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

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> }
     <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...(π)

   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).

A=B ::= 1. A≡A                 // A≡B means "strictly isomorphic"
        2. A≡B <=> A±x ≡ B±x   // The recursion steps must be finite
                               // The meaning of A,B,± is analogous to what in
                               // abstract algebra

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

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

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

         A= lim(n->∞) 1-3/10^n = lim 0.999... =1
         B= lim(n->∞) 1-2/2^n  = lim 0.999... =1
         C= lim(n->∞) 1-1/n    = lim 0.999... =1
         ...

         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 process of
         10*0.999.. changed the structure of the number. For infinite series,
         this is important, because the number defined by the infinite series
         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
         exclusive about such a ℝ. 'Completeness' is impossible.

   Note: This definition implies that repeating decimals are irrational number.
         Let's list a common magic proof in the way as a brief explanation:
           (1) x= 0.999...
           (2) 10x= 9+x  // 10x= 9.999...
           (3) 9x=9
           (4) x=1
         Ans: There is no axiom or theorem to prove (1) => (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.
         If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
         rational. Otherwise, x is irrational, because, if x is rational, the
         last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
         repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
         be pattern p(i). Therefore, repeating decimal is not rational (.i.e.
         irrational).

Theorem: x∈ℚ,x>0 iff there exist finite number of q's,q∈ℚ, 0<q<x, such that
         x=q1+q2+...
  Proof: Let prosition A=lhs, proposition B=rhs, then we can have a truth table:
      A B
      ----+---
      T T | T  // x∈ℚ,x>0 and, finite q's,...,such that x=q1+q2+...
      T F | F  // x∈ℚ,x>0 and, non-finite q's,...,such that x=q1+q2+...
      F T | F  // x∉ℚ,x>0 and, finite q's,...,such that x=q1+q2+...
      F F | T  // x∉ℚ,x>0 and, non-finite q's,...,such that x=q1+q2+...

Theorem: The number represented by finite length of n-ary <fixed_pointr_number>
  is rational, and that by infinite length of n-ary <fixed_pointr_number> is
  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 method
for finding it, nothing to do with what the real number is (otherwise, a
definition like the above must be defined in advance to avoid circular-reasoning).

+-------+
| Limit |
+-------+
Limit::=  lim(x->a) f(x)=L
    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-limits.html
  
    The essence of limit is a provision that allowing us to specify a number (
    i.e. L) not via equations. Limit says: When x approaches a (x≠a), the limit
    of f(x) is L (ε-δ description is satisfied), not "When x approaches a,
    finally, f(a)=L".

    Ex1: A= lim(n->∞) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
         B= lim(n->∞) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1

    Ex2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x)  // ℵ₀,ℵ₁ being proper or not is
                                         // another issue here. But problematic
                                         // for "finally equal" interpretation:
                                         // Are f(ℵ₀),f(ℵ₁) the same?

    Limit defines A=B, 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))= (lim(x->c) f(x))*(lim(x->c) g(x)):
    
          Let A=lim(n->∞) (1-1/n)= 1
          A*A*..*A= ... = lim(n->∞) (1-1/n)^n    // 1=1/e ?

    Note: Infinitesimal is not small in that roughly every 'list-able' interval
          [x,c) in the approaching sequence of limit might still be 1-1
          corresponding with ℝ.

+--------------------------------------+
| Restoring Interpretation of Calculus |
+--------------------------------------+
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
https://en.wikipedia.org/wiki/Derivative

Assume calculus is basically the area problem of a function: Let F compute the
the area of f. From the meaning of area, we can have:

    (F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2)  // h is a sufficiently small (test)offset
<=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2    // the limit(h->0) of rhs is f(x)

Expected property of F: (1)Error |lhs-rhs| strictly decreases (see Note1) with
    the tiny (test) offset h (2)When h=0, lhs=rhs.
    Because the h in the lhs cannot be 0, the basic problem of calculus is
    finding such a F (or f) that satisfies the expected porperty above...Thus,

    D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)

    Note1: What is in text book is a bit different. Expected property of F just
           needs the number (i.e. limit L) be uniquely defined, others.., don't
           know.
    Note2: Hope that this interpretation can temporarily separate the
           interpretation of infinity/infinitesimal, and provide more correct
           foundation for some other theories.

+-----------------+
| Infinite Series |
+-----------------+
Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
  a(n) is called the general term, a(0),a(1),... the addend, summand or just
  term. n is referred to as the index. Series S is the sum from the first term
  a(0) to the last term a(k). The sum of those first terms (n<k) is called the
  partial sum. "a(0)+...+a(k)" is called expanded form.

Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S is
  called an infinite series. Note that there are infinite(NEVER terminate)
  addends. I.e. basically, the addition of addends cannot be completed in
  finite steps by definition.

Convergent::= The sequence of the partial sum of infinite series has limit.

Operation Principle of Infinite Series: The last addend of the expanded form
  (the index is ∞) must be shown to indicate the general term.

  The arithmetic of the expanded form is the same as finite series:
  Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
    S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
    <=> S= 1+a*S-a^(∞+1)
    <=> S(1-a)=1-a^(∞+1)
    <=> S= (1-a^(∞+1))/(1-a)

  Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
    S= 1+2+3+...+n  // (1)
    S= n+...+3+2+1  // (2)
    2S= n*(n+1)     // (1)+(2)
    <=> S= n*(n+1)/2

  If the last addend is missing, the expanded form is prone to magic tricks,
  because the rearrangement of the expanded form may likely change the
  definition of the series:
  Ex1: S can be any number from a rearrangement:
       S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
       = Σ(n=1,∞) n+1  // S is modified
         (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)

  Ex2:
    S=1+2+4+8+...          // The last addend is omitted (ill-formed)
    <=> S=1+2(1+2+4+8+...) // Various possibilities of interpretation
    <=> S=1+2S
    <=> S=-1

  Last addend is shown:
    S=1+2+4+8+...+2^∞
    <=> S=1+2(1+2+4+...+2^(∞-1))
    <=> S=1+2S-2^(∞+1)
    <=> S=2^(∞+1)-1   // Lots of similar "magic calculation" deriving the result
                      // S=-1 can be found in youtube (from the omission of the
                      // term containing ∞).

Theorem1: s1=s2 <=> s1-s2=0

Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
                    = a(∞)+ Σ(n=0,∞-1) a(n)

Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
  Proof: Omitted (Can be derived from the expanded form. Trivial rules are also
         omitted)

  Basically, formula for finite series are also applicable to infinite series(
  but mathematical inducion cannot prove such formula because by definition,
  ∞ means 'the procedure never terminate' and the Peano axiom is only valid in
  finite steps).

  Note: Many 'equations' of infinite series (esp. about π,e) can be proved
        false by the theorems above. They are actually approximates (limits).
        Ex: Σ(n=1,∞) 1/n² ≒ π²/6
            Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
            Σ(n=0,∞) k^n/n! ≒ e^k

+----------+
| Appendix |
+----------+
Appendix1: Just as an example, the theoretical ℝ might be made (yes, mathematics
    ,truth are made up) like bellow for proving equations:
    Eℕ ::= {n| n is the number defined by Peano axioms (n∈ ℕ<0,S>), including
           axiom "n∈ℕ => S(n)∈ℕ" may be applied infinitely to define infinitely
           many infinities. }
    Eℤ ::= {n| n∈Eℕ or -n∈Eℕ }
    ℝ  ::= {p/q| p,q∈Eℤ, q>=2 }

Appendix2: No 1-1 correspondence procedure exists between set ℝ and ℕ.
    Proof: Let X= {x| x is the number defined by Peano axioms including those
    (defined by the successor) that the procedure does not terminate }. Then,
    no 1-1 correspondence (procedure) can be established between X and ℕ
    (simply because of termination/non-termination issues).
    And, because X and ℝ are isomophic, no 1-1 correspondence can be established
    ℝ and ℕ (This proof is sloop, but should be fine). This property indirectly
    explains that most numbers in ℝ are inexpressible by finite symbols.
    Note: Set X can be used to demonstrate that "repeating decimal 0.999..." (
          as 1-nary number) is able to define an isomorphic set of real number.
          Anyway, "repeating decimal 0.999..." actually can refer to a very
          large set of numbers.

Appendix3: I am not experienced with infinity ∞. Just provide several principles
    that are more sure about:
    1. From the definition, ℝ also contains infinity ∞, which share the same
       property with finite numbers in that they can compare and perform +-*/.
       There are infinite number of infinities. Take a common contradiction for
       instance: The concept of "1+∞ =∞" contradicts the concept of approach in
       "lim(x->∞) f(x)". Because if so, we cannot tell whether x+1 or x-1
       approach ∞ of leave ∞. The semantics of infinity must be unique.

       Note: If infinity does not in ℝ, the 'obsolete real' should not talk
             about infinite series. The 'obsolete real' cannot be both proven
             consistent and complete. E.g. A number of such number system(s) is
             defined on number (self-referential) (sorry, I am not even sure
             how many 'obsolete real' really are. Ex. Archimedean property
             asserts a number system that has no infinity and no infinitesimal,
             while allowing infinite additions and using lim(x->∞) ....? And, the
             Dedekind cut theory, except mentioned, declares "it can construct
             every real number", which is of course false). Therefore, if it
             were consistent, at most, such a number system was a subset of the
             Real defined in this file.
      
       In all, ℝ satisfies [1] Trichotomy [2] Dense property (infinite steps)
       [3] Close under +* (including the steps are infinite). Therefore, the
       very signature of ℝ is that it contains infinity.

    2. The semantics of infinity is basically about a procedural loop (as in
       Peano axioms). The characteristics is "never terminate" (might mean
       'undecidable','undefined' or 'controdiction' in procedural proofs).
       Another relative issue is about the dense property, which is different
       when applied to rational and real in that the procedure must terminate
       for the former and not necessarily for the latter (similar to Peano
       axioms).

    3. Infinity ∞ also contains Cantor's cardinal number, because the concept
       of 'number' is very primitive. Although cardinal number has its
       definition, its concept is still based on counting (and, op +* in ℝ is
       closed).

    4. If the meaning of '∞' is unique, the meaning of 'e' (base of natural
       logarithm) in the sense of 'number' of the following three are different:
       e::= lim(n->∞) (1+1/n)^n = (1+1/∞)^∞
       e^k= lim(n->∞) (1+k/n)^n = (1+k/∞)^∞
       e^k= Σ(n=0,∞) k^n/n!

    5. This example may help explain the concept of infinity ∞:
       Let A(0)= 0
           A(n)= (A(n-1)+1)/2
       Ask: A(∞)= ?
       Ans: The value 'A(∞)' does not exist in the range of the given problem.
            This answer also applies to the basic Zeno paradox and Supertask
            paradox.

Appendix4: 2D-number can express plane. In 2D-number, as long as the distance
    postulate (1. Distance between points is invariant by movement 2.The
    ratio of distance between points is invariant by scalar multiplication) are
    satisfied, Euclidean geometry system can be established. What is meant to
    say is that: Such a 'mass-point universe' is constructed based on our
    preset property. We are ultimately exploring the semantics of our own
    knowledge. And, as long as the logic holds, the respective reality should
    be expected. Inversely, exploring 'real number' by physics is basicly valid.
    In the digital era, universe (semantics) is a natural computer.

Appendix5: Convergent (that has limit) infinite series should use "lim" or '≒'
    (except definition) to be more precise and less prone to error, E.g:
    lim(k->∞) Σ(n=0,k) (-1^n)*(1/(2n+1)) = π/4, or
    Σ(n=0,∞) (-1^n)*(1/(2n+1))≒ π/4.
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