Ancient Egyptian and Russian Peasant

Russian peasant and Ancient Egyptian multiplication are both based on the principles of doubling and halving. They all work with powers of two, which makes the process simpler and more efficient. 

Revised: 

Russian Peasant Method:

Doubling and Halving: This method involves writing two numbers to be multiplied at the top of two columns. One column is repeatedly halved, ignoring any fractions, and the other is doubled.

Eliminating Even Numbers: Any row where the halved number is even is eliminated.

Summation: The remaining numbers in the doubled column are summed up to get the final product.

Example:

Multiplying 9 by 13:

Halve 9 and double 13: 4 (ignore the fraction) and 26.

Halve 4 and double 26: 2 and 52.

Halve 2 and double 52: 1 and 104.

Eliminate the rows with even numbers in the halved column (4 and 2).

Add the remaining numbers in the doubled column: 13 + 104 = 117.

Ancient Egyptian Method:

Doubling and Representation: Similar to the Russian method, it involves doubling one of the numbers. However, the other number is represented in terms of powers of 2.

Selecting Representations: The method identifies which powers of 2 can be combined to represent the second number.

Summation: The corresponding values from the doubled column are added to get the product.

Example:

Multiplying 9 by 13:

Write down 1 and double it to get 2, 4, 8 (stop at or before reaching the multiplier, which is 9).

Double 13 alongside: 26, 52, 104.

Since 9 can be represented as 8 + 1, select the rows with 8 and 1.

Add the corresponding numbers in the doubled column: 104 + 13 = 117.

Similarities and Differences:

Similarity: Both methods are based on the principle of doubling and halving. They utilize the decomposition of numbers into sums of powers of 2.

Difference: The Russian peasant method involves halving one number and eliminating rows based on even numbers, while the Ancient Egyptian method focuses on representing one number as a sum of powers of 2 and selecting the corresponding rows for summation.

Equivalence: They are equivalent because both methods essentially break down the multiplication process into additions of numbers that are powers of 2. This is a fundamental principle in binary arithmetic, where numbers are represented as sums of powers of 2.

In summary, while both methods use the concept of doubling and powers of 2, the Russian peasant method is more about halving and eliminating even rows, whereas the Ancient Egyptian method is about representing a number with powers of 2 and selecting the corresponding rows for summation.