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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:
Since the additions, subtractions, and digit shifts (multiplications by powers of B) in Karatsuba's basic step take time proportional to n, their cost becomes negligible as n increases. More precisely, if T(n) denotes the total number of elementary operations that the algorithm performs when multiplying two n-digit numbers, then
Another method is to simply multiply the number by 10, and add the original number to the result. For example: 17 × 11 = ? 17 × 10 = 170 170 + 17 = 187 17 × 11 = 187 One last easy way: If one has a two-digit number, take it and add the two numbers together and put that sum in the middle, and one can get the answer.
A small number is chosen, usually 2 through 9, by which to multiply the large number. In this example the small number by which to multiply the larger is 6. The horizontal row in which this number stands is the only row needed to perform the remaining calculations and may now be viewed in isolation. Second step of solving 6 x 425
In 1977, at Southern Methodist University, she computed the 23rd root of a 201-digit number in 50 seconds. [ 10 ] [ 15 ] Her answer, 546,372,891, was confirmed by calculations done at the US Bureau of Standards using the UNIVAC 1101 computer, for which a special program had to be written to perform such a large calculation. [ 17 ]
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.
Operation Input Output Algorithm Complexity Addition: Two -digit numbers : One +-digit number : Schoolbook addition with carry ()Subtraction: Two -digit numbers : One +-digit number