Dear calculator friends,

What do I see on internet about Mental Calculation and comparable activities?
Heaps of longer and shorter videos which have without exception, one thing
common: 100% sensation and 0% information.

I do not want to say anything negative about all these remarkable performances,
but I fear that non of these does any positive contribution to stimulate mental
calculation, possibly more they frighten the audience.

That's why I plead for visiting shools etc and give explanation about what we do
so that people can leanr from us. That we can give insight in structures etc
etc.

Reently I gave a colloquium Mathematics on the Technical University of
Eindhoven, in September I'll go to the Utrecht University and I hope also to get
the opportunity to speak on secundary schools.

Hereunder you find the report of the TUe.

Regards,

Willem Bouman

The presentation by Mental Calculator Willem Bouman,

June 23, 2009, General Mathematics Colloquium, Eindhoven University of
Technology


Mister Willem Bouman, well known mental calculator, gave a short demonstration
of his mental arithmetic abilities, and subsequently gave an elaborate
explanation of his techniques.

The demonstration of about 10 minutes consisted of:
- squares of numbers below 1000,
- cubes of numbers below 100,
- multiplication of 6 digit numbers,
- exact square root of 12 digit numbers,
- exact cube root of 18 digit numbers,
- approximate square root of a 6 digit number,
- exact division of 12 digit numbers by 6 digit numbers.

The demonstration was not completely without errors, because Bouman preferred to
spend his time on explanations, and therefore did not pause to check his
results.

The explanation started with the "cross method", a technique for smaller
multiplications that avoids having to write down intermediate results. This
method was explained for 2×2, 3×3 and 4×4 digits.
Bouman himself prefers to work with blocks of two digits, and multiplies up to 8
digit numbers with this technique.

Next, modular arithmetic was discussed, with the following moduli:
- 9, for checking multiplications,
- 11, as an improvement of this,
- 33, for computing cube roots,
- 37, for computing and checking divisions.

For example, when computing a cube root 37 cannot be used as modulus, since 1³,
10³ and 26³ all are 1 (mod 37). And 101 cannot be used for 5th powers, as 6, 14,
17, 65 and 100 to the 5th power all are 100 (mod 101).

Structures in squares were discussed: the "quadrant logic", partitioning the
squares below 100 into groups of 4 with equal last digits. For example, 7², 43²,
57² and 93² all end in 49. Similarly there is an "octant logic": 7², 243², 257²,
493², 507², 743², 757² and 993² all end in 049.

To compute cube roots he uses a combination of Newton's method and modular
arithmetic.

Willem Bouman easily filled an hour, entertaining and inspiring his audience. He
speaks about mental arithmetic with knowledge and authority, and with visible
enthusiasm and humour.

Dr. B.M.M. de Weger

General Mathematics Colloquium Committee



----------

The presentation by

Mental Calculator Willem Bouman,

June 23, 2009,

General Mathematics Colloquium,

Eindhoven University of Technology





Mister Willem Bouman, well known mental calculator, gave a short demonstration
of his mental arithmetic abilities, and subsequently gave an elaborate
explanation of his techniques.

The demonstration of about 10 minutes consisted of:
- squares of numbers below 1000,
- cubes of numbers below 100,
- multiplication of 6 digit numbers,
- exact square root of 12 digit numbers,
- exact cube root of 18 digit numbers,
- approximate square root of a 6 digit number,
- exact division of 12 digit numbers by 6 digit numbers.

The demonstration was not completely without errors, because Bouman preferred to
spend his time on explanations, and therefore did not pause to check his
results.

The explanation started with the “cross method”, a technique for smaller
multiplications that avoids having to write down intermediate results. This
method was explained for 2×2, 3×3 and 4×4 digits.
Bouman himself prefers to work with blocks of two digits, and multiplies up to 8
digit numbers with this technique.

Next, modular arithmetic was discussed, with the following moduli:
- 9, for checking multiplications,
- 11, as an improvement of this,
- 33, for computing cube roots,
- 37, for computing and checking divisions.

For example, when computing a cube root 37 cannot be used as modulus, since 1³,
10³ and 26³ all are 1 (mod 37). And 101 cannot be
used for 5th powers, as 6, 14, 17, 65 and 100 to the 5th power all are 100 (mod
101).

Structures in squares were discussed: the “quadrant logic”, partitioning the
squares below 100 into groups of 4 with equal last digits. For example, 7², 43²,
57² and 93² all end in 49. Similarly there is an “octant logic”: 7², 243², 257²,
493², 507², 743², 757² and 993² all end in 049.

To compute cube roots he uses a combination of Newton’s method and modular
arithmetic.

Willem Bouman easily filled an hour, entertaining and inspiring his audience. He
speaks about mental arithmetic with knowledge and authority, and with visible
enthusiasm and humour.

Dr. B.M.M. de Weger

General Mathematics Colloquium Committee




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