the trachtenberg system, wikipedia

19/08/2013 Trachtenberg system - Wikipedia, the free encyclopedia

en.wikipedia.org/wiki/Trachtenberg_system#Multiplying_by_5 1/9

Trachtenberg systemFrom Wikipedia, the free encyclopedia

The Trachtenberg System is a system of rapid mental calculation. The system consists of a number of readilymemorized operations that allow one to perform arithmetic computations very quickly. It was developed by theRussian Jewish engineer Jakow Trachtenberg in order to keep his mind occupied while being held in a Naziconcentration camp. The rest of this article presents some of the methods devised by Trachtenberg. The most

important algorithms are the ones for general multiplication, division and addition[citation needed]. Also, theTrachtenberg system includes some specialized methods for multiplying small numbers between 5 and 13.

The chapter on addition demonstrates an effective method of checking calculations that can also be applied tomultiplication.

Contents

1 General multiplication

2 General division

3 General addition4 Other multiplication algorithms

4.1 Multiplying by 11




4.5 Multiplying by 94.6 Multiplying by 8




5 Publications

6 Other systems

7 Software

8 References9 External links

General multiplication

The method for general multiplication is a method to achieve multiplications with low space complexity, i.e.as few temporary results as possible to be kept in memory. This is achieved by noting that the final digit iscompletely determined by multiplying the last digit of the multiplicands. This is held as a temporary result. To findthe next to last digit, we need everything that influences this digit: The temporary result, the last digit of times thenext-to-last digit of , as well as the next-to-last digit of times the last digit of . This calculation is performed,and we have a temporary result that is correct in the final two digits.



2 Finger method

In general, for each position in the final result, we sum for all : .

Ordinary people can learn this algorithm and thus multiply four digit numbers in their head - writing down only thefinal 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 onedigit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwisemultiplication, even fewer temporary results need to be held.

Example:

To find the first digit of the answer:The units digit of .

To find the second digit of the answer, start at the second digit of the multiplicand:The units digit of plus the tens digit of plusThe units digit of .

.

The second digit of the answer is and carry to the third digit.

To find the fourth digit of the answer, start at the fourth digit of the multiplicand:The units digit of plus the tens digit of plusThe units digit of plus the tens digit of plusThe units digit of plus the tens digit of .

carried from the third digit.The fourth digit of the answer is and carry to the next digit.

Professor Trachtenberg called this the 2 Finger Method. The calculationsfor 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 multiplicanddirectly above the digit of the answer you wish to find, with the otherarrows each pointing one digit to the right. Each arrow head points to aUT Pair, or Product Pair. The vertical arrow points to the product wherewe will get the Units digit, and the sloping arrow points to the productwhere we will get the Tens digits of the Product Pair. If an arrow pointsto a space with no digit there is no calculation for that arrow. As yousolve for each digit you will move each of the arrows over the multiplicand one digit to the left until all of the arrowspoint to prefixed zeros.

General division

Division in the Trachtenberg System is done much the same as in multiplication but with subtraction instead ofaddition. Splitting the dividend into smaller Partial Dividends, then dividing this Partial Dividend by only the left-mostdigit of the divisor will provide the answer one digit at a time. As you solve each digit of the answer you thensubtract Product Pairs (UT pairs) and also NT pairs (Number-Tens) from the Partial Dividend to find the nextPartial Dividend. The Product Pairs are found between the digits of the answer so far and the divisor. If asubtraction results in a negative number you have to back up one digit and reduce that digit of the answer by one.



Setting up for Division

With enough practice this method can be done in your head.

General addition

A method of adding columns of numbers and accurately checking the result without repeating the first operation. Anintermediate sum, in the form of two rows of digits, is produced. The answer is obtained by taking the sum of theintermediate results with an L-shaped algorithm. As a final step, the checking method that is advocated removesboth the risk of repeating any original errors and allows the precise column in which an error occurs to be identifiedat 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 thereis a risk of interference.

Other multiplication algorithms

When performing any of these multiplication algorithms the following "steps" should be applied.

The answer must be found one digit at a time starting at the least significant digit and moving left. The last calculationis on the leading zero of the multiplicand.

Each digit has a neighbor, i.e., the digit on its right. 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 thishalving process instantaneous. So instead of thinking "half of seven is three and a half, so three" it's suggested thatone thinks "seven, three". This speeds up calculation considerably. In this same way the tables for subtracting digitsfrom 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 upfor dropping 0.5 in the next digit's calculation.

Multiplying by 11

Rule: Add the digit to its neighbor. (By "neighbor" we mean the digit on the right.)

Example:

3 7 6 7 5



(=0+3) (=3+4) (=4+2) (=2+5) (=5+0)

To illustrate:

Thus,

Multiplying by 12

Rule: to multiply by 12:Starting from the rightmost digit, double each digit and add the neighbor. (By "neighbor" we mean the digit on theright.)

If the answer is greater than a single digit, simply carry over the extra digit (which will be a 1 or 2) to the nextoperation. The remaining digit is one digit of the final result.

Example:

Determine neighbors in the multiplicand 0316:

digit 6 has no right neighbordigit 1 has neighbor 6

digit 3 has neighbor 1

digit 0 (the prefixed zero) has neighbor 3

Multiplying by 6

Rule: to multiply by 6: Add half of the neighbor to each digit, then, if the current digit is odd, add 5.

Example:357 6 = 2142

Working right to left,7 has no neighbor, add 5 (since 7 is odd) = 12. Write 2, carry the 1.5 + half of 7 (3) + 5 (since the starting digit 5 is odd) + 1 (carried) = 14. Write 4, carry the 1.



3 + half of 5 (2) + 5 (since 3 is odd) + 1 (carried) = 11. Write 1, carry 1.0 + half of 3 (1) + 1 (carried) = 2. Write 2.

Multiplying by 7

Rule: to multiply by 7:

1. Double each digit.

2. Add half of its neighbor.3. If the digit is odd, add 5.

Example: 523 x 7 = 3,661.

3x2+0+5=11, 1.2x2+1+1=6.5x2+1+5=16, 6.0x2+2+1=3.

3661.

Multiplying by 9

Rule:

1. Subtract the right-most digit from 10.

1. Subtract the remaining digits from 9.

2. Add the neighbor.

3. For the leading zero, subtract 1 from the neighbor.

For rules 9, 8, 4, and 3 only the first digit is subtracted from 10. After that each digit is subtracted from nineinstead.

Example: 2,130 9 = 19,170

Working from right to left:

(10 - 0) + 0 = 10. Write 0, carry 1.

(9 - 3) + 0 + 1 (carried) = 7. Write 7.

(9 - 1) + 3 = 11. Write 1, carry 1.

(9 - 2) + 1 + 1 (carried) = 9. Write 9.2 - 1 = 1. Write 1.

Multiplying by 8

Rule:

1. Subtract right-most digit from 10.




2. Double the result.

3. Add the neighbor.

4. For the leading zero, subtract 2 from the neighbor.

Example: 456 x 8 = 3648


(10 - 6) x 2 + 0 = 8. Write 8.

(9 - 5) x 2 + 6 = 14, Write 4, carry 1.

(9 - 4) x 2 + 5 + 1 (carried) = 16. Write 6, carry 1.4 - 2 + 1 (carried) = 3. Write 3.

Multiplying by 4

Rule:

1. Subtract the right-most digit from 10.


2. Add half of the neighbor, plus 5 if the digit is odd.3. For the leading 0, subtract 1 from half of the neighbor.

Example: 346 * 4 = 1384


(10 - 6) + Half of 0 (0) = 4. Write 4.

(9 - 4) + Half of 6 (3) = 8. Write 8.

(9 - 3) + Half of 4 (2) + 5 (since 3 is odd) = 13. Write 3, carry 1.

Half of 3 (1) - 1 + 1 (carried) = 1. Write 1.

Multiplying by 3

Rule:

1. Subtract the rightmost digit from 10.

1. Subtract the remaining digits from 9.2. Double the result.

3. Add half of the neighbor, plus 5 if the digit is odd.

4. For the leading zero, subtract 2 from half of the neighbor.

Example: 492 x 3 = 1476


(10 - 2) x 2 + Half of 0 (0) = 16. Write 6, carry 1.

(9 - 9) x 2 + Half of 2 (1) + 5 (since 9 is odd) + 1 (carried) = 7. Write 7.

(9 - 4) x 2 + Half of 9 (4) = 14. Write 4, carry 1.



Half of 4 (2) - 2 + 1 (carried) = 1. Write 1.

Multiplying by 5

Rule: to multiply by 5: Take half of the neighbor, then, if the current digit is odd, add 5.

Example: 42x5=210Half of 2's neighbor, the trailing zero, is 0.Half of 4's neighbor is 1.Half of the leading zero's neighbor is 2.43x5=215Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.Half of 4's neighbor is 1.Half of the leading zero's neighbor is 2.93x5=465Half of 3's neighbor is 0, plus 5 because 3 is odd, is 5.Half of 9's neighbor is 1, plus 5 because 9 is odd, is 6.Half of the leading zero's neighbor is 4.

Publications

Rushan Ziatdinov, Sajid Musa. Rapid mental computation system as a tool for algorithmic thinking of

elementary school students development. European Researcher (http://erjournal.ru/en/index.html) 25(7):

1105-1110, 2012 [1] (http://erjournal.ru/journals_n/1342467174.pdf).

The Trachtenberg Speed System of Basic Mathematics by Jakow Trachtenberg, A. Cutler (Translator),

R. McShane (Translator), was published by Doubleday and Company, Inc. Garden City, New York in

1960.[1]

The book contains specific algebraic explanations for each of the above operations.

Most of the information in this article is from the original book.

The algorithms/operations for multiplication etc. can be expressed in other more compact ways that the book

doesn't specify, despite the chapter on algebraic description.[2] [3]

Other systems

There are many other methods of calculation in mental mathematics. The list below shows a few other methods ofcalculating, though they may not be entirely mental.

Bharati Krishna Tirtha's book "Vedic mathematics"

Abacus system - As students become used to manipulating the abacus with their fingers, they are typically

asked to do calculation by visualizing abacus in their head. Almost all proficient abacus users are adept at

doing arithmetic mentally.[citation needed]

Chisanbop



Software

Following are known programs and sources available as teaching tools

Web

Vedic Mathematics Academy [2] (http://vedicmaths.org/Other%20Material/Trachtenberg.asp)

PC

Trachtenberg Speed Math [3] (http://www.shermankeene.com/tracten.html)

Trachtenberg Mathematics Software [4] (http://www.shermankeene.com/tracten.html)

iPhone

Mercury Math [5] (https://itunes.apple.com/us/app/mercury-math-fast-mathematics/id318133094?mt=8)

References

Trachtenberg, J. (1960). The Trachtenberg Speed System of Basic Mathematics. Doubleday and Company,

Inc., Garden City, NY, USA.

., - . , 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 [6]

(http://erjournal.ru/journals_n/1342467174.pdf).

External links

Learn All about Mathematical Shortcuts (http://www.sapnaedu.in/category/mathematical-shortcuts)

1. ^ Trachtenberg, Jakow (1960). The Trachtenberg Speed System of Basic Mathematics. Translated by A. Cutler, R.McShane. Doubleday and Company, Inc. p. 270.

2. ^ All of this information is from an original book published and printed in 1960. The original book has seven fullChapters and is exactly 270 pages long. The Chapter Titles are as follows (the numerous sub-categories in eachchapter are not listed).

3. ^ The Trachtenberg speed system of basic mathematicsChapter 1 Tables or no tablesChapter 2 Rapid multiplication by the direct methodChapter 3 Speed multiplication-"two-finger" methodChapter 4 Addition and the right answerChapter 5 Division - Speed and accuracyChapter 6 Squares and square rootsChapter 7 Algebraic description of the method



"A revolutionary new method for high-speed multiplication, division, addition, subtraction and square root." (1960)"The best selling method for high-speed multiplication, division, addition, subtraction and square root - without acalculator." (Reprinted 2009)Multiplication is done without multiplication tables "Can you multiply 5132437201 times 4522736502785 in seventyseconds?" "One young boy (grammar school-no calculator) did--successfully--by using The Trachtenberg SpeedSystem of Basic Mathematics"Jakow Trachtenberg (its founder) escaped from Hitler's Germany from an active institution toward the close ofWWII. Professor Trachtenberg fled to Germany when the czarist regime was overthrown in his homeland Russiaand lived there peacefully until his mid-thirties when his anti-Hitler attitudes forced him to flee again. He was afugitive and when captured spent a total of seven years in various concentration camps. It was during these yearsthat Professor Trachtenberg devised the system of speed mathematics. Most of his work was done without pen orpaper. Therefore most of the techniques can be performed mentally.

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