This is a system of rapid calculation similar to Vedic mathematics. It was invented by Jakow Trachtenberg, a Russian engineer. Trachtenberg invented this system while trapped in a Nazi concentration camp in order to keep his mind off the horror of the camp. Fortunately, his wife was able to bribe the guards for Jakow's escape, thus these handy math shortcuts weren't lost to the Holocaust.

It's estimated that this system generally shortens computation time (versus longhand math) by 20% with 99% accuracy. Here is a summary of the multiplication shortcuts contained in the Trachtenberg system:

Multiplying by 12

Example: 12 x 345
First, add a zero to the front of the number: 0345
Each digit (except the last) has a neighbor, the number to it's right.
Starting on the right side, double each digit and add its neighbor:
(5 x 2) + 0 (there is no neighbor) = 10
The answer so far is xxx0 and the 1 will be carried to the next step.
(4 x 2) + 5 + the carried 1 = 14
The answer so far is xx40 and the 1 will be carried.
(3 x 2) + 4 + 1 = 11
The answer so far is x140 and the 1 will be carried.
(0 x 2) + 3 + 1 = 4
The answer is 4140

Multiplying by 11

Example: 11 x 2345
The method here is to recopy the last digit, then add the digits next to each other two by two, then recopy the first digit. Carry the one to the next step when necessary.
Recopy the last digit: now the answer we have is xxxx5
Add the last two digits: 5 + 4 = 9
Now the answer we have is xxx95
Add the 2nd from last and 3rd from last digits: 4 + 3 = 7
Now the answer we have is xx795
Add the 3rd from last and 4th from last digits: 3 + 2 = 5
Now the answer we have is x5795
Recopy the first digit. Now the answer we have is 25795
Thus, 11 x 2345 = 25795
You may have noticed that this is similar to the third sutra of Vedic mathematics.

Multiplying by 9

Example: 9 x 1234
First add a zero to the beginning of the number: 01234
Now subtract the last digit from 10: 10 - 4 = 6
So far the answer we have is: xxxx6
Subtract all of the middle digits from 9, proceeding right to left, then add the neighbor (the number to the right).
9 - 3 + 4 = 10
So far the answer we have is: xxx06, the 1 will be carried to the next step.
9 - 2 + 3 + the carried 1 = 11
So far the answer we have is: xx106, and the 1 will be carried to the next step.
9 - 1 + 2 + the carried 1 = 11
So far the answer we have is: x1106, and the 1 will be carried to the next step
For the leftmost digit of the answer, subtract 1 from it's neighbor:
1 - 1 + the carried 1 = 1
Thus, 9 x 1234 = 11106

Multiplying by 8

Example: 8 x 1234 First add a zero to the beginning of the number: 01234
Now subtract the last digit from 10 and double the result: (10 - 4) x 2 = 12
So far the answer we have is: xxxx2 and carry the 1
Subtract all of the middle digits from 9, proceeding right to left, double the result, then add the neighbor (the number to the right).
[(9 - 3) x 2] + 4 + the carried 1 = 17
So far the answer we have is: xxx72, the 1 will be carried to the next step.
[(9 - 2) x 2] + 3 + the carried 1 =
So far the answer we have is: xx872, and the 1 will be carried to the next step.
[(9 - 1) x 2] + 2 + the carried 1 = 19
So far the answer we have is: x9872, and the 1 will be carried to the next step
For the leftmost digit of the answer, subtract 2 from it's neighbor:
1 - 2 + the carried 1 = 0
Thus, 8 x 1234 = 9872
Note: this will not work when multiplying numbers in the 90s, such as 91, 92, etc.

Multiplying by 7

Example: 7 x 1234
Starting on the right side and moving left, double each digit and add half of the neighbor. If the digit of the number that is being doubled is odd, add 5. If the neighbor divided in half is not a whole number, always round down, not up
(4 x 2) + (0 / 2) = 8 (do not add 5 because 4 is even)
So the answer so far is: xxx8
(3 x 2) + (4 / 2) = 8 + 5 (because 3 is odd) = 13
So the answer so far is: xx38 and carry the 1
(2 x 2) + (3 / 2) = 5 + the carried 1 = 6
So the answer so far is: x638
(1 x 2) + (2 / 2) = 3 + 5 = 8
So the answer so far is: 8638
Thus, 7 x 1234 = 8638

Multiplying by 6

Example: 6 x 1234
This is calculated exactly the same as the method for multiplying by 7, except the digit is not doubled before half the neighbor is added:
4 + (0 / 2) = 4 (do not add 5 because 4 is even)
So the answer so far is: xxx4
3 + (4 / 2) + 5 = 10
So the answer so far is: xx04 and the 1 is carried.
2 + (3 / 2) + the carried 1 = 4 (do not add 5 because 4 is even)
So the answer so far is: x404
1 + (2 / 2) + 5 = 7
So the answer so far is: 7404
Thus, 6 x 1234 = 7404

The Trachtenberg system also has applications in division and addition problems, and in multiplying by other numbers than those listed above. To learn about those and other facets of the Trachtenburg system, check out the resources below:

http://hucellbiol.mdc-berlin.de/~mp01mg/oldweb/Tracht.htm
http://vedicmaths.org/Files/Trachtenberg.asp
http://mathforum.org/dr.math/faq/faq.trachten.html
http://en.wikipedia.org/wiki/Trachtenberg_system (the multiplication explaination for the number 8 is incorrect at this link)
http://poly.lausd.k12.ca.us/gate/mathfun.html#anchor615073

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