Tag Archives: Speed Mathematics

Speed Mathematics using Trachtenberg System

16 Apr

It is surprising what the human mind can achieve when it is pushed to extreme conditions. One such thing happened when a Russian Jew Jakow Trachtenberg evolved his own algorithms to do calculations speedily when he was incarcerated in the Nazi concentration camps.

Multiplication Algorithms:

2 Finger Method:

Suppose we have two numbers 120 and 131 and we want to multiply them quickly without the use of a calculator. The following figure will better illustrate what will happen:


Illustration of the 2 finger method.

Illustration of the 2 finger method.


The vertical arrow points to the product where we will get the units digit, and the sloping arrow points to the product where we will get the tens digits of the product pair. If an arrow points to a space with no digit there is no calculation for that arrow. As you solve for each digit you will move each of the arrows over the multiplicand one digit to the left until all of the arrows point to prefixed zeros.


Likewise, Trachtenberg also introduced a number of special algorithms to multiply any number by 3,4,5 and so on up till 13. For example suppose that we wanted to multiply a number by 11 then we’ll add every number to its neighbor i.e the number on it’s right side:

For example 11×37015 = 407165

3  7   0   1   5

(3+0)+1(carry from next)     (3+7)  (7+0)   (1+0)  (1+5)  (5+0)

= 4                                       0          7          1         6         5

Division Method:

The division method is pretty much the same as general multiplication method except that you have to subtract instead of adding. The division will be done in the same digit to digit way.

Addition Method:

The addition method is the usual one of writing the numbers one above the other and adding the columns together. The carry is taken over to the next digit which is something taught to us from the very beginning.


After the end of World War 2, Trachtenberg who survived the camps started to teach this method to students who did not have confidence in their abilities. It has been proven that the system can shorten calculation time by 20 percent and the results are accurate 99 percent of the time.