Re: Types of numerals; bases in natlangs.

On Thu, 12 Jan 2006 13:54:18 +0200, John Vertical
> <johnvertical@...> wrote:
>> [snip]
>> If a language does not have base three or base five, why
>> is "fifteen" a common numeral?
> Obviously it's "half-round" in base 10 and "quarter-round" in
> base 20.
I had thought of that; but I wasn't sure that was enough of a reason.

>> If a language has base five or base eight or base ten or
>> base twenty, why is "twelve" a common numeral?
> The majority of the languages surveyed are European, so the
> measure systems of 12 inch in a foot and 12 ounces in a pound
> may have influenced this.
This may be so.

>I can't think of a directly numerical explanation.
Ray Brown explained why the Romans liked to do their fractions in twelfths;
and Mark J. Reed, in an off-list reply Mon Jan 9, said "I would guess that
twelve is probably chosen for its factorability", that is, because it is
divisible by 2, 3, 4, and 6, which was essentially why Ray suggested the
Romans might have liked it.

Ray also told us how the Romans denoted fractions, which was new to me, and
helps explain why planting trees or bushes in a pattern like the spots on
the five-side of a die, is called "quincunx".

>> ObConLang; Human natlang multiplication-and-addition based
>> numeral systems, seem to have bases of seventy-four or less.
>> Would a conlang for a non-human language work well with a base
>> of eighty-four, ninety, ninety-six, one-hundred-eight,
>> one-hundred-twenty, or one-hundred-sixty-eight?
>> What sort of rationale would make this plausible?
> I've contemplated a double-base system, where there's a
> smallish number (say, 5) acting as a base for small numbers,
> and some exponent of it (say, 3125) acting as a base for larger
> numbers that would be "round" in the smaller base.
> For example, basic words would exist for 1, 2, 3, 4, 5, 25,
> 125; and for 625, 1250, 1875, 2500, 3125 and powers of 3125.
> Of course, there's no reason for the upper system to have the
> same amount of basic words if we choose a non-prime base, eg 6,
> for the lower system instead. In this case 3, 4, 12 etc. would
> also work.
> This is already done to a smaller extent in languages that
> shift from base 5 to base 10; the *illion system for naming
> certain powers of 10 could also be seen as a similar(ish)
> structure.
> [snip]
This is also something I intend to include in one of my first conlangs.

First let me digress to natlangs.
As for natlangs, there are several that have two bases, where the larger
base is some multiple -- not necessrily a power -- of the smaller.
"Computer-speak" uses K=2^10 and meg=K^2, often along with base-eight or
base-sixteen; K is not a power of either eight or sixteen, and although meg
is the fifth power of sixteen, it is not a power of eight.
The Mayan numeral system used base five for numbers up to twenty, and base
twenty thereafter.  So, two-hundred-sixty would have been expressed, by
them, as something like "(two fives and three) of twenties".
Our own hours, minutes, and seconds notation for time could be viewed as
fundamentally base ten for numbers up to sixty, and base sixty thereafter.
If I remember correctly, I have read of natlangs with two (or more) bases
in which a larger base is _not_ a multiple of a smaller base.

I was inspired by words like decillion and centillion and so on to wonder
why English's and other Standard Averge European languages' systems use ten-
to-the-sixth instead of ten-to-the-tenth.  If the base of the system is
ten, it would seem that the exponents in powers of the base should also be
expressed in base-ten.  I wondered why there are words for thousand and
million, instead of words for hundred (10^2) and lakh (10^5)and 10^10,
whatever that is, and so on up to googol (10^100).

I thought of systems like the following:
If the "first base" of the system is B, the system will have B bases.
The lowest will be B, used for numbers up to B^B.
The next will be B^B, used for numbers up to B^(B^2).
The next will be B^(B^2), used for numbers up to B^(B^3).
...
... and so on ...
...
until;
The Bth base will be B^(B^(B-1)), used for numbers up to B^(B^B).

Example for B=four;
Base four up until two-hundred-fifty-six (4^4).
Base two-hundred-fifty-six up until 4,294,967,296 (4^16).
Base 4,294,967,296 up until 4^64 (about 3.40282E+38).
Base 4^64 up until 4^256 (about 1.3408E+154).

As you can see, this system can go very high even for small "first bases".
In fact, even for base-three, the bases would already be 3, 27, and 19,683,
and could express any number less than 7,625,597,484,987 = 3^(3^3).  Since
nobody in Europe needed to count beyond one million before the Crusades, I
don't think any "primitive" conculture is going to neeed to count higher
than 3^27.

For B=five, as you propose, the first four bases will be
5,
3,125 = 5^5,
5^25 = about 2.98023223876953E+17,
5^125 = about 2.35098870164458E+87;
and then 5^625, whatever that is.

For my own conlang, with "first base" = twelve, the bases will be:
12;
12^12 which is 8,916,100,448,256;
12^144 which is about 2.52405858452707E+155;
12^1,728;
12^20,736;
12^248,832;
12^2,985,984;
12^35,831,808;
12^429,981,696;
12^5,159,780,352;
12^61,917,364,224;
12^743,008,370,688.
This will express any positive integer less than 12^8,916,100,448,256.

-----

Tom H.C. in MI

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