The Trachtenberg System is a system of rapid 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 very quickly.

The rest of this article presents some of the methods devised by Trachtenberg. These are for illustration only. To actually learn the method requires practice and a more complete treatment.

Multiplying by 12

Imagine your number to be pre-pended with a zero. Each digit (including the prepended zero ) but the last has a neighbor, i.e., the digit on its right.
Rule: to multiply by 12: Starting from the rightmost digit, double each digit and add the neighbor. 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: 0316 × 12 = 3,792:
In this example
the 3 is neighbor to the prepended zero.
the 1 is neighbor to the 3.
the 6 is neighbor to the 1.
the last digit 6 has no neighbor.
6 × 2 = 12 (2 carry 1)
1 × 2 + 6 + 1 = 9
3 × 2 + 1 = 7
0 × 2 + 3 = 3
Instead of pre-pending a zero, you can think that the first digit only needs to be copied, adding any carry.

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