Return-Path: <@FINHUTC.HUT.FI:LOJBAN@CUVMB.BITNET>
Received: from FINHUTC.hut.fi by xiron.pc.helsinki.fi  with smtp
	(Linux Smail3.1.28.1 #1) id m0rmooF-00007ZC; Thu, 9 Mar 95 22:27 EET
Message-Id: <m0rmooF-00007ZC@xiron.pc.helsinki.fi>
Received: from FINHUTC.HUT.FI by FINHUTC.hut.fi (IBM VM SMTP V2R2)
   with BSMTP id 9866; Thu, 09 Mar 95 22:27:43 EET
Received: from SEARN.SUNET.SE (NJE origin MAILER@SEARN) by FINHUTC.HUT.FI (LMail V1.1d/1.7f) with BSMTP id 9863; Thu, 9 Mar 1995 22:27:36 +0200
Received: from SEARN.SUNET.SE (NJE origin LISTSERV@SEARN) by SEARN.SUNET.SE (LMail V1.2a/1.8a) with BSMTP id 9956; Thu, 9 Mar 1995 21:23:35 +0100
Date:         Thu, 9 Mar 1995 14:38:56 EST
Reply-To:     jorge@PHYAST.PITT.EDU
Sender:       Lojban list <LOJBAN%CUVMB.bitnet@FINHUTC.hut.fi>
From:         jorge@PHYAST.PITT.EDU
Subject:      Numbers
To:         lojban@cuvmb.cc.columbia.edu
Content-Length: 2818
Lines: 76

The parser accepts any string of PAs as a number, but not all
combinations are meaningful (at least to me). Here is an attempt
to describe which are the meaningful combinations, written in
bnf-ish notation. (I believe this mostly agrees with the grammar
paper, with the exception of my treatment of {ji'i}.)

(The parser also accepts letterals as parts of numbers. I've
ignored them here.)


<digit> = no|pa|re|ci|vo|mu|xa|ze|bi|so|dau|fei|gai|jau|rei|vai|ki'o

<natural> = <digit> ... | xo | no'o

<sign> = (su'o|su'e|me'i|za'u) & (ma'u|ni'u)

<real0> = <natural> & (pi [<natural>] & (ra'e <natural>)) | pai | te'o | ci'i

<real1> = <sign> & <real0> & (ji'i <real0>)

<complex> = <real1> & (ka'o <real1>)

<fraction> = <complex> & (fi'u <complex>)

<quantifier> = (su'o|su'e|me'i|za'u|da'a) &
               (ro|so'a|so'e|so'i|so'o|so'u|rau|mo'a|du'e|ci'i|<natural>)

<fractionator> = pi <quantifier>

<percentage> = (<real1> | <quantifier>) ce'i

<general> = <fraction>|<quantifier>|<fractionator>|<percentage>|tu'o

<number> = <general> & (pi'e [<general>]) ...


Notes:

1- {ki'o} is a special digit. There has to be a number of digits
   multiple of three between {ki'o}s or between {ki'o} and {pi}.
   If there aren't three explicit digits then 0s are assumed
   implicitly as the higher order digits.
   If ki'o is the first digit, a 1 is assumed in front.
   e.g.  ki'ore = 1002  ; piciki'o = 0.003
   (I may be wrong about this convention, but there is some
   convention or other.)

2- Either {fi'u} or {ka'o} has to have higher scope than the other,
   otherwise {1fi'u2ka'o3} would be ambiguous between .5+3i and
   1/(2+3i).
   I prefer to give fi'u higher scope, because that allows
   {fi'u <complex>} to be the inverse of <complex>. The other
   possibility would not allow an easy way to express inverses,
   and things like "2/3 + i4/5" are not really as important as
   inverses.

3- My interpretation of {ji'i} allows to say everything that you can
   say with the one proposed in the grammar paper, and more.
   With my interpretation <number>ji'i<number> means a number
   between those two, or approximately that. So 20ji'i30 would
   be a number between 20 and 30, but could eventually be 19 or 31,
   it is approximate, and the difference between the numbers gives
   an idea of the uncertainty.

   With the interpretation of the paper, 20ji'i30 would be a number
   between 2010 and 2099, or something like that. To say that with
   my interpretation, I would say 2050ji'i or ji'i2050. The
   uncertainty is given by the last significant (non-zero) digit.
   {ji'i} would only say that the total number is not exact, not
   a particular digit. (The ji'i+ and ji'i- convention for rounding
   could still be kept.)


Comments and critics about all this are most welcome and solicited.

Jorge