8 • THE TRACHTENBERG SPEED SYSTEM OF BASIC MATHEMATICS. The Trachtenberg method is not only speedy but simple. Once one has mastered the . Download a free PDF copy of The Trachtenberg Speed System of Basic Mathematics. I am excited to see if even an old man can pick up this speed math. The Trachtenberg Speed System of Basic Mathematics is a system of mental speed math, created and access to hundreds of downloadable PDF worksheets.

The Trachtenberg Speed System Of Basic Mathematics Pdf

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The Trachtenberg speed system of basic mathematics, translated and adapted by Ann Cutler and Rudolph McShane. Pp £ ISBN The Trachtenberg Speed System of Basic Uploaded by All Rights Reserved. Download as DOC, PDF, TXT or read online from Scribd. The Trachtenberg system is a system of rapid mental calculation. The system consists of a .. The Trachtenberg Speed System of Basic Mathematics. Doubleday and Company, Inc., Garden City, NY, USA. Катлер Э., Мак-Шейн Р. Система быстрого.

Multiply by 2. Write down 4, carry 1. Write down 9. Write down 4.

Write down 2. Write down 0.

Trachtenberg system

Write down 8, carry 1. Write down 1.

Multiply by 3. Write down 1, carry 1. Write down 7, carry 1. Write down 3, carry 2. Write down 0, carry 2.

Write down 7. These rules allow one to dispense with memorizing multiplication tables, if that is desired. Even better, it gives a way to help memorize them, by allowing one to work out the answer by rule if one cannot remember it by rote. 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 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 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.

Trachtenberg system

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 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.

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. 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.

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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.

Trachtenberg Speed System Basic Mathematics

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.

Each digit has a neighbor , i. 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. 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.It has a short review of basic methods of algebra and then applies these basic methods to the procedures discussed earlier.

After seeing how neat Trachtenberg's system is, you may begin to develop the ability to simplify. Are there any more ways of doing 16 x 12? You don't have to physically trace your finger over the digits. Middle steps: subtract from 9, double, and add the neighbor. Write down 7.

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