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 2024-03-24 14:59:52 +0000, wij said:

> Note: This definition implies that repeating decimals are irrational number.

This contradicts the definition that an irrationa number is not rational.

> 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).

A theory of numbers should have enough axioms to determine what
the sum or product of any two numbers is.

--
Mikko

On 24/03/24 15:59, 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.

So the RATIO of 1 and 7 isn't RATIOnal?

On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> On 24/03/24 15:59, 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.
>
> So the RATIO of 1 and 7 isn't RATIOnal?

All rationals p/q are representable by q-ary fixed-point number.

On 25/03/24 00:05, wij wrote:
> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>> On 24/03/24 15:59, 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.
>>
>> So the RATIO of 1 and 7 isn't RATIOnal?
>
> All rationals p/q are representable by q-ary fixed-point number.
>

0.99999... is representable by the following 10-ary fixed-point number: 1.0.

On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
> On 25/03/24 00:05, wij wrote:
> > On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> > > On 24/03/24 15:59, 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.
> > >
> > > So the RATIO of 1 and 7 isn't RATIOnal?
> >
> > All rationals p/q are representable by q-ary fixed-point number.
> >
>
> 0.99999... is representable by the following 10-ary fixed-point number: 1..0.

How?

On 25/03/24 01:53, wij wrote:
> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>> On 25/03/24 00:05, wij wrote:
>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>> On 24/03/24 15:59, 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.
>>>>
>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>
>>> All rationals p/q are representable by q-ary fixed-point number.
>>>
>>
>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>
> How?

By the proof you already showed and rejected, they are two ways of
writing the same number. At least in the real numbers. If you make up a
new number system, it won't be the same as the real numbers and you
shouldn't call it the real numbers.

On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
> On 25/03/24 01:53, wij wrote:
> > On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
> > > On 25/03/24 00:05, wij wrote:
> > > > On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> > > > > On 24/03/24 15:59, 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.
> > > > >
> > > > > So the RATIO of 1 and 7 isn't RATIOnal?
> > > >
> > > > All rationals p/q are representable by q-ary fixed-point number.
> > > >
> > >
> > > 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
> >
> > How?
>
> By the proof you already showed and rejected, they are two ways of
> writing the same number. At least in the real numbers. If you make up a
> new number system, it won't be the same as the real numbers and you
> shouldn't call it the real numbers.

It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.

If you like to stick to the theory that the universe revolves around the earth, that is fine.
One thing to be sure, such theory won't survive.

On 25/03/24 02:31, wij wrote:
> On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
>> On 25/03/24 01:53, wij wrote:
>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>> On 25/03/24 00:05, wij wrote:
>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>> On 24/03/24 15:59, 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.
>>>>>>
>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>
>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>
>>>>
>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>
>>> How?
>>
>> By the proof you already showed and rejected, they are two ways of
>> writing the same number. At least in the real numbers. If you make up a
>> new number system, it won't be the same as the real numbers and you
>> shouldn't call it the real numbers.
>
> It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.
>
> If you like to stick to the theory that the universe revolves around the earth, that is fine.
> One thing to be sure, such theory won't survive.

If two real numbers defined by Dedekind cuts are different, there is a
number in between them. This is obvious from the Dedekind cut
definition. If you invent a different system, you can do that but it's a
different system.

On Mon, 2024-03-25 at 03:29 +0100, immibis wrote:
> On 25/03/24 02:31, wij wrote:
> > On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
> > > On 25/03/24 01:53, wij wrote:
> > > > On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
> > > > > On 25/03/24 00:05, wij wrote:
> > > > > > On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> > > > > > > On 24/03/24 15:59, 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.
> > > > > > >
> > > > > > > So the RATIO of 1 and 7 isn't RATIOnal?
> > > > > >
> > > > > > All rationals p/q are representable by q-ary fixed-point number..
> > > > > >
> > > > >
> > > > > 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
> > > >
> > > > How?
> > >
> > > By the proof you already showed and rejected, they are two ways of
> > > writing the same number. At least in the real numbers. If you make up a
> > > new number system, it won't be the same as the real numbers and you
> > > shouldn't call it the real numbers.
> >
> > It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.
> >
> > If you like to stick to the theory that the universe revolves around the earth, that is fine.
> > One thing to be sure, such theory won't survive.
>
> If two real numbers defined by Dedekind cuts are different, there is a
> number in between them. This is obvious from the Dedekind cut
> definition. If you invent a different system, you can do that but it's a
> different system.
>

Let's say yes (if you'd like me to put this way), I am inventing a system to replace your obsolete
system.

Op 25.mrt.2024 om 01:53 schreef wij:
> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>> On 25/03/24 00:05, wij wrote:
>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>> On 24/03/24 15:59, 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.
>>>>
>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>
>>> All rationals p/q are representable by q-ary fixed-point number.
>>>
>>
>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>
> How?

1.0 = 3*(1/3) = 3*0.33333... = 0.99999...

If not (if there is a difference), which number expresses the difference
between 1.0 and 0.99999... ?

Op 25.mrt.2024 om 04:11 schreef wij:
> On Mon, 2024-03-25 at 03:29 +0100, immibis wrote:
>> On 25/03/24 02:31, wij wrote:
>>> On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
>>>> On 25/03/24 01:53, wij wrote:
>>>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>>>> On 25/03/24 00:05, wij wrote:
>>>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>>>> On 24/03/24 15:59, 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.
>>>>>>>>
>>>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>>>
>>>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>>>
>>>>>>
>>>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>>>
>>>>> How?
>>>>
>>>> By the proof you already showed and rejected, they are two ways of
>>>> writing the same number. At least in the real numbers. If you make up a
>>>> new number system, it won't be the same as the real numbers and you
>>>> shouldn't call it the real numbers.
>>>
>>> It not 'make-up', instead, your math. theory is fabricated, in way worst than mine, at least.
>>>
>>> If you like to stick to the theory that the universe revolves around the earth, that is fine.
>>> One thing to be sure, such theory won't survive.
>>
>> If two real numbers defined by Dedekind cuts are different, there is a
>> number in between them. This is obvious from the Dedekind cut
>> definition. If you invent a different system, you can do that but it's a
>> different system.
>>
>
> Let's say yes (if you'd like me to put this way), I am inventing a system to replace your obsolete
> system.

A system becomes obsolete only if a new system has been worked out and
it is proved to have an advantage. As long as a new system is being
invented and no advantage has been proved, the old system is not obsolete.

On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
> Op 25.mrt.2024 om 01:53 schreef wij:
> > On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
> > > On 25/03/24 00:05, wij wrote:
> > > > On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> > > > > On 24/03/24 15:59, 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.
> > > > >
> > > > > So the RATIO of 1 and 7 isn't RATIOnal?
> > > >
> > > > All rationals p/q are representable by q-ary fixed-point number.
> > > >
> > >
> > > 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
> >
> > How?
>
> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
>
What is that? Refutation by assertion?

> If not (if there is a difference), which number expresses the difference
> between 1.0 and 0.99999... ?

Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)

On 2024-03-25 08:47:34 +0000, Fred. Zwarts said:

> Op 25.mrt.2024 om 04:11 schreef wij:
>> On Mon, 2024-03-25 at 03:29 +0100, immibis wrote:
>>> On 25/03/24 02:31, wij wrote:
>>>> On Mon, 2024-03-25 at 01:59 +0100, immibis wrote:
>>>>> On 25/03/24 01:53, wij wrote:
>>>>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>>>>> On 25/03/24 00:05, wij wrote:
>>>>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>>>>> On 24/03/24 15:59, 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.
>>>>>>>>>
>>>>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>>>>
>>>>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>>>>
>>>>>>>
>>>>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>>>>
>>>>>> How?
>>>>>
>>>>> By the proof you already showed and rejected, they are two ways of
>>>>> writing the same number. At least in the real numbers. If you make up a
>>>>> new number system, it won't be the same as the real numbers and you
>>>>> shouldn't call it the real numbers.
>>>>
>>>> It not 'make-up', instead, your math. theory is fabricated, in way
>>>> worst than mine, at least.
>>>>
>>>> If you like to stick to the theory that the universe revolves around
>>>> the earth, that is fine.
>>>> One thing to be sure, such theory won't survive.
>>>
>>> If two real numbers defined by Dedekind cuts are different, there is a
>>> number in between them. This is obvious from the Dedekind cut
>>> definition. If you invent a different system, you can do that but it's a
>>> different system.
>>>
>>
>> Let's say yes (if you'd like me to put this way), I am inventing a
>> system to replace your obsolete
>> system.
>
> A system becomes obsolete only if a new system has been worked out and
> it is proved to have an advantage.

And even then not always. What makes the new system better for some
purposes may make it worse for other purposes. For example, Riemannian
geometry covers everything Euclidean geometry covers and more but
Euclidean geometry is still used as it is easier to use where it
can be used.

--
Mikko

Op 25.mrt.2024 om 10:11 schreef wij:
> On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
>> Op 25.mrt.2024 om 01:53 schreef wij:
>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>> On 25/03/24 00:05, wij wrote:
>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>> On 24/03/24 15:59, 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.
>>>>>>
>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>
>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>
>>>>
>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>
>>> How?
>>
>> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
>>
> What is that? Refutation by assertion?
>
>> If not (if there is a difference), which number expresses the difference
>> between 1.0 and 0.99999... ?
>
> Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
>

I did not ask about a number between a and b, but about the difference
between a and b. How do you write a-b? Is that -0.00000...? Do you think
it is not equal to 0.0?

On 25/03/2024 08:43, Fred. Zwarts wrote:
> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...

Just to point out that all such "visual" proofs are flawed.
We normally multiply from the right, with the rules about "carries".
With infinite strings, we have to multiply from the left, meaning
that at any given stage, we have a range of possible answers. Thus
3 * 0.3xyz... lies between 0.90 and 1.02, and we can't write down
the start of the answer until we inspect x; if x < 3, then the
answer starts 0.9; if x > 3, it starts 1.0; sadly, if x = 3, then
we have made no progress until we inspect y and then z. If they are
all 3's, then we continually have to defer starting the answer, and
we never find out whether 3 * 0.333... is 0.999... or 1.000.... I
suspect that most people who are happy with the above would be less
happy with "1.0 = 7*(1/7) = 7*0.1428561428... = 0.99999..." where you
clearly have to worry about the carries.

[Of course, there are satisfactory proofs that in the reals
0.999... = 1 using standard results about limits. But these are much
harder to explain to non-mathematicians. Inevitably, they depend on
the Archimedean axiom, which is not (at least in the UK) routinely
taught to children as they learn about numbers.]

> If not (if there is a difference), which number expresses the
> difference between 1.0 and 0.99999... ?

In [for example] the surreals and the hyperreals, there are
numbers that are strictly less than 1 and strictly greater than any
member of 0.9, 0.99, 0.999, .... Some such system of numbers would
surely meet Wij's needs better than a half-baked attempt to describe
an incoherent extension to the reals.

[Wiki on both of the above is a decent introduction once you
are past the first couple of paragraphs; but I suspect that most
Wij-alikes give up on reading "In mathematics, the surreal number
system is a totally ordered proper class ...".]

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Bizet

wij <wyniijj5@gmail.com> writes:

> +-------------+
> | 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

So neither 0 nor 10 are Fixed_point numbers. OK. That's going to limit
the usefulness of these numbers.

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

That's ill-defined as there are two {0, <nzd>} parts in the grammar. I
think you intended only one of them to be permitted to be not finite.

(And as has been pointed out, this is not a model of set to the real
numbers. You need to choose a different name.)

As usual, you don't really care about the details, do you? It's all
about looking technical rather than being precise.

--
Ben.

On 3/25/24 5:11 AM, wij wrote:
> On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
>> Op 25.mrt.2024 om 01:53 schreef wij:
>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>> On 25/03/24 00:05, wij wrote:
>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>> On 24/03/24 15:59, 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.
>>>>>>
>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>
>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>
>>>>
>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>
>>> How?
>>
>> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
>>
> What is that? Refutation by assertion?
>
>> If not (if there is a difference), which number expresses the difference
>> between 1.0 and 0.99999... ?
>
> Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
>

But the density property requires a and b to be DIFFERENT numbers before
it asserts that there is a number between them, otherwise c is just the
same number as a and b.

On Mon, 2024-03-25 at 07:14 -0400, Richard Damon wrote:
> On 3/25/24 5:11 AM, wij wrote:
> > On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
> > > Op 25.mrt.2024 om 01:53 schreef wij:
> > > > On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
> > > > > On 25/03/24 00:05, wij wrote:
> > > > > > On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
> > > > > > > On 24/03/24 15:59, 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.
> > > > > > >
> > > > > > > So the RATIO of 1 and 7 isn't RATIOnal?
> > > > > >
> > > > > > All rationals p/q are representable by q-ary fixed-point number..
> > > > > >
> > > > >
> > > > > 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
> > > >
> > > > How?
> > >
> > > 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
> > >
> > What is that? Refutation by assertion?
> >
> > > If not (if there is a difference), which number expresses the difference
> > > between 1.0 and 0.99999... ?
> >
> > Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
> >
>
> But the density property requires a and b to be DIFFERENT numbers before
> it asserts that there is a number between them, otherwise c is just the
> same number as a and b.

The premise of this thread said "If not" (means IF 0.999...!=1)

On Mon, 2024-03-25 at 10:24 +0000, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
>
> > +-------------+
> > > 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
>
> So neither 0 nor 10 are Fixed_point numbers.  OK.  That's going to limit
> the usefulness of these numbers.
>
Good catch. I will fix these grammar things, latter.

> > Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point
> > number. The string of digits of x may be infinitely long }
>
> That's ill-defined as there are two {0, <nzd>} parts in the grammar.  I
> think you intended only one of them to be permitted to be not finite.
>
ditto

> (And as has been pointed out, this is not a model of set to the real
> numbers.  You need to choose a different name.)
>
IMO, these all boil down to the question "what is number?". I find it better
(so far) to define 'number' on symbols and the associated ops.
Because I feel computation (TM, algorithm,program,..) can and should be a large
part of the basics of mathematics. E.g. I feel infinity may be nothing but a
manifest of procedural loop. (This article tries to grasp this kind of idea
https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download )

> As usual, you don't really care about the details, do you?  It's all
> about looking technical rather than being precise.
>
“Premature optimization is the root of all evil”

On 25/03/2024 12:04, wij wrote:
> IMO, these all boil down to the question "what is number?".

Yes, but you persist in calling your answer "real number".
Sorry, but that concept is taken. Either "Wij-number" and "real
number" are the same [in which case a lot of what you say is plain
wrong, as wrong as a cartload of wrong things], or they differ, in
which case you need to find another name [and "surreal", "hyperreal",
"superreal" and others are also already taken].

> I find it better
> (so far) to define 'number' on symbols and the associated ops.
> Because I feel computation (TM, algorithm,program,..) can and should be a large
> part of the basics of mathematics.

Sure, but that has been true "forever", and has been largely
formalised and analysed [NA, symbolic computation, ...] for several
decades. I don't see anything new or even interesting in what you
have been writing.

> E.g. I feel infinity may be nothing but a
> manifest of procedural loop.

Lots of clever people have been grappling with infinities and
infinitesimals over the past 2500 or so years. You are going to have
to work a lot harder to produce anything interesting that evaded
Archimedes, Newton, Euler, ....

[Ben:]
>> As usual, you don't really care about the details, do you?  It's all
>> about looking technical rather than being precise.
> “Premature optimization is the root of all evil”

Yes, but arm-waving is little better.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Bizet

wij <wyniijj5@gmail.com> writes:

> On Mon, 2024-03-25 at 10:24 +0000, Ben Bacarisse wrote:
>> wij <wyniijj5@gmail.com> writes:
>>
>> > +-------------+
>> > > 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
>>
>> So neither 0 nor 10 are Fixed_point numbers.  OK.  That's going to limit
>> the usefulness of these numbers.
>>
> Good catch. I will fix these grammar things, latter.
>
>> > Real Nunmber(ℝ)::= {x| x is represented by n-ary fixed-point
>> > number. The string of digits of x may be infinitely long }
>>
>> That's ill-defined as there are two {0, <nzd>} parts in the grammar.  I
>> think you intended only one of them to be permitted to be not finite.
>>
> ditto
>
>> (And as has been pointed out, this is not a model of set to the real
>> numbers.  You need to choose a different name.)
>>
> IMO, these all boil down to the question "what is number?".

No, it boils down to what words are already taken for certain classes of
number. You know this. You did not pick "real number" because you
didn't know what it means. You picked it because you did know and you
wanted to imply that everyone else has been using it wrongly.

> I find it
> better (so far) to define 'number' on symbols and the associated ops.

That is fine way to do things but you have not even taken the first step
since your definition excludes 10.

> Because I feel computation (TM, algorithm,program,..) can and should
> be a large part of the basics of mathematics. E.g. I feel infinity may
> be nothing but a manifest of procedural loop. (This article tries to
> grasp this kind of idea
> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> )
>
>> As usual, you don't really care about the details, do you?  It's all
>> about looking technical rather than being precise.
>>
> “Premature optimization is the root of all evil”

Getting your base definition right so that it includes 0 as a number is
not and optimisation and it would not be premature. Telling people they
have been using the wrong concept of "real number" when you can can't
even get the basic syntax right for your proposed alternative is
premature publication.

--
Ben.

wij <wyniijj5@gmail.com> writes:

> On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
....
>> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
....
>> If not (if there is a difference), which number expresses the difference
>> between 1.0 and 0.99999... ?
>
> Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and
> b. (dense property)

Since you have not yet taken even the first step in defining your
numbers you have the luxury of pretending that (a+b)/2 might be anything
you'd like it to be. But the computational processes you want to
represent numbers don't permit what you are imagining. If + and / are
defined as processes on infinite strings then with a=0.999... and b=1

c=(a+b)/2 = 0.999... = a.

Stop pretending and do the work. (1) Define your numbers. (2) Define
=, +, -, * and / on them and only /then/ start telling the world that
they have been wrong for centuries.

--
Ben.

On Mon, 2024-03-25 at 12:49 +0000, Andy Walker wrote:
> On 25/03/2024 12:04, wij wrote:
> > IMO, these all boil down to the question "what is number?".
>
> Yes, but you persist in calling your answer "real number".
> Sorry, but that concept is taken.  Either "Wij-number" and "real
> number" are the same [in which case a lot of what you say is plain
> wrong, as wrong as a cartload of wrong things], or they differ, in
> which case you need to find another name [and "surreal", "hyperreal",
> "superreal" and others are also already taken].
>
> > I find it better
> > (so far) to define 'number' on symbols and the associated ops.
> > Because I feel computation (TM, algorithm,program,..) can and should be a large
> > part of the basics of mathematics.
>
> Sure, but that has been true "forever", and has been largely
> formalised and analysed [NA, symbolic computation, ...] for several
> decades.  I don't see anything new or even interesting in what you
> have been writing.
>
> >      E.g. I feel infinity may be nothing but a
> > manifest of procedural loop.
>
> Lots of clever people have been grappling with infinities and
> infinitesimals over the past 2500 or so years.  You are going to have
> to work a lot harder to produce anything interesting that evaded
> Archimedes, Newton, Euler, ....
>
> [Ben:]
> > > As usual, you don't really care about the details, do you?  It's all
> > > about looking technical rather than being precise.
> > “Premature optimization is the root of all evil”
>
> Yes, but arm-waving is little better.
>

I think there is no need to discuss much. Time will tell.

On 3/25/24 7:54 AM, wij wrote:
> On Mon, 2024-03-25 at 07:14 -0400, Richard Damon wrote:
>> On 3/25/24 5:11 AM, wij wrote:
>>> On Mon, 2024-03-25 at 09:43 +0100, Fred. Zwarts wrote:
>>>> Op 25.mrt.2024 om 01:53 schreef wij:
>>>>> On Mon, 2024-03-25 at 01:09 +0100, immibis wrote:
>>>>>> On 25/03/24 00:05, wij wrote:
>>>>>>> On Sun, 2024-03-24 at 23:35 +0100, immibis wrote:
>>>>>>>> On 24/03/24 15:59, 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.
>>>>>>>>
>>>>>>>> So the RATIO of 1 and 7 isn't RATIOnal?
>>>>>>>
>>>>>>> All rationals p/q are representable by q-ary fixed-point number.
>>>>>>>
>>>>>>
>>>>>> 0.99999... is representable by the following 10-ary fixed-point number: 1.0.
>>>>>
>>>>> How?
>>>>
>>>> 1.0 = 3*(1/3) = 3*0.33333... = 0.99999...
>>>>
>>> What is that? Refutation by assertion?
>>>
>>>> If not (if there is a difference), which number expresses the difference
>>>> between 1.0 and 0.99999... ?
>>>
>>> Let a=0.999..., b=1, c=(a+b)/2, c is the number between a and b. (dense property)
>>>
>>
>> But the density property requires a and b to be DIFFERENT numbers before
>> it asserts that there is a number between them, otherwise c is just the
>> same number as a and b.
>
> The premise of this thread said "If not" (means IF 0.999...!=1)
>

False premise -> Unsound conclusions.

If you want 0.999(9) to be different than 1.00 you need to move beyond
the Real Numbers into some form of extended Reals, and understand what
that does to what logic you can use.

If going into some alternate or extended number system IS your goal,
then you shouldn't call your numbers "The Reals", as that name is taken.