Ancient Egyptian multiplication
||It has been suggested that Egyptian multiplication and division be merged into this article. (Discuss) Proposed since March 2011.|
|This article does not cite any references or sources. (September 2010)|
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.
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.
The decomposition into a sum of powers of two was not intended as 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 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 it out and repeat until nothing remained. (The Egyptians did not make use of the number zero in mathematics.)
To find the largest power of 2 keep doubling your answer starting with number 1, for example
1 × 2 = 2 2 × 2 = 4 4 × 2 = 8 8 × 2 = 16 16 × 2 = 32
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 = 0 25 is thus the sum of the powers of two: 16, 8 and 1.
After the decomposition of the first multiplicand, it is necessary 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:
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 technique is that it makes use of only addition, subtraction, and multiplication by two.
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 238.
Since 238 = 2 + 4 + 8 + 32 + 64 + 128, distribution of multiplication over addition gives:
|238 × 13||= (128 + 64 + 32 + 8 + 4 + 2) × 13|
|= 128 × 13 + 64 × 13 + 32 × 13 + 8 × 13 + 4 × 13 + 2 × 13|
|= 1664 + 832 + 416 + 104 + 52 + 26|
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:|
|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:|
|The algorithm can be illustrated with the binary representation of the numbers:|
- Egyptian mathematics
- Multiplication algorithms
- Binary numeral system
- Egyptian multiplication and division
- Bharati Krishna Tirtha's Vedic mathematics
- Russian Peasant Multiplication
- The Russian Peasant Algorithm (pdf file)
- Peasant Multiplication from cut-the-knot
-  New and Old classifications of Ahmes Papyrus
- Egyptian Multiplication by Ken Caviness, The Wolfram Demonstrations Project.
- Russian Peasant Multiplication at The Daily WTF
- Michael S. Schneider explains how the Ancient Egyptians (and Chinese) and modern computers multiply and divide