Trachtenberg system
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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.
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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
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.
Multiplying by 5
- Rule: to multiply by 5:
- Take half of the neighbor
- Add numbers up, two by two
- Add 5 if number is odd
Multiplying by 6
- Rule: to multiply by 6:
- Add half of the neighbor to each digit.
- If the result is odd, add 5.
Multiplying by 7
- Rule: to multiply by 7:
- Double each digit.
- Add half of its neighbor.
- If the result is odd, add 5.
Multiplying by 8
- Rule: to multiply by 8:
- Subtract last digit from 10.
- Subtract 9 from the other digits.
- Add result to the neighboring digits on the right.
Multiplying by 9
- Rule: to multiply by 9: