Multiplying by 12



Multiplying by 11



Multiplying by 9



Multiplying by 8



Multiplying by 7



Multiplying by 6



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

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:Example: 12 x 345First, add a zero to the front of the number:Each digit (except the last) has a, 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) = 10The answer so far isand the 1 will be carried to the next step.(4 x 2) + 5 + the carried 1 = 14The answer so far isand the 1 will be carried.(3 x 2) + 4 + 1 = 11The answer so far isand the 1 will be carried.(0 x 2) + 3 + 1 = 4The answer isExample: 11 x 2345The 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 isAdd the last two digits: 5 + 4 = 9Now the answer we have isAdd the 2nd from last and 3rd from last digits: 4 + 3 = 7Now the answer we have isAdd the 3rd from last and 4th from last digits: 3 + 2 = 5Now the answer we have isRecopy the first digit. Now the answer we have isThus, 11 x 2345 = 25795You may have noticed that this is similar to the third sutra of Vedic mathematics Example: 9 x 1234First add a zero to the beginning of the number: 01234Now subtract the last digit from 10:So far the answer we have is:Subtract all of the middle digits from 9, proceeding right to left, then add the neighbor (the number to the right).9 - 3 + 4 = 10So far the answer we have is:, the 1 will be carried to the next step.9 - 2 + 3 + the carried 1 = 11So far the answer we have is:, and the 1 will be carried to the next step.9 - 1 + 2 + the carried 1 = 11So far the answer we have is:, and the 1 will be carried to the next stepFor the leftmost digit of the answer, subtract 1 from it's neighbor:1 - 1 + the carried 1 = 1Thus, 9 x 1234 =Example: 8 x 1234 First add a zero to the beginning of the number: 01234Now subtract the last digit from 10 and double the result:So far the answer we have is:and carry the 1Subtract 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 = 17So far the answer we have is:, the 1 will be carried to the next step.[(9 - 2) x 2] + 3 + the carried 1 =So far the answer we have is:, and the 1 will be carried to the next step.[(9 - 1) x 2] + 2 + the carried 1 = 19So far the answer we have is:, and the 1 will be carried to the next stepFor the leftmost digit of the answer, subtract 2 from it's neighbor:1 - 2 + the carried 1 = 0Thus, 8 x 1234 =Note: this will not work when multiplying numbers in the 90s, such as 91, 92, etc.Example: 7 x 1234Starting 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 (4 x 2) + (0 / 2) = 8 (do not add 5 because 4 is even)So the answer so far is:(3 x 2) + (4 / 2) = 8 + 5 (because 3 is odd) = 13So the answer so far is:and carry the 1(2 x 2) + (3 / 2) = 5 + the carried 1 = 6So the answer so far is:(1 x 2) + (2 / 2) = 3 + 5 = 8So the answer so far is:Thus, 7 x 1234 = 8638Example: 6 x 1234This 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:3 + (4 / 2) + 5 = 10So the answer so far is: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:1 + (2 / 2) + 5 = 7So the answer so far is:Thus, 6 x 1234 = 7404The 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: