Trachtenberg method

The Trachtenberg method is a mental calculation system, somewhat similar to the Vedic mathematics of Bharati Krishna Tirtha. It was developed by the Russian engineer Jakow Trachtenberg in order to keep his mind occupied when he was a prisoner in a Nazi concentration camp.

The system consists of a number of easily memorized patterns that allow one to perform arithmetic computations without the aid of pencil and paper.

The rest of this article presents some of the methods designed by Trachtenberg.

Multiply by 12

• Rule: to multiply by 12, double each digit before adding it to the digit to your right and then copy the first digit (note that if the number is greater than 9 you will have to add 1 in the next operation).

• Example: 314 × 12 = 3.768:

4 × 2 = 8

1 × 2 + 4 = 6
-

3 × 2 + 1 = 7
+

Back to copy 3

• Although it may not be practical or mental ease, this method also serves with numbers 12, 13, until 19. To do so, the step of doubling the digit to triple (in case of 13), quadruple (in case of 14) and so on until 19.
• In the case of larger numbers (from a thousand) the exact same technique is used:
• Example: 2739 x 12= 32.868:

9 x 2 = 18 (when the number is greater than 9 the

number that forms the tenth, in this

case the 1 is added to the next digit and you

you are left with only the number that forms the

unit, in this case the 8)

3 x 2 + 9 + 1 = 16

7 x 2 + 3 + 1 = 18

2 x 2 + 7 + 1 = 12

2 + 1 = 3

We copy the 2 again (the sum of any factor will be equal to 0)

Multiply by 11

• Rule: to multiply by 11, copy back the last digit. Then, two by two, add each digit to the digit to your right. Again copy the first digit (note that if the number is greater than 9 you will have to add 1 in the next operation.

• Example: 3.422 × 11 = 37.642

Back to copy 2

2 + 2 = 4

4 + 2 = 6

3 + 4 = 7

Back to copy 3

Multiply by 5

• Rule: By multiplying a digit by 5 you only have to multiply the number by 10 and divide the number between 2 regardless of whether the number is accurate or not.

• Example (with exact number): 240 x 5 = 1200

240 x 10 =2400

2400 ÷ 2 = 1200

• Example (with inaccurate number): 241 x 5 = 1205

241 x 10 = 2410

2410 ÷ 2 = 1205

Multiply by 6

• Rule: to multiply by 6:
  1. The last digit of the result is the same number of multiplying if this is pair, but yes the last digit is odd must be added five.
  2. Add half of the number on the right to each digit (if the digit on the right is odd, the result is rounded to the lowest integer).
  3. For the first and last digit of the result is considered zero.

• Example: 4.532 × 6 = 27.192
• So: 04532 x 6

2 + (0/2) = 2 digit par

3 (print) + 5 + (2 / 2) = 9

5 (print) + 5 + (3 / 2) = 11
(we take us 1)

4 + (5 / 2) + 1 = 7

0 + (4/2) = 2

//-->.

• Example: 657.831 × 6 = 3.946.986
• So: 0657831 x 6

1 + 5 + (0/2) = 6

3 + 5 + (1 / 2) = 83 is unstoppable 5

8 + (3 / 2) = 93 is odd is reduced to 2

7 + 5 + (8 / 2)= 167 is unstoppable 5, and 1

5 + 5 + (7 / 2) + 1 = 145 is unstoppable, 5 is added, and 1 is taken. 7 is odd is reduced to 6

6 + (5 / 2) + 1 = 9; adds 1 that was taken. 5 is odd is reduced to 4

0 + (6/2) = 3

Multiply by 7

• Rule: to multiply by 7:
  1. Multiply for two each digit.
  2. If the whole thing you're adding to is odd, add 5.
  3. Add half of the number on the right to each digit (if the digit on the right is odd, the result is rounded to the lowest integer).
  4. The first digit of the number to multiply is considered zero.

• Example: 657.832 × 7 = 4.604.824
• So: 0657832 x 7

2 × 2 = 4

3 × 2 + 5 + (2 / 2)= 123 is unstoppable 5

8 × 2 + (3 / 2) + 1 = 18; He adds 1 that he took. 3 is odd is reduced to 2.

7 × 2 + 5 + (8 / 2) + 1= 24; He adds 1 that he took. 7 is unstoppable, 5 and 2

5 × 2 + 5 + (7 / 2) + 2= 20They add 2 that they were taking. 5 is odd sum 5. 7 is odd is reduced to 6

6 × 2 + (5 / 2) + 2 = 16; they add 2 that they took. 5 is odd is reduced to 4

0 × 2 + (6/2) + 1 = 4

Multiply by 8

• Rule: to multiply by 8:
  1. Remove the last digit of 10 and double.
  2. Remove two to the right digit and add if you take it.
• multiply by 8 the first digit

• Example: 7.623.453 × 8 = 60,987,624

(10 - 3) x 2 = 14

(10 - 5) x 2 + (3 - 2) + 1= 12

(10 - 4) x 2 + (5 - 2) + 1= 16

(10 - 3) x 2 + (4 - 2) + 1= 17

(10 - 2) x 2 + (3 - 2) + 1= 18

(10 - 6) x 2 + (2 - 2) + 1= 9

(8 x 7)+ (6 - 2)= 60

Multiply by 9

• Rule: to multiply by 9:
  1. Substrate the last digit of 10. (Ex.: 10 - 3 = 7)
  2. Subtract the other 9 numbers and add to the right digit.
  3. Subtract one from the first digit.

• Example: 583.264 × 9 = 5.249.376

10 - 4 = 6

9 - 6 + 4 = 7

9 - 2 + 6 = 13It takes 1

9 - 3 + 2 + 1 = 9; Adds 1 to take

9 - 8 + 3 = 4

9 - 5 + 8 = 12It takes 1

5 - 1 + 1 = 5; Adds 1 to take