The Art and Science of Learning Craft  



Mental Calculators

Professor Aitken and Mr. Klein use very similar methods in their mental computation. Their speeds are quite comparable, and they are at least as fast as, and probably faster than, most of the prodigies, past and present. 

They know by heart the multiplication tables up to 100 x 100, all squares up to 1000 x 1000, And an enormous number of odd facts, such as 3937 x 127 = 499999, which are very useful to them, and seem to arise instantaneously in their minds when they are needed.

In addition Mr. Klein knows by heart the logarithms of all numbers less than 100, and of all prime numbers less than 10,000 (to twenty decimal places), So that he can work out sums like compound interest by "looking up" the logs in his head, after factorizing the numbers he is using, if need be. He learnt most of the Amsterdam telephone directory for fun.

Professor Aitken has neglected logarithms in favor of mathematical formulae, but nevertheless he learnt 802 places of p by heart in about fifteen minutes.

An example of Professor Aitkin's methods.

While he was mentally evaluating the square root of 567,which he finally checked by comparing his answer with 9V7.
His method is based upon the fact that if a is a first approximation to Vn then ½(a + n/a) is considerably closer. His calculations are facilitated by his astonishing familiarity with tables of reciprocals.

He noted 24 as a first approximation, and 23.8125 as a second (23.8125 = ½(24 + 567/24)). At the same moment he recalled that 1000/42 = 23.809523 the digits of which are close to 23.8125.

He performed 567 x 42 = 23,814 almost before he had thought what he was doing. Averaging 23.809523 ... and 23.8140 he had as a third approximation 23.81176190476. He recalled simultaneously that 1/84 = 0.0117619047619 ...

Finally he registered 23.811762 as the square root he wanted, and in extraordinarily less time than it takes to describe it, perhaps in three seconds at most.


Willem Klein, also known as Wim Klein or under his stage names Pascal and Willy Wortel, was a Dutch mathematician, famous for being able to carry out very complicated calculations in his head very fast. On 27 August 1976, he calculated the 73rd root of a 500-digit number in 2 minutes and 43 seconds. This feat was recorded by the Guinness Book of Records.

He worked in circuses in France and the Netherlands until he was hired by CERN (European Organization for Nuclear Research)in 1958 as calculator,a programmer and numerical analyst. In the early days Klein was in considerable demand at CERN, as"Computers were not very well developed, and the physicists did not yet program them themselves.

Jeremy Bernstein described, of CERN in Geneva, described, Klein "He must be one of the fastest human computers who has ever lived."

Mr. Klein multiplied

1388978361 x 5645418496 = 7841364129733165056

completely in his head, a calculation which involved twenty-five multiplications each of two two-digit numbers and twenty-four additions of four-digit numbers - forty-nine operations in all, in sixty-four seconds.

In June 1974, shortly before his departure from CERN, Klein became intrigued with the problem of extracting integer roots of large numbers. The 1974 edition of the Guinness Book of World Records reported that Herbert B. de Grote of Mexico City had extracted the 13th root of a hundred-digit number in 23 minutes.

Klein says: "What is the use of extracting the 13th root of 100 digits? Must be a bloody idiot you say. No. It puts you in the Guinness Book, of course.

I never came on the idea until I got this notice about this man in Mexico. I thought, hey, how interesting. I should have thought of that. First I had to find out how to tackle the problem. Then I needed material - I needed numbers raised to the wanted power. So they wrote a multi-precision program on the computer. And I was practicing like hell, like hell, like hell. Once you know the system for the first one, you have to learn another series of numbers by heart for the next one."

By October 8, 1974, Klein succeeded in extracting the 23rd root of a 200-digit number in 18 minutes, 7 seconds, and on March 5, 1975, in Lyon, he reduced the time to 10 minutes, 32 seconds.

Later, Klein went on to extract a variety of roots: the 19th root of 133 digits (1 m. 43 sec.), the seventh root of 63 digits (8 m. 27 sec.), the 73d root of 500 digits in 2 minutes time.

The ability to perform mentally comes from complete learning.

We all learned. by heart, multiplication tables up to 15 or twenty in our child hood. If asked to multiply 7 x 7 or 12 x 4 off hand, when we are not prepared, we can answer.

We can answer instantly, with out consciously thinking and will  never make mistakes. Because any thing completely learned is sublimated (entrusted to right brain), doesn’t require conscious effort to retrieve and is infallible. ( explained in detail elsewhere).

Though, we completely learned (committed to subliminal memory) only 10 tables or so, we could even mentally calculate 2 two digit numbers.

It is no surprise Aitken and Klein who learned multiplication tables up to 100 x 100 could mentally calculate two 3 or 5 digit multiplications in their head.

Thus the secret of mental calculation doesn’t lie in devising shortcuts; for, while using short cuts too, we are solving the problems consciously, a process which is always slow and fallible.

The trick is in complete learning. Enabling your mind to solve the problems. It will find its own short cuts. This is what mental math's is.

Only memorizing is required in mental calculations i.e. multiplications divisions etc.To be able to solve mentally, problems such as averages , percentages, trigonometry and applying  formulae, mere memorizing do not suffice.

As procedural memory is used while solving these problems, to be able to solve them mentally ‘Automaticity’ is to be invoked.