Trachtenberg system

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The Trachtenberg System is a system of rapid mental calculation, somewhat similar to Swami Bharati Krishna Tirtha's Vedic mathematics developed several decades earlier. It was developed by the Ukrainian engineer Jakow Trachtenberg in order to keep his mind occupied while being held in a Nazi concentration camp.

The system consists of a number of readily memorized operations that allow one to perform arithmetic computations very quickly.

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. Also, the method 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 to multiplication.

Contents 1 General multiplication 2 General division 3 General addition 4 Other multiplication algorithms 5 Multiplying by 12 6 Multiplying by 11 7 Multiplying by other numbers 7.1 Multiplying by 5 7.2 Multiplying by 6 7.3 Multiplying by 7 7.4 Multiplying by 8 7.5 Multiplying by 9 8 Book 9 See also 10 References

[edit] General multiplication

The method for general multiplication is a method to achieve multiplication of a*b 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 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.

In general, for each position n in the final result, we sum for all i: a (digit at i) *b (digit at(n-i)). Ordinary people can learn this algorithm and multiply 4 digit numbers in their heads, writing down the final result, with the last digit first.

Trachtenberg defined this algorithm with a kind of pairwise multiplication where 2 digits are multiplied by 1 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.

[edit] General division

Builds upon the multiplication method

[edit] General addition

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.

[edit] Other multiplication algorithms

When performing any of these multiplication algorithms, the multiplier should have as many zeros prefixed to it as there are digits in the multiplicand. This will provide room for any carrying operations. For instance, when multiplying 366 × 7, add one zero to the front of 366 (write it "0366"); when multiplying 985 × 12, prefix two zeroes to 985 ("00985").

Each digit but the last, including the prefixed zeros, has a neighbor, i.e., the digit on its right.

[edit] 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 the right.)

This gives one digit of the result. If the answer is greater than 1 digit simply carry over the 1 or 2 to the next operation.
Example: 316 × 12 = 3,792:
In this example:

• the last digit 6 has no neighbor.
• the 6 is neighbor to the 1.
• the 1 is neighbor to the 3.
• the 3 is neighbor to the second prefixed zero.
• the second prefixed zero is neighbor to the first.

6 × 2 = 12 (2 carry 1)
1 × 2 + 6 + 1 = 9
3 × 2 + 1 = 7
0 × 2 + 3 = 3
0 × 2 + 0 = 0

[edit] Multiplying by 11

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

Example: 3,425 × 11 = 37,675

0 3 4 2 5 x 11=

  3     7     6     7     5

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

To illustrate:
11=10+1

Thus,

3425 x 11 = 3425 x(10+1)

37675    = 34250 + 3425

[edit] Multiplying by other numbers

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

A digit's 'neighbor' is the digit to its right.

[edit] Multiplying by 5

• Rule: to multiply by 5:
  1. Take half of the neighbor
  2. Add 5 if number is odd

Example: 42x5=210
4=2,2=1
43x5=215
4=2,3=1, i.e. 210 and as 43 is odd, add 5 thus 215

[edit] Multiplying by 6

• Rule: to multiply by 6:
  1. Add half of the neighbor to each digit.
  2. If the starting digit is odd, add 5.

Example:
6 × 357 = 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.

[edit] 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.

[edit] Multiplying by 8

• Rule: to multiply by 8:
  1. Subtract last digit from 10 and double
  2. Subtract the other digits from 9 and double
  3. Add result to the neighboring digit on the right.
  4. For the last calculation (The leading Zero), subtract 2 from the neighbour.

[edit] Multiplying by 9

• Rule: to multiply by 9:
  1. Subtract the last digit from 10.
  2. For each of the remaining digits, subtract it from 9, and add its neighbor.
  3. Finally, prepend one less than the first digit.

• Example: 2,130 × 9 = 19,170
  1. 10 - 0 = 10. Write 0; carry 1.
  2. 9 - 3 = 6; 6 + 0 + 1 (carried) = 7. Write 7.
  3. 9 - 1 = 8; 8 + 3 = 11. Write 1; carry 1.
  4. 9 - 2 = 7; 7 + 1 + 1 (carried) = 9. Write 9.
  5. 2 - 1 = 1. Write 1.

[edit] Book

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

The book's copyright is by Ann Cutler, a journalist in New York City at the time. The other person involved in the translation to English, Rudolph McShane, is a mathematician who lived in New Orleans at the time of publication and also worked on restricted USA government projects. ^{[2] [3]}

[edit] See also

• Swami Bharati Krishna Tirtha's Vedic mathematics

[edit] References