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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.
Trachtenberg defined this algorithm with a kind of pairwise multiplication where two digits are multiplied by one digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held. Example:
If the sum contains more than one digit, the value of the tens place is carried into the next diagonal (see Step 2). Step 2. Numbers are filled to the left and to the bottom of the grid, and the answer is the numbers read off down (on the left) and across (on the bottom). In the example shown, the result of the multiplication of 58 with 213 is ...
The classical method of multiplying two n-digit numbers requires n 2 digit multiplications. Multiplication algorithms have been designed that reduce the computation time considerably when multiplying large numbers.
Two -digit numbers One +-digit ... One -digit number Schoolbook long multiplication () Karatsuba algorithm 3-way Toom–Cook multiplication ()-way Toom ...
To easily multiply any 2-digit numbers together a simple algorithm is as follows (where a is the tens digit of the first number, b is the ones digit of the first number, c is the tens digit of the second number and d is the ones digit of the second number): (+) (+)
So for example, for a promptuary capable of multiplying two five-digit numbers together, the strips should 6 times as long as they are wide, with 50 number strips and 50 mask strips. Napier's example specified strips 1 finger (19mm) wide and 11 fingers (209mm) long, enabling the device to multiply two 10-digits numbers to produce a 20-digit result.
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