mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...

methods used by scribes, is a systematic method for multiplying two numbers that does not require the

multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...

, only the ability to multiply and divide by 2, and to

add Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

. It decomposes one of the

multiplicand Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...

s (preferably the smaller) into a set of numbers of

powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative ...

and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication. This method may be called mediation and duplation, where

mediation Mediation is a structured, interactive process where an impartial third party neutral assists disputing parties in resolving conflict through the use of specialized communication and negotiation techniques. All participants in mediation are ...

means halving one number and duplation means doubling the other number. It is still used in some areas. The second Egyptian multiplication and division technique was known from the

hieratic Hieratic (; grc, ἱερατικά, hieratiká, priestly) is the name given to a cursive writing system used for Ancient Egyptian and the principal script used to write that language from its development in the third millennium BC until the ris ...

Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...

and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe

Ahmes Ahmes ( egy, jꜥḥ-ms “, a common Egyptian name also transliterated Ahmose) was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dyna ...

. Gunn, Battiscombe George. Review of The Rhind Mathematical Papyrus by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137. Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as

long multiplication A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the de ...

after the multiplier and multiplicand are converted to

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...

. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors.

Method

The ancient Egyptians had laid out tables of a great number of powers of two, rather than recalculating them each time. The decomposition of a number thus consists of finding the powers of two which make it up. The Egyptians knew empirically that a given power of two would only appear once in a number. For the decomposition, they proceeded methodically; they would initially find the largest power of two less than or equal to the number in question,

subtract Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

it out and repeat until nothing remained. (The Egyptians did not make use of the number

zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

in mathematics.) After the decomposition of the first multiplicand, the person would construct a table of powers of two times the second multiplicand (generally the smaller) from one up to the largest power of two found during the decomposition. The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand.

Example

25 × 7 = ? Decomposition of the number 25: : The largest power of two is 16 and the second multiplicand is 7. As 25 = 16 + 8 + 1, the corresponding multiples of 7 are added to get 25 × 7 = 112 + 56 + 7 = 175.

Russian peasant multiplication

In the Russian peasant method, the powers of two in the decomposition of the multiplicand are found by writing it on the left and progressively halving the left column, discarding any remainder, until the value is 1 (or −1, in which case the eventual sum is negated), while doubling the right column as before. Lines with

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...

numbers on the left column are struck out, and the remaining numbers on the right are added together.Cut the Knot - Peasant Multiplication
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Example

238 × 13 = ?

References

Other sources

* Boyer, Carl B. (1968) A History of Mathematics. New York: John Wiley. * Brown, Kevin S. (1995) The Akhmin Papyrus 1995 --- Egyptian Unit Fractions. * Bruckheimer, Maxim, and Y. Salomon (1977) "Some Comments on R. J. Gillings' Analysis of the 2/n Table in the Rhind Papyrus," Historia Mathematica 4: 445–52. * Bruins, Evert M. (1953) Fontes matheseos: hoofdpunten van het prae-Griekse en Griekse wiskundig denken. Leiden: E. J. Brill. * ------- (1957) "Platon et la table égyptienne 2/n," Janus 46: 253–63. *Bruins, Evert M (1981) "Egyptian Arithmetic," Janus 68: 33–52. * ------- (1981) "Reducible and Trivial Decompositions Concerning Egyptian Arithmetics," Janus 68: 281–97. * Burton, David M. (2003) History of Mathematics: An Introduction. Boston Wm. C. Brown. * Chace, Arnold Buffum, et al. (1927) The Rhind Mathematical Papyrus. Oberlin: Mathematical Association of America. * Cooke, Roger (1997) The History of Mathematics. A Brief Course. New York, John Wiley & Sons. * Couchoud, Sylvia. "Mathématiques égyptiennes". Recherches sur les connaissances mathématiques de l'Egypte pharaonique., Paris, Le Léopard d'Or, 1993. * Daressy, Georges. "Akhmim Wood Tablets", Le Caire Imprimerie de l'Institut Francais d'Archeologie Orientale, 1901, 95–96. * Eves, Howard (1961) An Introduction to the History of Mathematics. New York, Holt, Rinehard & Winston. * Fowler, David H. (1999) The mathematics of Plato's Academy: a new reconstruction. Oxford Univ. Press. * Gardiner, Alan H. (1957) Egyptian Grammar being an Introduction to the Study of Hieroglyphs. Oxford University Press. * Gardner, Milo (2002) "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term" in History of the Mathematical Sciences, Ivor Grattan-Guinness, B.C. Yadav (eds), New Delhi, Hindustan Book Agency:119-34. * -------- "Mathematical Roll of Egypt" in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Nov. 2005. * Gillings, Richard J. (1962) "The Egyptian Mathematical Leather Roll," Australian Journal of Science 24: 339–44. Reprinted in his (1972) Mathematics in the Time of the Pharaohs. MIT Press. Reprinted by Dover Publications, 1982. * -------- (1974) "The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It?" Archive for History of Exact Sciences 12: 291–98. * -------- (1979) "The Recto of the RMP and the EMLR," Historia Mathematica, Toronto 6 (1979), 442–447. * -------- (1981) "The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?" Historia Mathematica: 456–57. * Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum" Journal of Egyptian Archaeology 13, London (1927): 232–8 * Griffith, Francis Llewelyn. The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vols. 1, 2. Bernard Quaritch, London, 1898. * Gunn, Battiscombe George. Review of The Rhind Mathematical Papyrus by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137. * Hultsch, F. Die Elemente der Aegyptischen Theihungsrechmun 8, Übersicht über die Lehre von den Zerlegangen, (1895):167-71. * Imhausen, Annette. "Egyptian Mathematical Texts and their Contexts", Science in Context 16, Cambridge (UK), (2003): 367–389. * Joseph, George Gheverghese. The Crest of the Peacock/the non-European Roots of Mathematics, Princeton, Princeton University Press, 2000 * Klee, Victor, and Wagon, Stan. Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, 1991. * Knorr, Wilbur R. "Techniques of Fractions in Ancient Egypt and Greece". Historia Mathematica 9 Berlin, (1982): 133–171. * Legon, John A.R. "A Kahun Mathematical Fragment". Discussions in Egyptology, 24 Oxford, (1992). * Lüneburg, H. (1993) "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim: 81=85. * * Robins, Gay. and Charles Shute, The Rhind Mathematical Papyrus: an Ancient Egyptian Text" London, British Museum Press, 1987. * Roero, C. S. "Egyptian mathematics" Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences" I. Grattan-Guinness (ed), London, (1994): 30–45. * Sarton, George. Introduction to the History of Science, Vol I, New York, Williams & Son, 1927 * Scott, A. and Hall, H.R., "Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC", British Museum Quarterly, Vol 2, London, (1927): 56. * Sylvester, J. J. "On a Point in the Theory of Vulgar Fractions": American Journal of Mathematics, 3 Baltimore (1880): 332–335, 388–389. * Vogel, Kurt. "Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv für Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386-407 * van der Waerden, Bartel Leendert. Science Awakening, New York, 1963 * Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.

External links

*http://rmprectotable.blogspot.com/ RMP 2/n table *https://web.archive.org/web/20130625181118/http://weekly.ahram.org.eg/2007/844/heritage.htm *http://emlr.blogspot.com Egyptian Mathematical Leather Roll *https://web.archive.org/web/20120913011126/http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html *https://web.archive.org/web/20120606142257/http://planetmath.org/encyclopedia/RationalNumbers.html *http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0 Math forum and two ways to calculate 2/7
New and Old classifications of Ahmes Papyrus

The Russian Peasant Algorithm (pdf file)

Peasant Multiplication
from

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Egyptian Multiplication
by Ken Caviness,

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

.
Russian Peasant Multiplication
at

The Daily WTF The Daily WTF (also called Worse Than Failure from February to December 2007) is a humorous blog dedicated to "Curious Perversions in Information Technology". The blog, run by Alex Papadimoulis, "offers living examples of code that invites the ex ...

Michael S. Schneider explains how the Ancient Egyptians (and Chinese) and modern computers multiply and divide

Russian Multiplication - Numberphile
{{number theoretic algorithms Egyptian mathematics Multiplication Number theoretic algorithms Egyptian fractions Ancient Egyptian literature Mathematics manuscripts

Method

Example

Russian peasant multiplication

Example

See also

References

Other sources

External links