Trachtenberg system

The Trachtenberg system , also known as the Trachtenberg fast calculation method , is a system for fast mental arithmetic . It consists of several easy-to-recognize operations that allow arithmetic calculations to be performed quickly. The system was developed by the Russian-Jewish engineer Jakow Trachtenberg , who invented the system in a Nazi concentration camp while he was imprisoned there. The most important algorithms are those for general multiplying, dividing, and adding. In addition, the Trachtenberg system contains some special methods for the cases of multiplying small numbers between five and thirteen.

The addition method contains an effective method for checking calculations, which can also be applied to multiplication.

General multiplying

The method for general multiplication is a method for performing multiplications with little effort; that is, having to remember as few intermediate results as possible. This is achieved by establishing that the final digit of the multiplication is determined by the last digits of the factors. This is recorded as an interim result. To find the penultimate digit, you need everything that influences this digit: The intermediate result, the last digit from times the penultimate figure from , and the penultimate figure from times the last figure from . This calculation is carried out and we have an intermediate result whose last two digits are correct. ${\ displaystyle a \ times b}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle b}$

In general, for each position in the bottom line, we have for all of them : ${\ displaystyle n}$ ${\ displaystyle i}$

${\ displaystyle a \ (\ mathrm {digit \ at \} i) \ times \ b \ (\ mathrm {digit \ at \} (ni)).}$

You can learn this algorithm and then multiply four-digit numbers in your head just by writing down the final result. You start on the right and end with the leftmost digit.

Trachtenberg defined this algorithm as a kind of pairwise multiplication, in which two digits are multiplied by another digit, just keeping the middle digit of the result. When using this algorithm with pairwise multiplication, fewer and fewer intermediate results will have to be recorded.

Example: ${\ displaystyle 123456 \ times 789}$

To find the first digit of the result:

The one of ${\ displaystyle 9 \ times 6 = 4.}$

To find the second digit of the result, start with the second digit of the multiplicand:

The ones of plus the tens of plus ${\ displaystyle 9 \ times 5}$ ${\ displaystyle 9 \ times 6}$

the one of . ${\ displaystyle 8 \ times 6}$

${\ displaystyle 5 + 5 + 8 = 18}$ .

The second digit is and memorized for the third digit. ${\ displaystyle 8}$ ${\ displaystyle 1}$

To find the fourth digit, start with the fourth digit of the multiplicand:

The ones of plus the tens of plus ${\ displaystyle 9 \ times 3}$ ${\ displaystyle 9 \ times 4}$

the ones of plus the tens of plus ${\ displaystyle 8 \ times 4}$ ${\ displaystyle 8 \ times 5}$

the ones of plus the tens of . ${\ displaystyle 7 \ times 5}$ ${\ displaystyle 7 \ times 6}$

${\ displaystyle 7 + 3 + 2 + 4 + 5 + 4 = 25 + 1}$ noted by the third digit.

The fourth digit of the answer is up and noted for the next digit. ${\ displaystyle 6}$ ${\ displaystyle 2}$

Two-finger method

Trachtenberg called this the two-finger method. The calculations to find the fourth digit from the above example can be clearly seen in the graphic on the right:

The arrow from the nine will always point to the digit of the multiplicand just above the answer for which you want to calculate the value. The other arrows always point one to the right. The vertical arrow shows the number of which we need the units, the angled arrow shows the number of which we need the tens. If an arrow points to a field with no number, no calculation is required. When calculating a number, the accumulation of arrows slides one unit to the left until all arrows point to leading zeros.

General division

In the Trachtenberg system, division is very similar to Trachtenberg's multiplication, but with subtraction instead of addition.

literature

Ann Cutler, Rudolph Matas McShane, Jakow Trachtenberg: The Trachtenberg speed system of basic mathematics . Greenwood Press, Westport, Conn 1981, ISBN 0-313-23200-8 .
Trachtenberg, J. (1960): The Trachtenberg Speed System of Basic Mathematics. Doubleday and Company, Inc., Garden City, NY, US
Э. Катлер, Р. Мак-Шейн: Система быстрого счёта по Трахтенбергу , 1967.
Rushan Ziatdinov, Sajid Musa: Rapid mental computation system as a tool for algorithmic thinking of elementary school students development . European Researcher 25 (7): 1105-1110, 2012.