Trachtenberg system - Wikipedia
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 Russian engineer Jakow Trachtenberg in order to keep his mind occupied while being in a Nazi concentration camp.
The rest of this article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are ones 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.
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} :
