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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.
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.
[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.
Single-digit numbers are written in the bottom right triangle leaving the other triangle blank, while double-digit numbers are written with a digit on either side of the diagonal. If the tables are held on single-sided rods, 40 rods are needed in order to multiply 4-digit numbers – since numbers may have repeated digits, four copies of the ...
Operation Input Output Algorithm Complexity Addition: Two -digit numbers : One +-digit number : Schoolbook addition with carry ()Subtraction: Two -digit numbers : One +-digit number
For single digit numbers simply duplicate the number into the tens digit, for example: 1 × 11 = 11, 2 × 11 = 22, up to 9 × 11 = 99. The product for any larger non-zero integer can be found by a series of additions to each of its digits from right to left, two at a time.