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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Real Number --- Merely numbers whose digits can be infinitely long
Date: Mon, 29 Apr 2024 04:42:12 +0800
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The purpose this text is for establishing the bases for computational algor=
ithm.
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::=3D Number represented by a string of digits, the
   string may contain a minus sign or a point:

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

     Ex: 78, -12.345, 3.1414159...(=CF=80)

   Addition/subtraction of n-ary fixed-point numbers are the same as what i=
s=20
   taught in elementary schools (or on abacus). Any two n-ary fixed-point=
=20
   number (same n-ary) a,b are equal iff their <fixed_point_number>=20
   representation are identical. Otherwise, a>b or a<b, exactly one of thes=
e
   two holds (Law of trichotomy).

A=3DB ::=3D 1. A=E2=89=A1A                 // A=E2=89=A1B means "strictly i=
somorphic"
        2. A=E2=89=A1B <=3D> A=C2=B1x =E2=89=A1 B=C2=B1x   // The recursion=
 steps must be finite
                               // The meaning of A,B,=C2=B1 is analogous to=
 what in
                               // abstract algebra

   Note: 'Number' is an abstract concept. But, ultimately, the semantics of=
=20
         'abstract' must refer to objective symbols (or model, or something
         concrete). In this view, 'numbers' are  procedures of computation =
of=20
         0's and 1's that correspond to our operations. So, in the end,=20
         'numbers' (or, even mathematics) are mostly likely involved with, =
and
         maybe modeled by, computer programs or algorithms.

Real Nunmber (=E2=84=9D)::=3D {x| x is finitely represented by n-ary <fixed=
_point_number>
   and those that cannot be finitely represented (ref. Appendix)}. =E2=84=
=9D must be=20
   able to provide an algorithm to make the marks of a physical ruler with
   arbitrary precision. In this respect, =E2=84=9D is not entirely pure the=
oretical and
   the basic reason we can describe the universe.

   Note: Numbers that are not finitely representable cannot all be explicit=
ly
         defined because most of those numbers are inexpressible. This is t=
he
         property of real number based on discrete symbols (like quantum?).=
 E.g.

         A=3D lim(n->=E2=88=9E) 1-3/10^n =3D lim 0.999... =3D1
         B=3D lim(n->=E2=88=9E) 1-2/2^n  =3D lim 0.999... =3D1
         C=3D lim(n->=E2=88=9E) 1-1/n    =3D lim 0.999... =3D1
         ...

         IOW, by repeatedly multiplying 0.999... with 10, you can only see =
9,
         the structure of the rear end of 0.999... is not seen. (The proces=
s of
         10*0.999.. changed the structure of the number. For infinite serie=
s,
         this is important, because the number defined by the infinite seri=
es
         may thus have been implicitly modified)

         Since <fixed_point_number> is very definitely real and infinity is
         involved, theories that composed of finite words cannot be too=20
         exclusive about such a =E2=84=9D. 'Completeness' is impossible.

   Note: This definition implies that repeating decimals are irrational num=
ber.
         Let's list a common magic proof in the way as a brief explanation:
           (1) x=3D 0.999...
           (2) 10x=3D 9+x  // 10x=3D 9.999...
           (3) 9x=3D9
           (4) x=3D1
         Ans: There is no axiom or theorem to prove (1) =3D> (2).
              (2) is an interpretation of (1) from infinite possibilities.

   Note: To determine whether a repeating decimal x is rational or not, we
         can repeatedly subtract the repeating pattern p(i) from x.=20
         If x-p(1)-p(2)-...=3D0 can be verified in finite steps, then x is
         rational. Otherwise, x is irrational, because, if x is rational, t=
he
         last remaining piece r(i)=3D x-p(1)-p(2)-... must exactly be the=
=20
         repeating pattern p(i). But, by definition of 'repeating', r(i) ca=
nnot
         be pattern p(i). Therefore, repeating decimal is not rational (.i.=
e.
         irrational).

Theorem: x=E2=88=88=E2=84=9A,x>0 iff there exist finite number of q's,q=E2=
=88=88=E2=84=9A, 0<q<x, such that
         x=3Dq1+q2+...
  Proof: Let prosition A=3Dlhs, proposition B=3Drhs, then we can have a tru=
th table:
      A B
      ----+---
      T T | T  // x=E2=88=88=E2=84=9A,x>0 and, finite q's,...,such that x=
=3Dq1+q2+...
      T F | F  // x=E2=88=88=E2=84=9A,x>0 and, non-finite q's,...,such that=
 x=3Dq1+q2+...
      F T | F  // x=E2=88=89=E2=84=9A,x>0 and, finite q's,...,such that x=
=3Dq1+q2+...
      F F | T  // x=E2=88=89=E2=84=9A,x>0 and, non-finite q's,...,such that=
 x=3Dq1+q2+...

Theorem: The number represented by finite length of n-ary <fixed_pointr_num=
ber>
  is rational, and that by infinite length of n-ary <fixed_pointr_number> i=
s
  irrational.
  Proof: The main condition when the number is an infinitely long fraction =
can
         be proved by using the above theorem.

Real number is just this simple. Limit defines derivative and provides meth=
od
for finding it, nothing to do with what the real number is (otherwise, a=
=20
definition like the above must be defined in advance to avoid circular-reas=
oning).

+-------+
| Limit |
+-------+
Limit::=3D  lim(x->a) f(x)=3DL
    http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04=
..pdf
    http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
    https://en.wikipedia.org/wiki/Limit_(mathematics)
    https://en.wikipedia.org/wiki/Limit_of_a_function
    https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-li=
mits.html
  =20
    The essence of limit is a provision that allowing us to specify a numbe=
r (
    i.e. L) not via equations. Limit says: When x approaches a (x=E2=89=A0a=
), the limit
    of f(x) is L (=CE=B5-=CE=B4 description is satisfied), not "When x appr=
oaches a,=20
    finally, f(a)=3DL".

    Ex1: A=3D lim(n->=E2=88=9E) 1-1/n=3D lim(n->0=E2=81=BA) 1-n=3D lim 0.99=
9...=3D1
         B=3D lim(n->=E2=88=9E) 1+1/n=3D lim(n->0=E2=81=BA) 1+n=3D lim 1.00=
0..?=3D1

    Ex2: A=3Dlim(x->=E2=84=B5=E2=82=80) f(x), B=3Dlim(x->=E2=84=B5=E2=82=81=
) f(x)  // =E2=84=B5=E2=82=80,=E2=84=B5=E2=82=81 being proper or not is
                                         // another issue here. But problem=
atic
                                         // for "finally equal" interpretat=
ion:
                                         // Are f(=E2=84=B5=E2=82=80),f(=E2=
=84=B5=E2=82=81) the same?

    Limit defines A=3DB, does not mean the contents of the limit are equal.=
 If the
    "x approaches..., then equal" logic is adopted, lots of issues arise.

    Note: Limit is defined on existing number system, it cannot be used to
          define the number in the approaching sequence.

    Note: The equation of limit may be questionable
          lim(x->c) (f(x)*g(x))=3D (lim(x->c) f(x))*(lim(x->c) g(x)):
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