Re: Real Number --- Merely numbers whose digits can be infinitely long

On Mon, 2024-04-29 at 11:57 +0100, Ben Bacarisse wrote:

wij <wyniijj5@gmail.com> writes:
 

The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.
 
+-------------+

Real Number | ('computational' may be added to modify terms used in this file

+-------------+   if needed)
 
n-ary Fixed-Point Number::= Number represented by a string of digits, the
   string may contain a minus sign or a point:
 
     <fixed_point_number>::= [-] <wnum> [ . <frac> ]
     <wnum>::= 0 | <nzd> { 0, <nzd> }

 
Some readers will be confused by this.  When {}s are used for "zero or
more repetitions" what goes inside is the same kind of thing that goes
on the RHS of ::= and there (in production rules) you use | for
alternation.  Hence
 
  <wnum> ::= 0 | { 0 | <nzd> }
 
would be clearer.  Of course, if you want "," to denote alternation, you
could write
 
  <wnum> ::= 0, { 0, <nzd> }
 
instead.  It's the mixing up of the notation that's pointless and
potentially confusing.
 

     <frac>::= { 0, <nzd> } <nzd>
     <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary

 
Since you already have a notation for alternatives, it would be clearer
to write
 
  <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
 
otherwise, you need to say what that the brackets with commas mean.
 

     Ex: 78, -12.345, 3.1414159...(π)

 
The third example does not match the grammar.  Also, I would guess you
intended to use the digits of pi.
 

   Addition/subtraction of n-ary fixed-point numbers are the same as what is
   taught in elementary schools (or on abacus). Any two n-ary fixed-point
   number (same n-ary) a,b are equal iff their <fixed_point_number>
   representation are identical. Otherwise, a>b or a<b, exactly one of these
   two holds (Law of trichotomy).

 
So which of -0>0 and -0<0 holds?  I'll take what you write about limits
with a pinch of salt until the basics are clear.
 

Got your idea. I'll try use '|' exclusively. Thanks for the suggestions:

     <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
     <wnum>::= 0
     <wnum>::= <nzd> { 0 | <nzd> }
     <frac>::= { 0 | <nzd> } <nzd>
     <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary

    Ex: 78, -12.345, 3.1414159