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[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.
More formally, multiplying two n-digit numbers using long multiplication requires Θ(n 2) single-digit operations (additions and multiplications). When implemented in software, long multiplication algorithms must deal with overflow during additions, which can be expensive.
That forces the multiplication process to be adapted to handle two's complement numbers, and that complicates the process a bit more. Similarly, processors that use ones' complement, sign-and-magnitude, IEEE-754 or other binary representations require specific adjustments to the multiplication process.
Starting from the rightmost digit, double each digit and add the neighbor. (The "neighbor" is the digit on the right.) 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:
Napier performed multiplication and division on an abacus, as was common in his times. However, Egyptian multiplication gives an elegant way to carry out multiplication without tables using only doubling, halving and adding. Multiplying a single-digit number by another single-digit number is a simple process.
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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