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Ancient Egyptian multiplication

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In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, was a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (generally the larger) into a sum of powers of two and creates a table of doublings of the second multiplicand. This method may be called mediation and duplation, where mediation 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 Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.

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. 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.

aliens

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.

For example, to multiply 238 by 13, the smaller of the numbers (to reduce the number of steps), 13, is written on the left and the larger on the right. The left number is progressively halved (discarding any remainder) and the right one doubled, until the left number is 1:
 13  238
 6  (remainder discarded) 476
 3  952
 1  (remainder discarded) 1904
Lines with even numbers on the left column are struck out, and the remaining numbers on the right are added, giving the answer as 3094:
 13   238 
 6   476 
 3   952 
 1  + 1904 
 3094 
The algorithm can be illustrated with the binary representation of the numbers:
1101 (13) 11101110 (238)
110 (6) 111011100 (476)
11 (3) 1110111000 (952)
1 (1) 11101110000 (1904)
       
1 1 1 0 1 1 1 0 (238)
× 1 1 0 1 (13)
1 1 1 0 1 1 1 0 (238)
0 0 0 0 0 0 0 0 0 (0)
1 1 1 0 1 1 1 0 0 0 (952)
+ 1 1 1 0 1 1 1 0 0 0 0 (1904)
1 1 0 0 0 0 0 1 0 1 1 0 (3094)

See also

References

  1. Cut the Knot - Peasant Multiplication

Other sources

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  • Brown, Kevin S. (1995) The Akhmin Papyrus 1995 --- Egyptian Unit Fractions.
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  • -------- "Mathematical Roll of Egypt" in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Nov. 2005.
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  • -------- (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.
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  • Lüneburg, H. (1993) "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim: 81=85.
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External links

Number-theoretic algorithms
Primality tests
Prime-generating
Integer factorization
Multiplication
Euclidean division
Discrete logarithm
Greatest common divisor
Modular square root
Other algorithms
  • Italics indicate that algorithm is for numbers of special forms
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