The Trachtenberg method of fast addition
The Trachtenberg method of addition is a way to add several numbers quickly and then check, using a different procedure, that we have worked correctly. It was devised by the Russian mathematician Jakow Trachtenberg. We will show the method on an example. Let’s try to add these three numbers:
First, we write the numbers one under another like this:
Now we start adding the columns, beginning on the right. We are interested in whether the sum is greater than or equal to 11. In the first step we add the last column, that is, 1 + 5 + 2 = 8. This result is less than 11, so we just write the number under the line:
In the second step we add the second column from the right, that is, 5 + 7 + 6 = 18. The Trachtenberg method says that whenever the sum is greater than or equal to 11, we subtract 11 from the sum and remember how many times we had to do so. Since 18 is greater than 11, we subtract 18 − 11 = 7. We write the number seven under the line, and because we subtracted 11 once, we note that one row lower:
If we go back to the previous column, the one on the far right, we see that the sum was eight, so we did not have to subtract 11 at all. We therefore fill in a zero in the table:
We continue with the third column from the right and add 5 + 3 + 2 = 10. Don’t be put off by the number ten: it is less than 11, so we write it under the line together with a zero, because we did not have to subtract 11:
And now we add the fourth column from the right, that is, the first column. We see that 9 + 7 = 16, which is greater than 11. So we subtract 16 − 11 = 5, remember that we have already subtracted 11 once, and carry on with the number 5, which we add to the number in the third row, that is, we work out 5 + 9 = 14. Fourteen is also greater than 11, so we subtract 14 − 11 = 3. We write this result under the line together with the information that we had to subtract eleven twice:
Under the line we also add one more column with two zeros, which will come in handy later:
How to get the result with the Trachtenberg method
We draw another horizontal line and add the two rows of intermediate results. First we add 8 + 0 = 8 and write the result under the line:
From now on we have to proceed in a more complicated way and add the numbers in the shape of the letter “L”. So we add the two numbers from the second column from the right and add to them the second number from the previous column, that is, 7 + 1 + 0 = 8:
Next we add 10 + 0 + 1 = 11. Now we proceed as in ordinary addition: we write down only the last digit, 1, and the ten is carried to the next round.
In the next round we add 3 + 2 + 0 + 1 = 6, where the 1 at the end is the one that was carried from the previous round (that is, for each ten that is carried, we add a one here).
And now we finally use the column with two zeros. It is there purely so that we don’t forget to do the last sum, 0 + 0 + 2 = 2.
This is our result, obtained by the Trachtenberg method. So,
Checking the intermediate results
An advantage of the Trachtenberg method is that we can check relatively easily whether we have worked correctly, without doing the whole calculation again. Trachtenberg found a way to check both the intermediate results and the final result by a completely different procedure. So if you work out the sum using the Trachtenberg method and also do the check, you can be fairly sure that you have worked correctly. Let’s try to check that our result is right.
We can split our whole calculation into three parts. We have the numbers being added, the intermediate results and the result:
In the first phase we check whether we have worked out the intermediate results correctly. To do that we have to find the digital root of each column. We get the digital root of a number by adding all the digits of that number. For example, we get the digital root of 456 by adding 4 + 5 + 6 = 15. If the result is greater than 9, we repeat the process, that is, we also work out 1 + 5 = 6. The digital root of 456 is therefore 6.
Now we go back to the numbers being added and work out the digital root of the last column, that is, the digital root of the number 152: we work out 1 + 5 + 2 = 8.
Next we work out the digital root of the second column from the right, that is, of the number 576, which is 5 + 7 + 6 = 18, and then we get 1 + 8 = 9.
For the third column from the right we get 5 + 3 + 2 = 10, and because the number has two digits, we get 1 + 0 = 1:
and for the first column we have 9 + 7 + 9 = 25 and then 2 + 5 = 7:
The resulting digital roots, read from left to right, are 7, 1, 9, 8. Now we look at the intermediate results:
We copy the second row and add it at the end, so the second row appears twice:
Now we do the same as with the numbers being added. From each column we form a number and work out its digital root. Again we can start from the right, so we get 8 + 0 + 0 = 8, the next column gives 7 + 1 + 1 = 9, the next 10 + 0 + 0 = 10 and then 1 + 0 = 1, and finally 3 + 2 + 2 = 7:
At this point we compare the digital roots. The digital roots of the numbers being added gave 7, 1, 9, 8, and so did the digital roots of the intermediate results, so we worked correctly. If these results differed, we would have made a mistake somewhere, either during the check or during the addition itself.
Checking the result
Checking that we have worked out the result correctly is now simple. We work out the digital root of the result and the digital root of the digital roots, and they must agree. So we take our result 26,188 and work out its digital root:
Now we work out the digital root of our calculated digital roots, that is, of the digits 7, 1, 9, 8:
We see that both digital roots are equal to seven, so the check came out well and we worked correctly!
