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k 3 = b · (c + d) Real part = k 1 − k 3 Imaginary part = k 1 + k 2. This algorithm uses only three multiplications, rather than four, and five additions or subtractions rather than two. If a multiply is more expensive than three adds or subtracts, as when calculating by hand, then there is a gain in speed.
[1] [2] [3] It is a divide-and-conquer algorithm that reduces the multiplication of two n-digit numbers to three multiplications of n/2-digit numbers and, by repeating this reduction, to at most single-digit multiplications.
If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. The remaining digit is one digit of the final result. Example: Determine neighbors in the multiplicand 0316: digit 6 has no right neighbor; digit 1 has neighbor 6; digit 3 has neighbor 1
(For example, the sixth row is read as: 0 ⁄ 6 1 ⁄ 2 3 ⁄ 6 → 756). Like in multiplication shown before, the numbers are read from right to left and add the diagonal numbers from top-right to left-bottom (6 + 0 = 6; 3 + 2 = 5; 1 + 6 = 7). The largest number less than the current remainder, 1078 (from the eighth row), is found.
The next band from the right has three digits, 2, 1 and 8. These are added together to get 11. The units digit of this addition, 1, is written down as the next digit of the multiplication result. The tens digit, which is 1, is carried into the next band. The third band from the right has five digits, 2, 4, 3, 1 and 6 plus the carried 1.
Operation Input Output Algorithm Complexity Addition: Two -digit numbers : One +-digit number : Schoolbook addition with carry ()Subtraction: Two -digit numbers : One +-digit number
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