Ancient Egyptian Multiplication - The Decomposition

The Decomposition

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.

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.