On Mon, 2024-03-25 at 19:43 -0400, Richard Damon wrote:
> 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.

'real number' is already there. e.g. 'a line' or just a series of digits,
whatever had practically used world wide now and then.
Your 'real number' is also a theory which also suffers from revision.
If you need a specific number system, you should name it otherwise.

On 3/25/24 10:52 PM, wij wrote:
> On Mon, 2024-03-25 at 19:43 -0400, Richard Damon wrote:
>> 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.
>
> 'real number' is already there. e.g. 'a line' or just a series of digits,
> whatever had practically used world wide now and then.
> Your 'real number' is also a theory which also suffers from revision.
> If you need a specific number system, you should name it otherwise.
>

So, you are AGREEING that the term "Real Number" is already in use and
that your attempts to "redefine" it is thus just an attempt to be
confusiong?

Maybe you just don't understand how much actual Theory there is behind
the term "The Real Number System"?

It seems you are becoming schizophrenic.

On 03/25/2024 01:43 AM, 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... ?

You know it's kind of funny you ask, consider this,
the continuum limit, or infinite limit of these functions,
for non-negative or natural integers,

f(n) = n/d, 0 <= n <= d, d -> infinity

So, n the numerator ranges from zero to d,
and d the denominator goes to infinity.

It's so that 0/d = 0, and as n -> d that n/d = 1,
in the limit.

Then, it must have some constant difference, or
that it results being an equi-partition,
that f is constant monotone strictly increasing,
not that it's a constant, but that it's strictly
increasing, by a constant.

So, that establishes extent, [0,1], and density, in [0,1],
of ran(f), the range of f.

Completeness then is the existence of the least-upper-bound,
it's satisfied for any {f(m)}, that the LUB is f(max(m)+1).

Then, measure, is about the sigma-algebras, basically that
any initial segment 0, ..., m gets assigned a length f(m),
there are a variety of ways to equip ran(f) with a sigma-algebra.

So, that there's extent, density, completeness, and measure, [0,1].

So, if you assign these elements as to a Cantor space, like 2^w,
but in the same order as the elements of ran(f), then it's not
like the usual Cantor space, first of all because it's countable,
as "f is not a Cartesian function, can't be changed its contents
and order, not a real function, not a Cartesian function, yet
does have analytical character". Then this arrangement has a
beginning, 000..., and end, 111..., and the sequences are ordered
by their natural order.

Then, 1-f is sort of reverse that, 111... to 000..., and as a
matter of notation, it makes for that ".999...", is a notation,
and in this model of a continuous domain, which is different
than the usual standard model of a continuous domain as the
complete ordered field, or that this is "line-reals" for usual
"field-reals", then 1-f(1) is sort of like ".999...", helping to
reflect that in the course-of-passage, that's how it would start.

In the linear curriculum, at some point it's established
that 0.999... is, and, means, 1.000..., from the complete
ordered field, "R", this results the set of numbers called R.

The idea that the numbers roll over, is called clock arithmetic.
I.e, "to get to 1.0 it rolled over .9, .99, .999, .999...,
to get back it would roll back and first through .999...",
is a very usual notion, and, also it is formalizable, with
showing there are at least these two models of continuous domains,
as I've written so briefly above.

The other day I was reading some educator, and he said,
"when I surveyed the class whether .999... equals one
or is less than one, about half voted either way".

It's both, just by a matter of notation, and meaning,
it's one or the other.

The standard definition of R most surely has that
there are dual representations of numbers that
repeat b-1 in base b, and 0.999... = 1.0 in decimal,
in the complete ordered field, the usual interpretation.

That R is not the only model of a continuous domain,
is about real numbers and restoring interpretation of calculus.

On 03/26/2024 07:44 PM, Ross Finlayson wrote:
> On 03/25/2024 01:43 AM, 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... ?
>
> You know it's kind of funny you ask, consider this,
> the continuum limit, or infinite limit of these functions,
> for non-negative or natural integers,
>
> f(n) = n/d, 0 <= n <= d, d -> infinity
>
> So, n the numerator ranges from zero to d,
> and d the denominator goes to infinity.
>
> It's so that 0/d = 0, and as n -> d that n/d = 1,
> in the limit.
>
> Then, it must have some constant difference, or
> that it results being an equi-partition,
> that f is constant monotone strictly increasing,
> not that it's a constant, but that it's strictly
> increasing, by a constant.
>
> So, that establishes extent, [0,1], and density, in [0,1],
> of ran(f), the range of f.
>
> Completeness then is the existence of the least-upper-bound,
> it's satisfied for any {f(m)}, that the LUB is f(max(m)+1).
>
> Then, measure, is about the sigma-algebras, basically that
> any initial segment 0, ..., m gets assigned a length f(m),
> there are a variety of ways to equip ran(f) with a sigma-algebra.
>
> So, that there's extent, density, completeness, and measure, [0,1].
>
> So, if you assign these elements as to a Cantor space, like 2^w,
> but in the same order as the elements of ran(f), then it's not
> like the usual Cantor space, first of all because it's countable,
> as "f is not a Cartesian function, can't be changed its contents
> and order, not a real function, not a Cartesian function, yet
> does have analytical character". Then this arrangement has a
> beginning, 000..., and end, 111..., and the sequences are ordered
> by their natural order.
>
> Then, 1-f is sort of reverse that, 111... to 000..., and as a
> matter of notation, it makes for that ".999...", is a notation,
> and in this model of a continuous domain, which is different
> than the usual standard model of a continuous domain as the
> complete ordered field, or that this is "line-reals" for usual
> "field-reals", then 1-f(1) is sort of like ".999...", helping to
> reflect that in the course-of-passage, that's how it would start.
>
> In the linear curriculum, at some point it's established
> that 0.999... is, and, means, 1.000..., from the complete
> ordered field, "R", this results the set of numbers called R.
>
> The idea that the numbers roll over, is called clock arithmetic.
> I.e, "to get to 1.0 it rolled over .9, .99, .999, .999...,
> to get back it would roll back and first through .999...",
> is a very usual notion, and, also it is formalizable, with
> showing there are at least these two models of continuous domains,
> as I've written so briefly above.
>
> The other day I was reading some educator, and he said,
> "when I surveyed the class whether .999... equals one
> or is less than one, about half voted either way".
>
> It's both, just by a matter of notation, and meaning,
> it's one or the other.
>
> The standard definition of R most surely has that
> there are dual representations of numbers that
> repeat b-1 in base b, and 0.999... = 1.0 in decimal,
> in the complete ordered field, the usual interpretation.
>
> That R is not the only model of a continuous domain,
> is about real numbers and restoring interpretation of calculus.
>
>
>
>

So, the notions of number sense and relating .999... with
..999, .99, .9, and so on, and relating .999... with 1.0,
are two separate notions, but two combined notions,
the notion that there's both the infinitely-divisible
of the field, and, the infinitely-divided of the smoothly
continuous, vis-a-vis the completion of the complete
ordered field, and the completion of course-of-passage
of instants in time, from a beginning, to an end.

I'm reading a book called "The Number Sense", it's pretty
good, Dehaene's, so far it's mostly about integers and
quantities, and perception and perceptual modeling in
animals, infants, and across cultures, about that continuity
has a lot going on.

Integer Continuum
line-reals
field-reals <- 3 different models completeness/continuous domains
signal-reals
Long-Line Continuum

The Jordan measure in 1-D is like these "iota-values",
where "iota" is also the traditional word for "least bit",
then for things like Dirichlet function and metrizing ultrafilter,
it is like these signal reals, as for example also the
analyses like Fourier-style demonstrating convergence
of the sums in infinitesimally-narrow regions, demonstrate
completeness, with regards to things like Shannon-Nyquist
and super-sampling and re-construction, so, it goes to
show that there are these complementary and dual ways
to look at Zeno's situation, including one where the
continuous passage of time is what enables the establishment
of related rates problems, vis-a-vis the beginning and end,
of the runner and the archer, and what results finite
quantities of time.

That there's really an easier and clearly direct formalism
showing that these line-reals make a continuous domain,
than a usual formalism of equivalence classes of sequences
that are Cauchy, with Dedekind completeness and defined
measure 1.0, the complete ordered field or field-reals,
that fact that line-reals have no axioms while field-reals
in usual descriptive set theory have to axiomatize both
least-upper-bound and measure 1.0, help to show that
line-reals is even sort of part of the definition of
field-reals, then that the great utilitity of the field
makes for its centrality instead, in algebra.

So, if you didn't know how easy it was to formalize line-reals,
it's about as easy as the above, then with regards to that
the linear curriculum clearly skates around that for pre-calculus,
to establish the much more characterized delta-epsilonics,
as what results best for an overall linear curriculum,
where students are expected to know derivations of their
mathematics, thoroughly.