Trachtenberg method

The method Trachtenberg is a method of calculating mental invented by Jacow Trachtenberg.

This method makes it possible to quickly carry out complex arithmetic calculations by breaking up them into simpler calculations.

Opening remarks

Notice n°1: In a multiplication of two numbers ( factors ), the first factor is called multiplicand and the multiplying second . The result of the operation is the produced .

Notice n°2: Figure and Nombre is two distinct concepts. A number is written using figures. The easy ways of calculation described below are based on a simple technique which consists in considering each figure of the multiplicand as a number to which one applies the stated rule. By abuse language, but for simplifer the comprehension of the easy ways, the continuation of this article will employ only the term figure in its strict direction or to replace the term number.

Notice n°3: To apply the easy ways below, it is necessary to carry out calculation figures of the product of the right-hand side towards the left (since the units while going up towards the figures of increasingly strong weight) starting from the figures of the multiplicand in the same order.

Notice n°4: Multiplicand should be added on the left a number equal of zeros to the number of figures in the multiplier
Ex 1: by multiplying 325 * 6, one takes 0325 * 6. ex 2: by multiplying 325 * 12, one takes 00325 * 12

Notice n°5: When one divides by two an odd figure, one does not take account of the figures after the comma.
Ex 1: 3/2 = 1 (and not 1.5). Ex 2: 7/2 = 3 (and not 3.5)

Multiplication by small figures

Multiplication by 5

Regulate: half of the neighbor of right-hand side, + 5 if the figure is odd.

Stated in a different way: Each figure of the product is equal to half of the neighbor of right-hand side of the figure of the multiplicand occupying the same position, increased by 5 if this last is odd.

Example 1: 413 X 5 = 2065

4: 2 = 2 1: 2 = 0 3: 2 = 1 + 5 (because " 3" is the neighbor of right-hand side of " 1" who is odd) = 6 To recopy 5 (because the multiplicand is odd)

Example 2: 812 X 5 = 4060

8: 2 = 4 1: 2 = 0 2: 2 = 1 + 5 (because " 2" is the neighbor of right-hand side of " 1" who is odd) = 6 To recopy 0 (because the multiplicand is even)

Example 3: 5036 X 5 =?

5: 2 = 2 0: 2 = 0 + 5 (because " 0" is the neighbor of right-hand side of " 5" who is odd) = 5 3: 2 = 1 6: 2 = 3 + 5 (because " 6" is the neighbor of right-hand side of " 3" who is odd) = 8 To recopy 0 (because the multiplicand is even) Result: 25180

Multiplication by 6

Regulate: To add half of the neighbor of right-hand side to each figure, plus 5 if the figure is odd and the possible reserve of the immediately lower row.

Example 1: 5314 X 6 =?

One starts by adding one on the left 0 of 5314 => 05314 (n°4 notices) One starts with the line (n°3 notices) 4 (does not have neighbor of right-hand side) => 4 1 + (4: 2) + 5 (because " 1" ) = 8 is odd 3 + (1: 2) + 5 (because " 3" ) = 8 is odd 5 + (3: 2) + 5 (because " 5" is odd) = 11 => 1 and retained of 1 0 + (5: 2) + 1 (preceding reserve) = 3 result of 31884

Example 2: 3267 X 6 =?

One adds one on the left 0 of 3267 => 03267 (n°4 notices) 7 (does not have neighbor of right-hand side) + 5 (because " 7" is odd) = 12 => 2 and retained of 1 6 + (7: 2) + 1 (preceding reserve) = 10 => 0 and retained of 1 2 + (6: 2) + 1 (preceding reserve) = 6 3 + (2: 2) + 5 (because " 3" ) = 9 is odd 0 + (3: 2) = 1 result of 19602

Multiplication by 7

Regulate: Doubler each figure and to add half of the neighbor, + 5 if the figure is odd.

Example 1: 5314 X 7 =?

One starts by adding one on the left 0 of 5314 => 05314 (n°4 notices) 4 * 2 = 8 1 * 2 + (4: 2) + 5 (because " 1" ) = 9 is odd 3 * 2 + (1: 2) + 5 (because " 3" is odd) = 11 => 1 and retained of 1 5 * 2 + (3: 2) + 5 (because " 5" ) + 1 is odd (of reserve) = 17 => 7 and retained of 1 0 * 2 + (5: 2) + 1 (preceding reserve) = 3 result of 37198

Example 2: 3267 X 7 =?

One starts by adding one on the left 0 of 3267 => 03267 (n°4 notices) 7 * 2 + 5 (because " 7" is odd) = 19 => 9 and retained of 1 6 * 2 + (7: 2) + 1 (of reserve) = 16 => 6 and retained of 1 2 * 2 + (6: 2) + 1 (of reserve) = 8 3 * 2 + (2: 2) + 5 (because " 3" is odd) = 12 => 2 and retained of 1 0 * 2 + (3: 2) + 1 (preceding reserve) = 2 result of 22869

Multiplication by 8

Regulate: To withdraw the last figure of 10 and to double the result. To withdraw the other figures of 9. To double the result and to add to the neighbor of right-hand side. To remove 2 with the first figure.

Example 1: 5314 X 8 =?

(10 - 4) * 2 = 12 => 2 and retained of 1 (9 - 1) * 2 + 4 + 1 (reserve) = 21 => 1 and retained of 2 (9 - 3) * 2 + 1 + 2 (of reserve) = 15 => 5 and retained of 1 (9 - 5) * 2 + 3 + 1 (of reserve) = 12 => 2 and retained of 1 (5 - 2) + 1 (of reserve) = 4 result of 42512

Example 2: 3267 X 8 =?

(10 - 7) * 2 = 6 (9 - 6) * 2 + 7 = 13 => 3 and retained of 1 (9 - 2) * 2 + 6 + 1 (of reserve) = 21 => 1 and retained of 2 (9 - 3) * 2 + 2 + 2 (of reserve) = 16 => 6 and retained of 1 (3 - 2) + 1 (of reserve) = 2 result of 26136

Multiplication by 9

Regulate: To withdraw the last figure of 10, to withdraw the others of 9 and to add the result to the neighbor of right-hand side. To withdraw 1 from the first figure.

Example 1: 5314 X 9 =?

(10 - 4) = 6 (9 - 1) + 4 = 12 => 2 and retained of 1 (9 - 3) + 1 + 1 (of reserve) = 8 (9 - 5) + 3 = 7 (5 - 1) = 4 result of 47826

Example 2: 3267 X 9 =?

(10 - 7) = 3 (9 - 6) + 7 = 10 => 0 and retained of 1 (9 - 2) + 6 + 1 (of reserve) = 14 => 4 and retained of 1 (9 - 3) + 2 + 1 (of reserve) = 9 (3 - 1) = 2 result of 29403

Multiplication by 11

Regulate: To recopy the last figure. To add 2 by 2 close figures. To recopy the first figure of the multiplicand.

Example: 3.422 X 11 = 37.642

To recopy 2. 2 + 2 = 4 4 + 2 = 6 3 + 4 = 7 To recopy 3

Multiplication by 12

Regulate: To double each figure before adding it to its neighbor of right-hand side. To recopy the first figure (more possibly reserve).

Example 1: 314 X 12 = 3.768

4 X 2 = 8 1 X 2 + 4 = 6 3 X 2 + 1 = 7 To recopy 3

Example 2: 5267 X 12 =?

7 X 2 = 14 => 4 and retained of 1 6 X 2 + 7 + 1 (of reserve) = 20 => 0 and 2 of reserve 2 X 2 + 6 + 2 (of reserve) = 12 => 2 and 1 of reserve 5 * 2 + 2 + 1 (of reserve) = 13 => 3 and retained of 1 To recopy 5 + 1 (of reserve) = 6 Result of 63204

Multiplication by 13

Multiplication by 4

Regulate: Soustraire the last figure of 10 and to add 5 if it is odd. To withdraw the others of 9 and to add half of the neighbor of right-hand side and 5 if the figure is odd. To take motié of the figure of left minus 1.

Example 1: 5314 X 4 =?

(10 - 4) = 6 (9 - 1) + (4/2) + 5 (because 1 is odd) = 15 => 5 and retained of 1 (9 - 3) + (1/2) + 5 (because 3 is odd) + 1 (of reserve) = 12 => 2 and retained of 1 (9 - 5) + (3/2) + 5 (because 5 is odd) + 1 (of reserve) = 11 => 1 and 1 of reserve (5/2) - 1 + 1 (of reserve) = 2 result of 21256

Example 2: 3267 X 4 =?

(10 - 7) + 5 (because 7 is odd) = 8 (9 - 6) + (7/2) = 6 (9 - 2) + (6/2) = 10 => 0 and retained of 1 (9 - 3) + (2/2) + 5 (because 3 is odd) + 1 (of reserve) = 13 => 3 and 1 of reserve (3/2) - 1 + 1 (of reserve) = 1 result of 13068

Multiplication by 3

Regulate: Soustraire the last figure of 10, to multiply it by two and to add 5 if it is odd. To withdraw the others of 9, to multiply by two and to add half of the neighbor of right-hand side and 5 if the figure is odd. To take motié of the figure of left minus 2.

Example 1: 5314 X 3 =?

(10 - 4) * 2 = 12 => 2 and retained of 1 (9 - 1) * 2 + (4/2) + 5 (because 1 is odd) + 1 of reserve = 24 => 4 and retained of 2 (9 - 3) * 2 + (1/2) + 5 (because 3 is odd) + 2 (of reserve) = 19 => 9 and retained of 1 (9 - 5) * 2 + (3/2) + 5 (because 5 is odd) + 1 (of reserve) = 15 => 5 and 1 of reserve (5/2) - 2 + 1 (of reserve) = 1 result of 15942

Example 2: 3267 X 3 =?

(10 - 7) * 2 + 5 (because 7 is odd) = 11 => 1 and retained of 1 (9 - 6) * 2 + (7/2) + 1 (of reserve) = 10 => 0 and retained of 1 (9 - 2) * 2 + (6/2) + 1 (of reserve) = 18 => 8 and retained of 1 (9 - 3) * 2 + (2/2) + 5 (because 3 is odd) + 1 (of reserve) = 19 => 9 and 1 of reserve (3/2) - 2 + 1 (of reserve) = 0 result of 9801

See too

Technical of mental calculation

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