This updated file should solve most doubts I encountered. Hope, useful to y=
ou
(of course, not in official exam if that is your 'real') https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/do= wnload

+-------------+
| 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 [-] <dstr1> [ . <dstr2> ]
<dstr1>::=3D 0 | <nzd> { 0, <nzd> }
<dstr2>::=3D { 0, <nzd> } <nzd>
<nzd> ::=3D (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on=
n-ary

Two n-ary fixed-point number (same n-ary) x,y are equal iff their
<fixed_point_number> representation are identical.

Real Nunmber(=E2=84=9D)::=3D {x| x is finitely represented by n-ary <fixed_= point_number>
and those that cannot be finitely represented }

Note: Numbers that is not finitely representable cannot all be explicitl=
y
defined, this is the property of real number based on discrete sym= bols
(like quantum?). E.g.

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

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.

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 =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).

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)-...=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
repeating pattern p(i). But, by definition of 'repeating', r(i) ca= nnot
be pattern p(i). Therefore, repeating decimal is irrational.

Real number is just this simple. The limit theory is a methodology for find= ing
derivative, nothing to do with what the real number is (otherwise, a defini= tion
like the above must be defined in advance to avoid circular-reasoning).

+-------+
| 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

L is defined as the limit (a number) while x approaches a (f(a) may not
be defined, although, while f is continuous, L=3Df(a)). L is a defined = value,
not "if something infinitely close ... then equal" (no such logic).

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.

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

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)):

Let A=3Dlim(n->=E2=88=9E) (1-1/n)=3D 1
A*A*..*A=3D ... =3D lim(n->=E2=88=9E) (1-1/n)^n // 1=3D1/e ?

+--------------------------------------+
| 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)) =E2=89=92 (f(x+h)+f(x))*(h/2) // h is a sufficiently sma=
ll (test)offset
<=3D> (F(x+h)-F(x))/h =E2=89=92 (f(x+h)+f(x))/2 // the limit(h->0) of rh=
s is f(x)

Expected property of F: (1)Error |lhs-rhs| strictly decreases with the tiny
(test) offset h (2)When h=3D0, lhs=3Drhs.
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...Th= us,

D(f(x))=3D lim(h->0) (F(x+h)-F(x))/h =3D f(x)

Note: Hope that this interpretation can avoid the interpretation of inf= inity
/infinitesimal, and provide more correct foundation for some theo= ries
, e.g. Zeno paradoxes, repeating decimal,...,and more (exponienti= al,
Cantor set, infinite series...).

---------------------------------------------------------------------------= ----




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