The Trachtenberg system is a system of rapid mental calculation. The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly. It was developed by the Ukrainian-Jewish mathematician and engineer Jakow Trachtenberg in order to keep his mind occupied while being held prisoner in a German Nazi concentration camp. This article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are for general multiplication, division and addition. Also, the Trachtenberg system includes some specialised methods for multiplying small numbers between 5 and 13. The section on addition demonstrates an effective method of checking calculations that can also be applied to multiplication.
General multiplication The method for general multiplication is a method to achieve multiplications a × b {\displaystyle a\times b} with low space complexity, i.e. as few temporary results as possible to be kept in memory. This is achieved by noting that the final digit is completely determined by multiplying the last digit of the multiplicands. This is held as a temporary result. To find the next to last digit, we need everything that influences this digit: The temporary result, the last digit of a {\displaystyle a} times the next-to-last digit of b {\displaystyle b} , as well as the next-to-last digit of a {\displaystyle a} times the last digit of b {\displaystyle b} . This calculation is performed, and we have a temporary result that is correct in the final two digits. In general, for each position n {\displaystyle n} in the final result, we sum for all i {\displaystyle i} :
a (digit at i ) × b (digit at ( n − i ) ) . {\displaystyle a{\text{ (digit at }}i{\text{ )}}\times b{\text{ (digit at }}(n-i){\text{)}}.}
People can learn this algorithm and thus multiply four-digit numbers in their head – writing down only the final result. They would write it out starting with the rightmost digit and finishing with the leftmost. 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: 123456 × 789 {\displaystyle 123456\times 789}
To find the first (rightmost) digit of the answer, start at the first digit of the multiplicand
The units digit of 9 × 6 {\displaystyle 9\times 6} is 4. {\displaystyle 4.}
The first digit of the answer is 4 {\displaystyle 4} . The tens digit 5 {\displaystyle 5} is ignored.
To find the second digit of the answer, start at the second digit of the multiplicand:
The units digit of 9 × 5 {\displaystyle 9\times 5} plus the tens digit of 9 × 6 {\displaystyle 9\times 6} plus The units digit of 8 × 6 {\displaystyle 8\times 6} .
5 + 5 + 8 = 18 {\displaystyle 5+5+8=18} . The second digit of the answer is 8 {\displaystyle 8} and carry 1 {\displaystyle 1} to the third digit.
To find the third digit of the answer, start at the third digit of the multiplicand:
The units digit of 9 × 4 {\displaystyle 9\times 4} plus the tens digit of 9 × 5 {\displaystyle 9\times 5} plus The units digit of 8 × 5 {\displaystyle 8\times 5} plus the tens digit of 8 × 6 {\displaystyle 8\times 6} plus The units digit of 7 × 6 {\displaystyle 7\times 6}
1 + 6 + 4 + 0 + 4 + 2 = 17 {\displaystyle 1+6+4+0+4+2=17}
The third digit of the answer is 7 {\displaystyle 7} and carry 1 {\displaystyle 1} to the next digit. To find the fourth digit of the answer, start at the fourth digit of the multiplicand:
The units digit of 9 × 3 {\displaystyle 9\times 3} plus the tens digit of 9 × 4 {\displaystyle 9\times 4} plus The units digit of 8 × 4 {\displaystyle 8\times 4} plus the tens digit of 8 × 5 {\displaystyle 8\times 5} plus The units digit of 7 × 5 {\displaystyle 7\times 5} plus the tens digit of 7 × 6 {\displaystyle 7\times 6} .
1 + 7 + 3 + 2 + 4 + 5 + 4 = 26 {\displaystyle 1+7+3+2+4+5+4=26} carried from the third digit. The fourth digit of the answer is 6 {\displaystyle 6} and carry 2 {\displaystyle 2} to the next digit. Continue with the same method to obtain the remaining digits.
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