Using limit theory to prove "1/3=0.333..."?

[wij]

>>
>> 12 years-old kids know immediately what is meant by a single line:
>>
>> 1/3= 0.333... + non_zero_remaindera
>
>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.

Of course, fossil head !

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

CubicDiagnal= lim(x->√2) x=?

Pythagoreans might use your logic to 'believe' CubicDiagnal is a ratio number.
How would one refute Pythagoreans' logic using 'closer' theory without
jeopardizing one's own 'close' theory (that's tricky)?
Your Reals may not be too much different from Pythagoreans' Reals.
The sequence of x is { 1/2 +3/8 +15/64 +35/256 +... }
Note that BY DEFINITION, x can't be √2 !!!

---
lim(x->1) x=? (1?)

I don't think limit theory had ever said what x should be. A ratio sequence
can jump to rational/irrational conclusion whatever one feels fit.

But, ...wait, I heard a voice: "That's a limit"

lim(x->a) f(x)= f(a)

If the limit of f(x) when x 'approaches' a is equal to f(a), why we need limit?
If the limit f(a) (lim(x->a) f(a)) is different from f(a), why do we remove the
notation 'lim' latter to pretend (it is not the limit) it is 'f(a)'.

limit theory was developed to assist the development of calculus. It states
the principle of calculus in limit's view. Calculus is undeniably very
useful in many fields, but it depends on "close enough is identical to equal".
Whatever it is, using that theory to prove anything about "1/3=0.333..." is
simply invalid. Sadly, that is what fossil head would do.

[Chris M. Thomasson]

On 12/26/2021 10:35 PM, wij wrote:
>>>
>>> 12 years-old kids know immediately what is meant by a single line:
>>>
>>> 1/3= 0.333... + non_zero_remaindera
>>
>> And what non-zero number is that remainder? (in The Reals).

[...]

1/3 = .333... in base ten, therefore, .333... = 1/3 in base ten.

[Serg io]

On 12/27/2021 12:35 AM, wij wrote:
>>>
>>> 12 years-old kids know immediately what is meant by a single line:
>>>
>>> 1/3= 0.333... + non_zero_remaindera
>>

<snip crap>

try not to over think this.

1/3 and 0.333... and 3/9 and 1000/3000 are simply representations of the same number.

it is only a number, that's all.

[wij]

Try an algorithm converting decimal fraction input to rational

[Serg io]

very easy.

simply follow out the decimal until it repeats


https://www.wikihow.com/Convert-Repeating-Decimals-to-Fractions

[wij]

How do you know it repeats infinitely?
Assuming you don't know what algorithm is:
If I send a long list of 0.333..., can you determine the number I send?

[Serg io]

Dont you know that the three dots ... means it repeats forever.

=> *read the url*

your repeating digit is 3

take 3/9, simplify = 1/3

NOW so you say the repeating is 333 above, and not 3

then it is 333/999 = 3*111/(9*111) = 3/9 = 1/3 => CORRECTOMUNDO !!


=> ask yourself, when I divide by 9, what happens ? (*the digits repeat!*)

what about 0.35353535...

repeating digits are 35

so answer is 35/99


=> *read the url* again



[Extra credit: Show how you can do this in base7 ]

[wij]

On Tuesday, 28 December 2021 at 12:48:59 UTC+8, Serg io wrote:
> On 12/27/2021 8:32 PM, wij wrote:
> > On Tuesday, 28 December 2021 at 10:18:29 UTC+8, Serg io wrote:
> >> On 12/27/2021 7:50 PM, wij wrote:
> >>> On Tuesday, 28 December 2021 at 04:09:33 UTC+8, Chris M. Thomasson wrote:
> >>>> On 12/26/2021 10:35 PM, wij wrote:
> >>>>>>>
> >>>>>>> 12 years-old kids know immediately what is meant by a single line:
> >>>>>>>
> >>>>>>> 1/3= 0.333... + non_zero_remaindera
> >>>>>>
> >>>>>> And what non-zero number is that remainder? (in The Reals).
> >>>> [...]
> >>>>
> >>>> 1/3 = .333... in base ten, therefore, .333... = 1/3 in base ten.
> >>>
> >>> Try an algorithm converting decimal fraction input to rational
> >> very easy.
> >>
> >> simply follow out the decimal until it repeats
> >>
> >>
> >> https://www.wikihow.com/Convert-Repeating-Decimals-to-Fractions
> >
> > How do you know it repeats infinitely?
> > Assuming you don't know what algorithm is:
> > If I send a long list of 0.333..., can you determine the number I send?
> Dont you know that the three dots ... means it repeats forever.

"..." is context dependent.
You avoided the question. But OK, we can go back to the repeating decimal issue.

> => *read the url*
>
> your repeating digit is 3
>
> take 3/9, simplify = 1/3
>
> NOW so you say the repeating is 333 above, and not 3
>
> then it is 333/999 = 3*111/(9*111) = 3/9 = 1/3 => CORRECTOMUNDO !!
>
>
> => ask yourself, when I divide by 9, what happens ? (*the digits repeat!*)
>
> what about 0.35353535...
>
> repeating digits are 35
>
> so answer is 35/99
>
>
> => *read the url* again
>
>
>
> [Extra credit: Show how you can do this in base7 ]

35/99= 0.(232156036500663351424544112620)(base 7)... + non_zero_remainder

Yes, a finite string repeats infinitely.
But the infinite repeat is based on the existence of non_zero_remainder, is
always part of the equation. Otherwise, no infinite repeat can happen in long
division or whatever algorithm.

The point is that non_zero_remainder must exist to balance the equality.

[FromTheRafters]

wij expressed precisely :

Is Pi over three times Pi a rational number?

[wij]

Strange question. (π/3)*π= π^2/3?
IIRC, π^2 is an irrational number,

[Serg io]

On 12/28/2021 2:44 AM, wij wrote:
> On Tuesday, 28 December 2021 at 12:48:59 UTC+8, Serg io wrote:
>> On 12/27/2021 8:32 PM, wij wrote:
>>> On Tuesday, 28 December 2021 at 10:18:29 UTC+8, Serg io wrote:
>>>> On 12/27/2021 7:50 PM, wij wrote:
>>>>> On Tuesday, 28 December 2021 at 04:09:33 UTC+8, Chris M. Thomasson wrote:
>>>>>> On 12/26/2021 10:35 PM, wij wrote:
>>>>>>>>>
>>>>>>>>> 12 years-old kids know immediately what is meant by a single line:
>>>>>>>>>
>>>>>>>>> 1/3= 0.333... + non_zero_remaindera
>>>>>>>>
>>>>>>>> And what non-zero number is that remainder? (in The Reals).
>>>>>> [...]
>>>>>>
>>>>>> 1/3 = .333... in base ten, therefore, .333... = 1/3 in base ten.
>>>>>
>>>>> Try an algorithm converting decimal fraction input to rational
>>>> very easy.
>>>>
>>>> simply follow out the decimal until it repeats
>>>>
>>>>
>>>> https://www.wikihow.com/Convert-Repeating-Decimals-to-Fractions
>>>
>>> How do you know it repeats infinitely?
>>> Assuming you don't know what algorithm is:
>>> If I send a long list of 0.333..., can you determine the number I send?
>> Dont you know that the three dots ... means it repeats forever.
>
> "..." is context dependent.

no, it is not. that is where your mistake is.


> You avoided the question.

no, I corrected you.

> But OK, we can go back to the repeating decimal issue.
>
>> => *read the url*
>>
>> your repeating digit is 3
>>
>> take 3/9, simplify = 1/3
>>
>> NOW so you say the repeating is 333 above, and not 3
>>
>> then it is 333/999 = 3*111/(9*111) = 3/9 = 1/3 => CORRECTOMUNDO !!
>>
>>
>> => ask yourself, when I divide by 9, what happens ? (*the digits repeat!*)
>>
>> what about 0.35353535...
>>
>> repeating digits are 35
>>
>> so answer is 35/99
>>
>>
>> => *read the url* again
>>
>>
>>
>> [Extra credit: Show how you can do this in base7 ]
>
> 35/99= 0.(232156036500663351424544112620)(base 7)... + non_zero_remainder
>
> Yes, a finite string repeats infinitely.
> But the infinite repeat is based on the existence of non_zero_remainder,

no. It is because it is a rational number.

> is
> always part of the equation.

no.
In Math Everyone uses the three dots, or the over strike to indicate repeating decimal string, you fail to do so to say that part is variable and must
be your non_zero_remainder, which has less information that the three dots or over strike.

>Otherwise, no infinite repeat can happen in long
> division or whatever algorithm.

no, we are only talking about nomenclature, the representation of numbers, that is all.

>
> The point is that non_zero_remainder must exist to balance the equality.

no, you way is far less accurate.

[Serg io]

wrong, try again

[Ross A. Finlayson]

Delta-epsilonics and limit theory....

The delta-epsilonics or bounding rule is a great set of conditions to
more or less understand that limit theory and delta-epsilonics arrives
at a basis for real analysis, treated in the unbounded.

Calling it that "delta-epsilonics" for the proofs or cases in character
of delta and epsilon, small constants one or the other is smaller like
an infinitesimal or so established in the case to be no different than
zero as what results, limit theory and delta-epsilonics are a usual pre-calculus.

[Ross A. Finlayson]

The great point about delta-epsilonics is that according to induction,
it's sound. And: it results exactly and always in the corresponding
conditions the right result, not like these other inductions according
to an infinity in mathematics being un-sound.

Also there's also the importance of extending the framework to be
sound, in a usual sense like "ah, I've proven it finite, no, I've proven
it finite or zero".


Then the point is though that while limit theory and delta-epsilonics
result fundamental for resulting applications in real analysis, then also
there's about the fundamental geometry and foundations, that
higher-order applications of infinity, result from what results must have
been (or corresponds with solely) that the supertask in clock arithmetic
is the usual task of time.

Then, time is usually a continuous quantity.

Then, most people know modern science with the theory of atoms and
molecules, particles and waves, and their duality, superstring theory as
"superstrings are about a continuous quantity to these what are atomic
in our particle model of them the constituents of matter", relativity and
quantum mechanics, and about the Particle Zoo and the Standard Model,
about the four or five usual fundamental forces what result a unified field
theory and a gauge theory, and may have heard of scattering/tunneling and
wavepacket, soliton, instanton, ..., parallel transport and a usual geometric
algebra's model of linear systems, most people know about modern science
with what people tell them science tells them.

[Serg io]

really? do go on...

[Ross A. Finlayson]

Delta-epsilonics and limit theory totally sets up the rest of the curriculum,
after trigonometry which is used for basic results about the plane.

The identities of trigonometry, that I forget, include these usual ratios in
their constants root two, root three, and two, about usual quadrants and
octants, thus that sine and cosine about the circle with tangent their outside,
make all trigonometry, resulting for what is in the symbolic analysis, the
most usual example of orthogonal functions, that under their operators
are complement.

Given delta-epsilonics and limit theory and the usual fundamental theorems
of calculus, is that most all the usual Cauchy-Weierstrass forms in the Riemannian,
which includes the Stieltjes and Lebesgue as it were, and is usually for example
framed as included from Stieltjes or Lebesgue instead, the integral defined as
the sum of the areas under the curve, in the limit, makes sense in the abstraction
that the differential is arbitrarily small, while still summing in constant function
to one.

Points include "I forget this, it's the same: must be fundamental".

[Ross A. Finlayson]

... Which is cumbersome, the curriculum, what results restatement,
for results.

Then a general course of "this is a general course of mathematics",
is along the ideas of "here are for example twelve grades of mathematics
in school and what-all's a usual course that results abstractly, it's also
built formally later the same way in the course, general mathematics".

Then here that's algebra and geometry around seventh and eighth grade,
then trigonometry, then pre-calculus and calculus, though it's only my own
education in effect it's the one I know.

Then most of the crankiness and pottiness of "these are fundamental
elements in elements, and their sound definition cannot be conflated
except what results extensionality, formally", of course anyone here can
tell you about some sound mathematics quite all day: but any non-logical
object in a sense is thus proper, abstract, and profound.


Then it's easier later to build a model of continuous 0-1 though it results
all the time having to maintain a model in field continuity and also a model
in line continuity, writing results in field continuity.

I.e. it's easier, more direct, usually more understood, ..., "with the caveat
that all these numbers are also members of the complete ordered field".

I.e., when there are "different" sets or models that here establish the
properties of a continuous domain or continuity under definition, it's
entirely framed the usual sets of continuous functions, in effect. Here,
the extensionality that usually follows from sets by their models is that
"the models though the same objects are differet, ..., 'sets', these 'all the
real numbers in R bar and all the real numbers in R dots', sets in set theory".

I.e., there's more for a sense that "this line-drawing is fundamentally more
primitive or primary what results from instead the usually most primary
which is integers or two integers".

It's not without the caveat that "of course formally what are the models of
expectations ... are that my counting arguments are integer, in combinatorics",
is also for that probabilities are real values 0-1. ("Zero One")

It's probably more usual historically often to be this way for continuity,
then also what's key after modern formalism or since the Renaissance or
Enlightenment, also our modern includes after set comprehension for logic
as instead only the geometric also the (eventually...) combinatoric,
here it's the post-modern mathematical formalism after its critique the
1900's, these days is the 2000's, though it seems I put together all of
mathematics with logic and physics for pure theory.

Or, at least the beginning of this kind of thing is along these lines:
about resolving mathematical paradox by establishing all of it or
"foundations", in the sense that lightning bolts from heaven in a
sense are not the giant lightning bolt from the center of the universe.

Which is a thing....

Read over the Banach-Tarski sci.logic thread I'd say, if you might mechanize
a usual diagram of statement, my diagrammhea of a sort shows an example
of how the "paradox" is seated in geometry: that there's not a paradox about
Banach-Tarski yet it remains geometrical. That's an example: where I have
made such sorts of examples as an entire affront.

[Ross A. Finlayson]

It's just that after modern mathematics there were still the mathematical
"paradoxes" that are to get sorted out in a theory what result not them.
(While still resulting the rest of the theory or the theory, ..., i.e. some sound
complete theory that if G-d is a perfect mathematician has one.)

For which I had to write out "this is the equivalency function, here is an
entire catalog of statement to its effect, it's not paradoxical then indeed
in this theory it's shewn the mathematics' bridge from countable to uncountable
or infinity to finity for that matter".



That's all....

[mitchr...@gmail.com]

They are different quantities and different representations
but their difference is by the unlimited small that calculus
saw behaved the same to the no quantity.

Mitchell Raemsch

[Chris M. Thomasson]

You are making me think of the following question...

When does the following sequence equal one:

[0] = .0
[1] = .9
[2] = .99
[3] = .999
[4] = .9999
[5] = .99999
[...] = [...]

Or, perhaps, at what iteration does this _perfectly_ equal one?

[Ross A. Finlayson]

The difference between four or five or six or seven orders of magnitude,
is that the usual odds or "five 9's is in the time of the company while
six or seven 9's might be out to the time of the dinosaurs", is that
it equals 9/10 at least for all of those.

I.e., here it also when is .1, .01, .001, or 1-[], that is in odds for what is
an effective term, an example, of that 1-[] is less than .1, less than .01,
....

Basically the difference then is where there is insurance and risk.

That the sequence with n-many 9's is [n], is about that it looks like
those array dimensions are the same, compared to say n and 2^n,
or whatever results under combinatorics, counting-arguments,
combinatorial enumeration of course, here for example one notices
for the term that the offset and length are same.

Then, it's how or why those things apply for example make expressions.


So, it's usual then that it's seven 9's where it's no different than not not one,
in an example that is a function in numbers in arithmetic,
infinite-precision arithmetic.

A large scalar word like 1024, makes for that estimating 2^10 and 10^3 this
way, makes for a usual means of estimating binary and decimal about same
after making for converting kibibyte and kilobytes, figuring for example that
often people write out kilobytes when they mena kibibytes, in a general sense
of the order of magnitude that results, where 1024 bytes is one kilobyte.

[Chris M. Thomasson]

[...]

Delta would be the target, minus the iteration value, so:

delta[0] = 1 - 0 = 0
delta[1] = 1 - .9 = .1
delta[2] = 1 - .99 = .01
delta[3] = 1 - .999 = .001
delta[4] = 1 - .9999 = .0001
delta[5] = 1 - .99999 = .00001
...

Each delta added to the iteration equals one.

[Ross A. Finlayson]

About, graphics, or, the floating-point unit of the usual general
purpose CPU often after Intel 32 and AMD 64 bit architecture,
there is from the i586 the SIMD registers, multimedia extensions
or what is making use of the floating point unit and the arithmetic
logic unit with respect to register transfer logic, the point being
we could talk about computer architecture in general, C and C++ in
detail, still mostly I am just a coder and for ARM and RISC besides CISC,
besides, thinking about the usual organization of the PC platform,
yes until I write an executive and OS for example for PC 2000 platform,
or before, what results from some usual machine bootstrap an
environment, I am just a coder with usual ideas of how everything works.

I.e. totally all the code I write these days is in basically configuration
or is some framework or tooling, sure a more direct platform might
be to target the machine for its usually routines of scheduling all the
units, now I am thinking about what I can do to turn my Java routines
into native high-performance code, that exercises more or less the
entirety of the processor's and coprocessors' execution, or for example
makes for simple accessors then a more or less linear and compiled routine,
and for the "very long instruction word".

So, I see how you can write out there a ". ... 01", as much writing a ". 999 ... 99".

It's like, given a floating point number, in usual say IEEE 754 or I don't know
the internals, but where float is say 32 or 80 and double 32 or 80, digits in bits,
not to get into sizeof so much though it's >= sizeof int, if it's for a quotient
of two integers, and a rational number, if the denominator is not very large,
then at least looking for ".3, .33, .333., ..., in the first so many digits at least"
is an example for, besides delta-epsilonics, also treating floating point values
as rational value holders, under some caveat that the denominator is small.

I.e. otherwise for example it's often for matters of accumulation or rounding,
that it's mostly otherwise usually a fixed-point routine, here with an example
of an idea of some bounded denominator, which under the bounds of the rational
number's numerators would be extractable from the floating point representation,
or machine format, anyways it's in these systems of bounds, what more or less,
make for "according to concrete mathematics, this is enough 9's".

[Chris M. Thomasson]

Try convergents of continued fractals to hit a desired precision of a
.333.....

[Ross A. Finlayson]

Arriving at "convergents", you say fractals, continued, and when I think
of fractals it's mostly, parameter, like, Lindenmeyer, or, cellular, for example
Wolfram, what I associate what a "fractal, continued", could be is really in
their continuation and as the converge or under each others' boundary conditions,
i.e. their boundaries whatever conditions they are, it seems you are also
describing basically some resonant systems in what also make fractals that
arrive at having a value that starts the same as 1/3's or some other given number.

I.e., I imagine with shader tools and such for example you might have in mind,
I know Chris has many examples of graphics, and in fractals, basically what
arrive at that the convergents are basically the result of iteration, that happens
to be in the world of shader tools and dedicated graphics, here then where I'd
wonder why he would care, ".333, ...".

I would see though why I would care where basically it makes for a graphically-driven
analytics solver of a sort, for whatever inputs arrive at what resolves, usually
according to mouse, or pointer input.

I.e. why it would be desired, is for under whatever other conditions it guarantees,
given it has the same value generated from arithmetic as from the fractals
(that whatever "convergent, continued fractals continued and converged
that instead of the fractal dimension the arithmetic result happens to be 1/3").

I.e. if there's some conditions where that's a thing I'd wonder, how to care.

[Chris M. Thomasson]

13/(3*13)=1/3
1/(3*1)=1/3
42/(3*42)=1/3
pi/(3*pi)=1/3

;^)

[wij]

You have presented lots of irrelevant smokes (to blind viewers including yourself
or to scare kids whose brain are not yet fossilized?).
The question is simple: Does 1/3 equals exactly to 0.333... or not?

The "1/3=0.333..." question is a fundamental thing, using any theory higher
level than basic arithmetic to prove "1/3=0.333..." (right or wrong) is simply INVALID.

No one has a valid proof that 1/3 exactly equals "0.333..." but "theory"

[FromTheRafters]

on 12/29/2021, Chris M. Thomasson supposed :

Rational numbers are ordered pairs of integers. Relying upon algorithms
would mean he would have to complete calculating pi before multiplying
the result by three and the numerator and denominator are still not
integers. However, pi can be divided by pi to reduce the fraction to a
rational number.

[FromTheRafters]

on 12/29/2021, wij supposed :

Wrong, see "Series" and "Summation".

[FromTheRafters]

Chris M. Thomasson formulated on Tuesday :

Never. This is an infinite sequence of partial sums. Summing this
sequence (which is a series) does not yield one.

https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

[Serg io]

wrong. and very old stale troll food.

same can be said about 1/2 exactly equals "0.5000..." how do you know those 0's continue ?? or are you ASSUMING they do ?

so then using your logic, 1/2 is not equal to 0.500...

and WHAT ABOUT THE PRECEEDIGN 0's ?

1/3 = ...000.333... not 0.333... who knows what came before the 0 !!

nomenclature.

silly trolls argue about nomenclature, and not math, cause they do not know any math.

[Ross A. Finlayson]

Yeah, if someone tried to tell me I thought that, that there wasn't a
.333... and it was 1/3, I would become entirely outraged and more-or-less
thus reject the entire agen-da.

I think it's clear these days that when introducing numeric identities
that result after division about .999, that besides that one that reads
"carry .9, carry .09, carry .009, ..., all carry" is the one that reads "..., 1,
..., 2, ..., ..., 9, carry".

I.e. it's ignorant not to notice that writing .999... is _both_ a notation
for the overload of ellipsis "..., continue" with "carry on modulus in
the field" and "carry with clock in time". (The partial results that
accumulate and that both ways there exists a limit and it's 1.)

I.e., it should be taught as about the fact that there are mutiple
representations of rational numbers in expansions that are written
like .123..., that a notation that results "yes this does mean .9, then .99,
..., forever, and .999..., for example as 1/3 * 3", is as clear as reading off
the value of what its limit is, then about clock arithmetic there's that
where there exists a limit of n/d as n->d and d->oo and it happens to
exist and its one that each of _its_partial terms also _builds the same
digits_, though as it results .000 ... 1 to .999 ... 9 the clock as modulus
instead of radix as modulus, it results that the sweep of the clock's
hand crosses the paths of all the . ... . expansion in the modulus.

I.e. saying that "besides the usual notation where .999... = 1 as its
limit is so as built from the radix" is "this notation .999(...) < 1 reflects
that according to any scale 1: . 000 ... (1) that [0,1] includes it as a
value by notation", i.e. it's as well the case that the limit exists where
the sum _doesn't_, while in the usual case, it's built incrementally.


So, it's possible this way to much better sort out what ".999" and "..."
mean, for what they are that happens to write them as that in the
limit i.e. when the writing is both unbounded and exhausted, here
is for the various expressions and that the modulus is of the field,
and, that the modulus is of the space, in the sense of algebra's field
and geometry's space. (Their definitions.)

Yeah, after all the hard work to learn long definition,
".333..." means "1/3". But, since I learned to count to ten,
it's still "1 2 3, ..., 9, 10".

[Serg io]

3/4 of the world do not use math at all, and will never understand what 0.333... even means.

>
> I think it's clear these days that when introducing numeric identities
> that result after division about .999, that besides that one that reads
> "carry .9, carry .09, carry .009, ..., all carry" is the one that reads "..., 1,
> ..., 2, ..., ..., 9, carry".

it is shift right, not carry.

[Ross A. Finlayson]

In tally there is only carry.

Most people read and write numbers in base 10 but there also 2, e,
and for that matter base 1, tally arithmetic, which is pretty much the
same as having only one digit that ranges the infinity of numbers, base infinity.

That logarithms are natural they're also general, roots.

Truth is regular....

[Chris M. Thomasson]

When I see the symbol pi, I think this is pi in all of its infinite
glory. :^) Now, when calculating pi as a rational, one way is get a
rational up to a certain precision using convergents of continued
fractions. Implement atan from scratch and go:

atan(1) * 4 = pi

Perhaps using CORDIC.

[Serg io]

did you know COBAL has an ATAN function ?

and there is the base16 Bailey–Borwein–Plouffe formula that computes the n-th digit of pi very cool

[Chris M. Thomasson]

Never used COBAL, but, it should have one! I am more of a C/C++ guy.

>
> and there is the base16 Bailey–Borwein–Plouffe formula that computes the
> n-th digit of pi  very cool

Indeed!