In many emails I receive from blog readers, I am often asked for advice on techniques to simplify and accelerate the learning of basic arithmetic.
How to Multiply Fast – The Trachtenberg Method
Well, I like to learn and write about these techniques too! Thus, researching a bit on the internet, I discovered the so-called ” Trachtenberg method “. Jakow Trachtenberg was a Ukrainian mathematician who, despite being imprisoned in a Nazi concentration camp during World War II, managed to develop a mathematical system based on learning speed, no doubt also to help preserve his sanity in that hell.
The Trachtenberg method is an extremely effective calculation system and allows for multiplication of a certain size using small numbers (although its theories have a much broader approach). For those wishing to buy it, the book that explains his mathematical theories is available on Amazon .
The core of the Trachtenberg method demonstrates how it is possible to easily multiply any number by 12 . The mathematician, to explain this theory, uses the term ” next “ , ie the figure to the right of the figure for which the technique is being applied. Now let’s see what it consists of (read also 10 Mathematics Tips and Tricks ).
The basic theory for multiplying a number by 12 is to double the digit and add its neighbor to it.
Example 1
For example, let’s multiply 34 x 12
Let’s start with the rightmost digit, ie 4. After doubling it (4 + 4), we need to add its neighbor on the right (if any). So we will have double 4, ie 8. We write the 8 as the rightmost digit of the answer.
Now let’s move to the next digit, that is 3. Let’s double it to get 6 and then add its neighbor (4) to get 10. We write the 0 of 10 and report the 1.
We’re done with the figures, but now we have to go left one last time. We have double the non-existent digit, which we will call 0, add its neighbor (3) and add the carry (1). Hence, we write 4 as the final digit.
Done! We wrote at the bottom, from right to left: 8-0-4. So, our final answer is 408 .
Example 2
Now let’s take another example: let’s calculate 346 x 12
Let’s start with the rightmost digit, ie 6. Double it and add the next one (none in this case). We get 12. We only write the 2 of the 12 and report the 1.
Now let’s move to the left for the next digit, that is 4. Double it to get 8, now add the next (6) to get 14, and add the carry to get 15. Write down the 5 and return the 1.
Let’s move to the left for the next digit, ie 3. Double it to get 6. We add its neighbor (4) and we get 10. Now add the carry to get 11. Note the 1 and report the other 1.
Let’s always move to the left for the “nonexistent figure”. Double it to get 0 and add the neighbor (3), which gives us 3. Finally, add the carry to get 4. Note the 4.
So, our answer is 4152 .
Example 3
Let’s take a final example, increasing the difficulty coefficient with a larger number.
We calculate 123456 x 12.
The first digit is 6. Double 6 + no neighbor = 12. Write down the 2 and bring back the 1.
The next digit is 5. Double 5 + his next 6 + carry 1 = 17. Write down the 7 and bring back 1.
The next digit is 4. Double 4 + next 5 + carry 1 = 14. Write down 4 and carry over 1.
The next digit is 3. Double 3 + next 4 + carry 1 = 11. Write 1 and bring back 1.
The next digit is 2. Double 2 + his next 3 + carry over 1 = 8. Write down 8 and bring nothing back.
The next digit is 1. Double 1 + the next 2 + (no carry) = 4. Note 4 and do not report anything.
The next digit does not exist. We have double 0 + its neighbor 1 + (no carry) = 1. Note 1.
So 123456 x 12 = 1481472.
Really interesting this math trick, don’t you think? We can therefore say that its most interesting features are:
- allows you to write the answer without writing down and without adding partial results;
- with practice, you can do this type of arithmetic very quickly;
- it contributes to improving the self-esteem of all students, especially those who are notoriously not fond of mathematics;
- improves the ability to view partial results, which are stored more easily;
- the practice will help and improve the ability to concentrate
