Question

Compute the product 2135 x 4014 using Karatsuba multiplication algorithm.

[Note: You need to bring it down to 1-digit multiplication]
In large integer multiplication algorithms, why did we not include multiplication by 10ⁿ in the multiplication count?

Answer 1

To calculate the product 2135 x 4014 using the Karatsuba multiplication algorithm, split both numbers into two parts, perform three multiplications, multiply the results by the corresponding powers of 10, and add them together.

Step-by-step explanation:

To calculate the product 2135 x 4014 using the Karatsuba multiplication algorithm, we can follow these steps:

1. Split both numbers into two parts: 2135 = 21 * 100 + 35 and 4014 = 40 * 100 + 14
2. Calculate three multiplications: First, multiply the first parts: 21 x 40 = 840. Then, multiply the second parts: 35 x 14 = 490. Finally, multiply the sum of the first parts and the sum of the second parts: (21 + 35) * (40 + 14) = 56 * 54 = 3024.
3. Multiply the resulting numbers by the corresponding powers of 10: 840 * 100^2 = 84000, 490 * 100 = 49000, and 3024.
4. Add the three results together: 84000 + 49000 + 3024 = 137024.

Therefore, the product of 2135 and 4014 using the Karatsuba multiplication algorithm is 137024.

In large integer multiplication algorithms, multiplication by 10^n is not included in the multiplication count because it only involves shifting the digits to the left or right, which is a relatively simple operation and does not require extensive calculations.