Numbers in Qthen|gai (and in Tyl Sjok) [long]

Hi!

I'd like to give the promised introduction to the number system of
Qthen|gai.

Basic Grammar of Numbers
------------------------
Numbers are suffixed in Qthen|gai.

There are two basic types of number suffixes:

   a) grammaticalised number suffixes

      Like singular and plural in English, but there are a lot
      more of them.  These can be attached to any word.
      So this works as follows:

         person + <trial>      = 3 persons
         person + <known>      = a known number of persons
         person + <nullar>     = no persons
         person + <plural>     = many persons
         person + <paucal>     = some persons
         etc.

      Number suffixes can be applied more than once:

         person + <collective> + <dual> = 2 groups of persons

      For mass nouns, they are interpreted as amounts, not counts,
      i.e., plural then means ,many':

         beer + <known>     = a known amount of beer
         beer + <nullar>    = no beer
         beer + <plural>    = much beer
         beer + <paucal>    = some beer

      Those suffixes that represent exact numbers, namely singular
      (1), dual (2) and trial (3) will count typical amounts, like in
      English: one beer = one glass/bottle of beer:

         beer + <singular>  = one (glass/bottle of) beer

   b) fully structured numbers
      These are words on their own in Qthen|gai.  The structure of
      them is equal to that in Tyl Sjok, which I will introduce
      below.  The difference to a) is that these numbers cannot
      simply be attached to a word.  Instead, they can be suffixed
      to any grammaticalised number suffix, i.e., first you
      attach a grammatical number, then you can specify the exact
      amount.  It is typical to use the grammatical number 'known
      amount' and then add the full number.

      E.g.
         person + <known> + [432]  = 432 persons

      The number in [] has an internal structure defined below.  What
      is important here is that you always need a grammatical number
      suffix to introduce a stream of morphemes representing the exact
      number.  As a grammatical number suffix, you can choose whatever
      you like, as long as it makes sense, e.g. to stress that you are
      counting entities, you could use:

         person + <singular> + [432] = 432 single persons

      or you could count groups:

         person + <collective> + [432] = 432 groups of persons

Structure of Numbers
--------------------
The basic idea I had for Tyl Sjok was to solve two problems:

   1) Different languages use different widths of blocks of digits
      to encode numbers.

      E.g. in English, you have words for 10,100,1000 and then reuse
      the smaller number to form 10000 (ten thousand).  For this
      reason, separators are inserted every three digits (as in
      10,000) to make reading easier.  The larger numbers in English
      are all multiples of 1000.

      In Chinese, Korean and Japanese, however, the major structuring
      uses *four* digits instead of *three* in English.  So there is
      a word for 10 (shi), 100 (bai), 1000 (quan), 10000 (wan), and then
      100000 is encoded as '10 10000' (shi wan).  And 1 million
      is '100 10000' (bai wan).

      Therefore, it is quite hard to translate large numbers from
      Chinese to English and vice versa.

      And there are even more complex systems like Hindi, which uses
      a mixed two and three digit system.

      I wanted to make it reasonably easy to use Tyl Sjok regardly of
      your L1 system.  The only chance I saw was to use the smallest
      denominator, i.e., the largest basic number is *ten* in Tyl
      Sjok.

   2) Like in Chinese, I wanted number bases to be very similar to
      units.

      I.e. 50 = 5 10 (wu shi) in Mandarin.  Here, 10 is the base and 5
      is the coefficient.

      Further, '5 minutes' (wu fen) has exactly the same structure in
      Chinese as 50.  I wanted the same unification for Tyl Sjok.

      The consequence is that there are be no fused number words like
      'fifty', but only 'five' and 'ten'.

   3) The system should be usable for science as well, so very large
      and very small numbers should fit into the system without needing
      changes.

   4) The system should feel appropriate and easy to normal speakers.
      This might collide with 3), of course.

I don't know whether I solved 4), but I think I solved the other three
ones. :-)

So the system I came up with works as this: for each digit of the
number you want to say, use the sequence 'exponent base coefficient'
and join them with the word 'and'.  Any trivial things can be left out
(like coefficient = 0 or exponent = 1).  E.g:

   500 = 2 10 5  in Tyl Sjok (that is 10^2 * 5 = 100 * 5 = 500)
   50  = 10 5    (short for 1 10 5)
   51  = 10 5 and 1
   520 = 2 10 5 and 10 2
   502 = 2 10 5 and 2
   532 = 2 10 5 and 10 3 and 2

The order is 'large exponent before small exponent', like in English,
Chinese and probably many languages (I don't know whether there are
some that *systematically* reverse the whole sequence of digits
(German and others swap two digits: at 10 and 1, but not the whole
number)).

Because this form can become very long and explicit and since the
exponent typically decreases by one in each step, there is a
simplified form where you can give coefficients after you first
defined at what exponent to start.  E.g. instead of
'2 10 5 and 10 3 and 2' you can say:

   532 = 2 10 5 3 2

And
   520 = 2 10 5 2

As you can see here, you need not give all digits at the end if they
are zeros.

You may need zeros now:

   502 = 2 10 5 0 2

If there are too many zeros in a row, you can use a mixed system:

   56,000,023 = 7 10 5 6 and 10 2 3

I hope you are still listening. :-)

Different Bases
---------------

With this system, you need quite a minimal set of basic words for
numbers, namely 0 .. 10 for a base 10 system plus the word 'and',
making 12 basic words.

To add a bit more, Tyl Sjok supports different basis as well.  The
smallest is 2 and the largest native base is 16.

Large Numbers
-------------

For very large numbers, the system is recursively applied.  E.g.

5.000.000.000.000  is  10 1 2 10 5

I.e. the exponent is 12, which is '10 1 2' and then this is put in
front of the base of 10 which is then multiplied by 5.

Units
-----

So that's the basic system.  I will post the incorporation of
units later.  If you are interesting, look at

    http://www.theiling.de/projects/s2/grammar2.ps.gz
or
    http://www.theiling.de/projects/s2/grammar2.pdf

page 75.


Now, Qthen|gai works exactly the same (of course, the number words are
totally different).  You can see some samples at:

    http://www.theiling.de/projects/s7/s_05#06_02

If you are still reading, please make some comments! :-)

**Henrik

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