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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 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. For example: 24 x 11 = 264 because 2 + 4 = 6 and the 6 is placed in between the 2 and the 4. Second example: 87 x 11 = 957 because 8 + 7 = 15 so the 5 goes in between the 8 and the 7 and the 1 is carried to the 8.
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.
The standard procedure for multiplication of two n-digit numbers requires a number of elementary operations proportional to , or () in big-O notation. Andrey Kolmogorov conjectured that the traditional algorithm was asymptotically optimal , meaning that any algorithm for that task would require Ω ( n 2 ) {\displaystyle \Omega (n^{2 ...
The corresponding bones to the leading number are placed in the board. For this example, the bones 8, 2, and 5 were placed in the proper order as shown below. First step of solving 825 × 913. To multiply by a multi-digit number, multiple rows are reviewed. For this example, the rows for 9, 1, and 3 have been removed from the board for clarity.
It is an extension of Napier's Bones, using two sets of rods to achieve multi-digit multiplication without the need to write down intermediate results, although some mental addition is still needed to calculate the result. The rods for the multiplicand are similar to Napier's Bones, with repetitions of the values.
Mental calculation is said to improve mental capability, increases speed of response, memory power, and concentration power. Many veteran and prolific abacus users in China, Japan, South Korea, and others who use the abacus daily, naturally tend to not use the abacus any more, but perform calculations by visualizing the abacus.
A mental calculator or human calculator is a person with a prodigious ability in some area of mental calculation (such as adding, subtracting, multiplying or dividing large numbers). In 2005, a group of researchers led by Michael W. O'Boyle, an American psychologist previously working in Australia and now at Texas Tech University , has used MRI ...
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