Galley division
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In
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, the galley method, also known as the batello or the scratch method, was the most widely used method of
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
in use prior to 1600. The names galea and batello refer to a boat which the outline of the work was thought to resemble. An earlier version of this method was used as early as 825 by
Al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
. The galley method is thought to be of
Arab The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...
origin and is most effective when used on a sand
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
. However,
Lam Lay Yong Lam Lay Yong (maiden name Oon Lay Yong, ; born 1936) is a retired Professor of Mathematics. Academic career From 1988 to 1996 she was Professor at the Department of Mathematics, National University of Singapore (NUS). She graduated from the Uni ...
's research pointed out that the galley method of division originated in the 1st century AD in ancient China. The galley method writes fewer figures than
long division In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps ...
, and results in interesting shapes and pictures as it expands both above and below the initial lines. It was the preferred method of division for seventeen centuries, far longer than long division's four centuries. Examples of the galley method appear in the 1702 British-American cyphering book written by Thomas Prust (or Priest).


How it works

Set up the problem by writing the dividend and then a bar. The quotient will be written after the bar. Steps: :(a1) Write the divisor below the dividend. Align the divisor so that its leftmost digit is directly below the dividend's leftmost digit (if the divisor is 594, for instance, it would be written an additional space to the right, so that the "5" would appear below the "6", as shown in the illustration). :(a2) Dividing 652 by 594 yields the quotient 1 which is written to the right of the bar. Now multiply each digit of the divisor by the new digit of the quotient and subtract the result from the left-hand segment of the dividend. Where the subtrahend and the dividend segment differ, cross out the dividend digit and write if necessary the subtrahend digit and next vertical empty space. Cross out the divisor digit used. :(b) Compute 6 − 5×1 = 1. Cross out the 6 of the dividend and above it write a 1. Cross out the 5 of the divisor. The resulting dividend is now read off as the topmost un-crossed digits: 15284. :(c) Using the left-hand segment of the resulting dividend we get 15 − 9×1 = 6. Cross out the 1 and 5 and write 6 above. Cross out the 9. The resulting dividend is 6284. :(d) Compute 62 − 4×1 = 58. Cross out the 6 and 2 and write 5 and 8 above. Cross out the 4. The resulting dividend is 5884. :(e) Write the divisor one step to the right of where it was originally written using empty spaces below existing crossed out digits. :(f1) Dividing 588 by 594 yields 0 which is written as the new digit of the quotient. :(f2) As 0 times any digit of the divisor is 0, the dividend will remain unchanged. We therefore can cross out all the digits of the divisor. :(f3) We write the divisor again one space to the right :(omitted) Dividing 5884 by 594 yields 9 which is written as the new digit of the quotient. 58 − 5×9 = 13 so cross out the 5 and 8 and above them write 1 and 3. Cross out the 5 of the divisor. The resulting dividend is now 1384. 138 − 9×9 = 57. Cross out 1,3, and 8 of the dividend and write 5 and 7 above. Cross out the 9 of the divisor. The resulting dividend is 574. 574 − 4×9 = 538. Cross out the 7 and 4 of the dividend and write 3 and 8 above them. Cross out the 4 of the divisor. The resulting dividend is 538. The process is done, the quotient is 109 and the remainder is 538.


Other versions

The above is called the cross-out version and is the most common. An erasure version exists for situations where erasure is acceptable and there is not need to keep track of the intermediate steps. This is the method used with a sand abacus. Finally, there is a printers' method that uses neither erasure or crossouts. Only the top digit in each column of the dividend is active with a zero used to denote an entirely inactive column.


Modern usage

Galley division was the favorite method of division with arithmeticians through the 18th century and it is thought that it fell out of use due to the lack of cancelled types in printing. It is still taught in the
Moorish The term Moor, derived from the ancient Mauri, is an exonym first used by Christian Europeans to designate the Muslim inhabitants of the Maghreb, the Iberian Peninsula, Sicily and Malta during the Middle Ages. Moors are not a distinct or se ...
schools of
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and other parts of the
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.


Origin

Lam Lay Yong, mathematics professor of
National University of Singapore The National University of Singapore (NUS) is a national public research university in Singapore. Founded in 1905 as the Straits Settlements and Federated Malay States Government Medical School, NUS is the oldest autonomous university in the c ...
, traced the origin of the galley method to the ''
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still ...
'' written about 400AD. The division described by
Al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
in 825 was identical to the Sunzi algorithm for division.Lam Lay Yong, The Development of Hindu-Arabic and Traditional Chinese Arithmetic, Chinese Science, 13 1996, 35–54


See also

*
Group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
*
Field (algebra) In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which ...
*
Division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
*
Division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
*
Long division In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (Positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps ...
* Vinculum


References

{{Reflist


External links


Galley or Scratch Method of Division
at The Math Forum Division (mathematics) History of mathematics simple:Division