Ancient Egyptian multiplication
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The technique of Ancient Egyptian multiplication rests on the decomposition of one of the multiplicands (generally the larger) into a sum of powers of two and the creation of a table of doublings of the second multiplicand.
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.
The earliest known indication of Egyptian multiplication, in the form of the Ishango bone, was discovered along the headwaters of the Nile River (located on the northeastern edge of the Congo), dating to 20,000 BC.
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 of that time 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 proceded 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).
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,
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 mutliplying 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.
| ✔ | 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 = 3094.
See also
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