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Re: The Versatility of NumbersPart II
Posted:
Jun 14, 1996 1:24 AM


First, a quick question. What actually ARE *very* versatile numbers. You defined versatile, but NOT (that I could see) "very" versatile. Help?!
Now:
wlauritzen@aol.com (WLauritzen) writes:
> I believe we've been surrounded by versatile numbers for > so long (months, seconds, minutes, hours, dozens, feet, sixpacks, > twelvepacks, twelve notes, twelve pence, etc.) that we have > forgotten about them. And even if one does discover them, they are > an embarrassment to a society that uses "ten," not a versatile > number, as the core of its number system.
This reminds me of a good old "Punch" joke. A shop had a counter with a pile of "decimal converter" cardboard sliderule thingies on it. The chap at the counter was saying  "I`ll take a dozen, please".
> There are probably several methods of teaching these numbers. First, > one would want the students to know how to factor a number. There > are certain helpful rules that are found in many mathematics > textbooks, or which, I believe could be derived by the students with > proper guidance. Some examples of these rules are: 1) all even > numbers can be divided by two, 2) all numbers that end in 5 or 0 can > be divided by five, 3) all numbers whose digits add to a number > divisible by 3 can be divided by 3. There are other rules, and one > could go into as much depth as one wished.
Well, if you realy want to go off the decimal numbering system, (and aren't enough of a computer freak to want an 8 or 16base system), certainly 12 has great merit. However, I would humbly suggest that 6 is better still !

Firstly, the addition and multiplication tables are MUCH smaller, while the base itself is still big enough for millionybilliony numbers to have a manageable length, (unlike in base 2).

Secondly, the quickfactor rules you mention above are MUCH better in base 6.
In base 10 we have quick rules for 2,3,5,11, (and also 9; 4,8,16 etc and a few more esoteric ones, e.g. 101). But esentially, four distinct small ones.
In base 12 we would have quick rules for 2,3,11,13; also four, but not quite as good it, seems to me.
But in base 6, we would have quick rules for 2,3,5,7. Still just four, but much better! The first four primes, without any gaps, thus more efficient for checking medium sized numbers. Why isn't this fact more wellknown!? =================================== 
Now here's something else! We still limp on with the remnant of the Babylonian base60 system; GOK how this was foisted onto us!! (I think we could probably blame Regiomontanus for this.) Of course, it still has the useful smallpartitions properties that it had for the Babylonians; and occasionally this is even handy! (See e.g. my recent comment on the recurring thread with the triangle with diagonals at altitudes 60 and 70 degrees.)
However this purely geometrical efficiency is hardly enough to disrupt our digital systems. (In fact a "big angle" unit of 15 degrees would be much handier!) Nor should we go to a base60 numeral system; so many tables to learn! But if we WERE to go to such a vast number, I suggest we should go just a logstep further, and use a base120 system!!
Why?
Well  it would have quick factortests for 2,3,5,7 & 11 !!! A WINNER !
However, I must regretfully observe that there would be just too many digits...
So then... LET'S HEAR IT FOR BASE 6 !! LET'S HEAR IT FOR BASE 6 !! LET'S HEAR IT FOR BASE 6 !!
LET'S HEAR IT FOR BASE 6 !! LET'S HEAR IT FOR BASE 6 !! LET'S HEAR IT FOR BASE 6 !!
 Don't sweat the petty things, just pet the sweaty things.  Bill Taylor wft@math.canterbury.ac.nz  If this post seems fishy, then put it in your krillfile. 



