Ancient Egyptian multiplication
Encyclopedia
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
.
then were unaware of such concepts and had to resort to much simpler methods. The ancient Egypt
ians 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.
Example:
1 x 2 = 2
2 x 2 = 4
4 x 2 = 8
8 x 2 = 16
16 x 2 = 32
Example of the decomposition of the number 25:
25 is thus the sum of the powers of two: 16, 8 and 1.
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 main advantage of this technique is that it makes use of only addition, subtraction, and multiplication by two.
Since 13 = 8 + 4 + 1, distribution of multiplication over addition gives 13 × 238 = (8 + 4 + 1) × 238 = 8 x 238 + 4 × 238 + 1 × 238 = 1904 + 952 + 238 = 3094.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Multiplication table
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
, only the ability to multiply and divide by 2, and to add
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
. 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
Division by two
In mathematics, division by two or halving has also been called mediation or dimidiation. The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its...
means halving one number and duplation means doubling the other number. It is still used in some areas.
The second Egyptian multiplication and division
Egyptian multiplication and division
Since the 1880s, as formalized in the 1920s, an incomplete view has defined Egyptian multiplication. Springer's on-line encyclopedia summarizes the 1920s view this way :...
technique was known from the hieratic
Hieratic
Hieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
Moscow
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
and Rhind Mathematical Papyri
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
written in the seventeenth century B.C. by the scribe Ahmes
Ahmes
Ahmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
.
The decomposition
The decomposition into a sum of powers of two was not intended as a change from base ten to base two; the EgyptiansEgyptians
Egyptians are nation an ethnic group made up of Mediterranean North Africans, the indigenous people of Egypt.Egyptian identity is closely tied to geography. The population of Egypt is concentrated in the lower Nile Valley, the small strip of cultivable land stretching from the First Cataract to...
then were unaware of such concepts and had to resort to much simpler methods. The ancient Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
ians 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.
Example:
1 x 2 = 2
2 x 2 = 4
4 x 2 = 8
8 x 2 = 16
16 x 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.
The table
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:
- 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 technique 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.| ✓ | 1 | 238 | |||||
| 2 | 476 | ||||||
| ✓ | 4 | 952 | |||||
| ✓ | 8 | 1904 | |||||
| |
|||||||
| 13 | 3094 | ||||||
Since 13 = 8 + 4 + 1, distribution of multiplication over addition gives 13 × 238 = (8 + 4 + 1) × 238 = 8 x 238 + 4 × 238 + 1 × 238 = 1904 + 952 + 238 = 3094.
See also
- Egyptian mathematicsEgyptian mathematicsEgyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...
- Multiplication algorithmMultiplication algorithmA multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are in use...
s - Binary numeral systemBinary numeral systemThe binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
- Egyptian multiplication and divisionEgyptian multiplication and divisionSince the 1880s, as formalized in the 1920s, an incomplete view has defined Egyptian multiplication. Springer's on-line encyclopedia summarizes the 1920s view this way :...
External links
- Russian Peasant Multiplication
- The Russian Peasant Algorithm (pdf file)
- Peasant Multiplication from cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
- http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html New and Old classifications of Ahmes Papyrus
- Egyptian Multiplication by Ken Caviness, The Wolfram Demonstrations Project.
- Russian Peasant Multiplication at The Daily WTFThe Daily WTFThe Daily WTF is a humorous blog dedicated to “Curious Perversions in Information Technology”...