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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
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 binary operation, operation for an algebraic system. The decimal multiplication table was traditionally tau ...
, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (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 the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hi ...
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, voluntary process for resolving disputes, facilitated by a neutral third party known as the mediator. It is a structured, interactive process where an independent third party, the mediator, assists disputing parties ...
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 (; ) 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 BCE until the rise of Demotic in the mid-first millennium BCE ...
Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...
and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe
Ahmes Ahmes ( “, a common Egyptian name also transliterated Ahmose (disambiguation), Ahmose) was an ancient Egyptian scribe who lived towards the end of the 15th Dynasty, Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of t ...
. 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 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 values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...
. 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 Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
ians had laid out tables of a great number of powers of two, rather than recalculating them each time. To decompose a number, they identified 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 (which is signified by the minus sign, –) is one of the four arithmetic operations along with addition, multiplication and division. Subtraction is an operation that represents removal of objects from a collection. For example, ...
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. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
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. Because mathematically speaking, multiplication of natural numbers is just "exponentiation in the additive
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
", this multiplication method can also be recognised as a special case of the
Square and multiply In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some ...
algorithm for exponentiation.


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 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 = ?


See also

*
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from eac ...
*
Egyptian mathematics Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counti ...
*
Multiplication algorithm A multiplication algorithm is an algorithm (or method) to multiplication, multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much resea ...
s *
Binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" ( zero) and "1" ( one). A ''binary number'' may als ...


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 The ''British Museum Quarterly'' was a peer-reviewed academic journal published by the British Museum. It described recent acquisitions and research concerning the museum's collections and was published from 1926 to 1973. It is available electron ...
'', 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


RMP 2/n table

Egyptian Mathematical Leather Roll

Math forum and two ways to calculate 2/7





The Russian Peasant Algorithm (pdf file)

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

Egyptian Multiplication
by Ken Caviness,
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.
Russian Peasant Multiplication
at
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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