TRACHTENBERG METHOD: WHAT IT CONSISTS OF, EXAMPLES - MATHS - 2025

The Trachtenberg method is a system to perform arithmetic operations, mainly multiplication, in an easy and fast way, once its rules are known and mastered.

It was devised by the Russian-born engineer Jakow Trachtenberg (1888-1953) when he was a prisoner of the Nazis in a concentration camp, as a form of distraction to maintain sanity while he continued in captivity.

Figure 1. Multiplication tables. Source: Wikimedia Commons. Taulacat

What does it consist of, advantages and disadvantages

The advantage of this method is that to perform multiplication it is not necessary to memorize the multiplication tables, at least in part, it is enough to know how to count and add, as well as to divide a digit by two.

The downside is that there is no universal rule for multiplying by any number, rather the rule varies according to the multiplier. However, the patterns are not difficult to memorize and in principle allow operations to be carried out without the aid of paper and pencil.

Throughout this article we will focus on the rules for multiplying quickly.

Examples

To apply the method, it is necessary to know the rules, so we are going to present them one by one and with examples:

- Multiply a number by 10 or by 11

Rule for multiplying by 10

-To multiply any number by 10, simply add a zero to the right. For example: 52 x 10 = 520.

Rules for multiplying by 11

-A zero is added to the beginning and end of the figure.

-Each digit is added with its neighbor to the right and the result is placed below the corresponding digit of the original figure.

-If the result exceeds nine, then the unit is noted and a dot is placed on it to remember that we have a unit that will be added in the sum of the next figure with its neighbor on the right.

Detailed example of multiplication by 11

Multiply 673179 by 11

0 673 179 0 x 11 =

-----

= 7404969

The steps required to arrive at this result, illustrated by colors, are as follows:

-The 1 of the unit of the multiplier (11) was multiplied by the 9 of the multiplicand (0 673179 0) and 0 was added. The unit digit of the result was obtained: 9.

-Then multiply 1 by 7 and add nine to 16 and carry 1, place the ten digit: 6.

-After multiplying 1 by 1, adding the neighbor on the right 7 plus 1 that he had, resulting in 9 for the hundred.

-The next figure is obtained by multiplying 1 by 3 plus neighbor 1, resulting in 4 for the thousands digit.

-You multiply 1 by 7 and add the neighbor 3, resulting in 10, place zero (0) as the ten-thousand digit and take one.

-Then 1 times 6 plus neighbor 7 results in 13 plus a 1 that led to 14, the 4 is placed as a digit of the hundred-thousand and 1 is taken.

-Finally, 1 is multiplied by the zero that was added at the beginning, giving zero plus the neighbor 6 plus one that was taken. It is finally 7 for the digit corresponding to the millions.

- Multiplication by numbers from 12 to 19

To multiply any number by 12:

-A zero is added at the beginning and another zero at the end of the figure to be multiplied.

-Each digit of the number to be multiplied is doubled and added with its neighbor on the right.

-If the sum exceeds 10, a unit is added to the next duplication operation and sum with the neighbor.

Example of multiplication by 12

Multiply 63247 by 12

0 63 247 0 x 12 =

---–

758964

The details to reach this result, strictly following the stated rules, are shown in the following figure:

Figure 2. Trachtenberg's method to multiply any number by 12. Source: F. Zapata.

- Extension of the rules for multiplication by 13,… up to 19

The method of multiplication by 12 can be extended to multiplication by 13, 14 through 19 simply by changing the rule of doubling by tripling for the case of thirteen, quadrupling for the case of 14 and so on until reaching 19.

Rules for products by 6, 7 and 5

- Multiplication by 6

-Add zeros to the beginning and end of the figure to multiply by 6.

-Add half of its neighbor to the right to each digit, but if the digit is odd add 5 additionally.

Figure 3. Multiplication of a figure by 6, following the Trachtenberg method. Source: F. Zapata.

- Multiplication by 7

-Add zeros to the beginning and end of the number to multiply.

-Duplicate each digit and add the lower whole half of the neighbor, but if the digit is odd additionally add 5.

Example of multiplication by 7

-Multiply 3412 by 7

-The result is 23884. To apply the rules it is advisable to first recognize the odd digits and place a small 5 above them to remember to add this figure to the result.

Figure 4. Example multiplication of a figure by 7, according to the Trachtenberg method. Source: F. Zapata.

- Multiplication by 5

-Add zeros to the beginning and end of the number to multiply.

-Place the lower whole half of the neighbor to the right under each digit, but if the digit is odd, add additionally 5.

Example

Multiply 256413 by 5

Figure 5. Example multiplication of a figure by 5, according to the Trachtenberg method. Source: F. Zapata.

Rules for products by 9

-A zero is added at the beginning and another at the end of the figure to be multiplied by nine.

-The first digit on the right is obtained by subtracting the corresponding digit from the figure to multiply from 10.

-Then the next digit is subtracted from 9 and the neighbor is added.

-The previous step is repeated until we reach the zero of the multiplicand, where we subtract 1 from the neighbor and the result is copied below zero.

Example of multiplication by 9

Multiply 8769 by 9:

087690 x 9 =

-----

78921

Operations

10 - 9 = 1

(9-6) + 9 = 1 2 (copy 2 and carry 1)

(9-7) + 1 + 6 = 9

(9-8) +7 = 8

(8-1) = 7

Multiplication by 8, 4, 3 and 2

-Add zeros to the beginning and end of the number to multiply.

-For the first digit on the right subtract from 10 and the result is doubled.

-For the following digits subtract from 9, the result is doubled and the neighbor is added.

-When reaching zero, subtract 2 from the neighbor on the right.

- Multiplication by 8

Example of multiplication by 8

-Multiply 789 by 8

Figure 6. Example of multiplication of a figure by 8, according to the Trachtenberg method. Source: F. Zapata.

- Multiplication by 4

-Add zeros to the right and left of the multiplicand.

-Subtract the corresponding digit of the unit from 10 by adding 5 if it is an odd digit.

-Subtract from 9 in the form of each digit of the multiplicand, adding half of the neighbor on the right and if it is an odd digit add 5 additionally.

-When reaching the zero of the beginning of the multiplicand, place half of the neighbor minus one.

Example of multiplication by 4

Multiply 365187 x 4

Figure 7. Example multiplication of a figure by 4, according to the Trachtenberg method. Source: F. Zapata.

- Multiplication by 3

-Add zero to each end of the multiplicand.

-Subtract 10 minus the unit digit and add 5 if it is an odd digit.

-For the other digits, subtract 9, double the result, add half of the neighbor and add 5 if it is odd.

-When you reach the zero of the header, place the whole lower half of the neighbor minus 2.

Example of multiplication by 3

Multiply 2588 by 3

Figure 8. Example multiplication of a number by 3, according to the Trachtenberg method. Source: F. Zapata.

- Multiplication by 2

-Add zeros at the ends and double each digit, if it exceeds 10 add one to the next.

Example

Multiply 2374 by 2

0 2374 0 x 2

04748

Multiply by composite figures

The rules listed above apply, but the results are run to the left by the number of places corresponding to tens, hundreds, and so on. Let's look at the following example:

Exercise

1. Cutler, Ann. 1960 The Trachtenberg speed system of basic mathematics. Doubleday & CO, NY.
2. Dialnet. Quick basic math system. Recovered from: dialnet.com
3. Mathematical corner. Rapid multiplication by the Trachtenberg method. Recovered from: rinconmatematico.com
4. The Trachtenberg Speed ​​System of Basic Mathematics. Recovered from: trachtenbergspeedmath.com
5. Wikipedia. Trachtenberg method. Recovered from: wikipedia.com