Egyptian multiplication works through a process of repeated doubling and addition, utilizing a system akin to binary arithmetic to perform multiplication. It doesn't require memorizing multiplication tables.
Here's a breakdown of the process:
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Set up the Columns: Create two columns. The first column starts with 1. The second column starts with the number you want to multiply (the multiplicand).
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Repeated Doubling:
- In the first column, repeatedly double the number until the next doubling would exceed the multiplier (the number you are multiplying by).
- In the second column, simultaneously double the multiplicand the same number of times you doubled the first column.
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Identify Key Rows: Look at the numbers in the first column. Identify which of these numbers, when added together, equal the multiplier.
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Sum Corresponding Multiplicands: Add up the numbers in the second column that correspond to the numbers you selected in the first column. The result is the product.
Example: Let's multiply 13 by 17 using Egyptian multiplication.
| Column 1 (Doubling of 1) | Column 2 (Doubling of 17) |
|---|---|
| 1 | 17 |
| 2 | 34 |
| 4 | 68 |
| 8 | 136 |
- We stop at 8 in the first column because the next doubling (16) would be greater than the multiplier, 13.
- Now, find the numbers in the first column that add up to 13. In this case, 1 + 4 + 8 = 13.
- Finally, add the corresponding numbers in the second column: 17 + 68 + 136 = 221.
Therefore, 13 x 17 = 221.
Why it Works (Binary Connection):
Egyptian multiplication is essentially based on representing the multiplier as a sum of powers of 2 (binary representation). In the example above, 13 is represented as (1 x 2^0) + (0 x 2^1) + (1 x 2^2) + (1 x 2^3) = 1 + 0 + 4 + 8. By doubling the multiplicand and adding the appropriate doublings, they were effectively multiplying by the different powers of two present in the binary representation of the multiplier.
In summary, Egyptian multiplication, dating back to the Old Kingdom, provides a method of multiplication based on repeated doubling and addition, reflecting a practical understanding of binary principles.
