Judging by the comments I've received on the blog there's a good amount of interest in techniques for simplifying and speeding up basic arithmetic. This is great because I enjoy learning and writing about these techniques.
Jakow Trachtenberg was a Russian Jewish mathematician who, while imprisoned in a Nazi concentration camp during World War II, developed a system of speed mathematics, no doubt to help preserve his sanity. The Trachtenberg system is particularly good at allowing one to multiply big numbers by small numbers although it teaches a number of other techniques as well.
Amazon and other online booksellers have multiple editions available, at quite different prices, so shop around if you want to own a copy of the book. Here's one edition.
As a gentle introduction to the Trachtenberg system I'll demonstrate how to multiply any number by 12. Trachtenberg has the notion of neighbor, which is the digit to the right of the digit you're applying a technique to. Also, when multiplying with Trachtenberg we move from right to left and keep track of carries just as we do with the approach to multiplication most of us are familiar with.
The basic recipe for multiplying by 12 is to double the digit and add its neighbor.
Let's multiply 34 by 12.
We start with the right-most digit, 4. We double it and would add the neighbor to the right if there were one, so we just double 4 and get 8. Write down the 8 as the right-most digit of the answer.
We move over to the next digit, 3. Double the 3 to get 6 then add the neighbor, 4, to get 10. Write down the 0 from 10 and carry the 1.
We're done with the digits but we have to move to the left one final time. We double the non-existent digit, which we'll call 0, add the neighbor, 3, and add the carry, 1. So, we write down 4 as the final digit.
We're done. We wrote down, from right to left: 8-0-4. Our answer is 408.
Let's try another example: 346x12
Start with the right-most digit, 6. Double the 6 and add the neighbor (none in this case). We get 12. Write down 2 from 12, and carry the 1.
Move left to the next digit, 4. Double the 4 to get 8, add the neighbor, 6, to get 14, and add the carry, to get 15. Write down the 5, and carry the 1.
Move left to the next digit, 3. Double the 3 to get 6. Add the neighbor, 4, and we get 10. Add the carry, to get 11. Write down the 1, and carry the 1.
Move left to the non-existent digit. Double it to get 0, and add the neighbor, 3, which gives us 3. Finally, add the carry to get 4. Write down 4.
Our answer is 4152.
One final example. Let's calculate 123456x12.
First digit is 6. Double 6 plus no neighbor = 12. Write down 2 and carry 1.
Next digit is 5. Double 5 plus neighbor 6 plus carry 1 = 17. Write down 7 and carry 1.
Next digit is 4. Double 4 plus neighbor 5 plus carry 1 = 14. Write down 4 and carry 1.
Next digit is 3. Double 3 plus neighbor 4 plus carry 1 = 11. Write down 1 and carry 1.
Next digit is 2. Double 2 plus neighbor 3 plus carry 1 = 8. Write down 8 and no carry.
Next digit is 1. Double 1 plus neighbor 2 plus no carry = 4. Write down 4 and no carry.
Next digit doesn't exist. We double 0 plus neighbor 1 plus no carry = 1. Write down 1.
Our answer is 1481472.
Pretty cool, huh? There are some nice features of Trachtenberg:
- For techniques like this one you can write the answer without writing down and adding partial results.
- With practice you can do this kind of arithmetic very quickly.
- The techniques build self-esteem in students who are not comfortable with Math
- The techniques build visualization skills as partial results are stored in one's head.
- One learns to concentrate better with practice.