On Friday, 24 December 2021 at 20:10:59 UTC+8, malcolm.ar...@gmail.com wrote:
> On Friday, 24 December 2021 at 09:14:45 UTC, wij wrote:
> > On Friday, 24 December 2021 at 09:55:52 UTC+8, Keith Thompson wrote:
> > > wij <wyn...@gmail.com> writes:
> > > > On Friday, 24 December 2021 at 07:20:02 UTC+8, Keith Thompson wrote:
> > > >> wij <wyn...@gmail.com> writes:
> > > >> [...]
> > > >> > 1/3= 0.333... + non_zero_remainder
> > > >> > 1/3-0.333...= non_zero_remainder
> > > >> [...]
> > > >>
> > > >> Please explain, precisely and rigorously, what you mean by the notation
> > > >> "0.333...".
> > > >
> > > > Number notation formed by "..." is mostly indeterminate (see [Ref])..
> > > I'm not at all sure what that means.
> > >
> > > [...]
> > > > By "0.333...", I mean exactly, infinite repeating decimal 3.
> > > That's not as rigorous an answer as I was looking for.
> > >
> > > First, the phrase "infinite repeating decimal 3" by itself doesn't
> > > say that the infinite sequence of 3s follows "0.".
> > >
> > > And you've only defined "0.333..." in terms of another notation.
> > > > So, yes, various definitions of 0.333... is not exactly equal to 1/3.
> > > That's incorrect if you're talking about real numbers and the usual
> > > meaning of the "0.333..." notation -- and I still don't know how your
> > > meaning of "0.333..." differs from the usual one in a way that explains
> > > what you're talking about.
> > > >> If 0.333... is not equal to 1/3, then you are using that notation to
> > > >> mean something different from what most mathematicians mean by it.
> > > >
> > > > By "0.333...", I mean exactly, infinite repeating decimal 3.
> > > > So, yes, various definitions of 0.333... is not exactly equal to 1/3.
> > > > "0.333..." is with most people is indeterminate to me.
> > > What does "with most people" have to do with anything?
> > >
> > > If you're saying that the value represented by "0.333..." is
> > > indeterminate, then why are you using that notation?
> > >
> > > If "0.333..." is "infinite repeating decimal 3", then what is the
> > > mathemtical value of "infinite repeating decimal 3"? Is that value a
> > > real number, yes or no?
> > > >> Is 0.333... a real number? If not, what is it? If so, is your
> > > >> definition of "real number" consistent with the definition used by most
> > > >> mathematicians?
> > > >
> > > > I think, for most people, 'real number' means the number that can be the mark of
> > > > a physical, straight ruler (or X-Axis). Note that my idea is that non-zero length
> > > > interval cannot be stuffed by points (see Ref).
> > > > I did not define what the real number is (seemingly not required by program, yet).
> > > I cannot extract either a yes or a no from that sequence of words.
> > >
> > > Again, is your definition of "real number" consistent with the
> > > definition used by mosh mathematicians? (Most mathematicians *do not*
> > > identify real numbers with marks on some physical object.)
> > > > [Ref] The basic idea is in NumberView-en.txt. The short article is aimed to
> > > > try out programming concepts, not to explicitly explain 'repeating decimal'.
> > > > https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> > > That doesn't sound relevant, so I'm not going to bother reading it.
> > > --
> > > Keith Thompson (The_Other_Keith) Keith.S.T...@gmail.com
> > > Working, but not speaking, for Philips
> > > void Void(void) { Void(); } /* The recursive call of the void */
> > 12 years-old kids know immediately what is meant by a single line:
> >
> > 1/3= 0.333... + non_zero_remainder
> >
> > That the density property can be applied infinitely is not difficult for 14
> > years-old kids to understand. If 0.999...=1, the density property is broken.
> >
> > If you are not familiar with such issues, that is all I should say.
> >
> The rule that 0.999... equals unity is a bit counter-intuitive, and high school
> students often have trouble with it.

High school students' intuition is correct. It is invalid to apply the limit
theory to prove anything more basic than itself, such as 1=0.999... or not.

> As and others have said, the rules of notation
> are human-made. So you can say that 0.999... equals 1 minus an infinitesimal, if
> you wish. However then you've got to introduce infinitesimals into your system.
> That has all sorts of other implications.

'∞' (infinity):: 1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

'∞' is simply a symbol capable of algebraic operation, not much different from
the symbol for imaginary number 'i'. '∞' is not arbitrarily defined. The two
rules above should be acceptable, no fancy things.

Therefore, there are infinite number of infinitesimals, e.g. r/∞, r∈ℝ.

> Another implication is that you no longer have a notation for a third. Because
> 0.333... equals 1/3 minus an infinitesimal. So the ellipsis becomes less
> practically useful.

No exact fractional notation for 1/3, because the fractional notion is equivalent to
a infinite series.

> Now you can say that the ... notation introduces infinities in any case. That's a
> bit beyond my pay grade.

"..." is not that difficult, and useful for expanded infinite series, but, use with caution:
+-----------------+
| Infinite Series |
+-----------------+
Rule1: Infinite series A,B are equal iff the summand and the index are equal
Rule2: The expanded form of infinite series must list the last addend.
Otherwise, the expanded form is ill-formed (obscure semantics).

Ex.1 (the last addend is omitted):
A=1+2+3+4+5+...
=(1+2)+(3+4)+5+...
=3+7+5+... // ill-formed, obsecure semantics.

Last addend listed:
A=1+2+3+4+5+...+∞ // well-formed, the exanded form of Σ(n=1,∞) {n}

Ex.2:
S=1+2+4+8+... // ill-formed
<=> S=1+2(1+2+4+8+...)
<=> S=1+2S
<=> S=-1

Last addend listed:
S=1+2+4+8+...+2^∞
<=> S=1+2(1+2+4+...+2^(∞-1))
<=> S=1+2S-2^(∞+1)
<=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
// S=-1 can be found in youtube. (the term containing the
// last addend ∞ is ignored)

Ex.3:
"f(n)= Σ(k=0,n) 1/k! => f(∞)=e(The base of natural logarithm)"?
We know for sure ∀n∈ℕ, f(n)∈ℚ. To get the result f(n)=e (f(n)∉ℚ), the only
current option is n=∞. But the issue whether or not f(∞)=e (exact equal by
definition) can only be decided via definition, e.g. e≡f(∞). Otherwise, we
can only say f(∞)≈e. (In considering the definition of the equal sign '=',
other forms of e are likely not mutually replaceable with f(∞))

Ex.4: x= Σ(n=1,∞) 1/n²
A common expression is x= Σ(n=1,∞) 1/n²= π²/6, therefore, π=√(6*x)
The issue here is: Lots of π can be derived from various infinite serieses.
But, according to the definition of '=', the result of mutual substitution
may become inconsistent.
For now, the uncontroversial definition of π is the ratio of the
circumference of a circle to its diameter, it is more correct to use '≈'.
Therefore, Σ(n=1,∞) 1/n² ≈ π²/6 is what it is.

Note] "..." in expression is normally indeterminant, of vague semantic.
"0.999..." is also indeterminant before the "..." is eliminated, the
number "0.999..." represents is uncertain.

Ex1: Let x=0.999...
10*x= 9+x // This is the reult of x after interpreted, not necessarily
// the result followed from "x=0.999..."
// This equation must be given to define x (eliminate the
// ambiguous "...")
Ex2: Let x=√(2+√(2+√(2+...))). Then, possible interpretation of x are:
x=√(2+x)
x=√(2+√(2+x))
x=√(2+√(2+√(2+x)))
...

On Friday, 24 December 2021 at 21:39:55 UTC+8, Andy Walker wrote:
> On 24/12/2021 00:58, wij wrote:
> [Richard:]
> >> The Reals only deal with FINITE values.
> > Then, show us in your "The Real Number System", how A1 sums up to
> > exactly one
> > (note that you said n is FINITE)?:
> > A1= Σ(n=1,∞) 9/10^n
> I hope no-one said "A1 sums up to exactly one" [but I don't
> intend to check back through previous articles to make sure]; in
> conventional mathematics "Σ(n=1,∞)" does not mean the sum of
> infinitely many terms, but rather the /limit/ [if it exists] of the
> finite sums as "n" tends to infinity. By the usual proofs, given
> in this thread, that limit does exist for "A1", and is 1. "Tends
> to infinity" is a phrase with a mathematically well-defined meaning,
> which does not, again in conventional mathematics, require of any
> number used that it be either infinite or infinitesimal. Every use
> of "infinity", yet again in conventional mathematical analysis, is
> couched in that sort of language and has a conventional meaning
> related to limits.
>
> Note that in geometry, there are terms such as "point at
> infinity" and "line at infinity" which are not limits, and can in
> real use [eg in architecture] be placed more or less where you
> like. [Think of the horizon in a picture.]
>
> [...]
> >>> By "0.333...", I mean exactly, infinite repeating decimal 3.
> Then your meaning is not well defined, until you explain
> what you mean what /you/ mean by the unconventional term "infinite
> repeating".
> >>> I think, for most people, 'real number' means the number that can
> >>> be the mark of a physical, straight ruler (or X-Axis).
> "Most people" do not learn rigorous mathematics. That
> definition is fine for introducing children to the concept of a
> "number line" and similar concepts; it doesn't get you from the
> rationals to the reals.
> >>> Note that
> >>> my idea is that non-zero length interval cannot be stuffed by
> >>> points (see Ref).
> Define "stuffed". There are obviously an "infinity" [ie,
> more than any (finite) count] of points in any proper interval;
> but it is elementary that any such set of points that you can
> construct in sequence has measure zero. This is not the sort of
> thing you can explain to "most people".
>
> Basically, you can either stick to conventional maths
> [which has developed the way it did for excellent reasons, even
> if those reasons are incomprehensible to "most people"], or you
> can switch to one or other of the "well known" [though, again,
> not well-known to "most people", nor even to most people with a
> decent mathematical background] alternative systems, such as
> one that introduces infinite and infinitesimal numbers; or you
> can go down your own route. But if you insist on going down
> your own route without understanding what is already known,
> you are doomed to fail.
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Palmgren

I skipped some 'sub-topic' to post the passage below first. I think I am successful
in interpretation of differentiation. Hope it says all.

+---------------------------------------------------------------------------+
| Differentiation as the finding of the primitive function (divide-by-zero) |
+---------------------------------------------------------------------------+

Set of function expression 𝔽::= {f| f is an average mathematical function (not
using ∞)}. For clarity reason, this sloppiness should be fine for the moment.

Set of infinitesimal ℍ::= {h| ∀q∈ℚ, |h|<|q|}

Neighborhood::= Set of neighborhood of a point x is defined as U(x)≡ {x+h| h∈ℍ}

Deleted neighborhood::= Set of deleted neborhood is defined as U'(x)≡ U(x) - {x}

Undefined-Point primitive function::= Function g' is a UP-primitive function of
g <=> g is undefined at point a but g' is defined, beside this,
∀x (x≠a), g'(x)=g(x).
IOW, g'(x,h) is a function of g(x,U'(x)). Because g is often undefined at
point 0, g' may be referred to as f's Zero-Point primitive function.

Ex: Let f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h, h∈U'(x), // two argument example
If g'(x,h) is the UP-primitive of g(x,h) <=> g(x,0) is undefined but
g'(x,0) is defined, beside this, g'(x,h)=g(x,h).

Note: The result of g'(x,0) is similar to lim(h->0) g(x,h), but the theory
of limit is very inconsistent (self-contradictory). Computers just
do not have that kind of free will (or not allowed).

Derivative::= Generalized notion of the slope of a tangent line.
Let function f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h. I..e. g(x,h) is the slope of all
the straight lines passing point (x,f(x)) and (U'(x),f(U'(x))).
If g'(x,h), the ZP-primitive of g(x,h), exists, g'(x,0) is the derivative of
f(x).

Note: In the ZP-privitive definition, h must be 0 to ensure the result is of
a tagent line instead of as a secant line. This is very different from
average text books.

Derivative function::= The function of derivative, noted as D(f(x)) or f'(x).
Derivative function is sometimes referred to as derivative or
differentiation, noted as (d/dx)f(x).

----
Ex1: Power function f(x)= x^a
From definition, g(x,h)= ((x+h)^a - x^a)/h, h∈U'(x)
= (x^a + C(a,1)*x^(a-1)*h + C(a,2)*x^(a-2)*h^2 + ... + h^n - x^a)/h
// Binomial expansion
= C(a,1)*x^(a-1) + C(a,2)*x^(a-2)*h^1 + ... + h^(n-1)

Because the denominator (zero) is eliminated, the ZP-primitive g' of g
is the same: g'(x,h)= g(x,h).
Therefore, the derivative of f(x)= f'(x)= g'(x,0)=C(a,1)*x^(a-1)= a*x^(a-1)

Ex2: Exponential function f(x)= a^x
From definition, g(x,h)= (a^(x+h)-a^x)/h = (a^x)*((a^h-1)/h), h∈U'(x)
Assume g'(x,h)= s(x)+j(x,h) is the ZP primitive.
(a^x)*[(a^h-1)/h]= s(x)+j(x,h) // ZP term in square brackets
<=> (a^x)*[(a^h-1)/h] - s(x) = j(x,h)
While h=0, LHS has to be 0, therefore,
(a^x)*[(a^h-1)/h] = s(x)
Because [(a^h-1)/h] is a term that "converges" depending on a, we can
"assume the ZP primitive of [(a^h-1)/h]" is in form of C+remainder
(C=lim(h->0)(a^h-1)/h), then, the derivative of a^x= g'(x,0)= (a^x)*C

∴ g'(x,0)= C*a^x depends on the assumption of convergence

[Note] The result indicates the previous definition of Euler's Constant
e≡(1+1/∞)^∞ is still problematic, because this is the result of
assuming h=1/∞, not h=0.
"Value converges" is a confusing notion, fine for engineering
mathematics but not in logic.

Ex3: Trigonometric function f(x)= sin(x)
From definition g(x,h)= (sin(x+h)-sin(x))/h, h∈U'(x)
I am not able to deduce the ZP primitive, so assume g'(x,h)= cos(x)+j(x,h)
<=> ((sin(x+h)-sin(x))/h) - cos(x)= j(x,h)
<=> sin(x)*cos(h)+ cos(x)*sin(h) -sin(x) - h*cos(x) = h*j(x,h)
<=> sin(x)*(cos(h)-1) + cos(x)*(sin(h)-h) = h*j(x,h)
While h=0, LHS has to be 0, therefore,
sin(x)*(1-cos(h)= cos(x)*(sin(h)-h)
<=> tan(x)= [(sin(h)-h)/(1-cos(h))]
<=> false (LHS and RHS are unrelated) // may mean something different.

∴ Assumption g'(x,h)= cos(x) + j(x,h) is false.

[note] In numerics, lim(h->0) (sin(x+h)-sin(x))/h ≈ cos(x) + h*j(x,h).
The numerical differences of LHS and RHS of '≈' is infinitesimal.

On 12/24/21 4:14 AM, wij wrote:
> On Friday, 24 December 2021 at 09:55:52 UTC+8, Keith Thompson wrote:
>> wij <wyn...@gmail.com> writes:
>>> On Friday, 24 December 2021 at 07:20:02 UTC+8, Keith Thompson wrote:
>>>> wij <wyn...@gmail.com> writes:
>>>> [...]
>>>>> 1/3= 0.333... + non_zero_remainder
>>>>> 1/3-0.333...= non_zero_remainder
>>>> [...]
>>>>
>>>> Please explain, precisely and rigorously, what you mean by the notation
>>>> "0.333...".
>>>
>>> Number notation formed by "..." is mostly indeterminate (see [Ref]).
>> I'm not at all sure what that means.
>>
>> [...]
>>> By "0.333...", I mean exactly, infinite repeating decimal 3.
>> That's not as rigorous an answer as I was looking for.
>>
>> First, the phrase "infinite repeating decimal 3" by itself doesn't
>> say that the infinite sequence of 3s follows "0.".
>>
>> And you've only defined "0.333..." in terms of another notation.
>>> So, yes, various definitions of 0.333... is not exactly equal to 1/3.
>> That's incorrect if you're talking about real numbers and the usual
>> meaning of the "0.333..." notation -- and I still don't know how your
>> meaning of "0.333..." differs from the usual one in a way that explains
>> what you're talking about.
>>>> If 0.333... is not equal to 1/3, then you are using that notation to
>>>> mean something different from what most mathematicians mean by it.
>>>
>>> By "0.333...", I mean exactly, infinite repeating decimal 3.
>>> So, yes, various definitions of 0.333... is not exactly equal to 1/3.
>>> "0.333..." is with most people is indeterminate to me.
>> What does "with most people" have to do with anything?
>>
>> If you're saying that the value represented by "0.333..." is
>> indeterminate, then why are you using that notation?
>>
>> If "0.333..." is "infinite repeating decimal 3", then what is the
>> mathemtical value of "infinite repeating decimal 3"? Is that value a
>> real number, yes or no?
>>>> Is 0.333... a real number? If not, what is it? If so, is your
>>>> definition of "real number" consistent with the definition used by most
>>>> mathematicians?
>>>
>>> I think, for most people, 'real number' means the number that can be the mark of
>>> a physical, straight ruler (or X-Axis). Note that my idea is that non-zero length
>>> interval cannot be stuffed by points (see Ref).
>>> I did not define what the real number is (seemingly not required by program, yet).
>> I cannot extract either a yes or a no from that sequence of words.
>>
>> Again, is your definition of "real number" consistent with the
>> definition used by mosh mathematicians? (Most mathematicians *do not*
>> identify real numbers with marks on some physical object.)
>>> [Ref] The basic idea is in NumberView-en.txt. The short article is aimed to
>>> try out programming concepts, not to explicitly explain 'repeating decimal'.
>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>> That doesn't sound relevant, so I'm not going to bother reading it.
>> --
>> Keith Thompson (The_Other_Keith) Keith.S.T...@gmail.com
>> Working, but not speaking, for Philips
>> void Void(void) { Void(); } /* The recursive call of the void */
>
> 12 years-old kids know immediately what is meant by a single line:
>
> 1/3= 0.333... + non_zero_remainder

And what non-zero number is that remainder? (in The Reals).

Any number you quote, and I can tell you how many digits you need to
include to make the number closer than that.

That is your problem, you are using notations from 'The Reals' and
trying to make statements about systems which are not in it.

>
> That the density property can be applied infinitely is not difficult for 14
> years-old kids to understand. If 0.999...=1, the density property is broken.
>

If it 0.999... ISN'T 1 then the densitiy property is broken!!!

If 0.999... is one, then they are the same number so we DON'T expect
there to be a number between it and itself.

If they aren't the same number, then what number can be between them,
are you going to try to invent a number like:

0.999...5? And what would that mean?

> If you are not familiar with such issues, that is all I should say.

On 12/24/21 9:17 AM, wij wrote:

> I skipped some 'sub-topic' to post the passage below first. I think I am successful
> in interpretation of differentiation. Hope it says all.
>
> +---------------------------------------------------------------------------+
> | Differentiation as the finding of the primitive function (divide-by-zero) |
> +---------------------------------------------------------------------------+
>
> Set of function expression 𝔽::= {f| f is an average mathematical function (not
> using ∞)}. For clarity reason, this sloppiness should be fine for the moment.
>
> Set of infinitesimal ℍ::= {h| ∀q∈ℚ, |h|<|q|}
>
> Neighborhood::= Set of neighborhood of a point x is defined as U(x)≡ {x+h| h∈ℍ}
>
> Deleted neighborhood::= Set of deleted neborhood is defined as U'(x)≡ U(x) - {x}
>
> Undefined-Point primitive function::= Function g' is a UP-primitive function of
> g <=> g is undefined at point a but g' is defined, beside this,
> ∀x (x≠a), g'(x)=g(x).
> IOW, g'(x,h) is a function of g(x,U'(x)). Because g is often undefined at
> point 0, g' may be referred to as f's Zero-Point primitive function.
>
> Ex: Let f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h, h∈U'(x), // two argument example
> If g'(x,h) is the UP-primitive of g(x,h) <=> g(x,0) is undefined but
> g'(x,0) is defined, beside this, g'(x,h)=g(x,h).
>
> Note: The result of g'(x,0) is similar to lim(h->0) g(x,h), but the theory
> of limit is very inconsistent (self-contradictory). Computers just
> do not have that kind of free will (or not allowed).
>
> Derivative::= Generalized notion of the slope of a tangent line.
> Let function f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h. I.e. g(x,h) is the slope of all
> the straight lines passing point (x,f(x)) and (U'(x),f(U'(x))).
> If g'(x,h), the ZP-primitive of g(x,h), exists, g'(x,0) is the derivative of
> f(x).
>
> Note: In the ZP-privitive definition, h must be 0 to ensure the result is of
> a tagent line instead of as a secant line. This is very different from
> average text books.
>
> Derivative function::= The function of derivative, noted as D(f(x)) or f'(x).
> Derivative function is sometimes referred to as derivative or
> differentiation, noted as (d/dx)f(x).
>
> ----
> Ex1: Power function f(x)= x^a
> From definition, g(x,h)= ((x+h)^a - x^a)/h, h∈U'(x)
> = (x^a + C(a,1)*x^(a-1)*h + C(a,2)*x^(a-2)*h^2 + ... + h^n - x^a)/h
> // Binomial expansion
> = C(a,1)*x^(a-1) + C(a,2)*x^(a-2)*h^1 + ... + h^(n-1)
>
> Because the denominator (zero) is eliminated, the ZP-primitive g' of g
> is the same: g'(x,h)= g(x,h).
> Therefore, the derivative of f(x)= f'(x)= g'(x,0)=C(a,1)*x^(a-1)= a*x^(a-1)
>
> Ex2: Exponential function f(x)= a^x
> From definition, g(x,h)= (a^(x+h)-a^x)/h = (a^x)*((a^h-1)/h), h∈U'(x)
> Assume g'(x,h)= s(x)+j(x,h) is the ZP primitive.
> (a^x)*[(a^h-1)/h]= s(x)+j(x,h) // ZP term in square brackets
> <=> (a^x)*[(a^h-1)/h] - s(x) = j(x,h)
> While h=0, LHS has to be 0, therefore,
> (a^x)*[(a^h-1)/h] = s(x)
> Because [(a^h-1)/h] is a term that "converges" depending on a, we can
> "assume the ZP primitive of [(a^h-1)/h]" is in form of C+remainder
> (C=lim(h->0)(a^h-1)/h), then, the derivative of a^x= g'(x,0)= (a^x)*C
>
> ∴ g'(x,0)= C*a^x depends on the assumption of convergence
>
> [Note] The result indicates the previous definition of Euler's Constant
> e≡(1+1/∞)^∞ is still problematic, because this is the result of
> assuming h=1/∞, not h=0.
> "Value converges" is a confusing notion, fine for engineering
> mathematics but not in logic.
>
> Ex3: Trigonometric function f(x)= sin(x)
> From definition g(x,h)= (sin(x+h)-sin(x))/h, h∈U'(x)
> I am not able to deduce the ZP primitive, so assume g'(x,h)= cos(x)+j(x,h)
> <=> ((sin(x+h)-sin(x))/h) - cos(x)= j(x,h)
> <=> sin(x)*cos(h)+ cos(x)*sin(h) -sin(x) - h*cos(x) = h*j(x,h)
> <=> sin(x)*(cos(h)-1) + cos(x)*(sin(h)-h) = h*j(x,h)
> While h=0, LHS has to be 0, therefore,
> sin(x)*(1-cos(h)= cos(x)*(sin(h)-h)
> <=> tan(x)= [(sin(h)-h)/(1-cos(h))]
> <=> false (LHS and RHS are unrelated) // may mean something different.
>
> ∴ Assumption g'(x,h)= cos(x) + j(x,h) is false.
>
> [note] In numerics, lim(h->0) (sin(x+h)-sin(x))/h ≈ cos(x) + h*j(x,h).
> The numerical differences of LHS and RHS of '≈' is infinitesimal.

So, in short, you are working on some extension to The Reals, and not
talking about things in The Reals without giving proper notice.

Note, this is really off topic here. You want some more general
Mathematics newsgroup, as computation theory is stretched a bit to even
talk about things in the Reals, but tends to restrict itself to the at
most the countably infinite spaces, like the Natural Numbers.

I don't know these extensions well enough to know if you are describing
something already defined before.

On Friday, 24 December 2021 at 22:45:11 UTC+8, richar...@gmail.com wrote:
....
> > 12 years-old kids know immediately what is meant by a single line:
> >
> > 1/3= 0.333... + non_zero_remainder
> And what non-zero number is that remainder? (in The Reals).

Depending on how many steps n is taken, non_zero_remainder equals 3/10^n
non_zero_remainder can never be Reals except 3/10^∞

> Any number you quote, and I can tell you how many digits you need to
> include to make the number closer than that.

Of course! Pythagoreans can also argue that ratio can be close as desired.

--- Pythagoreans' Logic ---
Infinitely approaching means equal. Number too small equals zero.
Real number can be approached by ratio number.

> That is your problem, you are using notations from 'The Reals' and
> trying to make statements about systems which are not in it.

Your Reals does not define ∞, but use it almost everywhere. And you called it DEFINED.
Your Reals does not clarify "...", also use it ambiguously everywhere (like in infinite series)
to compose 'valid' proof.

> >
> > That the density property can be applied infinitely is not difficult for 14
> > years-old kids to understand. If 0.999...=1, the density property is broken.
> >
> If it 0.999... ISN'T 1 then the densitiy property is broken!!!
How, if 0.999...≠1, the density property is broken?

> If 0.999... is one, then they are the same number so we DON'T expect
> there to be a number between it and itself.
>
> If they aren't the same number, then what number can be between them,
> are you going to try to invent a number like:
>
> 0.999...5? And what would that mean?

If "0.999..." is defined as A(∞)=Σ(n=1,∞) 9/10^n, then the number between A(∞)
and 1 is PRECISELY (A(∞)+1)/2, no ambiguity, the derivation is based on
algebraic rule. This notation(number) can be used every where in algebraic
equations consistently. (What the limit says is BY DEFINITION, no valid reason
and lots of inconsistency. That is your Reals)

On Friday, 24 December 2021 at 23:45:54 UTC+8, richar...@gmail.com wrote:
> On 12/24/21 9:17 AM, wij wrote:
>
> > I skipped some 'sub-topic' to post the passage below first. I think I am successful
> > in interpretation of differentiation. Hope it says all.
> >
> > +---------------------------------------------------------------------------+
> > | Differentiation as the finding of the primitive function (divide-by-zero) |
> > +---------------------------------------------------------------------------+
> >
> > Set of function expression 𝔽::= {f| f is an average mathematical function (not
> > using ∞)}. For clarity reason, this sloppiness should be fine for the moment.
> >
> > Set of infinitesimal ℍ::= {h| ∀q∈ℚ, |h|<|q|}
> >
> > Neighborhood::= Set of neighborhood of a point x is defined as U(x)≡ {x+h| h∈ℍ}
> >
> > Deleted neighborhood::= Set of deleted neborhood is defined as U'(x)≡ U(x) - {x}
> >
> > Undefined-Point primitive function::= Function g' is a UP-primitive function of
> > g <=> g is undefined at point a but g' is defined, beside this,
> > ∀x (x≠a), g'(x)=g(x).
> > IOW, g'(x,h) is a function of g(x,U'(x)). Because g is often undefined at
> > point 0, g' may be referred to as f's Zero-Point primitive function.
> >
> > Ex: Let f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h, h∈U'(x), // two argument example
> > If g'(x,h) is the UP-primitive of g(x,h) <=> g(x,0) is undefined but
> > g'(x,0) is defined, beside this, g'(x,h)=g(x,h).
> >
> > Note: The result of g'(x,0) is similar to lim(h->0) g(x,h), but the theory
> > of limit is very inconsistent (self-contradictory). Computers just
> > do not have that kind of free will (or not allowed).
> >
> > Derivative::= Generalized notion of the slope of a tangent line.
> > Let function f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h. I..e. g(x,h) is the slope of all
> > the straight lines passing point (x,f(x)) and (U'(x),f(U'(x))).
> > If g'(x,h), the ZP-primitive of g(x,h), exists, g'(x,0) is the derivative of
> > f(x).
> >
> > Note: In the ZP-privitive definition, h must be 0 to ensure the result is of
> > a tagent line instead of as a secant line. This is very different from
> > average text books.
> >
> > Derivative function::= The function of derivative, noted as D(f(x)) or f'(x).
> > Derivative function is sometimes referred to as derivative or
> > differentiation, noted as (d/dx)f(x).
> >
> > ----
> > Ex1: Power function f(x)= x^a
> > From definition, g(x,h)= ((x+h)^a - x^a)/h, h∈U'(x)
> > = (x^a + C(a,1)*x^(a-1)*h + C(a,2)*x^(a-2)*h^2 + ... + h^n - x^a)/h
> > // Binomial expansion
> > = C(a,1)*x^(a-1) + C(a,2)*x^(a-2)*h^1 + ... + h^(n-1)
> >
> > Because the denominator (zero) is eliminated, the ZP-primitive g' of g
> > is the same: g'(x,h)= g(x,h).
> > Therefore, the derivative of f(x)= f'(x)= g'(x,0)=C(a,1)*x^(a-1)= a*x^(a-1)
> >
> > Ex2: Exponential function f(x)= a^x
> > From definition, g(x,h)= (a^(x+h)-a^x)/h = (a^x)*((a^h-1)/h), h∈U'(x)
> > Assume g'(x,h)= s(x)+j(x,h) is the ZP primitive.
> > (a^x)*[(a^h-1)/h]= s(x)+j(x,h) // ZP term in square brackets
> > <=> (a^x)*[(a^h-1)/h] - s(x) = j(x,h)
> > While h=0, LHS has to be 0, therefore,
> > (a^x)*[(a^h-1)/h] = s(x)
> > Because [(a^h-1)/h] is a term that "converges" depending on a, we can
> > "assume the ZP primitive of [(a^h-1)/h]" is in form of C+remainder
> > (C=lim(h->0)(a^h-1)/h), then, the derivative of a^x= g'(x,0)= (a^x)*C
> >
> > ∴ g'(x,0)= C*a^x depends on the assumption of convergence
> >
> > [Note] The result indicates the previous definition of Euler's Constant
> > e≡(1+1/∞)^∞ is still problematic, because this is the result of
> > assuming h=1/∞, not h=0.
> > "Value converges" is a confusing notion, fine for engineering
> > mathematics but not in logic.
> >
> > Ex3: Trigonometric function f(x)= sin(x)
> > From definition g(x,h)= (sin(x+h)-sin(x))/h, h∈U'(x)
> > I am not able to deduce the ZP primitive, so assume g'(x,h)= cos(x)+j(x,h)
> > <=> ((sin(x+h)-sin(x))/h) - cos(x)= j(x,h)
> > <=> sin(x)*cos(h)+ cos(x)*sin(h) -sin(x) - h*cos(x) = h*j(x,h)
> > <=> sin(x)*(cos(h)-1) + cos(x)*(sin(h)-h) = h*j(x,h)
> > While h=0, LHS has to be 0, therefore,
> > sin(x)*(1-cos(h)= cos(x)*(sin(h)-h)
> > <=> tan(x)= [(sin(h)-h)/(1-cos(h))]
> > <=> false (LHS and RHS are unrelated) // may mean something different..
> >
> > ∴ Assumption g'(x,h)= cos(x) + j(x,h) is false.
> >
> > [note] In numerics, lim(h->0) (sin(x+h)-sin(x))/h ≈ cos(x) + h*j(x,h).
> > The numerical differences of LHS and RHS of '≈' is infinitesimal.
> So, in short, you are working on some extension to The Reals, and not
> talking about things in The Reals without giving proper notice.
>
> Note, this is really off topic here. You want some more general
> Mathematics newsgroup, as computation theory is stretched a bit to even
> talk about things in the Reals, but tends to restrict itself to the at
> most the countably infinite spaces, like the Natural Numbers.

I know this too tough for you. How about changing the subject to
"No TM can convert numbers between 1/3 and 0.333... exactly."
1/3= 0.333... Really? How does the TM says ...

> I don't know these extensions well enough to know if you are describing
> something already defined before.

I don't know any source described pretty much all I said here. My post is original.

On 12/24/21 1:11 PM, wij wrote:
> On Friday, 24 December 2021 at 22:45:11 UTC+8, richar...@gmail.com wrote:
> ...
>>> 12 years-old kids know immediately what is meant by a single line:
>>>
>>> 1/3= 0.333... + non_zero_remainder
>> And what non-zero number is that remainder? (in The Reals).
>
> Depending on how many steps n is taken, non_zero_remainder equals 3/10^n
> non_zero_remainder can never be Reals except 3/10^∞

Except that you have DEFINED you notation that ... is an INFINITE number
of digits, so the only n that matters is infinite.

For The Reals, this changes to a limit. and that value in the limit is ZERO.

You seem to be talking some strange number theory that both HAS values
that are infinites and infintesimals, but also doesn't accept infinite
series.

>
>> Any number you quote, and I can tell you how many digits you need to
>> include to make the number closer than that.
>
> Of course! Pythagoreans can also argue that ratio can be close as desired.
>
> --- Pythagoreans' Logic ---
> Infinitely approaching means equal. Number too small equals zero.
> Real number can be approached by ratio number.

Well yes. One property of The Reals is that all reals can be expressed
as a limit of rational numbers.

Which does say that

>
>> That is your problem, you are using notations from 'The Reals' and
>> trying to make statements about systems which are not in it.
>
> Your Reals does not define ∞, but use it almost everywhere. And you called it DEFINED.
> Your Reals does not clarify "...", also use it ambiguously everywhere (like in infinite series)
> to compose 'valid' proof.

The reals DO define what the concept of infinite means, via the rule of
limit. To the Reals, Infinity is NOT 'A Number' but a limit you can
approach with a number sequence.

If you can not not understand how that works, then the limitation is in
your own mind.

>
>>>
>>> That the density property can be applied infinitely is not difficult for 14
>>> years-old kids to understand. If 0.999...=1, the density property is broken.
>>>
>> If it 0.999... ISN'T 1 then the densitiy property is broken!!!
>
> How, if 0.999...≠1, the density property is broken?

What number is between 0.999... and 1?

>
>> If 0.999... is one, then they are the same number so we DON'T expect
>> there to be a number between it and itself.
>>
>> If they aren't the same number, then what number can be between them,
>> are you going to try to invent a number like:
>>
>> 0.999...5? And what would that mean?
>
> If "0.999..." is defined as A(∞)=Σ(n=1,∞) 9/10^n, then the number between A(∞)
> and 1 is PRECISELY (A(∞)+1)/2, no ambiguity, the derivation is based on
> algebraic rule. This notation(number) can be used every where in algebraic
> equations consistently. (What the limit says is BY DEFINITION, no valid reason
> and lots of inconsistency. That is your Reals)

But what number is that? It sounds like you have decided that you need
to abandon the concept that numbers can be expressed with an infinite
decimal representaton.

Also, how do you represent the A(∞)+1)/2, is it A(∞+1) ?

How much of the rules of math does ∞ hold?

That is the big issue with the TransFinite number systems, once you let
∞ be an actual value, some of the basic rules fail to hold, or have much
weaker guarentees.

There is nothing inherently wrong with TransFinite number systems, they
do add some useful properties, but they also loose some useful properties.

This is why it is important to make clear that you are using some
alternate definition of numbers, as what you can assume is true changes.

On 12/24/21 1:17 PM, wij wrote:
> On Friday, 24 December 2021 at 23:45:54 UTC+8, richar...@gmail.com wrote:
>> On 12/24/21 9:17 AM, wij wrote:
>>
>>> I skipped some 'sub-topic' to post the passage below first. I think I am successful
>>> in interpretation of differentiation. Hope it says all.
>>>
>>> +---------------------------------------------------------------------------+
>>> | Differentiation as the finding of the primitive function (divide-by-zero) |
>>> +---------------------------------------------------------------------------+
>>>
>>> Set of function expression 𝔽::= {f| f is an average mathematical function (not
>>> using ∞)}. For clarity reason, this sloppiness should be fine for the moment.
>>>
>>> Set of infinitesimal ℍ::= {h| ∀q∈ℚ, |h|<|q|}
>>>
>>> Neighborhood::= Set of neighborhood of a point x is defined as U(x)≡ {x+h| h∈ℍ}
>>>
>>> Deleted neighborhood::= Set of deleted neborhood is defined as U'(x)≡ U(x) - {x}
>>>
>>> Undefined-Point primitive function::= Function g' is a UP-primitive function of
>>> g <=> g is undefined at point a but g' is defined, beside this,
>>> ∀x (x≠a), g'(x)=g(x).
>>> IOW, g'(x,h) is a function of g(x,U'(x)). Because g is often undefined at
>>> point 0, g' may be referred to as f's Zero-Point primitive function.
>>>
>>> Ex: Let f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h, h∈U'(x), // two argument example
>>> If g'(x,h) is the UP-primitive of g(x,h) <=> g(x,0) is undefined but
>>> g'(x,0) is defined, beside this, g'(x,h)=g(x,h).
>>>
>>> Note: The result of g'(x,0) is similar to lim(h->0) g(x,h), but the theory
>>> of limit is very inconsistent (self-contradictory). Computers just
>>> do not have that kind of free will (or not allowed).
>>>
>>> Derivative::= Generalized notion of the slope of a tangent line.
>>> Let function f∈𝔽, g(x,h)≡ (f(x+h)-f(x))/h. I.e. g(x,h) is the slope of all
>>> the straight lines passing point (x,f(x)) and (U'(x),f(U'(x))).
>>> If g'(x,h), the ZP-primitive of g(x,h), exists, g'(x,0) is the derivative of
>>> f(x).
>>>
>>> Note: In the ZP-privitive definition, h must be 0 to ensure the result is of
>>> a tagent line instead of as a secant line. This is very different from
>>> average text books.
>>>
>>> Derivative function::= The function of derivative, noted as D(f(x)) or f'(x).
>>> Derivative function is sometimes referred to as derivative or
>>> differentiation, noted as (d/dx)f(x).
>>>
>>> ----
>>> Ex1: Power function f(x)= x^a
>>> From definition, g(x,h)= ((x+h)^a - x^a)/h, h∈U'(x)
>>> = (x^a + C(a,1)*x^(a-1)*h + C(a,2)*x^(a-2)*h^2 + ... + h^n - x^a)/h
>>> // Binomial expansion
>>> = C(a,1)*x^(a-1) + C(a,2)*x^(a-2)*h^1 + ... + h^(n-1)
>>>
>>> Because the denominator (zero) is eliminated, the ZP-primitive g' of g
>>> is the same: g'(x,h)= g(x,h).
>>> Therefore, the derivative of f(x)= f'(x)= g'(x,0)=C(a,1)*x^(a-1)= a*x^(a-1)
>>>
>>> Ex2: Exponential function f(x)= a^x
>>> From definition, g(x,h)= (a^(x+h)-a^x)/h = (a^x)*((a^h-1)/h), h∈U'(x)
>>> Assume g'(x,h)= s(x)+j(x,h) is the ZP primitive.
>>> (a^x)*[(a^h-1)/h]= s(x)+j(x,h) // ZP term in square brackets
>>> <=> (a^x)*[(a^h-1)/h] - s(x) = j(x,h)
>>> While h=0, LHS has to be 0, therefore,
>>> (a^x)*[(a^h-1)/h] = s(x)
>>> Because [(a^h-1)/h] is a term that "converges" depending on a, we can
>>> "assume the ZP primitive of [(a^h-1)/h]" is in form of C+remainder
>>> (C=lim(h->0)(a^h-1)/h), then, the derivative of a^x= g'(x,0)= (a^x)*C
>>>
>>> ∴ g'(x,0)= C*a^x depends on the assumption of convergence
>>>
>>> [Note] The result indicates the previous definition of Euler's Constant
>>> e≡(1+1/∞)^∞ is still problematic, because this is the result of
>>> assuming h=1/∞, not h=0.
>>> "Value converges" is a confusing notion, fine for engineering
>>> mathematics but not in logic.
>>>
>>> Ex3: Trigonometric function f(x)= sin(x)
>>> From definition g(x,h)= (sin(x+h)-sin(x))/h, h∈U'(x)
>>> I am not able to deduce the ZP primitive, so assume g'(x,h)= cos(x)+j(x,h)
>>> <=> ((sin(x+h)-sin(x))/h) - cos(x)= j(x,h)
>>> <=> sin(x)*cos(h)+ cos(x)*sin(h) -sin(x) - h*cos(x) = h*j(x,h)
>>> <=> sin(x)*(cos(h)-1) + cos(x)*(sin(h)-h) = h*j(x,h)
>>> While h=0, LHS has to be 0, therefore,
>>> sin(x)*(1-cos(h)= cos(x)*(sin(h)-h)
>>> <=> tan(x)= [(sin(h)-h)/(1-cos(h))]
>>> <=> false (LHS and RHS are unrelated) // may mean something different.
>>>
>>> ∴ Assumption g'(x,h)= cos(x) + j(x,h) is false.
>>>
>>> [note] In numerics, lim(h->0) (sin(x+h)-sin(x))/h ≈ cos(x) + h*j(x,h).
>>> The numerical differences of LHS and RHS of '≈' is infinitesimal.
>> So, in short, you are working on some extension to The Reals, and not
>> talking about things in The Reals without giving proper notice.
>>
>> Note, this is really off topic here. You want some more general
>> Mathematics newsgroup, as computation theory is stretched a bit to even
>> talk about things in the Reals, but tends to restrict itself to the at
>> most the countably infinite spaces, like the Natural Numbers.
>
> I know this too tough for you. How about changing the subject to
> "No TM can convert numbers between 1/3 and 0.333... exactly."
> 1/3= 0.333... Really? How does the TM says ...

Turing Maciness do not work on 'Real' Numbers, as they do not have easy
to use finite representations.

As I have said, Computation Theory works in domains with Countably
Infinite Members, like the Natural Numbers or the Rationals, as Turing
Machines need to work in finite time. For Turing Machines a real number
would need to be expressed as a limit or an infinite computation, and
dealing with all real numbers as just a limit expression for them isn't
very useful to see what the numbers are, and computations that take
infinite time aren't considered useful.

YOU were the one positing the message to the wrong group, and people
have been answering you honestly within the knowledge we have based on
the general principles of Mathematics.
>
>> I don't know these extensions well enough to know if you are describing
>> something already defined before.
>
> I don't know any source described pretty much all I said here. My post is original.

Never said it wasn't. Just pointing out that if you take it to a group
with people who DO work with such things, someone may recognize this as
a known about TransFinite number system, perhaps with know problems or
uses.

I do know of some examples of TransFinite number systems, but never
really studied them much (and some get into whole hierarchies of
infinites and infinitesimals). Maybe you are just afraid to talk with
people who actually KNOW well how this things work, because they will
show you a gaping hole in your formulation.

wij <wyniijj@gmail.com> writes:

> I know this too tough for you. How about changing the subject to
> "No TM can convert numbers between 1/3 and 0.333... exactly."
> 1/3= 0.333... Really? How does the TM says ...

There is a finite algorithm (so also a TM) that can convert, either way,
between any repeating decimal and its fractional equivalent.

--
Ben.

On Friday, 24 December 2021 at 18:17:35 UTC, wij wrote:
> .
> > I don't know these extensions well enough to know if you are describing
> > something already defined before.
> I don't know any source described pretty much all I said here. My post is original.
>
That's why if you do a formal academic project, it's normal to start off with
a literature review. It's relatively easy to come up with an original idea in the sense
that it is original to you. However most ideas turn out to be reformulations of
something which has been considered before. It neeeds a real familiarity with the
literature to be sure that what you are proposing is new.

One way is to do what you are doing, and post the idea on Usenet. No-one expects
a Usenet post to have the weight of an academic monograph, and you might get
some pointers to previous literature. But comp.theory isn't really the right place for
pure mathematics.

On 12/24/21 4:00 PM, Ben Bacarisse wrote:
> wij <wyniijj@gmail.com> writes:
>
>> I know this too tough for you. How about changing the subject to
>> "No TM can convert numbers between 1/3 and 0.333... exactly."
>> 1/3= 0.333... Really? How does the TM says ...
>
> There is a finite algorithm (so also a TM) that can convert, either way,
> between any repeating decimal and its fractional equivalent.
>

It depends on how you expect to represent the number 0.333...

If you mean actually making a tape with the infinite 3s, then it can't
be done.

If you mean using a representation that just INDICATES the infinite 3s,
then yes it can be done. (The ... notation is actually ambigous, so
can't be used, for example 0.13... could be 0.131313... or 0.133333...).

Maybe something like 0.R3

where everything after the R begins the pattern that repeats forever.

Richard Damon <Richard@Damon-Family.org> writes:

> On 12/24/21 4:00 PM, Ben Bacarisse wrote:
>> wij <wyniijj@gmail.com> writes:
>>
>>> I know this too tough for you. How about changing the subject to
>>> "No TM can convert numbers between 1/3 and 0.333... exactly."
>>> 1/3= 0.333... Really? How does the TM says ...
>> There is a finite algorithm (so also a TM) that can convert, either way,
>> between any repeating decimal and its fractional equivalent.
>
> It depends on how you expect to represent the number 0.333...
>
> If you mean actually making a tape with the infinite 3s, then it can't
> be done.

Yes, and you can't write a TM that adds two whole numbers together if
the numbers have to be represented with /their/ infinite digits! It's
always possible to come up with notations that thwart a computation.

The number 0.333... has been represented as a finite string throughout
these posts. There's no conceivable reason to stop doing that now. Of
course I'd /actually/ use the less well-known bracket notation: 0.(3),
0.(142857), 0.34(35) etc. because it's not (as you point out about
"...") ambiguous.

--
Ben.

On Saturday, 25 December 2021 at 05:00:13 UTC+8, Ben Bacarisse wrote:
> wij <wyn...@gmail.com> writes:
>
> > I know this too tough for you. How about changing the subject to
> > "No TM can convert numbers between 1/3 and 0.333... exactly."
> > 1/3= 0.333... Really? How does the TM says ...
> There is a finite algorithm (so also a TM) that can convert, either way,
> between any repeating decimal and its fractional equivalent.
>
> --
> Ben.

What I wanted to express is that "No algorithm can convert between 1/3 and the
fractional "0.333...", because the output or input of "0.333..." is INFINITE long.
TM sees this problem as an 'undecidable' problem (It can't answer).
1/3 and 0.333... denote different numbers.

On Saturday, 25 December 2021 at 05:00:35 UTC+8, malcolm.ar...@gmail.com wrote:
> On Friday, 24 December 2021 at 18:17:35 UTC, wij wrote:
> > .
> > > I don't know these extensions well enough to know if you are describing
> > > something already defined before.
> > I don't know any source described pretty much all I said here. My post is original.
> >
> That's why if you do a formal academic project, it's normal to start off with
> a literature review. It's relatively easy to come up with an original idea in the sense
> that it is original to you. However most ideas turn out to be reformulations of
> something which has been considered before. It neeeds a real familiarity with the
> literature to be sure that what you are proposing is new.
>
> One way is to do what you are doing, and post the idea on Usenet. No-one expects
> a Usenet post to have the weight of an academic monograph, and you might get
> some pointers to previous literature. But comp.theory isn't really the right place for
> pure mathematics.

I am doing a pre-program task.
But I declared succeeded in the interpretation of differentiation (The definition in
text books is definitely wrong). This is significant very different.
I know this is too mathematics, I won't post too much about this.

wij <wyniijj@gmail.com> writes:
[...]
> 12 years-old kids know immediately what is meant by a single line:
>
> 1/3= 0.333... + non_zero_remainder
>
> That the density property can be applied infinitely is not difficult for 14
> years-old kids to understand. If 0.999...=1, the density property is broken.
>
> If you are not familiar with such issues, that is all I should say.

A lot of 12 or 14 year old kids know that

1/3= 0.333... + non_zero_remainder

is false. Since you refuse to explain clearly what you mean and to
answer straightforward yes/no questions, and since you've resorted to
insults, I'm done here.

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

wij <wyniijj@gmail.com> writes:

> On Saturday, 25 December 2021 at 05:00:13 UTC+8, Ben Bacarisse wrote:
>> wij <wyn...@gmail.com> writes:
>>
>> > I know this too tough for you. How about changing the subject to
>> > "No TM can convert numbers between 1/3 and 0.333... exactly."
>> > 1/3= 0.333... Really? How does the TM says ...
>> There is a finite algorithm (so also a TM) that can convert, either way,
>> between any repeating decimal and its fractional equivalent.
>>
>
> What I wanted to express is that "No algorithm can convert between 1/3 and the
> fractional "0.333...", because the output or input of "0.333..." is
> INFINITE long.

What digit is missing from the output of a TM that generates 0. and then
goes:

state x: on ' ' write '3' move right goto state x

?

> TM sees this problem as an 'undecidable' problem (It can't answer).

That's not what undecidable means.

> 1/3 and 0.333... denote different numbers.

Not when using the usual meaning of those symbols.

--
Ben.

On 12/24/21 6:07 PM, wij wrote:
> On Saturday, 25 December 2021 at 05:00:13 UTC+8, Ben Bacarisse wrote:
>> wij <wyn...@gmail.com> writes:
>>
>>> I know this too tough for you. How about changing the subject to
>>> "No TM can convert numbers between 1/3 and 0.333... exactly."
>>> 1/3= 0.333... Really? How does the TM says ...
>> There is a finite algorithm (so also a TM) that can convert, either way,
>> between any repeating decimal and its fractional equivalent.
>>
>> --
>> Ben.
>
> What I wanted to express is that "No algorithm can convert between 1/3 and the
> fractional "0.333...", because the output or input of "0.333..." is INFINITE long.
> TM sees this problem as an 'undecidable' problem (It can't answer).
> 1/3 and 0.333... denote different numbers.
>

Except that if you represent the repeating decimal with a compact
notation, like my 0.R3 or Bens 0.(3) which mark the repeating decimals,
then the Rational numbers, even in 'Decimal' form are manageable by a
Turing Machine.

Note, this handles the RATIONAL numbers, which are countably infinite.

It is NOT difficult to convert 0.R3 or 0.(3) to 1/3 with a finite
machine, in fact, I even mostly described the process previously.

On Friday, 24 December 2021 at 23:07:08 UTC, wij wrote:
> On Saturday, 25 December 2021 at 05:00:13 UTC+8, Ben Bacarisse wrote:
> > wij <wyn...@gmail.com> writes:
> >
> > > I know this too tough for you. How about changing the subject to
> > > "No TM can convert numbers between 1/3 and 0.333... exactly."
> > > 1/3= 0.333... Really? How does the TM says ...
> > There is a finite algorithm (so also a TM) that can convert, either way,
> > between any repeating decimal and its fractional equivalent.
> >
>
> What I wanted to express is that "No algorithm can convert between 1/3 and the
> fractional "0.333...", because the output or input of "0.333..." is INFINITE long.
> TM sees this problem as an 'undecidable' problem (It can't answer).
> 1/3 and 0.333... denote different numbers.
>
A naive converter might ideed run forever stamping the symbol "3" on the infinite
tape. But we can and do get round this problem with the ellipsis.

0.333 is a different value to 1/3, though only by a little bit.
0.333... means 1/3 exactly, in the conventional notation. You can say that it
means "1/3 minus an infinitesimal" if you want, but it would be better to use
another symbol rather than an ellipsis to make it clear that you are being
unconventional. Then you've got to work out rules for when an infinitesimal is
treated differently to zero. Whilst I'm not personally familiar with this work,
I'm pretty sure that other people have done this. It's of theoretical interest
only, which is why I'm not very familiar with it. The reason is probably that
infinitesimals in the set of reals create a mass of special cases that require
more and more elaborate rules to keep the system consistent.

But by all means try to read the literature yourself. It might not be accessible
to you, which is ofren a problem. I have to read mathematical papers at work,
and I often spend a long time looking at an equation, only to realise that I've
misunderstood a symbol and in fact the equation is saying something quite
simple.

On 25/12/2021 01:59, Richard Damon wrote:
> Except that if you represent the repeating decimal with a compact
> notation, like my 0.R3 or Bens 0.(3) which mark the repeating
> decimals, then the Rational numbers, even in 'Decimal' form are
> manageable by a Turing Machine.

Or, of course, by a simple computer program.

> Note, this handles the RATIONAL numbers, which are countably
> infinite.
> It is NOT difficult to convert 0.R3 or 0.(3) to 1/3 with a finite
> machine, in fact, I even mostly described the process previously.

The [much more] interesting thing is that once you have
this notation, you can invent new "numbers" [not members of ℝ,
FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
and many others. You can easily write programs that process
strings in such a way that ordinary arithmetic works on strings
that correspond to rationals and produces consistent results on
strings that don't. It works more smoothly if you write your
numbers in "balanced binary" [digits representing +1 and -1, no
zero], as you can interpret such numbers as games -- cf Conway's
"On Numbers and Games" -- that you can actually /play/.

Merry Christmas [or other non-religion-specific festival
of your choice] to all our readers.

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

Andy Walker <anw@cuboid.co.uk> writes:

> The [much more] interesting thing is that once you have
> this notation, you can invent new "numbers" [not members of ℝ,
> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
> and many others.

I didn't want to bring that up, but the cat's out of the bag now! (You
can put them on the other side as well: 1(0)10.(0) for example.)

--
Ben.

On 12/25/21 9:10 AM, Ben Bacarisse wrote:
> Andy Walker <anw@cuboid.co.uk> writes:
>
>> The [much more] interesting thing is that once you have
>> this notation, you can invent new "numbers" [not members of ℝ,
>> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
>> and many others.
>
> I didn't want to bring that up, but the cat's out of the bag now! (You
> can put them on the other side as well: 1(0)10.(0) for example.)
>

One reason I gave a notation without the redundant end marker.

With 0.R3 there is less opportunity for abuse, Though you could put
multiple markers (or multiple decimal points).

Richard Damon <news.x.richarddamon@xoxy.net> writes:

> On 12/25/21 9:10 AM, Ben Bacarisse wrote:
>> Andy Walker <anw@cuboid.co.uk> writes:
>>
>>> The [much more] interesting thing is that once you have
>>> this notation, you can invent new "numbers" [not members of ℝ,
>>> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
>>> and many others.
>> I didn't want to bring that up, but the cat's out of the bag now! (You
>> can put them on the other side as well: 1(0)10.(0) for example.)
>
> One reason I gave a notation without the redundant end marker.

Some would say it's not redundant! 0.R3R4 is no more obviously
forbidden than 0.(3)(4) and 0.(3(4)) but it is ambiguous.

> With 0.R3 there is less opportunity for abuse, Though you could put
> multiple markers (or multiple decimal points).

Both can be specified so as to include only rationals.

--
Ben.

On 12/25/21 2:40 PM, Ben Bacarisse wrote:
> Richard Damon <news.x.richarddamon@xoxy.net> writes:
>
>> On 12/25/21 9:10 AM, Ben Bacarisse wrote:
>>> Andy Walker <anw@cuboid.co.uk> writes:
>>>
>>>> The [much more] interesting thing is that once you have
>>>> this notation, you can invent new "numbers" [not members of ℝ,
>>>> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
>>>> and many others.
>>> I didn't want to bring that up, but the cat's out of the bag now! (You
>>> can put them on the other side as well: 1(0)10.(0) for example.)
>>
>> One reason I gave a notation without the redundant end marker.
>
> Some would say it's not redundant! 0.R3R4 is no more obviously
> forbidden than 0.(3)(4) and 0.(3(4)) but it is ambiguous.
>
>> With 0.R3 there is less opportunity for abuse, Though you could put
>> multiple markers (or multiple decimal points).
>
> Both can be specified so as to include only rationals.
>

Yes, you can mis use it, but the trailing ) can really only 'normally'
exist as the last character of the number, and always and only occurs if
there is a previous ( in the number, so actually contains zero
information, so is, by definition, redundant.

Why add a symbol to your tape set tht doesn't acutally contain any
information.

On 25/12/2021 20:18, Richard Damon wrote:
[I wrote:]
>>>>>     The [much more] interesting thing is that once you have
>>>>> this notation, you can invent new "numbers" [not members of ℝ,
>>>>> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
>>>>> and many others.
[Ben:]
>>>> I didn't want to bring that up, but the cat's out of the bag now!

It's OK, no-one is reading this thread.

[...]
> Yes, you can mis use it, but the trailing ) can really only
> 'normally' exist as the last character of the number, and always and
> only occurs if there is a previous ( in the number, so actually
> contains zero information, so is, by definition, redundant.

You see it as "misuse"; I see it as introducing new forms
of number [different from ℝ and ℚ] that have interesting properties
worthy of study, and more accessible to computer analysis than ℝ, as
all relevant strings are intrinsically finite. As previously noted,
these numbers correspond to [a subset of] games, some of which are
perfectly playable [eg extensions/diversions of "Hackenbush"].

> Why add a symbol to your tape set tht doesn't acutally contain any
> information.

Lots of symbols are technically redundant. After all,
everything we do on our computers is representable entirely with
nothing but on/off bits. We add new ones not because they are
absolutely necessary but because they are interesting, or useful,
or give insight.

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

On Saturday, 25 December 2021 at 22:53:51 UTC, Andy Walker wrote:
> On 25/12/2021 20:18, Richard Damon wrote:
> [I wrote:]
> >>>>> The [much more] interesting thing is that once you have
> >>>>> this notation, you can invent new "numbers" [not members of ℝ,
> >>>>> FTAOD], including 0.(9)5, 0.(0)1, (9).0, 123.(45)(67), 0.(5(7))
> >>>>> and many others.
> [Ben:]
> >>>> I didn't want to bring that up, but the cat's out of the bag now!
> It's OK, no-one is reading this thread.
>
Well with 0.(0)1 you've introduced an infinitesimal.