The Trachtenberg System of Speed Math

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The Trachtenberg Speed System Basic Mathematics is a system of mental calculation, similar to Vedic mathematics. Sometimes presented as an alternative way of learning math, the system consists of many math shortcuts and mental math tricks, particularly multiplication tricks.

It was developed by the Russian engineer Jakow Trachtenberg to keep his mind busy when he was imprisoned in a Nazi concentration camp and is known to be one of the fastest speed math systems.

The system consists of a number of easily memorized patterns that allow one to perform arithmetic computations without the help of pen and paper. The system primarily focuses on multiplication tricks but with further practice and study you can learn division, addition, subtraction and square root.

A Quick Overview of Basic Multiplication

If you want to learn either how to multiply without a calculator effectively or how to teach multiplication effectively, then keep reading.

The process for calculation using the Trachtenberg method is rather simple and involves calculating the answer to your problem one digit at a time. The steps you must follow:

  • Write a zero ‘0’ before the number you are multiplying (the multiplicand) and underline it.
  • Apply the relevant rule for the number you are calculating (see below for rules).
  • Work from the right to the left of the multiplicand, individually applying the rule to each number in turn.
  • Record the result of each calculation below the number you applied the rule to.

Definition of Terms

The Multiplicand – The ‘Multiplicand’ is the number you are multiplying. In the photos above, and on all of our examples, it is the number on the right of the multiplier (eg: in the sum11 x 28344″ 28344 would be the multiplicand). The multiplicand will always be underlined and our answers will go beneath it, one digit at at time.

The ‘Number – When we talk about “adding the number to its neighbour” etc, the number is referencing to the current number in the multiplicand you are applying the rule to.  We will always start from the right most number in the multiplicand and work our way backwards, to the left.

The ‘Neighbour – The ‘neighbour‘ is the digit to the right of the number’. If my multiplicand was 345, then 5 would not have a neighbour (our we would use 0), the neighbour of 4 would be 5 and the neighbour of 3 would be 4.

Carried 10’s – When we do calculations which give us double digit answers above 10 we need to carry the 10’s. We do this by adding a dot to represent the carried ten (or two dots if the answer is 20-29) to the current number we are working on to remind us to add the dot (1 dot = 1) onto the next answer. In the below example, the 7’s recorded in are answer are actually  ’17’ but we record only the 7 and add a dot. Two dots are added if the resulting answer of the number rule is 20-29. (You will never have an answer which is above 29.)

    \begin{align*} 3 \times \underline{\textcolor{blue}{02}588} \\ \textcolor{red}{0}\dot{7}\dot{7}64 \end{align*}

Rounding Down Fractions – When using the Trachtenberg System you will deal with rules which involve halving the numbers or neighbours, when we ‘halve’ an odd number we will ignore any fractions and round down to the nearest whole number. Half of 5 would be recorded simply as 2, half of 7 would be 3 and half of 8 would be 4 (even numbers are halved as normal).

Jump to number: 11, 12, 6, 7, 5, 9, 8, 4, 3

Multiplication of 11 – Rules & Example

Multiply by 11. Quick math trickVisit our page on multiplying by 11 HERE and see more worked examples, tips & tricks, clearer steps, video tutorials, further resources and help.

We start by learning how to multiply by 11 because it is the easiest rule to learn. If you learn only one thing from this site, it should be this. It’s one of the best math tricks for kids and adults.

The Rule: Multiply by 11

  1. Add the neighbour to the number

Example: 11 x 633

  11 x 0633
=      6963
---
3 + 0 = 3       Step 1
3 + 3 = 6       Step 2
6 + 3 = 9       Step 3
0 + 6 = 6       Step 4

Description

When multiplying by 11 you simply add the number to the neighbour. Starting at the right hand number, 3, we see that it doesn’t have a neighbour so there is nothing to add (or we imagine 0 is its neighbour). We are then finished with our calculation and record 3.

Moving onto the next digit, 3, we add this to it’s right hand neighbour, another 3, to get the result 6. We record 6 below.

The next two digits we do the same. 6 + 3 = 9. We record the 9. 0 + 6 is 6, we record the 6. And we have our answer: 6963.

~~~

This is a very basic example of how to multiply by 11 using low numbers and thus doesn’t involve dealing with any carried 10’s. To learn more about how to multiply by 11 and see more complicated examples please visit our dedicated multiplying by 11 page.

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Multiplication of 12 – Rules & Example

Number 12 trachtenberg systemOur page on multiplying by 12. Here you will find more worked examples, tips & tricks, in-depth explanations, downloadable PDF’s, worksheets, exercises, video tutorials, further resources and help.

The Rule: Multiply by 12

  1. Double each number in turn and add its neighbor.

Example: 12 x 413

  12 x 0413
=      4956
---
(3 x 2) + 0 = 6       Step 1
(1 x 2) + 3 = 5       Step 2
(4 x 2) + 1 = 9       Step 3
(0 x 2) + 4 = 4       Step 4

Description

The rule for multiplying by 12 is very similar to that of multiplying by 11 shown above.

Starting from the right hand number of the multiplicand we double 3 to get 6. The first digit doesn’t have a neighbour so we add 0. The first digit in our answer is 6.

The next digit, 1, is then doubled to get 2. We then add its neighbour, 3, to get our second digit in our answer, 5.

We continue these steps to the left with the next two digits and get 9 and 4 respectively for the next two digits in our answer. Our answer: 4956.

~~~

This is a simple example of how to multiply by 12 which doesn’t involve carrying 10’s. To learn how to do this and see more worked examples, please visit our  dedicated “multiplying by 12” page.

Multiplication of 6 – Rules & Example

Number 6 maths trick trachtenbergBelow I will demonstrate how to multiply by 6 using the Trachtenberg System but note that this is a basic example and whilst it covers the entire rule for the number 6, it does’t show how to effectively multiply larger numbers and the theory behind doing so. Please visit our dedicated “Multiplying by 6” page for more info and multiplication tricks.

The Rule: Multiply by 6

  1. Add half the neighbour to the number; Add 5 if the “number” is odd.

Example: 6 x 613

  6 x 0613
=     3678
---
3 + (0 x 0.5) + 5 = 8       Step 1
1 + (3 x 0.5) + 5 = 7       Step 2
6 + (1 x 0.5)     = 6       Step 3
0 + (6 x 0.5)     = 3       Step 4

Description

The first digit on the right hand side of our multiplicand is 3. It has no neighbour so we don’t add anything to it. It is however an odd digit so we will add 5 to it to get the first digit in our answer, 8.

Moving along to the left, we add the number, 1, to half of its neighbour (3 x 0.5) and get 2. Next we add 5 beause “1” is odd. The second number in our answer is 7.

The next two numbers in our multiplicand, 6 and 0 are both even so we don’t have to add the additional 5 for odd numbers, we simply add half of their right hand neighbours to the numbers themselves to get the answers 6 and 3 respectively.

The final answer: 3678.

~~~

The above example is a very simple explanation of how to multiply by 6 using the Trachtenberg Method.  It uses small digits and doesn’t involve carrying any 10’s. To learn how to do this and see more worked examples, please visit our  dedicated “multiplying by 6” page.


Multiplication of 7 – Rules & Example

number 7 maths trachtenbergMultiplying by 7 involves using a combination of rules and can be a little trickier to remember and implement. Detailed instructions can be found on our “Multiplying by 7” page.

The Rule: Multiply by 7

  1. Double the number plus half of its neighbour; add 5 if the number is odd.

Example: 7 x 358

  7 x 0358
=     2506
---
(8 x 2) + (0 x 0.5)          = 16     Step 1
(5 x 2) + (8 x 0.5) + 5 + 1  = 20     Step 2
(3 x 2) + (5 x 0.5) + 5 + 2  = 5      Step 3
(0 x 2) + (3 x 0.5) + 1      = 2      Step 4

Description

First start with the right hand number of the multiplicand, 8. Double it to get 16 and as it has no neighbour we don’t have to add anything else.

The next number, 5, is doubled to get 10. Then its neighbour, 8, is halved to get 4 and added to the running sum to make 9. Because our number 5 is odd, we add 5. And finally we must remember to add the carried one from the previous digits result. This gives us a result of 20. We record the 0 and remember to carry 2 onto our next number.

The above steps are repeated for the next number. 3 doubled is 6, plus halve of its neighbour, 5, gives us a total of 8. Because 3 is odd, we add an additional 5 and then remember to add 2 from our previous answer. Our answer is 15 so we record the 5 and carry the 1.

The final digit is a 0. Double 0 is 0 so we can effectively ignore this step. Next we halve its neighbour, 3, to get 1 and add the carried one from our previous answer. We record our final digit to our answer 2.

Our answer: 2506

~~~

The above example is a simple explanation of how to multiply by 7 using the Trachtenberg System. To see more worked examples and find the most efficient way of multiplying by 7 please visit our  dedicated “multiplying by 7” page.


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Multiplication of 5 – Rules & Example

number 5 calculationsThe method for multiplying by 5 using the Trachtenberg System is very similar to that of 7, in fact, it’s probably a little easier. It involves using multiple steps within the rule. Detailed instructions can be found on our “Multiplying by 5” page.

The Rule: Multiply by 5

  1. Half the neighbour; add 5 if the number is odd.

Example: 5 x 271

  5 x 0271
=     1355
---
(0 x 0.5) + 5   = 5     Step 1
(1 x 0.5) + 5   = 5     Step 2
(7 x 0.5)       = 3     Step 3
(2 x 0.5)       = 1     Step 4

Description

The main difference between multiplying by 5 and 7 is that we are not directly manipulating or using the number we are on at the time. Calculations are made using the numbers neighbour, then we add 5 to the total if the number is odd.

Starting from the right, we have a 1. It has no neighbour but it is odd so we add 5 to the result of half its neighbour (0.5 x 0) to get 5.

The next digit, 7 has a neighbour of 1. Half of 1 is 0. 7 is an odd number so we add 5 to get the next digit in our answer: 5.

2 is our next digit. Half of its neighbour, 7 is 3. Add our number 2 is odd we do not add an additional 5. We record 3 as the next digit in our answer.

The final digit of the multiplicand is 0. Half of its neighbour, 2, is 1. Zero is odd so we do not add 5. We record 1.

Our answer: 1355

~~~

The above example is a simple example & explanation of how to multiply by 5 using the Trachtenberg System. To see more worked examples and find the most efficient way of multiplying by 5 please visit our  dedicated “multiplying by 5” page.


Multiplication of 9 – Rules & Example

Number 9 Trachtenberg SystemMultiplying by 9 using the Trachtenberg System involves 3 steps and separate rules are used for the beginning, last and middle numbers. Detailed instructions can be found on our “Multiplying by 9” page.

The Rule: Multiply by 9

  • Step 1: (First Digit Only): Subtract the right-hand figure of the multiplicand from 10.
  • Step 2: (For Middle Digits Only): Subtract number from 9Add its neighbour.
  • Step 3: (Last Digit, ‘0’): Subtract 1 from its neighbour.

Example: 9 x 879

  9 x 0879
=     7911
---
10 - 9            = 1      Step 1
(9 - 7) + 9       = 11     Step 2a
(9 - 8) + 7 + 1   = 9      Step 2b
(8 - 1)           = 7      Step 3

Description

The number 9 is the first number we see which has three separate rules in one. Whilst this might seem complicated at first, it is in fact an easy rule as a whole to remember as the first and last digits, in spite of having separate rules, are easy subtractions. Common sense and a basic understanding of numerology certainly helps when using rules with multiple steps.

The first number in the sequence, 9, must be subtracted from 10 to get our result: 1.

The middle numbers we subtract from 9 and then add its neighbour. So, 7 subtracted from 9 is 2, plus it’s neighbour, 9, equals 11. We record the 1 and carry the 10.

The second number we deal with is 8. Subtract 8 from 9 to get 1. Then add its neighbour, 7, to get 8. then we add on the carried one from the last calculation to get 9. We record 9.

The last step is simple, we subtract one from the neighbour of 0 (which is 8) to get 7. We record 7.

Our answer: 7911

~~~

Multiplying by 9 using the Trachtenberg Method is simple, but in order to improve your speed you need to practice. There are also a few more multiplication tricks and techniques you can use to help improve your accuracy and speed, these can be found on our dedicated “multiplying by 9” page.


Multiplication of 8 – Rules & Example

Number 8 Trachtenberg SystemSimilar to the rule for multiplying by 9, multiplying by 8 involves using a 3-step process.

Detailed instructions can be found on our “Multiplying by 8” page.

The Rule: Multiply by 8

  • Step 1: (First Digit Only): Subtract the right-hand figure of the multiplicand from 10 then double.
  • Step 2: (For Middle Digits Only): Subtract number from 9, then doubleAdd its neighbour.
  • Step 3: (Last Digit, ‘0’): Subtract 2 from its neighbour.

Example: 8 x 789

  8 x 0789
=     6312
---
(10 - 9) x 2             = 2      Step 1
((9 - 8) x 2) + 9        = 11     Step 2a
((9 - 7)) x 2) + 8 + 1   = 13     Step 2b
(7 - 2) + 1              = 7      Step 3

Description

Step 1 of our rule: Our first number, 9, is subtracted from 10 to give us 1. Then it is doubled to get 2. This is the first number in our answer.

The middle steps are slightly more complex. We move to the next number of the multiplicand, 8 and apply the rule for step 2 (the step for the middle numbers). We must first subtract 8 from 9 then double the answer to get 2. Next we add its neighbour, 9 to get our result, 11. We record the 1 and carry the 10.

The same rule is applied to the third number in our sequence, a 7. We first subtract it from 9, then double the answer to get 4. Next we add its neighbour, 8, and add the carried 10 from the last step to get 13. We record the 3 and carry the 10.

The final digit in our multiplicand is 0, and the last digit has its own rule, this is to simply subtract 2 from its neighbour (and add any carried digits). Its neighbour is a 7, so we subtract 2 from it to get 5 then add the carried 10 from our last step to get the final digit in our answer, 6.

Thus, 8 x 789 = 6312

~~~

Multiplying by 8 using the Trachtenberg Method is simple once the rules are memorized, but in order to improve your speed you need to practice. There are also a number of tricks and techniques you can use to help improve your accuracy and speed, these can be found on our dedicated “multiplying by 8” page.


Multiplication of 4 – Rules & Example

Trachtenberg System - Number 4The rule for multiplying by 4 involves using 3 steps just like the rules for number 8 & 9.

Full instructions can be found on the “Multiplying by 4” page but here is an overview.

The Rule for multiplying by 4:

  • Step 1 (for the first digit): Subtract from 10; add 5 if number is odd
  • Step 2 (middle digits): Subtract number from 9; add 5 if digit is odd; add half the neighbour.
  • Step 3 (last digit, 0): half the neighbour; minus 1.

Example: 4 x 9385

  4 x 09385
=     37940
---
(10 - 5) + 5           = 10     Step 1
(9 - 8) + 2 + 1        = 4      Step 2a
(9 - 3) + 5 + 4        = 15     Step 2b
(9 - 9) + 5 + 1 + 1    = 7      Step 2c
(0.5 x 9) - 1          = 3      Step 3

Description

Step 1 of our rule: Our first number, 5, is subtracted from 10 to give us 5. Then it we add 5 to get 10. This is the first number in our answer. We record the 0 and remember to carry the 10.

The middle steps in our sequence require that we subtract the number from 9, add 5 to the number if the number is odd, then add half its neighbour and carry any remaining 10’s from our previous answer.

The final digit in our multiplicand is 0, and the last digit has its own rule; the rule is to halve its neighbour (9) to get 4, and minus 1, to get our final digit in our answer, 4.

Thus, 4 x 9385 = 37940

~~~

Multiplying by 4 uses many of the same methods and rules as we have seen in previous numbers but applied slightly differently. One remembered it is just as easy to remember as all other numbers. For more information, further worked examples and clarification on points please visit our “multiplying by 4” page.

Multiplication of 3 – Rules & Example

Number 3 Speed System TrachtenbergThe rule for multiplying by 3 is similar to that of multiplying by 8 except that this time we’re only adding half of the neighbour. The rule for number 3 also has 3 steps.

Full instructions can be found on the “Multiplying by 3” page but here is an overview.

The Rule: Multiply by 3

  • Step 1: (First Digit Only): Subtract number from 10 then double.
  • Step 2: (For Middle Digits Only): Subtract number from 9, then double. Add half of its neighbour. Add 5 if the number is odd.
  • Step 3: (Last Digit, ‘0’): Subtract 2 from half of the neighbor.

Example: 3 x 2588

  3 x 2588
=     7764
---
(10 - 8) x 2                   = 4      Step 1
((9 - 8) x 2) + (8 x 0.5)      = 6      Step 2a
((9 - 5) x 2) + (0.5 x 8) + 5  = 17     Step 2b
((9 - 2) x 2) + (0.5 x 5) + 1  = 17     Step 2c
(0.5 x 5) - 2                  = 0      Step 3

Description

Step 1 of our rule: Our first number, 8, is subtracted from 10 to give us 2. Then we double it. This is the first number in our answer.

The middle steps in our sequence require that we subtract the number from 9, multiply the result by 2, add 5 to the number if the number is odd, then add half its neighbour and carry any remaining 10’s from our previous answer.

The final digit in our multiplicand is 0, and the last digit has its own rule; the rule is to halve its neighbour (5) to get 2, and minus 2, to get our final digit in our answer, 0.

Thus, 3 x 2588 = 7764

~~~

Multiplying by 4 uses many of the same methods and rules as we have seen in previous. One remembered it is just as easy to remember as all other numbers. For more information, further worked examples and clarification on points please visit our “multiplying by 3” page.

Multiply Larger Numbers

The Trachtenberg Method isn’t just limited to multiplying by single digits 1-12, once you have learnt the rules for each number you can start using the system to multiply any number by any other number. ie: 498 x 37654 :

498 x    37654
--------------
        301232
       338886-
      150616--
--------------
      18751692

To learn how to do this correctly, download the pdf download the pdf.

Questions, Problems or Comments

If you have any questions, problems, comments or advice to offer others then please leave them in the comments section below. All feedback is welcome and there’s never a silly question!!

3 comments

  1. A fantastic introduction into the Trachtenberg method. I remember my father trying to teach me this when I was younger but didn’t have the interest in it, 20 years later and I’m back!

    Thanks for taking the time to put together the instructions.

    Hannah

  2. Hello “Jakow Trachtenberg”!
    I’m a student Latin Math in secondary school and I need to write a paper. I choose the subject ‘Alternative math’. I found this beautiful system and I very interested in this. For my paper I need ofcourse some mathematical background. Can you help me giving the mathemathical proof behind these rules? Like the multiplication of 6. Why do you need to add 5 to the odd? I need to be able to answer such questions.

    Thanks you in advance and I hope you can help me out!

    Greets

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