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Topic: The Versatility of Numbers--Part II
Replies: 1   Last Post: Jun 14, 1996 1:24 AM

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Bill Taylor

Posts: 1,909
Registered: 12/8/04
Re: The Versatility of Numbers--Part II
Posted: Jun 14, 1996 1:24 AM
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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, six-packs,
|> twelve-packs, 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 slide-rule 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 16-base 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 milliony-billiony numbers to have a
manageable length, (unlike in base 2).

-------

Secondly, the quick-factor 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 well-known!?
===================================
-------

Now here's something else! We still limp on with the remnant of the
Babylonian base-60 system; GOK how this was foisted onto us!! (I think we
could probably blame Regiomontanus for this.) Of course, it still has the
useful small-partitions 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 base-60 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 log-step further, and use a base-120 system!!

Why?

Well - it would have quick factor-tests 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 !!

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Don't sweat the petty things, just pet the sweaty things.
-------------------------------------------------------------------------------
Bill Taylor wft@math.canterbury.ac.nz
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If this post seems fishy, then put it in your krillfile.
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