Jakow Trachtenberg spent years in a Nazi concentration camp and to escape the horrors he found refuge in his mind developing these methods. Some of the. Trachtenberg Speed Math software based on Jakow Trachtenberg’s speed math algorithms developed during WW II while imprisoned in Nazi camps. The Trachtenberg Speed System of Basic Mathematics. $ Paperback. Books by Jakow Trachtenberg. Showing 1 Result Books: Advanced Search.

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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 Jewish engineer Jakow Trachtenberg in trachtenverg 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 trcahtenberg 5 and The section on addition demonstrates an effective method of checking calculations that can also be applied to multiplication.

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: 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 trzchtenberg digit and finishing with the jakw. Trachtenberg defined this algorithm with a kind of pairwise trachtenbreg 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. Trachtenberg called this the 2 Finger Method. The calculations for finding the fourth digit from the example above are illustrated at right. The arrow from the nine will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right.

The vertical arrow points to the product where we will get the Units digit, and the sloping arrow points to the product where we will get the Tens digits of the Product Pair. If an arrow points to a space with no digit there is no calculation for that arrow. 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 Trqchtenberg System trachfenberg 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.

The Product Pairs are found between the digits of the answer so far and the divisor. If a subtraction results in a negative number you have to back up one digit and reduce that digit of the answer by one.

Trachtenberg Speed Math: Learn mental math and master mathematics with this maths software

With enough practice this method can be done in your head. A method of adding columns of numbers and accurately checking the result without repeating the first operation.

An intermediate sum, in the form of two rows of digits, is produced. The answer is obtained by taking the sum of the intermediate results with an L-shaped algorithm. 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.

It is based on a check or digit sums, such as the nines-remainder method. For the procedure to be effective, the different operations used in each stage must be kept distinct, otherwise there is a risk of interference. The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculation is on the leading zero of the multiplicand.

The Trachtenberg Speed System of Basic Mathematics by Jakow Trachtenberg

Each digit has a neighbori. The rightmost digit’s neighbor is the trailing zero. The ‘halve’ operation has a particular meaning to the Trachtenberg system. It is intended tracutenberg mean “half the digit, rounded down” but for speed reasons people following the Trachtenberg system are encouraged to make this halving process instantaneous.

So instead of thinking “half of seven is three and a half, so three” it’s suggested that one thinks “seven, three”. This speeds up calculation considerably.

In this same way the tables for subtracting digits from 10 or 9 are to be memorized.

Trachtenberg system

And whenever the rule calls for adding half of the neighbor, always add 5 if the current digit is odd. This makes up for dropping 0. For jalow 9, 8, 4, and 3 only the first digit is subtracted from After that each digit is subtracted from nine instead.


Starting from the rightmost digit, double each digit and add jkow neighbor. The “neighbor” is the digit on the right. 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. The remaining digit is one digit of the final result. The American film Gifted revolves around a child prodigy who at the age of 7 impresses her teacher by doing calculations in her head using the Trachtenberg system. 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.

From Wikipedia, the free encyclopedia. This section’s use of external links may not follow Wikipedia’s policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references.

May Learn how and when to remove this template message. Doubleday and Company, Inc. The original book has seven full Trachtenbwrg and is pages long. The chapter titles are as follows. The numerous sub-categories in each chapter are not listed.

Professor Trachtenberg fled to Germany when the czarist regime was overthrown jakoww his homeland, Russia, and lived there peacefully trachtenebrg his mid-thirties when his anti-Hitler attitudes forced him to flee again.

He was a fugitive and when captured spent a total of seven years in various concentration camps. It was during these years that Professor Trachtenberg devised the system of speed mathematics. Most of his work was done without pen or paper. Therefore most of the techniques can be performed mentally.

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