From: L' Ermit (lhermit@hotmail.com)
Date: Tue Jan 08 2002 - 22:13:28 MST
ben said:
I'm a fairly lacking mathematician, but here's my 2 cents worth:
Base 16 (hex) is nearly just as easy to picture mentally, and you can count
on your fingers in binary with a little practise (with a limit of 2047, not
10!) Base16 converts easily to binary (and vice-versa), which is why it's
used in programming.
<snip>
Hermit comments:
/me partially agrees with ben - only it is 1023, not 2047, unless you have
11 fingers.
Explanation: You can calculate the maximum value of any number of binary
digits as 2<sup>n</sup>-1 (because 0 is not a value but takes one
permutation to store) - in this case 2<sup>10</sup> - 1 ie 1024-1 = 1023.
Imagine that each finger is a 1 - like this: 512+256 + 128+64+32+16 +
8+4+2+1 [<- equal to all ten fingers at work] and sum the result to validate
it.
Once one has learnt the bit patterns for one nibble (i.e. 4 bits) there are
a <em>lot</em> of binary shortcuts available which make "finger arithmetic"
quite easy and very fast (e.g. subtraction by adding complements and
multiplication and division by shifting and summing).
Another trick, Chinese I suspect, is to use base five, and treat your hands
as an abacus, one hand and the thumb of the other providing the exponent and
the other the mantissa. e.g.
Left hand
160 80 40 20 10
Right thumb
5
Balance of right hand
4 3 2 1
This allows faster and simpler conversions to denary, the use of all the
abacus tricks (and there are thousands) and much easier "counting
arithmetic" than binary does- and allows you to count to 325 using all your
fingers. With a little practice, you can learn to "set" and "read the
result" by inspection.
However, if you learn to visualize numbers, mental arithmetic can be faster
yet, as thoughts move faster than fingers. I also suspect that base 12 and
base 60 actually make more sense from all perspectives except that of the
binary or 8 bit worlds, due to the very much higher number of prime factors
(which is why the Babylonians used base 60). What a pity we were not born
with six fingers on either hand.
Despite, or possibly because of, all the above, I would suggest that trying
to "unify" letters and numbers back together again would be a mistake - as,
in all cultures I know of, it took dropping this "unified" methodology and
the development of a positional system before advanced numeracy could
develop. Why throw away 800 to 2000 years of progress when there is no
obvious benefit to doing this?
Regards
Hermit
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