Bugfix and added another (polynomial) example to show that "repeating decimal is irrational"
(I personally is not very fond of such topic. But, like alway, it seems
many people are stuck there)

==============================================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>::= 0 | <nzd> { 0, <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 <fixed_point_number>
representation are identical.

Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
digits 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).

Note: Let b∈ℕ,b>=2,c=b-1, f(b)= Σ(i=1,∞) c*b^(-i)
From the "0.999...=1" theory, f(2)=f(3)=f(4)=f(5)=...= 1 (zero degree)
Obvious, this controdicts the definition of polynomial (at least):
"Two polynomials are the same for all values of the variable if and
only if they have the same degree and all the coefficients in
corresponding terms are the same."

https://mathspace.co/textbooks/syllabuses/Syllabus-453/topics/Topic-8422/subtopics/Subtopic-111666/

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 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

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...).

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

wij <wyniijj5@gmail.com> writes:

> Bugfix

You are welcome.

> and added another (polynomial) example to show that "repeating
> decimal is irrational"

> ===============================================
> 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>::= 0 | <nzd> { 0, <nzd> }

I thought you were using comma for alternation. Do

<dstr1>::= 0 | <nzd> { 0, <nzd> }
<dstr1>::= 0 , <nzd> { 0, <nzd> }
<dstr1>::= 0 | <nzd> { 0 | <nzd> }
<dstr1>::= 0 | <nzd> { 0 | <nzd> }

all mean the same thing?

> <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 <fixed_point_number>
> representation are identical.

Interesting. Presumably -0 < 0 < +0 though we'll only know if (when?)
you define the ordering on Wij-numbers.

> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> digits may be infinitely long }

So 1000...0 and 1000...0.25 are both Wij-real numbers?

> 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).

There are no axioms at all so neither you nor I can even prove that
1+1=2 in Wij-numbers!

How are +, *, /, - and = defined? Then you need to define the ordering
as well. Remember, you need to come up with definitions that satisfy
your fantasy that when a=1, b=0.999... the (a+b)/2 is between a and b.
The usual "school" definitions make the result 0.999...

> Real number is just this simple.

I see you are determined to keep using the wrong name for your, as yet,
useless number system.

--
Ben.

On Mon, 2024-03-25 at 23:55 +0000, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
>
> > Bugfix
>
> You are welcome.
>
> > and added another (polynomial) example to show that "repeating
> > decimal is irrational"
>
> > ==============================================> > 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>::= 0 | <nzd> { 0, <nzd> }
>
> I thought you were using comma for alternation.  Do
>
>       <dstr1>::= 0 | <nzd> { 0, <nzd> }
>       <dstr1>::= 0 , <nzd> { 0, <nzd> }
>       <dstr1>::= 0 | <nzd> { 0 | <nzd> }
>       <dstr1>::= 0 | <nzd> { 0 | <nzd> }
>
> all mean the same thing?
>
> >      <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 <fixed_point_number>
> >    representation are identical.
>
> Interesting.  Presumably -0 < 0 < +0 though we'll only know if (when?)
> you define the ordering on Wij-numbers.
>
> > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> >    digits may be infinitely long }
>
> So 1000...0 and 1000...0.25 are both Wij-real numbers?
>
> >    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).
>
> There are no axioms at all so neither you nor I can even prove that
> 1+1=2 in Wij-numbers!
>
> How are +, *, /, - and = defined?  Then you need to define the ordering
> as well.  Remember, you need to come up with definitions that satisfy
> your fantasy that when a=1, b=0.999... the (a+b)/2 is between a and b..
> The usual "school" definitions make the result 0.999...
>
> > Real number is just this simple.
>
> I see you are determined to keep using the wrong name for your, as yet,
> useless number system.
>

Sorry for the delay, this email should not be sent. It was sent by accident, 
and then my network was down immediately.