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Ancient Egyptian multiplication is an algorithm for multiplication that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. Also known as Egyptian multiplication and Peasant multiplication, 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. Wikipedia does not have an article with this exact name. ... This article or section is in need of attention from an expert on the subject. ... Flowcharts are often used to graphically represent algorithms. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... Division by two is simple in even-numbered bases. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

This technique is known in particular thanks to the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes. Development of hieratic script from hieroglyphs; after Champollion. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts discovered. ... The Moscow and Rhind Mathematical Papyri are two of the oldest mathematical texts and perhaps our best indication of what ancient Egyptian mathematics might have been like near 2000 BC. They are both written on papyrus. ... Ahmes (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. ...

The decomposition

The decomposition into a sum of powers of two is not, in fact, a change from base ten to base two; the Egyptians then were unaware of such concepts and had to resort to much simpler methods. The ancient Egyptians had laid out tables of a great number of powers of two so as not to be obliged to recalculate 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 power of two would appear but 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 it out and repeat until nothing remained. (The Egyptians did not make use of the number zero in mathematics). Khafres Pyramid (4th dynasty) and Great Sphinx of Giza (c. ...

Example of the decomposition of the number 25:

• the largest power of two less than or equal to 25 is 16,
• 25 – 16 = 9,
• the largest power of two less than or equal to 9 is 8,
• 9 – 8 = 1,
• the largest power of two less than or equal to 1 is 1,
• 1 – 1 = nil,

25 is thus the sum of the powers of two: 16, 8 and 1.

The table

After the decomposition of the first multiplicand, it suffices to 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. In the table, a line is obtained by multiplying the preceding line by two.

For example, if the largest power of two found during the decomposition is 16, and the second multiplicand is 7, the table is created as follows:

• 1; 7
• 2; 14
• 4; 28
• 8; 56
• 16; 112

The result

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.

The main advantage of this techniques is that it makes use of only addition, subtraction and multiplication by two.

Example

Here, in actual figures, is how 238 is multiplied by 13. The lines are multiplied by two, from one to the next. A check mark is placed by the powers of two in the decomposition of 13.

Since 13 = 8 + 4 + 1, distribution of multiplication over addition gives 13 × 238 = (8 + 4 + 1) × 238 = 8 x 238 + 4 × 238 + 1 × 238 = 3094.

Peasant multiplication

Peasant multiplication, or Russian peasant multiplication uses an algorithm similar to Egyptian multiplication. Flowcharts are often used to graphically represent algorithms. ...

• Write the two numbers (A and B) you wish to multiply, each at the head of a column.
• Starting with A, divide by 2, discarding any fractions, until there is nothing left to divide. Write the series of results under A.
• Starting with B, keep doubling until you have doubled it as many times as you divided the first number. Write the series of results under B.
• Add up all the numbers in the B-column that are next to an odd number in the A-column. This gives you the result.

Example: 27 times 82

The method works because multiplication is distributive, so: In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

See also

• Egyptian mathematics
• Multiplication algorithms
• Binary numeral system

This article or section is in need of attention from an expert on the subject. ... A multiplication algorithm is an algorithm (or method) to multiply two numbers. ... The binary numeral system (base 2 numerals) represents numeric values using two symbols, typically 0 and 1. ...

External links

• Russian Peasant Multiplication
• The Russian Peasant Algorithm (pdf file)
• Peasant Multiplication from cut-the-knot