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 |
+-------------+

n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a plus/minus sign or a point:

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

Two n-ary fixed-point number x,y are equal iff their form as mentioned above
are identical.

Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
of digits of x may be infinitely long }

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).

Real number is just this simple. The limit theory is a methodology for finding
derivative, nothing to do with what the real number is (otherwise, a definition
like the above must be defined in advance. Otherwise, latter dedution will be
difficult not to contain 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

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

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.

Limit defines A=B, 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))= (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 ?

+--------------------------------------+
| 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 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)

Note: Hope that this interpretation can avoid the interpretation of infinity
/infinitesimal, and provide more correct foundation for some theories
, e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
Cantor set, infinite series...).

-------------------------------------------------------------------------------

On 03/24/2024 08:02 AM, wij wrote:
> 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 |
> +-------------+
>
> n-ary Fixed-Point Number::= Number represented by a string of digits, the
> string may contain a plus/minus sign or a point:
>
> <fixed_point_number>::= [-,+] <dstr1> [ . <dstr2> ]
> <dstr1>::= <nzd> [{ 0, <nzd> } <nzd>]
> <dstr2>::= { 0, <nzd> } <nzd>
> <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
>
> Two n-ary fixed-point number x,y are equal iff their form as mentioned above
> are identical.
>
> Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point number. The string
> of digits of x may be infinitely long }
>
> 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).
>
> Real number is just this simple. The limit theory is a methodology for finding
> derivative, nothing to do with what the real number is (otherwise, a definition
> like the above must be defined in advance. Otherwise, latter dedution will be
> difficult not to contain 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
>
> L is defined as the limit (a number) while x approaches a (f(a) may not
> be defined, although, while f is continuous, L=f(a)). L is a defined value,
> not "if something infinitely close ... then equal" (no such logic).
>
> 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.
>
> Limit defines A=B, 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))= (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 ?
>
> +--------------------------------------+
> | 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 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)
>
> Note: Hope that this interpretation can avoid the interpretation of infinity
> /infinitesimal, and provide more correct foundation for some theories
> , e.g. Zeno paradoxes, repeating decimal,...,and more (exponiential,
> Cantor set, infinite series...).
>
> -------------------------------------------------------------------------------
>
>

Hope and a bottle of ketchup
isn't a hamburger, ....

I see you've found the trapezoid rule there.

If you look into "Differential Geometry" ,
they sort of get into some similar notions
about finding closed forms.

It's not "all of real analysis" though,
it's just a sub-field with some usual applications.
What they do is re-define "function" then
that the "functional analysis" is simplified.

The definition of "function" is among the most
varying of definitions, over time. With
geometry and arithmetic, and,
algebra and arithmetic, then
function theory and topology,
distributions and probability
yet the operator calculus,
specifying "function" variously
sees changes over time.

What you got these is usually
called "methods of finite differences".

Also look into "interval arithmetic", which
is a usual idea how to establish bounds when
numeric representations in the fixed-width
word or even the unbounded yet finite,
don't establish "zero-interval arithmetic".

Numerical methods are great forms for approximations,
of course everybody knows that approximations have
formally non-zero error terms.

This is that analytical methods are closed-form,
while they may involve infinite series, and don't.

There already is a "real analysis", there's even
standard and non-standard, about infinities and
infinitesimals. Why worry? It's fine to use
methods of finite differences and numerical methods
and interval arithmetic: if you know their limits.

It's not like the Pythagorans are going to say
that root two is a real number. So, if you want
to be a sort of retro-Pythagoran, keep in mind
that while Euclid's geometry is still a thing,
there's been a few thousand years of development.

Now, if you wanted to deconstruct Euler's identity,
and figure out how to re-visit the Eulerian-Gaussian
complex analysis, that has some things going on in it,
because, a lot of the associated conjectures, are
actually independent number theory and complex number theory,
then you might even find new mathematics,
instead of forgetting current mathematics.

If you want to work some numerical methods that's fine,
if you want to know foundations of all mathematics,
it's all involved.

Hope and a bottle of ketchup, ....