Trachtenberg system
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 a prisoner 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.
with low space complexity, i.e. as few temporary results as possible to be kept in memory.
This calculation is performed, and we have a temporary result that is correct in the final two digits.
: 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.
By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held.
The calculations for finding the fourth digit from the example above are illustrated at right.
As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left until all of the arrows point to prefixed zeros.
Division in the Trachtenberg System is done much the same as in multiplication but with subtraction instead of addition.
Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time.
As you solve each digit of the answer you then subtract Product Pairs (UT pairs) and also NT pairs (Number-Tens) from the Partial Dividend to find the next Partial Dividend.
The Product Pairs are found between the digits of the answer so far and the divisor.
When performing any of these multiplication algorithms the following "steps" should be applied.
The last calculation is on the leading zero of the multiplicand.
The rightmost digit's neighbor is the trailing zero.
The 'halve' operation has a particular meaning to the Trachtenberg system.
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.
In this same way the tables for subtracting digits from 10 or 9 are to be memorized.
And whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd.
The number T consists of n digits cn ... c1.
Example: 2,130 × 9 = 19,170 Working from right to left: Add 0 (zero) as the rightmost digit.
The book contains specific algebraic explanations for each of the above operations.
The algorithms/operations for multiplication, etc., can be expressed in other more compact ways that the book does not specify, despite the chapter on algebraic description.
[a] There are many other methods of calculation in mental mathematics.
The list below shows a few other methods of calculating, though they may not be entirely mental.
