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.