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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 Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp.
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 times the next-to-last digit of b , as well as the next-to-last digit of a times the last digit of b.
This calculation is performed, and we have a temporary result that is correct in the final two digits. Ordinary 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. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held.
The second digit of the answer is 8 and carry 1 to the third digit. The fourth digit of the answer is 6 and carry 2 to the next digit.
Professor 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. 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 stages 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. 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. 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.
This makes up for dropping 0. Rule: to multiply by Starting from the rightmost digit, double each digit and add the neighbor. By "neighbor" we mean 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. Write 2, carry the 1. Write 4, carry the 1. Write 1, carry 1.
Write 2. For rules 9, 8, 4, and 3 only the first digit is subtracted from After that each digit is subtracted from nine instead. Half of 4's neighbor is 1. Half of the leading zero's neighbor is 2. Half of 9's neighbor is 1, plus 5 because 9 is odd, is 6. Half of the leading zero's neighbor is 4. From Memory Techniques Wiki. Jump to: navigation , search. Translated by A. Cutler, R. Doubleday and Company, Inc.. The original book has seven full Chapters and is exactly 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 in his homeland Russia and lived there peacefully until 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. Category : Mental Calculation. Navigation menu Personal tools Log in. Namespaces Page Discussion. Views Read View source View history.
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The Trachtenberg Speed System of Basic Mathematics
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The Trachtenberg Speed System of Basic Mathematics is a system of mental mathematics which in part did not require the use of multiplication tables to be able to multiply. The method was created over seventy years ago. The main idea behind the Trachtenberg Speed System of Basic Mathematics is that there must be an easier way to do multiplication, division, squaring numbers and finding square roots, especially if you want to do it mentally. When solving equations it is not enough to just find the answer you also should prove to yourself you found the right answer.
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 Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp. 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 times the next-to-last digit of b , as well as the next-to-last digit of a times the last digit of b.
Math Dr. Math Home The Trachtenberg Speed System of Basic Mathematics This constitutes a system of performing high-speed multiplication, division, addition, subtraction, and square root, all in one's head. While it is impossible to capture the contents of this page book in one Web page, below you will find a part of the content of this system. Multiplication Finding the product of multiplicands of any length by small multipliers is accomplished using a set of rules. One works from right to left from digit to digit of the multiplicand, writing down the digits of the product. The rules use only addition, subtraction, doubling, and halving.