![]() ![]() As a final step, the checking method that is advocated removes both the risk of repeating any original errors and allows the precise column in which an error occurs to be identified at once. The answer is obtained by taking the sum of the intermediate results with an L-shaped algorithm. ![]() An intermediate sum, in the form of two rows of digits, is produced. ![]() The fourth digit of the answer is 6 and carry 2 to the next digit.Ī method of adding columns of numbers and accurately checking the result without repeating the first operation. The units digit of 7 5 plus the tens digit of 7 6.ħ + 3 + 2 + 4 + 5 + 4 = 25 + 1 carried from the third digit. The units digit of 8 4 plus the tens digit of 8 5 plus The units digit of 9 3 plus the tens digit of 9 4 plus To find the fourth digit of the answer, start at the fourth digit of the multiplicand: The second digit of the answer is 8 and carry 1 to the third digit. The units digit of 9 5 plus the tens digit of 9 6 plus To find the second digit of the answer, start at the second digit of the multiplicand: By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held. 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. They would write it out starting with the rightmost digit and finishing with the leftmost. Ordinary people can learn this algorithm and thus multiply four digit numbers in their head - writing down only the final result. In general, for each position n in the final result, we sum for all i: This calculation is performed, and we have a temporary result that is correct in the final two digits. To find the next to last digit, we need everything that influences this digit: The temporary result, the last digit of a times the next-to-last digit of b, as well as the next-to-last digit of a times the last digit of b. This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplicands. as few temporary results as possible to be kept in memory. The method for general multiplication is a method to achieve multiplications a*b with low space complexity, i.e. It was developed by Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. The Trachtenberg System is a system of rapid mental calculation. ![]()
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