Trachtenberg system

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The Trachtenberg System is a system of mental calculation, somewhat similar to vedic mathematics. It was developed by the Russian 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 patterns that allow one to perform arithmetic computations without the aid of pencil and paper.

The rest of this article presents some of the methods devised by Trachtenberg.

Table of contents

1 Multiplying by 12 2 Multiplying by 11 3 Multiplying by other numbers 3.1 Multiplying by 5 3.2 Multiplying by 6 3.3 Multiplying by 7 3.4 Multiplying by 8 3.5 Multiplying by 9 4 External links

Multiplying by 12

Rule: to multiply by 12, double each digit before adding it to the digit on its right, then recopy the first digit:

Example: 314 × 12 = 3,768:

4 × 2 = 8
1 × 2 + 4 = 6
3 × 2 + 1 = 7
Recopy 3

Multiplying by 11

Rule: to multiply by 11, recopy the last digit. Then, two by two, add digits next to each other. Recopy the first digit.

Example: 3,422 × 11 = 37,642

Recopy 2
2 + 2 = 4
4 + 2 = 6
3 + 4 = 7
Recopy 3

Multiplying by other numbers

Multiplying by 5

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

Multiplying by 6

• Rule: to multiply by 6:
  1. Add half of the neighbor to each digit.
  2. If the digit is odd, round off to the lowest whole number.
  3. If the result is odd, add 5.

Multiplying by 7

• Rule: to multiply by 7:
  1. Double each digit.
  2. Add half of its neighbor.
  3. If the result is odd, add 5.

Multiplying by 8

• Rule: to multiply by 8:
  1. Subtract last digit from 10.
  2. Subtract 9 from the other digits.
  3. Add result to the neighboring digits on the right.

Multiplying by 9

• Rule: to multiply by 9:
  1. Subtract the last digit from 10. (Ex.: 10 - 3 = 7)
  2. Subtract the other numbers from 9, and add to the number to the right.
  3. Take away 2 from the first number.

External links

• http://hucellbiol.mdc-berlin.de/~mp01mg/oldweb/Tracht.htm
• http://www.speed-math.com/

fr:Méthode_Trachtenberg

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