes or paper money, and the first Chinese government to establish a...
(960-1279) might contain a suanpan beside an account book and doctor's prescriptions on the counter of an apothecary
Apothecary
Apothecary is a historical name for a medical professional who formulates and dispenses materia medica to physicians, surgeons and patients — a role now served by a pharmacist and some caregivers....
). However, the identification of the object as an abacus is a matter of some debate.
A 5+1 suanpan appeared in Ming dynasty, an illustration in a 1573 book on suanpan showed a suanpan with one bead on top and five beads at the bottom.
The similarity of the Roman abacus
Roman abacus
The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors...
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
Roman Empire
The Roman Empire was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....
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 and Chinese model (like most modern Japanese) has 4 plus 1 bead per decimal place, the old version of the Chinese suanpan has 5 plus 2, allowing less challenging arithmetic algorithms, and also allowing use with a hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
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 place value decimal system
Decimal system
Decimal system may refer to:* The decimal number system, used in mathematics for writing numbers and performing arithmetic.* The Dewey Decimal System, a subject classification system used in libraries....
with empty spot as zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
.
Beads
There are two types of beads on the suanpan, those in the lower deck, below the separator beam, and those in the upper deck above it. The ones in the lower deck are sometimes called earth beads or water beads, and carry a value of 1 in their column. The ones in the upper deck are sometimes called heaven beads and carry a value of 5 in their column. The columns are much like the places in Arabic numerals: one of the columns, usually the rightmost, represents the ones place; to the left of it are the tens, hundreds, thousands place, and so on, and if there are any columns to the right of it, they are the tenths place, hundredths place, and so on.The suanpan is a 2:5 abacus: two heaven beads and five earth beads. If one compares the suanpan to the soroban which is a 1:4 abacus, one might think there are two "extra" beads in each column. In fact, to represent decimal numbers and add or subtract such numbers, one strictly needs only one upper bead and four lower beads on each column. So these extra beads might be used to represent hexadecimal numbers on the suanpan and add or subtract them. Also, some "old" methods to multiply or divide decimal numbers use those extra beads like the "Extra Bead technique" or "Suspended Bead technique".
At the end of a decimal calculation on a suanpan, it is never the case that all five beads in the lower deck are moved up; in this case, the five beads are pushed back down and one carry bead in the top deck takes their place. Similarly, if two beads in the top deck are pushed down, they are pushed back up, and one carry bead in the lower deck of the next column to the left is moved up. In hexadecimal calculation, all seven beads on each column are used. The result of the computation is read off from the beads clustered near the separator beam between the upper and lower deck.
There exist different methods to perform division on the suanpan. Some of them require the use of the so-called "Chinese division table".
| Chinese Division Table | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| | 一 1 | | 二 2 | | 三 3 | | 四 4 | | 五 5 | | 六 6 | | 七 7 | | 八 8 | | 九 9 |
|
| 一 1 |
进一 advance 1 |
Cycle repeats | |||||||
| 二 2 |
添作五 replace by 5 |
进一 advance 1 |
Cycle repeats | ||||||
| 三 3 |
三十一 31 |
六十二 62 |
进一 advance 1 |
Cycle repeats | |||||
| 四 4 |
二十二 22 |
添作五 replace by 5 |
七十二 72 |
进一 advance 1 |
Cycle repeats | ||||
| 五 5 |
添作二 replace by 2 |
添作四 replace by 4 |
添作六 replace by 6 |
添作八 replace by 8 |
进一 advance 1 |
Cycle repeats | |||
| 六 6 |
下加四 below add 4 |
三十二 32 |
添作五 replace by 5 |
六十四 64 |
八十二 82 |
进一 advance 1 |
Cycle repeats | ||
| 七 7 |
下加三 below add 3 |
下加六 below add 6 |
四十二 42 |
五十五 55 |
七十一 71 |
八十四 84 |
进一 advance 1 |
Cycle repeats | |
| 八 8 |
下加二 below add 2 |
下加四 below add 4 |
下加六 below add 6 |
添作五 replace by 5 |
六十二 62 |
七十四 74 |
八十六 86 |
进一 advance 1 |
Cycle repeats |
| 九 9 |
下加一 below add 1 |
下加二 below add 2 |
下加三 below add 3 |
下加四 below add 4 |
下加五 below add 5 |
下加六 below add 6 |
下加七 below add 7 |
下加八 below add 8 |
进一 advance 1 |
The two most extreme beads, the bottommost earth bead and the topmost heaven bead, are usually not used in addition and subtraction. They are essential (compulsory) in some of the multiplication methods (two of three methods require them) and division method (special division table, Qiuchu 九歸 , one amongst three methods). When the intermediate result (in multiplication and division) is larger than 15 (fifteen), the lowest of the upper beads is moved halfway to represent ten (xuanchu, suspended). Thus the same rod can represent up to 19 (compulsory as intermediate steps in tradition suanpan multiplication and division).
The mnemonics/readings of the Chinese division method [Qiuchu] has its origin in the use of bamboo sticks [Chousuan], which is one of the reasons that many believe the evolution of suanpan is independent of the Roman abacus.
This Chinese division method (i.e. with division table) was not in use when the Japanese changed their abacus to one upper bead and four lower beads in about the 1920s.
The beads and rods are often lubricated to ensure quick, smooth motion.
Decimal system
This device works as a bi-quinaryBi-quinary coded decimal
Bi-quinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, including the Colossus. The term bi-quinary indicates that the code comprises both a two-state and a five-state component...
based number system in which carries and shifting are similar to the decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
number system. Since each rod represents a digit in a decimal number, the computation capacity of the suanpan is only limited by the number of rods on the suanpan. When a mathematician runs out of rods, another suanpan can be added to the left of the first. In theory, the suanpan can be expanded indefinitely in this way.
Hexadecimal system
Traditional Chinese weighing units was a hexadecimalHexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
system. One jin (斤) equals sixteen liang (兩). Suanpans were commonly used in market places to perform calculations with these hexadecimal units. When all the beads in the suanpan are used, each column can be used to represent numbers between 0 and 15 (0 and F in modern hexadecimal notation). Computation in decimal and hexadecimal is quite similar except one extra bead from both the upper and lower deck are used.
Decline in modern usage
Suanpan arithmetic was still being taught in school in Hong KongHong Kong
Hong Kong is one of two Special Administrative Regions of the People's Republic of China , the other being Macau. A city-state situated on China's south coast and enclosed by the Pearl River Delta and South China Sea, it is renowned for its expansive skyline and deep natural harbour...
as recently as the late 1960s, and in Republic of China
Republic of China
The Republic of China , commonly known as Taiwan , is a unitary sovereign state located in East Asia. Originally based in mainland China, the Republic of China currently governs the island of Taiwan , which forms over 99% of its current territory, as well as Penghu, Kinmen, Matsu and other minor...
into the 1990s. However, when hand held calculator
Calculator
An electronic calculator is a small, portable, usually inexpensive electronic device used to perform the basic operations of arithmetic. Modern calculators are more portable than most computers, though most PDAs are comparable in size to handheld calculators.The first solid-state electronic...
s became readily available, school children's willingness to learn the use of the suanpan decreased dramatically. In the early days of hand held calculators, news of suanpan operators beating electronic calculators in arithmetic competitions in both speed and accuracy often appeared in the media. Early electronic calculators could only handle 8 to 10 significant digits, whereas suanpans can be built to virtually limitless precision. But when the functionality of calculators improved beyond simple arithmetic operations, most people realized that the suanpan could never compute higher functions – such as those in trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
– faster than a calculator. Nowadays, as calculators have become more affordable, suanpans are not commonly used in Hong Kong or Taiwan, but many parents still send their children to private tutors or school- and government- sponsored after school activities to learn bead arithmetic as a learning aid and a stepping stone to faster and more accurate mental arithmetic, or as a matter of cultural preservation. Speed competitions are still held. Suanpans are still being used elsewhere in China and in Japan, as well as in some few places in Canada and the United States.
In mainland China, formerly accountants and financial personnel had to pass certain graded examinations in bead arithmetic before they were qualified. Starting from about 2002 or 2004, this requirement has been entirely replaced by computer accounting.
Miscellanea
- The suanpan is closely tied to the Chinese "huāmǎ" numbering systemSuzhou numeralsThe Suzhou numerals or huama is a numeral system used in China before the introduction of Arabic numerals.-History:The Suzhou numeral system is the only surviving variation of the rod numeral system. The rod numeral system is a positional numeral system used by the Chinese in mathematics...
. - Many Taoist temples in ChinaMainland ChinaMainland China, the Chinese mainland or simply the mainland, is a geopolitical term that refers to the area under the jurisdiction of the People's Republic of China . According to the Taipei-based Mainland Affairs Council, the term excludes the PRC Special Administrative Regions of Hong Kong and...
and TaiwanTaiwanTaiwan , also known, especially in the past, as Formosa , is the largest island of the same-named island group of East Asia in the western Pacific Ocean and located off the southeastern coast of mainland China. The island forms over 99% of the current territory of the Republic of China following...
hang large suanpans to remind followers against being scheming since the calculation of Man never beats that of HeavenHeavenHeaven, the Heavens or Seven Heavens, is a common religious cosmological or metaphysical term for the physical or transcendent place from which heavenly beings originate, are enthroned or inhabit...
. - The suanpan abacus were being used in calculation of engineering data for the development of China's first atomic bomb596 (nuclear test)596 is the codename of the People's Republic of China's first nuclear weapons test, detonated on October 16, 1964 at the Lop Nur test site. It was a uranium-235 implosion fission device and had a yield of 22 kilotons...
.
External links
- Suanpan Tutor - See the steps in addition and subtraction
- A Traditional Suan Pan Technique for Multiplication