Write 2, carry the 1.ĥ + half of 7 (3) + 5 (since the starting digit 5 is odd) + 1 (carried) = 14. Rule: to multiply by 6: Add half of the neighbor to each digit, then, if the current digit is odd, add 5.ħ has no neighbor, add 5 (since 7 is odd) = 12. The remaining digit is one digit of the final result.ĭetermine neighbors in the multiplicand 0316:ĭigit 0 (the prefixed zero) has neighbor 3 If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the next operation. (By “neighbor” we mean the digit on the right.) Multiplying by 12ĭouble each digit and add the neighbor. This makes up for dropping 0.5 in the next digit’s calculation. In this same way the tables for subtracting digits from 10 or 9 are to be memorized.Īnd whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd. So instead of thinking “half of seven is three and a half, so three” it’s suggested that one thinks “seven, three”. It is intended to mean “half the digit, rounded down” but for speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous. The ‘halve’ operation has a particular meaning to the Trachtenberg system. The rightmost digit’s neighbor is the trailing zero. The last calculation is on the leading zero of the multiplicand.Įach digit has a neighbor, i.e., the digit on its right. The answer must be found one digit at a time starting at the least significant digit and moving left. When performing any of these multiplication algorithms the following “steps” should be applied. It is based on a check (or digit) sums, such as the nines-remainder method.įor the procedure to be effective, the different operations used in each stages must be kept distinct, otherwise there is a risk of interference. 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.
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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.