The Trachtenberg System of Speed Mathematics
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THE Trachtenberg System of Basic Mathematics is a method of doing high-speed multiplication, division, addition, subtraction and square roots without a calculator ...

There are plenty of these 'Speed-Maths' systems around. But try to use most to do more than a select few 'party pieces' and you find that at best the so called 'miracle method' relies to a great extent on a considerable amount of tedious effort on your part. At worst, the whole thing is just too complicated to be of any practical or frivolous use.

The Trachtenberg method is different. Yes there is some effort required. But it is such an easy and enjoyable system to learn and you get such instant results, that it is no problem to put that effort in to further progress.


The story of the Trachtenberg method starts in Odessa in 1888 with the birth of Jakow Trachtenberg. Graduating with the highest honours from the prestigious Berginstitut of St Petersburg, he quickly rose to Chief Engineer at the equally prestigious Obuschoff shipyards. He was an energetic principled man of many talents. One of his defining features was his pacifism. It was a most active pacifism. For example, when World War I broke out, he set up his own training school to train students to care for the wounded. Then, when revolution swept Russia, he spoke out fearlessly against injustice and brutality. So much so that he became a marked man, and he had to flee to Germany.There he enjoyed a new life in Berlin's intellectual quarter, gaining great kudos as an editor & writer and devised a method of teaching languages that is still used in many schools. He became the leader of the bitter young intellectuals of a city that reminded him of his beloved St Petersburg. With the emergence of fascism, Trachtenberg was once again prepared to speak out. He chose Hitler as his target.

At first, Hitler chose to ignore the esteemed intellectual but as the attacks grew - Trachtenburg was earmarked to be silenced. Again Trachtenberg had to run for his life. This time to Vienna. As war loomed, he wrote 'Das Friedensministerium', (The Ministry of Peace), which brought praise from Roosevelt among others. But he was eventually captured, and ended up in a concentration camp. To keep sane he took to manipulating numbers. There was very little to write with or on and so it was mostly done in his mind. He set himself the task of devising a simplified system for performing hugely complex calculations. He spent all his time honing this system and it brought him comfort from the horrors of the camp. At one stage he was even told he was to be executed. So he worked even harder to finish his system and entrusted his work to a fellow prisoner. The essentials were written on the few scraps of paper he could find and hide. Luckily, his wife managed to bribe the right people and got him transferred to another camp in Leipzig . As the war bombing created chaos he escaped from this camp, but was caught and sent to Triest. But he made his final escape and with his wife escaped to Switzerland.

As he recovered, he perfected his system. Typically he chose to teach disadvantaged children who were doing badly at school. They loved the speed system as for the first time they had a sense of achievement. They could see results. Their other studies improved with their new found confidence. In 1950 he founded the Mathematical Institute in Zurich. It taught children by day and adults in the evening. To this day the Swiss use the Trachtenberg method in all their banks, in most large firms and in their tax department! One of the reasons for this is that in addition to speed, then method gives great accuracy and double checks are quick to make.

Unlike tricks and devices, the Trachtenberg method is a complete system that gives anyone a new confidence in the 'most hated' of school subjects. Essentially it turns Mathematics into a language that we can all enjoy and understand. The book starts with a bang ... Unexpectedly the treasured 'Times Tables' are thrown out. The only short-cut of conventional Maths teaching is taken away from us!. So the book starts by teaching us how to manually multiply any of the sums we learn from the 'tables'. It goes through each number from 2 to 12 giving us a method to multiply with them for each number in turn. It may seem stupid and almost impossible to learn how to multiply 6 by 6 for example until you realise that it is even more odd to use memorised tables to do this. And then the chapter goes on so show how it is just as easy to multiply 6 by 6 as to multiply 6 by 124,535,676. The method really does require no special skills.

Go forth and Multiply

For example, the book starts with Multiplication by 11.

1342 x 11

Believe it or not you can write down the answer to this instantly with no workings out, using the following 3 rules:

1. The last number of the multiplicand (1342) is 2 and you can write this down as the right hand figure of the answer, so far so good!


2. Each successive number of the multiplicand is added to its neighbor at the right. So for 1342 we start by adding 4 to 2 and we can write down the second figure of the answer.


and again for 1342 we can then add 3+4 to give the third


and again for 1342 we can then add 1+3 to give the fourth


And now having added all the numbers the last step

3. The first number of the multiplicand becomes the left-hand number of the answer.

So the answer is


Sometimes you will get a two figure number when you add the numbers. ie 8+8 will give 16 in which case you write down the 6 and carry the 1. Just like in conventional maths. But in the Trachtenberg method you will never have to carry any number higher then a 2 for even the most complex cases.

The book then goes on to Multiplication by all the other numbers from 2 to 12. They all have a set of rules to learn. 11 is the easiest. For example, when multiplying by five, six or seven you have extra steps to learn including 'halving and rounding down' numbers but they are all very easy to pick up with a bit of practice.

After establishing these building blocks the book moves on to multiplication of any size of number by another. This requires a lot more practice but the key point is that you will always be able to do each step, it is just the speed that you build over time.

Adding up

After multiplication, the book looks at adding up columns of numbers and ways of double checking your answer quickly. I personally find this aspect of the book more useful than the multiplication. It is more common to add up a list of figures in daily life than to multiply huge numbers together. And how often have you had to check 3 or 4 times as you get a different answer each time!? With the Tractenberg method it is far easier and you have a true double check, not just another attempt at doing it the same way!

Then it is division and square roots and some rather esoteric algebraic proofs of the methods in the book which are rather more hard work but will interest some readers.

As you learn the techniques, they will become natural and your speed and accuracy will build. If you're anything like me you will end up using this technique mostly for adding up figures but there is always the chance to show off at Christmas when the party tricks come out. You can be sure that the amazing Jakob Trachtenberg would approve - his system was designed to both educate and amaze.


The Trachtenberg Speed System of Basic Mathematics by Jakow Trachtenberg. Can you multiply 5132437201 by 452736502785 in 70 seconds without reaching for your calaulator? Well you could if you used the techniques in this revolutionary book for high-speed multiplication, division, addition, subtraction and square root. £6.39 + P&P £2.75 = £9.14
• Delivery Time 2-4 days.


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