Trachtenberg system
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The Trachtenberg System is a system of rapid mental calculation, somewhat similar to Vedic mathematics. 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 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.
The most important algorithms are the ones for general multiplication, division and addition. In addition, the method includes some specialized methods for multiplying small numbers between 5 and 13.
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[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 a 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 to sum columns of numbers. Creates an intermediate sum, in the form of two rows of digits. The final step is to sum the intermediate results with an L-shaped algorithm
[edit] Other multiplication algorithms
When performing any of these multiplication algorithms, the multiplier should have as many zeroes prepended 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, prepend two zeroes to 985 ("00985").
Each digit but the last, including the prepended zeroes, 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 neighbour 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 prepended zero.
- the second prepended 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 neighbour.(By neighbour 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)
Proof: 11=10+1
Thus,
3425 x 11 = 3425 x(10+1)
= 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.
[edit] Multiplying by 5
- Rule: to multiply by 5:
- Take half of the neighbor
- Add 5 if number is odd
Example:42x5=210 4=2,2=1 43x5=2
[edit] Multiplying by 6
- Rule: to multiply by 6:
- Add half of the neighbor to each digit.
- 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:
- Double each digit.
- Add half of its neighbor.
- If the digit is odd, add 5.
[edit] Multiplying by 8
- Rule: to multiply by 8:
- Subtract last digit from 10 and double
- Subtract the other digits from 9 and double
- Add result to the neighboring digit on the right.
- For the last calculation (The leading Zero), subtract 2 from the neighbour.
[edit] Multiplying by 9
- Rule: to multiply by 9:
[edit] Book
The Trachtenberg Speed System of Basic Mathematics by Jakow Trachtenberg, A. Cutler (Translator), R. McShane (Translator), Rudolph Mcshane (Translator)
[edit] Proofs
There are specific algebra rules for each of the above to be found in the complete book.
