On Thursday, 16 December 2021 at 12:06:13 UTC, richar...@gmail.com wrote:
> On 12/16/21 6:15 AM, wij wrote:
> > On Thursday, 16 December 2021 at 09:13:54 UTC+8, richar...@gmail.com wrote:
> >> On 12/15/21 5:18 PM, wij wrote:
> >>> On Thursday, 16 December 2021 at 06:16:09 UTC+8, wij wrote:
> >>>> On Wednesday, 15 December 2021 at 21:21:00 UTC+8, richar...@gmail.com wrote:
> >>>>> On 12/15/21 7:01 AM, wij wrote:
> >>>>>> On Wednesday, 15 December 2021 at 09:55:45 UTC+8, richar...@gmail.com wrote:
> >>>>>>> On 12/14/21 12:13 PM, wij wrote:
> >>>>>>>> On Tuesday, 14 December 2021 at 19:58:14 UTC+8, richar...@gmail.com wrote:
> >>>>>>>>> On 12/14/21 5:10 AM, wij wrote:
> >>>>>>>>>> On Tuesday, 14 December 2021 at 11:34:59 UTC+8, Richard Damon wrote:
> >>>>>>>>>>> On 12/13/21 10:25 AM, wij wrote:
> >>>>>>>>>>>> On Monday, 13 December 2021 at 20:53:29 UTC+8, richar...@gmail.com wrote:
> >>>>>>>>>>>>> On 12/13/21 7:27 AM, wij wrote:
> >>>>>>>>>>>>>> On Monday, 13 December 2021 at 02:38:48 UTC+8, Mike Terry wrote:
> >>>>>>>>>>>>>>> On 12/12/2021 18:15, Richard Damon wrote:
> >>>>>>>>>>>>>>>> On 12/12/21 10:24 AM, Mike Terry wrote:
> >>>>>>>>>>>>>>>>> On 12/12/2021 08:48, wij wrote:
> >>>>>>>>>>>>>>>>>> Example 1:
> >>>>>>>>>>>>>>>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999....
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> ok. So A = 1
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> = 999.../1000...= (3*3*(11...1))/(5*2)^n
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Does not compute. 999... and 1000... are not numbers.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> No, of course not. (Does not compute)
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, what is the 3rd digit after the decimal point of A?
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Real numbers may have one or two decimal representations, a bit like
> >>>>>>>>>>>>>>>>> +0 = -0, or 3/5 = 6/10. A has two representations: 0.999... and
> >>>>>>>>>>>>>>>>> 1.000.... The 3rd digit after the decimal point of representation
> >>>>>>>>>>>>>>>>> 0.999 is 9, while for the representation 1.000 it is 0.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> What density property is that? (And how do you think it is broken?)
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> I believe he means the property that between any two members of the
> >>>>>>>>>>>>>>>> Real, or the Rationals, there will ALWAYS be another member of that set
> >>>>>>>>>>>>>>>> between them. I.E., there is NOT a 'next' value from a given value.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Yes, I thought he might mean that. I wouldn't call that "density"
> >>>>>>>>>>>>>>> myself, as "density" has a different meaning. Perhaps the "denseness"
> >>>>>>>>>>>>>>> property?
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Amazing! What a phenomenon!
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> google "density property".
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Density Property::=
> >>>>>>>>>>>>>> For any two different numbers, there exists another different number in
> >>>>>>>>>>>>>> between. Take interval [a,b] for instance: ∀i,j∈[a,b], i<j such that ∃k,
> >>>>>>>>>>>>>> i<k<j.
> >>>>>>>>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> But if A=1, how does this break the property? It doesn't - the property
> >>>>>>>>>>>>>>> breaks if A != 1, so that would make wij's claim plain Wrong, like
> >>>>>>>>>>>>>>> everything else he said. :) [no surprise, I guess.]
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Mike.
> >>>>>>>>>>>>>>>> One value between x and y will be (x+y)/2
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> The problem with thinking of 0.9999.... as something distinct from 1 is
> >>>>>>>>>>>>>>>> THAT breaks the density property, as there can be no number bigger than
> >>>>>>>>>>>>>>>> 0.9999... and less than 1.0000
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Mike.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Let A(0)=0, A(n)=(A(n-1)+1)/2, n∈ℕ
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Given two different numbser A(n), and 1, there always exists another different
> >>>>>>>>>>>>>> number A(n+1) such that A(n)<A(n+1)<1
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> When A(n)=1? Infinity?
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>> There is no FINITE n where A(n) is equal to 1
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> Neither a FINITE n is in limit.
> >>>>>>>>>>>> What is the n in "lim(n->∞) A(n)=1"? Finite, infinite, or not a number?
> >>>>>>>>>>> Each n is a finite number.
> >>>>>>>>>>>
> >>>>>>>>>>> The key is that the limit of a sequence doesn't need to be a member of
> >>>>>>>>>>> the sequence, and in fact, normally isn't.
> >>>>>>>>>>>>
> >>>>>>>>>>>>> Note, for the Reals, Naturals, etc., 'Infinity' isn't a value, only a
> >>>>>>>>>>>>> 'limiting case'
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Thus the limit(n->infinity) A(n) is 1, even though no individual A(n) is 1.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> This is a common property of limits.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> "No individual A(n) is 1. But limit(n->infinity) A(n) is 1".
> >>>>>>>>>>>> So limit theory turns 'approaching' to 'equal' in term of the limit smoke.
> >>>>>>>>>>>> Where I can find evidence that A(∞)=1 but from the 'approaching is equal'
> >>>>>>>>>>>> theory is the problem.
> >>>>>>>>>>> Right, the terms approach the limit.
> >>>>>>>>>>>
> >>>>>>>>>>> The limit is that value that terms get arbitraryily close to.
> >>>>>>>>>>>
> >>>>>>>>>>> A(infinity) isn't a proper notation, as A is a sequnce with Natural
> >>>>>>>>>>> Number indexes, and infinity isn't a Natural Number.
> >>>>>>>>>>>
> >>>>>>>>>>> One definition of 'The Limit' of a sequence is the number L, that for
> >>>>>>>>>>> any given arbirary positive value e, there is some N where all elements
> >>>>>>>>>>> of the seqence A(n), for all n > N, that |A(n) - L| < e
> >>>>>>>>>>>
> >>>>>>>>>>> i.e, for any arbitrarily chosen precision, we can find a point in the
> >>>>>>>>>>> sequence where it stays inside that bound.
> >>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Just like 0.9999... for any finite number of 9s isn't equal to 1, but
> >>>>>>>>>>>>> the limiting case with the endless 9s is.
> >>>>>>>>>>>>
> >>>>>>>>>>>> I am not talking about 'limiting case'. Limit theory is full of inconsistency.
> >>>>>>>>>>>> (Every one learned 'limit method' should have a sense of this. I do not what to dig into this shit deep)
> >>>>>>>>>>>> We should be interested in the case that 0.999... equal to 1 or not, not the "limiting case".
> >>>>>>>>>>> Maybe you should look at it again.
> >>>>>>>>>>>
> >>>>>>>>>>> If you aren't going to use the right definition of Limit, and the range
> >>>>>>>>>>> of the Natural, Rational, and Real number, don't use those terms.
> >>>>>>>>>>>>
> >>>>>>>>>>>> And, here, right now, the density property in this thread.
> >>>>>>>>>>>> Does not 'density property' mean to hold infinitely?
> >>>>>>>>>>>> The problem is: The density property procedure can go on infinitely. Can not?
> >>>>>>>>>>>>
> >>>>>>>>>>> Not sure what you mean by 'infinitely' here, especially if you reject
> >>>>>>>>>>> the concept of a limit. It can be done an unbounded number of times.
> >>>>>>>>>>>
> >>>>>>>>>>> Remember, when we are talking about counting with Natural numbers, there
> >>>>>>>>>>> is NO infinity. Infinity is just a limit we can approach.
> >>>>>>>>>>
> >>>>>>>>>> Right. "Infinity is just a limit we can approach". So the following:
> >>>>>>>>>> 0.999... can never reach the limit 1.
> >>>>>>>>> Wrong, by that logic we can't have a number like 0.9999.... or 0.3333...
> >>>>>>>>> because they are only that value 'in the limit' when we get to the
> >>>>>>>>> infinite number of digits.
> >>>>>>>>
> >>>>>>>> By what logic? I have no problem you can't have the number like 0.999... or
> >>>>>>>> 0.333... in mind or practice. As said, you just keep fixating on LIMIT theory
> >>>>>>>> and fabricating stories from the copy in brain to fool yourself again and again
> >>>>>>>> (like PO?). But I won't say lying.
> >>>>>>> Except that we KNOW that a number like 0.3333.... does exist in
> >>>>>>> practice. So we need some notation to handle it, or do you just want to
> >>>>>>> 'give up' and say that the only rationals that (in reduced form) have a
> >>>>>>> denominator consisting only of powers of 2 and 5 exist in decimal form?
> >>>>>>>
> >>>>>>> If 0.3333.... doesn't exist, then does 1/3? (Its the same number) Or is
> >>>>>>> 1/3 just not expressible as a decimal?
> >>>>>>
> >>>>>> If you insist changing the subject to the limit theory,
> >>>>>> as said, I don't want to dig deep into the shit deep of limit theory.
> >>>>>> If it is no problem to you, a possible reason is that you don't really
> >>>>>> use it, understand it, never encountered the contradictory, you just reciting
> >>>>>> the shallow memory imprint.
> >>>>> Except that the MEANING of any INFINITE series, which is what the ....
> >>>>> notation implies, is derived via limit theory (or related concepts)..
> >>>>>
> >>>>> Just like I tell PO, if you won't follow the DEFINITIONS of the system
> >>>>> you claim to talk about, you aren't talking about that system, but
> >>>>> something else.
> >>>>>
> >>>>> The sets of Natural, Rational, and Real numbers do NOT have a member
> >>>>> that represents 'Infinity', but only have it as an auxilary concept that
> >>>>> corresponds to limits.
> >>>> It is you not talking in the topic the thread "0.999...=1 or not and the
> >>>> density p", and insist I claimed I was talking about your "an established
> >>>> field of Mathematics" (what is that is also debatable).
> >>>>
> >>>>> Series APPROACH infinity.
> >>>>
> >>>> And, APPROACH means EQUAL (Pythagoreans' logic)
> >>>> Why not limit theory admit this plainly straight?
> >>>>>>
> >>>>>> E.g. given an interval [0,1/3), question: does 1/3 in [0,1/3) or not?
> >>>>>> (0.333... is an irrational number, no exact rational p/q form.)
> >>>>> Nope. 0.33333..... is the EXACT value of 1/3.
> >>>>>
> >>>>> Of course 1/3 is NOT in the open interval that ends at 1/3, that is the
> >>>>> DEFINITION of the open interval.
> >>>> Typo, it should be: Given an interval [0,1/3), question: does 0.333.... in [0,1/3) or not?
> >>>>>>
> >>>>>>>>
> >>>>>>>>> No finite number of 9s in 0.99999 will be equal to 1, but IN THE LIMIT,
> >>>>>>>>> when we imagine that we reach that infinite end, it is.
> >>>>>>>>
> >>>>>>
> >>>>>> Yes, BY IMAGINE we reach the infinite end, not by proof.
> >>>>> No, by the proof using the property of limits, which is how these system
> >>>>> DEFINE dealing with the infinite value.
> >>>> You said "when we imagine that we reach that infinite end, it is."
> >>>> lim(x->1) x=1
> >>>> limit theory explicitly says x approaches 1 but never be exactly 1.
> >>>> (but in the end, x is 1)
> >>>>>>
> >>>>>>>> Where is the evidence 0.999... WILL be equal to 1 (without breaking the density
> >>>>>>>> property).
> >>>>>>>> Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
> >>>>>>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
> >>>>>>>> yet still not exactly 1. Not a number? Or just too small(or too great)?
> >>>>>>> You seem to be stuck on the finite. Yes, No FINITE listing of the digits
> >>>>>>> of 0.999... will be equal to 1.
> >>>>>>
> >>>>>> You said: "...when we imagine that we reach that infinite end, it is."
> >>>>>>
> >>>>>>> The equality ONLY happens in the limit
> >>>>>>> when we allow for there to be the INFINITE number of digits. (and no
> >>>>>>> finite number is infinite).
> >>>>>>
> >>>>>> I have shown 0.999... can be INFINITE long, and yet not 1.
> >>>>> Nope, you have shown that for an unbounded length, it is differnt, not
> >>>>> for an INFINITE length.
> >>>> What are you talking about? I used the word "INFINITELY" explicitly, and
> >>>> it became "unbounded" in you eye.
> >>>>
> >>>> [quote] ...Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
> >>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
> >>>> yet still not exactly 1 ...
> >>>>>>
> >>>>>>>
> >>>>>>> The 'proof' is in the definition of the limit.
> >>>>>>
> >>>>>> limit has no valid proof. It has 'definition' and 'explanation' and smoke.
> >>>>>> But, definition is a lowest level of understanding --- limit don't understand
> >>>>>> what 0.999.... is but must use it as non-1 at the beginning and use it as
> >>>>>> exactly 1 latter.
> >>>>> LIMIT is how infinite is defined to exist in the discussion.
> >>>>>>
> >>>>>>> Give me any positive real
> >>>>>>> number, no matter how small, and I can find the finite number of 9's
> >>>>>>> that will make that value, and all those pass it closer to 1 than that.
> >>>>>>>
> >>>>>>
> >>>>>> And, the number you use to approach is still finite. Isn't it?
> >>>>> Right, for every FINITE error, there is a FINITE length you need to
> >>>>> acheive, which means that in the LIMIT, the error goes to ZERO when you
> >>>>> include the INFINITE length.
> >>>>
> >>>> "Give me any positive real number, no matter how small, and I can find the finite number of 9's
> >>>> that will make that value, and all those pass it closer to 1 than that."
> >>>> Where can you find 1 in the sequence 0.999..., where every element in the sequence
> >>>> has non-zero error.
> >>>>>>
> >>>>>>> This is the way the Reals, et all, handle infinite series.
> >>>>>>
> >>>>>> Your Real is Q plus numbers that have finite notation.
> >>>>> Nope, Sqrt(2) has no finite decimal notation, but is a Real.
> >>>>>
> >>>>> 0.333... is a non-finite notation, but is in Q.
> >>>> you need to prove 0.333...=1 (NOT BY your favorite DEFINITION)
> >>>>>>
> >>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Let f(n)= (2*n+1)!/((n!)^2*2^(3*k+1))
> >>>>>>>>>> S= Σ(n=0,∞) f(n) = √2
> >>>>>>>>>> S can never reach the limit √2 (albeit infinitely approaching, and, all the
> >>>>>>>>>> instances of the sequence and the sum are rational).
> >>>>>>>>> Wrong, No S(k) = Σ(n=0,k) f(n) will equal √2, but IN THE LIMIT, S does.
> >>>>>>>>>>
> >>>>>>>>>> This is the blind spot of Pythagoreans:
> >>>>>>>>>> --- Infinitely approaching means equal. Number too small equals zero. ---
> >>>>>>>>>>
> >>>>>>>>>> Yes, we should remember, "Infinity is just a limit we can approach".
> >>>>>>>>> And thus, 'In the limit', we reach it.
> >>>>>>>>> Not for any finite step, but in the limit.
> >>>>>>>>
> >>>>>>>> What is in discussion is whether 0.999...=1 or not without breaking the density property.
> >>>>>>>> Not the limit theory.
> >>>>>>>>
> >>>>>>>> --- Pythagoreans' Code ---
> >>>>>>>> Infinitely approaching means equal. Number too small equals zero..
> >>>>>>>>
> >>>>>>> Except that on common definition of what the ... notation means is based
> >>>>>>> on limit theory.
> >>>>>>
> >>>>>> Nope. I would say infinite series.
> >>>>> Whose value is based on Limit Theory.
> >>>>>>
> >>>>>>>
> >>>>>>> We actually don't need limit theory to handle 0.9999.... as being e
> >>>>>>> equal to 1.
> >>>>>>>
> >>>>>>
> >>>>>> Really? this showed you don't even really understand limit.
> >>>>> Do you deny the property is true? Can you find a counter example?
> >>>>>>
> >>>>>>> There is the other property, that any repeating fraction 0.xyzxyzxyz...
> >>>>>>> can be also expressed as a fraction of the unit xyz divided by the
> >>>>>>> number of 9's of the repeat cycle (in this case xyz/999)
> >>>>>>>
> >>>>>>> For example: 1/7 = 0.142857 142857 .... = 142857 / 999999
> >>>>>>>
> >>>>>>> If there are some leading decimal digits that aren't part of the repeat,
> >>>>>>> put those as a fraction over the right power of 10 and then add the same
> >>>>>>> number of 0s after the 9's.
> >>>>>>>
> >>>>>>> By this property 0.333... = 3/9 = 1/3, and 0.9999.... = 9/9 = 1.
> >>>>>>>
> >>>>>>> (Note, this also works for ANY base >= 2)
> >>>>>>
> >>>>>> Show me the whole derivation/argument.
> >>>>> Its been a while, let me look it up.
> >>>
> >>> Typo:
> >>> you need to prove 0.333...=1/3 (NOT BY your favorite DEFINITION)
> >> Just do the math by long division.
> >>
> >> If you think it isn't, do you think that not all rational number have
> >> decimal representations when including repeating representations?
> >>
> >> 1/3 gives us 0 remainder 1, so 0 integal part.
> >> Going down a digit we multiply the remainder by ten (the base) and
> >> divide again.
> >>
> >> This gives us 10/3 = 3 remainder 1, so the tenths digit is 1.
> >>
> >> Since we previously had a 1 remained, we can just repeat this pattern:
> >>
> >> 1/3 = 0.3...
> >
> > You have just confirmed/proved the conversion of 1/3 to 0.333... always leave a '1'
> > remainder behind? To be exact, 1/3= 0.333... + non_zero_remainder.
> > So, the conclusion that "1/3 = 0.333..." exactly is wrong.
> Wrong. That is the difference between any FINITE representation an the
> infinite representation.
>
> You seem to be using a different definition of the number system then
> assumed by convention, which means you need to state that or you are
> just being deceptive.
>
> That is as bad as just insisting that 1 + 1 = 10 without giving a hint
> that instead of using decimal numbers we are using binary.
> >
> >>
> >> Or to flip things on you, how do YOU define the meaning of a decimal
> >> representation which ends in a repeating digit sequence.
> >
> > I would use infinite series.
> >
> >> Conventional Real Number theory tends to use the limit definition, if
> >> you reject limits, then you don't have any way to actaully deal with
> >> infinite series and the like.
> >
> > Pythagoreans' Code:
> > "Infinitely approaching means equal. Number too small equals zero.
> > All 'real' numbers are in form of p/q." >
> > People have no problem using 'real' number this way, lasted for >1500 years.
> > Modern Pythagoreans use just a little bit more different numbers and no idea
> > they are the essentially the same as ancient Pythagoreans also have no problem
> > using 'real' numbers.
> This just PROVES that you are not using the term 'The Reals' as used by
> Modern Mathematics, but are working under an alternate theory of
> Mathematics that works under different principles.
>
> This is fine, such alternate systems do exist and people do productively
> work in them. The issue is that to say you ARE working on the same thing
> that Modern Mathematics calls 'The Reals' when you are not, is just
> being a Liar.
>
> I don't know that variation well enough to know what its 'proper' name
> would be, but to call it 'The Reals' in a context that implies the
> modern definition, when it is not, just shows either ignorance or
> malice, I will let you determine which you are guilty of.
>
'real' is in scare quotes. Which means that the right to use the term is
being disputed (like someone might say creation 'science'). Whilst this
can have a point, generally it bogs down the discussion in pointless spats
over semantics.
It's not quite clear what a real number means physically. I suggested that
you have a line, mark off a point with a pencil, and that's your real number,
but this was rejected by the professional mathematicians (of whom I am
not one). As for "real" meaning "the opposite of imaginary", it's a commonplace
that "imaginary" is a bad term for a square root of a negative number, but
it has stuck and we must use it if we don't want to casue confusion.

On 16/12/2021 12:23, Malcolm McLean wrote:
> On Thursday, 16 December 2021 at 12:06:13 UTC, richar...@gmail.com wrote:
>> On 12/16/21 6:15 AM, wij wrote:

[Much snipped -- 500+ line articles are just /too long/.]

>>> Pythagoreans' Code:
>>> "Infinitely approaching means equal. Number too small equals zero.
>>> All 'real' numbers are in form of p/q."

Whatever else Wij may be thinking of, this is not the
"Pythagoreans' Code". The existence of irrational numbers was
[notoriously] known to the Pythagoreans themselves. The fact
that ℝ does not contain infinitesimals is usually attributed to
Archimedes and/or Eudoxus. Note that this is an /axiom/ of ℝ,
not a fact about numbers in general. "Infinitely approaching
means equal" is not a sensible statement. It's literally
nonsense in talking about ℝ; if you, Wij, want to avoid the
use of limits, then you need to spell out more carefully what
you are actually talking about.

>>> People have no problem using 'real' number this way, lasted for >1500 years.

They did have problems. You can do some simple calculus
by intuition, largely as was done from Archimedes to Newton, but
that approach was always controversial and ran into real problems
in the 18thC. Solved by axiomatic constructions, including limits,
in the 19thC plus developments/generalisations in the 20thC.

>>> Modern Pythagoreans use just a little bit more different numbers and no idea
>>> they are the essentially the same as ancient Pythagoreans also have no problem
>>> using 'real' numbers.

I try to make all possible allowances when discussing with
non-native writers of English, but the above has defeated me. I
can't make any sense of it, even in what I understand to be Wij's
version of mathematics.

>> This just PROVES that you are not using the term 'The Reals' as used by
>> Modern Mathematics, but are working under an alternate theory of
>> Mathematics that works under different principles.

Either that, or else Wij is simply wrong. We need Wij to
set out more clearly what his conception of "real" numbers is.

[...]
> It's not quite clear what a real number means physically. I suggested that
> you have a line, mark off a point with a pencil, and that's your real number,
> but this was rejected by the professional mathematicians (of whom I am
> not one).

Well, quite. "A line"? Any line? A straight line? What
points are on this line [without begging questions!]? How do you
"mark off" a point? With another line? What will you do about
the vast majority of members of ℝ that are not constructible --
indeed not even describable.

> As for "real" meaning "the opposite of imaginary", it's a commonplace
> that "imaginary" is a bad term for a square root of a negative number, but
> it has stuck and we must use it if we don't want to casue confusion.

Well, quite. Again.

> I'd advise wij not to fuss too much about what is really a real number, and
> to try to understand the conventional rules and definitions, but also to understand
> that, as you say, people can and do propose alternative systems, most of which
> never get anywhere, but a few of which prove productive.

I suspect that Wij isn't interested in the conventional rules.
But without knowing the Wij proposals in more detail, then as above
we can't know whether this is something different, and perhaps
interesting, or simply an ill-expressed list of misconceptions.

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

On 12/12/2021 2:48 AM, wij wrote:
> Example 1:

WRONG

Are Repeating Decimals Rational?
Repeating or recurring decimals are decimal representations of numbers
with infinitely repeating digits. Numbers with a repeating pattern of
decimals are rational because when you put them into fractional form,
both the numerator a and denominator b become non-fractional whole numbers.

For example, when you use long division to divide 1 by 3, the resultant
quotient is 0.33333…. However, when put it into fractional form, it's
made of positive integers that don’t have decimal points:
https://tutorme.com/blog/post/are-repeating-decimals-rational/

> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> If A=1, what is the 3rd digit after the decimal point of A?
> If A=1, density property (of ℚ and ℝ) is broken (false).
>
> Example 2:
> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
> always exists as the prerequisite of 'infinite' repeating)
>
> Note that A is DEFINED exactly 0.999..., not 1.
> Since the non-zero fractional pattern of the number (repeating decimal) is
> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
> decimal) is irrational.
>
> ----
> [Tip] More about "repeating decimals":
> A0= 0.9 9 9 9 ...
> A1= 0.99 99 99 99 ...
> A2= 0.9 99 999 9999 ...
> A3= 0.999 9 9999 9 99999 ...
> A4= lim(n->∞) 1-1/n
> A5= lim(n->∞) 1-2/n
> A6= lim(n->∞) 1-3/10^n
> A7= lim(n->∞) n/(n+1)
> ...
> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On Thursday, 16 December 2021 at 21:37:25 UTC+8, Andy Walker wrote:
> On 16/12/2021 12:23, Malcolm McLean wrote:
> > On Thursday, 16 December 2021 at 12:06:13 UTC, richar...@gmail.com wrote:
> >> On 12/16/21 6:15 AM, wij wrote:
> [Much snipped -- 500+ line articles are just /too long/.]
> >>> Pythagoreans' Code:
> >>> "Infinitely approaching means equal. Number too small equals zero.
> >>> All 'real' numbers are in form of p/q."
> Whatever else Wij may be thinking of, this is not the
> "Pythagoreans' Code". The existence of irrational numbers was
> [notoriously] known to the Pythagoreans themselves. The fact
> that ℝ does not contain infinitesimals is usually attributed to
> Archimedes and/or Eudoxus. Note that this is an /axiom/ of ℝ,
> not a fact about numbers in general. "Infinitely approaching
> means equal" is not a sensible statement. It's literally
> nonsense in talking about ℝ;

"Pythagoreans' Code" is a title like in news paper to draw attention.
The contents is a slightly exaggerated statement.

> if you, Wij, want to avoid the
> use of limits, then you need to spell out more carefully what
> you are actually talking about.

https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

This link has been shown many times. All the basics from bottom up are there,
but I don't think people had really read it.

> >>> People have no problem using 'real' number this way, lasted for >1500 years.
> They did have problems. You can do some simple calculus
> by intuition, largely as was done from Archimedes to Newton, but
> that approach was always controversial and ran into real problems
> in the 18thC. Solved by axiomatic constructions, including limits,
> in the 19thC plus developments/generalisations in the 20thC.
> >>> Modern Pythagoreans use just a little bit more different numbers and no idea
> >>> they are the essentially the same as ancient Pythagoreans also have no problem
> >>> using 'real' numbers.
> I try to make all possible allowances when discussing with
> non-native writers of English, but the above has defeated me. I
> can't make any sense of it, even in what I understand to be Wij's
> version of mathematics.

[Revised passage] (If I understand what you mean. Sorry for the sloppiness)
Modern Pythagoreans use just a few more different numbers(π,e,√,...) and yet,
have no idea they are essentially the same as ancient Pythagoreans. They also
have no problem using 'real' numbers.

Thank all people tolerated my English.

> >> This just PROVES that you are not using the term 'The Reals' as used by
> >> Modern Mathematics, but are working under an alternate theory of
> >> Mathematics that works under different principles.
> Either that, or else Wij is simply wrong. We need Wij to
> set out more clearly what his conception of "real" numbers is.
>
> [...]
> > It's not quite clear what a real number means physically. I suggested that
> > you have a line, mark off a point with a pencil, and that's your real number,
> > but this was rejected by the professional mathematicians (of whom I am
> > not one).
> Well, quite. "A line"? Any line? A straight line? What
> points are on this line [without begging questions!]? How do you
> "mark off" a point? With another line? What will you do about
> the vast majority of members of ℝ that are not constructible --
> indeed not even describable.
> > As for "real" meaning "the opposite of imaginary", it's a commonplace
> > that "imaginary" is a bad term for a square root of a negative number, but
> > it has stuck and we must use it if we don't want to casue confusion.
> Well, quite. Again.
> > I'd advise wij not to fuss too much about what is really a real number, and
> > to try to understand the conventional rules and definitions, but also to understand
> > that, as you say, people can and do propose alternative systems, most of which
> > never get anywhere, but a few of which prove productive.
> I suspect that Wij isn't interested in the conventional rules.
> But without knowing the Wij proposals in more detail, then as above
> we can't know whether this is something different, and perhaps
> interesting, or simply an ill-expressed list of misconceptions.
>
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Sinding

On Thursday, 16 December 2021 at 20:06:13 UTC+8, richar...@gmail.com wrote:
> On 12/16/21 6:15 AM, wij wrote:
> > On Thursday, 16 December 2021 at 09:13:54 UTC+8, richar...@gmail.com wrote:
> >> On 12/15/21 5:18 PM, wij wrote:
> >>> On Thursday, 16 December 2021 at 06:16:09 UTC+8, wij wrote:
> >>>> On Wednesday, 15 December 2021 at 21:21:00 UTC+8, richar...@gmail.com wrote:
> >>>>> On 12/15/21 7:01 AM, wij wrote:
> >>>>>> On Wednesday, 15 December 2021 at 09:55:45 UTC+8, richar...@gmail.com wrote:
> >>>>>>> On 12/14/21 12:13 PM, wij wrote:
> >>>>>>>> On Tuesday, 14 December 2021 at 19:58:14 UTC+8, richar...@gmail.com wrote:
> >>>>>>>>> On 12/14/21 5:10 AM, wij wrote:
> >>>>>>>>>> On Tuesday, 14 December 2021 at 11:34:59 UTC+8, Richard Damon wrote:
> >>>>>>>>>>> On 12/13/21 10:25 AM, wij wrote:
> >>>>>>>>>>>> On Monday, 13 December 2021 at 20:53:29 UTC+8, richar...@gmail.com wrote:
> >>>>>>>>>>>>> On 12/13/21 7:27 AM, wij wrote:
> >>>>>>>>>>>>>> On Monday, 13 December 2021 at 02:38:48 UTC+8, Mike Terry wrote:
> >>>>>>>>>>>>>>> On 12/12/2021 18:15, Richard Damon wrote:
> >>>>>>>>>>>>>>>> On 12/12/21 10:24 AM, Mike Terry wrote:
> >>>>>>>>>>>>>>>>> On 12/12/2021 08:48, wij wrote:
> >>>>>>>>>>>>>>>>>> Example 1:
> >>>>>>>>>>>>>>>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999....
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> ok. So A = 1
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> = 999.../1000...= (3*3*(11...1))/(5*2)^n
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Does not compute. 999... and 1000... are not numbers.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> No, of course not. (Does not compute)
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, what is the 3rd digit after the decimal point of A?
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Real numbers may have one or two decimal representations, a bit like
> >>>>>>>>>>>>>>>>> +0 = -0, or 3/5 = 6/10. A has two representations: 0.999... and
> >>>>>>>>>>>>>>>>> 1.000.... The 3rd digit after the decimal point of representation
> >>>>>>>>>>>>>>>>> 0.999 is 9, while for the representation 1.000 it is 0.
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
> >>>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> What density property is that? (And how do you think it is broken?)
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> I believe he means the property that between any two members of the
> >>>>>>>>>>>>>>>> Real, or the Rationals, there will ALWAYS be another member of that set
> >>>>>>>>>>>>>>>> between them. I.E., there is NOT a 'next' value from a given value.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Yes, I thought he might mean that. I wouldn't call that "density"
> >>>>>>>>>>>>>>> myself, as "density" has a different meaning. Perhaps the "denseness"
> >>>>>>>>>>>>>>> property?
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Amazing! What a phenomenon!
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> google "density property".
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Density Property::=
> >>>>>>>>>>>>>> For any two different numbers, there exists another different number in
> >>>>>>>>>>>>>> between. Take interval [a,b] for instance: ∀i,j∈[a,b], i<j such that ∃k,
> >>>>>>>>>>>>>> i<k<j.
> >>>>>>>>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> But if A=1, how does this break the property? It doesn't - the property
> >>>>>>>>>>>>>>> breaks if A != 1, so that would make wij's claim plain Wrong, like
> >>>>>>>>>>>>>>> everything else he said. :) [no surprise, I guess.]
> >>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>> Mike.
> >>>>>>>>>>>>>>>> One value between x and y will be (x+y)/2
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>> The problem with thinking of 0.9999.... as something distinct from 1 is
> >>>>>>>>>>>>>>>> THAT breaks the density property, as there can be no number bigger than
> >>>>>>>>>>>>>>>> 0.9999... and less than 1.0000
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>>>> Mike.
> >>>>>>>>>>>>>>>>
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Let A(0)=0, A(n)=(A(n-1)+1)/2, n∈ℕ
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> Given two different numbser A(n), and 1, there always exists another different
> >>>>>>>>>>>>>> number A(n+1) such that A(n)<A(n+1)<1
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> When A(n)=1? Infinity?
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>> There is no FINITE n where A(n) is equal to 1
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> Neither a FINITE n is in limit.
> >>>>>>>>>>>> What is the n in "lim(n->∞) A(n)=1"? Finite, infinite, or not a number?
> >>>>>>>>>>> Each n is a finite number.
> >>>>>>>>>>>
> >>>>>>>>>>> The key is that the limit of a sequence doesn't need to be a member of
> >>>>>>>>>>> the sequence, and in fact, normally isn't.
> >>>>>>>>>>>>
> >>>>>>>>>>>>> Note, for the Reals, Naturals, etc., 'Infinity' isn't a value, only a
> >>>>>>>>>>>>> 'limiting case'
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Thus the limit(n->infinity) A(n) is 1, even though no individual A(n) is 1.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> This is a common property of limits.
> >>>>>>>>>>>>>
> >>>>>>>>>>>>
> >>>>>>>>>>>> "No individual A(n) is 1. But limit(n->infinity) A(n) is 1".
> >>>>>>>>>>>> So limit theory turns 'approaching' to 'equal' in term of the limit smoke.
> >>>>>>>>>>>> Where I can find evidence that A(∞)=1 but from the 'approaching is equal'
> >>>>>>>>>>>> theory is the problem.
> >>>>>>>>>>> Right, the terms approach the limit.
> >>>>>>>>>>>
> >>>>>>>>>>> The limit is that value that terms get arbitraryily close to.
> >>>>>>>>>>>
> >>>>>>>>>>> A(infinity) isn't a proper notation, as A is a sequnce with Natural
> >>>>>>>>>>> Number indexes, and infinity isn't a Natural Number.
> >>>>>>>>>>>
> >>>>>>>>>>> One definition of 'The Limit' of a sequence is the number L, that for
> >>>>>>>>>>> any given arbirary positive value e, there is some N where all elements
> >>>>>>>>>>> of the seqence A(n), for all n > N, that |A(n) - L| < e
> >>>>>>>>>>>
> >>>>>>>>>>> i.e, for any arbitrarily chosen precision, we can find a point in the
> >>>>>>>>>>> sequence where it stays inside that bound.
> >>>>>>>>>>>>
> >>>>>>>>>>>>>
> >>>>>>>>>>>>> Just like 0.9999... for any finite number of 9s isn't equal to 1, but
> >>>>>>>>>>>>> the limiting case with the endless 9s is.
> >>>>>>>>>>>>
> >>>>>>>>>>>> I am not talking about 'limiting case'. Limit theory is full of inconsistency.
> >>>>>>>>>>>> (Every one learned 'limit method' should have a sense of this. I do not what to dig into this shit deep)
> >>>>>>>>>>>> We should be interested in the case that 0.999... equal to 1 or not, not the "limiting case".
> >>>>>>>>>>> Maybe you should look at it again.
> >>>>>>>>>>>
> >>>>>>>>>>> If you aren't going to use the right definition of Limit, and the range
> >>>>>>>>>>> of the Natural, Rational, and Real number, don't use those terms.
> >>>>>>>>>>>>
> >>>>>>>>>>>> And, here, right now, the density property in this thread.
> >>>>>>>>>>>> Does not 'density property' mean to hold infinitely?
> >>>>>>>>>>>> The problem is: The density property procedure can go on infinitely. Can not?
> >>>>>>>>>>>>
> >>>>>>>>>>> Not sure what you mean by 'infinitely' here, especially if you reject
> >>>>>>>>>>> the concept of a limit. It can be done an unbounded number of times.
> >>>>>>>>>>>
> >>>>>>>>>>> Remember, when we are talking about counting with Natural numbers, there
> >>>>>>>>>>> is NO infinity. Infinity is just a limit we can approach.
> >>>>>>>>>>
> >>>>>>>>>> Right. "Infinity is just a limit we can approach". So the following:
> >>>>>>>>>> 0.999... can never reach the limit 1.
> >>>>>>>>> Wrong, by that logic we can't have a number like 0.9999.... or 0.3333...
> >>>>>>>>> because they are only that value 'in the limit' when we get to the
> >>>>>>>>> infinite number of digits.
> >>>>>>>>
> >>>>>>>> By what logic? I have no problem you can't have the number like 0.999... or
> >>>>>>>> 0.333... in mind or practice. As said, you just keep fixating on LIMIT theory
> >>>>>>>> and fabricating stories from the copy in brain to fool yourself again and again
> >>>>>>>> (like PO?). But I won't say lying.
> >>>>>>> Except that we KNOW that a number like 0.3333.... does exist in
> >>>>>>> practice. So we need some notation to handle it, or do you just want to
> >>>>>>> 'give up' and say that the only rationals that (in reduced form) have a
> >>>>>>> denominator consisting only of powers of 2 and 5 exist in decimal form?
> >>>>>>>
> >>>>>>> If 0.3333.... doesn't exist, then does 1/3? (Its the same number) Or is
> >>>>>>> 1/3 just not expressible as a decimal?
> >>>>>>
> >>>>>> If you insist changing the subject to the limit theory,
> >>>>>> as said, I don't want to dig deep into the shit deep of limit theory.
> >>>>>> If it is no problem to you, a possible reason is that you don't really
> >>>>>> use it, understand it, never encountered the contradictory, you just reciting
> >>>>>> the shallow memory imprint.
> >>>>> Except that the MEANING of any INFINITE series, which is what the ....
> >>>>> notation implies, is derived via limit theory (or related concepts)..
> >>>>>
> >>>>> Just like I tell PO, if you won't follow the DEFINITIONS of the system
> >>>>> you claim to talk about, you aren't talking about that system, but
> >>>>> something else.
> >>>>>
> >>>>> The sets of Natural, Rational, and Real numbers do NOT have a member
> >>>>> that represents 'Infinity', but only have it as an auxilary concept that
> >>>>> corresponds to limits.
> >>>> It is you not talking in the topic the thread "0.999...=1 or not and the
> >>>> density p", and insist I claimed I was talking about your "an established
> >>>> field of Mathematics" (what is that is also debatable).
> >>>>
> >>>>> Series APPROACH infinity.
> >>>>
> >>>> And, APPROACH means EQUAL (Pythagoreans' logic)
> >>>> Why not limit theory admit this plainly straight?
> >>>>>>
> >>>>>> E.g. given an interval [0,1/3), question: does 1/3 in [0,1/3) or not?
> >>>>>> (0.333... is an irrational number, no exact rational p/q form.)
> >>>>> Nope. 0.33333..... is the EXACT value of 1/3.
> >>>>>
> >>>>> Of course 1/3 is NOT in the open interval that ends at 1/3, that is the
> >>>>> DEFINITION of the open interval.
> >>>> Typo, it should be: Given an interval [0,1/3), question: does 0.333.... in [0,1/3) or not?
> >>>>>>
> >>>>>>>>
> >>>>>>>>> No finite number of 9s in 0.99999 will be equal to 1, but IN THE LIMIT,
> >>>>>>>>> when we imagine that we reach that infinite end, it is.
> >>>>>>>>
> >>>>>>
> >>>>>> Yes, BY IMAGINE we reach the infinite end, not by proof.
> >>>>> No, by the proof using the property of limits, which is how these system
> >>>>> DEFINE dealing with the infinite value.
> >>>> You said "when we imagine that we reach that infinite end, it is."
> >>>> lim(x->1) x=1
> >>>> limit theory explicitly says x approaches 1 but never be exactly 1.
> >>>> (but in the end, x is 1)
> >>>>>>
> >>>>>>>> Where is the evidence 0.999... WILL be equal to 1 (without breaking the density
> >>>>>>>> property).
> >>>>>>>> Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
> >>>>>>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
> >>>>>>>> yet still not exactly 1. Not a number? Or just too small(or too great)?
> >>>>>>> You seem to be stuck on the finite. Yes, No FINITE listing of the digits
> >>>>>>> of 0.999... will be equal to 1.
> >>>>>>
> >>>>>> You said: "...when we imagine that we reach that infinite end, it is."
> >>>>>>
> >>>>>>> The equality ONLY happens in the limit
> >>>>>>> when we allow for there to be the INFINITE number of digits. (and no
> >>>>>>> finite number is infinite).
> >>>>>>
> >>>>>> I have shown 0.999... can be INFINITE long, and yet not 1.
> >>>>> Nope, you have shown that for an unbounded length, it is differnt, not
> >>>>> for an INFINITE length.
> >>>> What are you talking about? I used the word "INFINITELY" explicitly, and
> >>>> it became "unbounded" in you eye.
> >>>>
> >>>> [quote] ...Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
> >>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
> >>>> yet still not exactly 1 ...
> >>>>>>
> >>>>>>>
> >>>>>>> The 'proof' is in the definition of the limit.
> >>>>>>
> >>>>>> limit has no valid proof. It has 'definition' and 'explanation' and smoke.
> >>>>>> But, definition is a lowest level of understanding --- limit don't understand
> >>>>>> what 0.999.... is but must use it as non-1 at the beginning and use it as
> >>>>>> exactly 1 latter.
> >>>>> LIMIT is how infinite is defined to exist in the discussion.
> >>>>>>
> >>>>>>> Give me any positive real
> >>>>>>> number, no matter how small, and I can find the finite number of 9's
> >>>>>>> that will make that value, and all those pass it closer to 1 than that.
> >>>>>>>
> >>>>>>
> >>>>>> And, the number you use to approach is still finite. Isn't it?
> >>>>> Right, for every FINITE error, there is a FINITE length you need to
> >>>>> acheive, which means that in the LIMIT, the error goes to ZERO when you
> >>>>> include the INFINITE length.
> >>>>
> >>>> "Give me any positive real number, no matter how small, and I can find the finite number of 9's
> >>>> that will make that value, and all those pass it closer to 1 than that."
> >>>> Where can you find 1 in the sequence 0.999..., where every element in the sequence
> >>>> has non-zero error.
> >>>>>>
> >>>>>>> This is the way the Reals, et all, handle infinite series.
> >>>>>>
> >>>>>> Your Real is Q plus numbers that have finite notation.
> >>>>> Nope, Sqrt(2) has no finite decimal notation, but is a Real.
> >>>>>
> >>>>> 0.333... is a non-finite notation, but is in Q.
> >>>> you need to prove 0.333...=1 (NOT BY your favorite DEFINITION)
> >>>>>>
> >>>>>>>>
> >>>>>>>>>>
> >>>>>>>>>> Let f(n)= (2*n+1)!/((n!)^2*2^(3*k+1))
> >>>>>>>>>> S= Σ(n=0,∞) f(n) = √2
> >>>>>>>>>> S can never reach the limit √2 (albeit infinitely approaching, and, all the
> >>>>>>>>>> instances of the sequence and the sum are rational).
> >>>>>>>>> Wrong, No S(k) = Σ(n=0,k) f(n) will equal √2, but IN THE LIMIT, S does.
> >>>>>>>>>>
> >>>>>>>>>> This is the blind spot of Pythagoreans:
> >>>>>>>>>> --- Infinitely approaching means equal. Number too small equals zero. ---
> >>>>>>>>>>
> >>>>>>>>>> Yes, we should remember, "Infinity is just a limit we can approach".
> >>>>>>>>> And thus, 'In the limit', we reach it.
> >>>>>>>>> Not for any finite step, but in the limit.
> >>>>>>>>
> >>>>>>>> What is in discussion is whether 0.999...=1 or not without breaking the density property.
> >>>>>>>> Not the limit theory.
> >>>>>>>>
> >>>>>>>> --- Pythagoreans' Code ---
> >>>>>>>> Infinitely approaching means equal. Number too small equals zero..
> >>>>>>>>
> >>>>>>> Except that on common definition of what the ... notation means is based
> >>>>>>> on limit theory.
> >>>>>>
> >>>>>> Nope. I would say infinite series.
> >>>>> Whose value is based on Limit Theory.
> >>>>>>
> >>>>>>>
> >>>>>>> We actually don't need limit theory to handle 0.9999.... as being e
> >>>>>>> equal to 1.
> >>>>>>>
> >>>>>>
> >>>>>> Really? this showed you don't even really understand limit.
> >>>>> Do you deny the property is true? Can you find a counter example?
> >>>>>>
> >>>>>>> There is the other property, that any repeating fraction 0.xyzxyzxyz...
> >>>>>>> can be also expressed as a fraction of the unit xyz divided by the
> >>>>>>> number of 9's of the repeat cycle (in this case xyz/999)
> >>>>>>>
> >>>>>>> For example: 1/7 = 0.142857 142857 .... = 142857 / 999999
> >>>>>>>
> >>>>>>> If there are some leading decimal digits that aren't part of the repeat,
> >>>>>>> put those as a fraction over the right power of 10 and then add the same
> >>>>>>> number of 0s after the 9's.
> >>>>>>>
> >>>>>>> By this property 0.333... = 3/9 = 1/3, and 0.9999.... = 9/9 = 1.
> >>>>>>>
> >>>>>>> (Note, this also works for ANY base >= 2)
> >>>>>>
> >>>>>> Show me the whole derivation/argument.
> >>>>> Its been a while, let me look it up.
> >>>
> >>> Typo:
> >>> you need to prove 0.333...=1/3 (NOT BY your favorite DEFINITION)
> >> Just do the math by long division.
> >>
> >> If you think it isn't, do you think that not all rational number have
> >> decimal representations when including repeating representations?
> >>
> >> 1/3 gives us 0 remainder 1, so 0 integal part.
> >> Going down a digit we multiply the remainder by ten (the base) and
> >> divide again.
> >>
> >> This gives us 10/3 = 3 remainder 1, so the tenths digit is 1.
> >>
> >> Since we previously had a 1 remained, we can just repeat this pattern:
> >>
> >> 1/3 = 0.3...
> >
> > You have just confirmed/proved the conversion of 1/3 to 0.333... always leave a '1'
> > remainder behind? To be exact, 1/3= 0.333... + non_zero_remainder.
> > So, the conclusion that "1/3 = 0.333..." exactly is wrong.
> Wrong. That is the difference between any FINITE representation an the
> infinite representation.
>
> You seem to be using a different definition of the number system then
> assumed by convention, which means you need to state that or you are
> just being deceptive.
>
> That is as bad as just insisting that 1 + 1 = 10 without giving a hint
> that instead of using decimal numbers we are using binary.

On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
> On 12/12/2021 2:48 AM, wij wrote:
> > Example 1:
>
> WRONG
>
> Are Repeating Decimals Rational?
> Repeating or recurring decimals are decimal representations of numbers
> with infinitely repeating digits. Numbers with a repeating pattern of
> decimals are rational because when you put them into fractional form,
> both the numerator a and denominator b become non-fractional whole numbers.
>
> For example, when you use long division to divide 1 by 3, the resultant
> quotient is 0.33333…. However, when put it into fractional form, it's
> made of positive integers that don’t have decimal points:
> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>
> > Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
> > If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> > If A=1, what is the 3rd digit after the decimal point of A?
> > If A=1, density property (of ℚ and ℝ) is broken (false).
> >
> > Example 2:
> > 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
> > always exists as the prerequisite of 'infinite' repeating)
> >
> > Note that A is DEFINED exactly 0.999..., not 1.
> > Since the non-zero fractional pattern of the number (repeating decimal) is
> > defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
> > decimal) is irrational.
> >
> > ----
> > [Tip] More about "repeating decimals":
> > A0= 0.9 9 9 9 ...
> > A1= 0.99 99 99 99 ...
> > A2= 0.9 99 999 9999 ...
> > A3= 0.999 9 9999 9 99999 ...
> > A4= lim(n->∞) 1-1/n
> > A5= lim(n->∞) 1-2/n
> > A6= lim(n->∞) 1-3/10^n
> > A7= lim(n->∞) n/(n+1)
> > ...
> > This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
> > Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
> > https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>
>
> --
> Copyright 2021 Pete Olcott
>
> Talent hits a target no one else can hit;
> Genius hits a target no one else can see.
> Arthur Schopenhauer

Then, try to explain the density property

A(0)=0
A(n)=(A(n-1)+1)/2

When and how A(n)=1 and the density property still holds.

On 12/16/2021 10:24 AM, wij wrote:
> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
>> On 12/12/2021 2:48 AM, wij wrote:
>>> Example 1:
>>
>> WRONG
>>
>> Are Repeating Decimals Rational?
>> Repeating or recurring decimals are decimal representations of numbers
>> with infinitely repeating digits. Numbers with a repeating pattern of
>> decimals are rational because when you put them into fractional form,
>> both the numerator a and denominator b become non-fractional whole numbers.
>>
>> For example, when you use long division to divide 1 by 3, the resultant
>> quotient is 0.33333…. However, when put it into fractional form, it's
>> made of positive integers that don’t have decimal points:
>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>>
>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>> If A=1, what is the 3rd digit after the decimal point of A?
>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>
>>> Example 2:
>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
>>> always exists as the prerequisite of 'infinite' repeating)
>>>
>>> Note that A is DEFINED exactly 0.999..., not 1.
>>> Since the non-zero fractional pattern of the number (repeating decimal) is
>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
>>> decimal) is irrational.
>>>
>>> ----
>>> [Tip] More about "repeating decimals":
>>> A0= 0.9 9 9 9 ...
>>> A1= 0.99 99 99 99 ...
>>> A2= 0.9 99 999 9999 ...
>>> A3= 0.999 9 9999 9 99999 ...
>>> A4= lim(n->∞) 1-1/n
>>> A5= lim(n->∞) 1-2/n
>>> A6= lim(n->∞) 1-3/10^n
>>> A7= lim(n->∞) n/(n+1)
>>> ...
>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>
>>
>> --
>> Copyright 2021 Pete Olcott
>>
>> Talent hits a target no one else can hit;
>> Genius hits a target no one else can see.
>> Arthur Schopenhauer
>
> Then, try to explain the density property
>
> A(0)=0
> A(n)=(A(n-1)+1)/2
>
> When and how A(n)=1 and the density property still holds.

Any number that can be expressed as the ratio of two positive integers
is a rational number. That is all there is to it.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On Friday, 17 December 2021 at 00:33:11 UTC+8, olcott wrote:
> On 12/16/2021 10:24 AM, wij wrote:
> > On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
> >> On 12/12/2021 2:48 AM, wij wrote:
> >>> Example 1:
> >>
> >> WRONG
> >>
> >> Are Repeating Decimals Rational?
> >> Repeating or recurring decimals are decimal representations of numbers
> >> with infinitely repeating digits. Numbers with a repeating pattern of
> >> decimals are rational because when you put them into fractional form,
> >> both the numerator a and denominator b become non-fractional whole numbers.
> >>
> >> For example, when you use long division to divide 1 by 3, the resultant
> >> quotient is 0.33333…. However, when put it into fractional form, it's
> >> made of positive integers that don’t have decimal points:
> >> https://tutorme.com/blog/post/are-repeating-decimals-rational/
> >>
> >>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
> >>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> >>> If A=1, what is the 3rd digit after the decimal point of A?
> >>> If A=1, density property (of ℚ and ℝ) is broken (false).
> >>>
> >>> Example 2:
> >>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
> >>> always exists as the prerequisite of 'infinite' repeating)
> >>>
> >>> Note that A is DEFINED exactly 0.999..., not 1.
> >>> Since the non-zero fractional pattern of the number (repeating decimal) is
> >>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
> >>> decimal) is irrational.
> >>>
> >>> ----
> >>> [Tip] More about "repeating decimals":
> >>> A0= 0.9 9 9 9 ...
> >>> A1= 0.99 99 99 99 ...
> >>> A2= 0.9 99 999 9999 ...
> >>> A3= 0.999 9 9999 9 99999 ...
> >>> A4= lim(n->∞) 1-1/n
> >>> A5= lim(n->∞) 1-2/n
> >>> A6= lim(n->∞) 1-3/10^n
> >>> A7= lim(n->∞) n/(n+1)
> >>> ...
> >>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
> >>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
> >>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >>
> >>
> >> --
> >> Copyright 2021 Pete Olcott
> >>
> >> Talent hits a target no one else can hit;
> >> Genius hits a target no one else can see.
> >> Arthur Schopenhauer
> >
> > Then, try to explain the density property
> >
> > A(0)=0
> > A(n)=(A(n-1)+1)/2
> >
> > When and how A(n)=1 and the density property still holds.
> Any number that can be expressed as the ratio of two positive integers
> is a rational number. That is all there is to it.
> --
> Copyright 2021 Pete Olcott
>
> Talent hits a target no one else can hit;
> Genius hits a target no one else can see.
> Arthur Schopenhauer

When you use long division to divide 1 by 3, the resultant quotient is
accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
This fact forms the equality:

1/3= 0.333... + non_zero_remainder

Is this correct? If so, from the equation above we can deduce
1/3 - 0.333...= non_zero_remainder

Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
1/3 ≠ 0.333...

Is there any flaw?

On 12/16/2021 2:40 PM, wij wrote:
> On Friday, 17 December 2021 at 00:33:11 UTC+8, olcott wrote:
>> On 12/16/2021 10:24 AM, wij wrote:
>>> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
>>>> On 12/12/2021 2:48 AM, wij wrote:
>>>>> Example 1:
>>>>
>>>> WRONG
>>>>
>>>> Are Repeating Decimals Rational?
>>>> Repeating or recurring decimals are decimal representations of numbers
>>>> with infinitely repeating digits. Numbers with a repeating pattern of
>>>> decimals are rational because when you put them into fractional form,
>>>> both the numerator a and denominator b become non-fractional whole numbers.
>>>>
>>>> For example, when you use long division to divide 1 by 3, the resultant
>>>> quotient is 0.33333…. However, when put it into fractional form, it's
>>>> made of positive integers that don’t have decimal points:
>>>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>>>>
>>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
>>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>>>> If A=1, what is the 3rd digit after the decimal point of A?
>>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>>>
>>>>> Example 2:
>>>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
>>>>> always exists as the prerequisite of 'infinite' repeating)
>>>>>
>>>>> Note that A is DEFINED exactly 0.999..., not 1.
>>>>> Since the non-zero fractional pattern of the number (repeating decimal) is
>>>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
>>>>> decimal) is irrational.
>>>>>
>>>>> ----
>>>>> [Tip] More about "repeating decimals":
>>>>> A0= 0.9 9 9 9 ...
>>>>> A1= 0.99 99 99 99 ...
>>>>> A2= 0.9 99 999 9999 ...
>>>>> A3= 0.999 9 9999 9 99999 ...
>>>>> A4= lim(n->∞) 1-1/n
>>>>> A5= lim(n->∞) 1-2/n
>>>>> A6= lim(n->∞) 1-3/10^n
>>>>> A7= lim(n->∞) n/(n+1)
>>>>> ...
>>>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
>>>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>>>
>>>>
>>>> --
>>>> Copyright 2021 Pete Olcott
>>>>
>>>> Talent hits a target no one else can hit;
>>>> Genius hits a target no one else can see.
>>>> Arthur Schopenhauer
>>>
>>> Then, try to explain the density property
>>>
>>> A(0)=0
>>> A(n)=(A(n-1)+1)/2
>>>
>>> When and how A(n)=1 and the density property still holds.
>> Any number that can be expressed as the ratio of two positive integers
>> is a rational number. That is all there is to it.
>> --
>> Copyright 2021 Pete Olcott
>>
>> Talent hits a target no one else can hit;
>> Genius hits a target no one else can see.
>> Arthur Schopenhauer
>

When-so-ever any repeated decimal can be represented as the ratio
between two integers then this number is a rational number.

This is the same sort of thing as saying then when-so-ever a cat is an
animal then it is not an office building.

> When you use long division to divide 1 by 3, the resultant quotient is
> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> This fact forms the equality:
>
> 1/3= 0.333... + non_zero_remainder
>
> Is this correct? If so, from the equation above we can deduce
> 1/3 - 0.333...= non_zero_remainder
>
> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> 1/3 ≠ 0.333...
>
> Is there any flaw?

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On Friday, 17 December 2021 at 05:15:05 UTC+8, olcott wrote:
> On 12/16/2021 2:40 PM, wij wrote:
> > On Friday, 17 December 2021 at 00:33:11 UTC+8, olcott wrote:
> >> On 12/16/2021 10:24 AM, wij wrote:
> >>> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
> >>>> On 12/12/2021 2:48 AM, wij wrote:
> >>>>> Example 1:
> >>>>
> >>>> WRONG
> >>>>
> >>>> Are Repeating Decimals Rational?
> >>>> Repeating or recurring decimals are decimal representations of numbers
> >>>> with infinitely repeating digits. Numbers with a repeating pattern of
> >>>> decimals are rational because when you put them into fractional form,
> >>>> both the numerator a and denominator b become non-fractional whole numbers.
> >>>>
> >>>> For example, when you use long division to divide 1 by 3, the resultant
> >>>> quotient is 0.33333…. However, when put it into fractional form, it's
> >>>> made of positive integers that don’t have decimal points:
> >>>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
> >>>>
> >>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
> >>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
> >>>>> If A=1, what is the 3rd digit after the decimal point of A?
> >>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
> >>>>>
> >>>>> Example 2:
> >>>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
> >>>>> always exists as the prerequisite of 'infinite' repeating)
> >>>>>
> >>>>> Note that A is DEFINED exactly 0.999..., not 1.
> >>>>> Since the non-zero fractional pattern of the number (repeating decimal) is
> >>>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
> >>>>> decimal) is irrational.
> >>>>>
> >>>>> ----
> >>>>> [Tip] More about "repeating decimals":
> >>>>> A0= 0.9 9 9 9 ...
> >>>>> A1= 0.99 99 99 99 ...
> >>>>> A2= 0.9 99 999 9999 ...
> >>>>> A3= 0.999 9 9999 9 99999 ...
> >>>>> A4= lim(n->∞) 1-1/n
> >>>>> A5= lim(n->∞) 1-2/n
> >>>>> A6= lim(n->∞) 1-3/10^n
> >>>>> A7= lim(n->∞) n/(n+1)
> >>>>> ...
> >>>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
> >>>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
> >>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> >>>>
> >>>>
> >>>> --
> >>>> Copyright 2021 Pete Olcott
> >>>>
> >>>> Talent hits a target no one else can hit;
> >>>> Genius hits a target no one else can see.
> >>>> Arthur Schopenhauer
> >>>
> >>> Then, try to explain the density property
> >>>
> >>> A(0)=0
> >>> A(n)=(A(n-1)+1)/2
> >>>
> >>> When and how A(n)=1 and the density property still holds.
> >> Any number that can be expressed as the ratio of two positive integers
> >> is a rational number. That is all there is to it.
> >> --
> >> Copyright 2021 Pete Olcott
> >>
> >> Talent hits a target no one else can hit;
> >> Genius hits a target no one else can see.
> >> Arthur Schopenhauer
> >
> When-so-ever any repeated decimal can be represented as the ratio
> between two integers then this number is a rational number.
>
> This is the same sort of thing as saying then when-so-ever a cat is an
> animal then it is not an office building.

Useless statement.

> > When you use long division to divide 1 by 3, the resultant quotient is
> > accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> > This fact forms the equality:
> >
> > 1/3= 0.333... + non_zero_remainder
> >
> > Is this correct? If so, from the equation above we can deduce
> > 1/3 - 0.333...= non_zero_remainder
> >
> > Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> > 1/3 ≠ 0.333...
> >
> > Is there any flaw?
> --
> Copyright 2021 Pete Olcott
>
> Talent hits a target no one else can hit;
> Genius hits a target no one else can see.
> Arthur Schopenhauer

Why do you skip this question asked while bothering answering another?
I would like to hear your answer/opinion to the question above.

On 12/16/21 11:22 AM, wij wrote:
> On Thursday, 16 December 2021 at 20:06:13 UTC+8, richar...@gmail.com wrote:
>> On 12/16/21 6:15 AM, wij wrote:
>>> On Thursday, 16 December 2021 at 09:13:54 UTC+8, richar...@gmail.com wrote:
>>>> On 12/15/21 5:18 PM, wij wrote:
>>>>> On Thursday, 16 December 2021 at 06:16:09 UTC+8, wij wrote:
>>>>>> On Wednesday, 15 December 2021 at 21:21:00 UTC+8, richar...@gmail.com wrote:
>>>>>>> On 12/15/21 7:01 AM, wij wrote:
>>>>>>>> On Wednesday, 15 December 2021 at 09:55:45 UTC+8, richar...@gmail.com wrote:
>>>>>>>>> On 12/14/21 12:13 PM, wij wrote:
>>>>>>>>>> On Tuesday, 14 December 2021 at 19:58:14 UTC+8, richar...@gmail.com wrote:
>>>>>>>>>>> On 12/14/21 5:10 AM, wij wrote:
>>>>>>>>>>>> On Tuesday, 14 December 2021 at 11:34:59 UTC+8, Richard Damon wrote:
>>>>>>>>>>>>> On 12/13/21 10:25 AM, wij wrote:
>>>>>>>>>>>>>> On Monday, 13 December 2021 at 20:53:29 UTC+8, richar...@gmail.com wrote:
>>>>>>>>>>>>>>> On 12/13/21 7:27 AM, wij wrote:
>>>>>>>>>>>>>>>> On Monday, 13 December 2021 at 02:38:48 UTC+8, Mike Terry wrote:
>>>>>>>>>>>>>>>>> On 12/12/2021 18:15, Richard Damon wrote:
>>>>>>>>>>>>>>>>>> On 12/12/21 10:24 AM, Mike Terry wrote:
>>>>>>>>>>>>>>>>>>> On 12/12/2021 08:48, wij wrote:
>>>>>>>>>>>>>>>>>>>> Example 1:
>>>>>>>>>>>>>>>>>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> ok. So A = 1
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> = 999.../1000...= (3*3*(11...1))/(5*2)^n
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Does not compute. 999... and 1000... are not numbers.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> No, of course not. (Does not compute)
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> If A=1, what is the 3rd digit after the decimal point of A?
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Real numbers may have one or two decimal representations, a bit like
>>>>>>>>>>>>>>>>>>> +0 = -0, or 3/5 = 6/10. A has two representations: 0.999... and
>>>>>>>>>>>>>>>>>>> 1.000.... The 3rd digit after the decimal point of representation
>>>>>>>>>>>>>>>>>>> 0.999 is 9, while for the representation 1.000 it is 0.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> What density property is that? (And how do you think it is broken?)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> I believe he means the property that between any two members of the
>>>>>>>>>>>>>>>>>> Real, or the Rationals, there will ALWAYS be another member of that set
>>>>>>>>>>>>>>>>>> between them. I.E., there is NOT a 'next' value from a given value.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Yes, I thought he might mean that. I wouldn't call that "density"
>>>>>>>>>>>>>>>>> myself, as "density" has a different meaning. Perhaps the "denseness"
>>>>>>>>>>>>>>>>> property?
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Amazing! What a phenomenon!
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> google "density property".
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Density Property::=
>>>>>>>>>>>>>>>> For any two different numbers, there exists another different number in
>>>>>>>>>>>>>>>> between. Take interval [a,b] for instance: ∀i,j∈[a,b], i<j such that ∃k,
>>>>>>>>>>>>>>>> i<k<j.
>>>>>>>>>>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> But if A=1, how does this break the property? It doesn't - the property
>>>>>>>>>>>>>>>>> breaks if A != 1, so that would make wij's claim plain Wrong, like
>>>>>>>>>>>>>>>>> everything else he said. :) [no surprise, I guess.]
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>>>>>>> One value between x and y will be (x+y)/2
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> The problem with thinking of 0.9999.... as something distinct from 1 is
>>>>>>>>>>>>>>>>>> THAT breaks the density property, as there can be no number bigger than
>>>>>>>>>>>>>>>>>> 0.9999... and less than 1.0000
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Let A(0)=0, A(n)=(A(n-1)+1)/2, n∈ℕ
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Given two different numbser A(n), and 1, there always exists another different
>>>>>>>>>>>>>>>> number A(n+1) such that A(n)<A(n+1)<1
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> When A(n)=1? Infinity?
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> There is no FINITE n where A(n) is equal to 1
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Neither a FINITE n is in limit.
>>>>>>>>>>>>>> What is the n in "lim(n->∞) A(n)=1"? Finite, infinite, or not a number?
>>>>>>>>>>>>> Each n is a finite number.
>>>>>>>>>>>>>
>>>>>>>>>>>>> The key is that the limit of a sequence doesn't need to be a member of
>>>>>>>>>>>>> the sequence, and in fact, normally isn't.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Note, for the Reals, Naturals, etc., 'Infinity' isn't a value, only a
>>>>>>>>>>>>>>> 'limiting case'
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Thus the limit(n->infinity) A(n) is 1, even though no individual A(n) is 1.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> This is a common property of limits.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> "No individual A(n) is 1. But limit(n->infinity) A(n) is 1".
>>>>>>>>>>>>>> So limit theory turns 'approaching' to 'equal' in term of the limit smoke.
>>>>>>>>>>>>>> Where I can find evidence that A(∞)=1 but from the 'approaching is equal'
>>>>>>>>>>>>>> theory is the problem.
>>>>>>>>>>>>> Right, the terms approach the limit.
>>>>>>>>>>>>>
>>>>>>>>>>>>> The limit is that value that terms get arbitraryily close to.
>>>>>>>>>>>>>
>>>>>>>>>>>>> A(infinity) isn't a proper notation, as A is a sequnce with Natural
>>>>>>>>>>>>> Number indexes, and infinity isn't a Natural Number.
>>>>>>>>>>>>>
>>>>>>>>>>>>> One definition of 'The Limit' of a sequence is the number L, that for
>>>>>>>>>>>>> any given arbirary positive value e, there is some N where all elements
>>>>>>>>>>>>> of the seqence A(n), for all n > N, that |A(n) - L| < e
>>>>>>>>>>>>>
>>>>>>>>>>>>> i.e, for any arbitrarily chosen precision, we can find a point in the
>>>>>>>>>>>>> sequence where it stays inside that bound.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Just like 0.9999... for any finite number of 9s isn't equal to 1, but
>>>>>>>>>>>>>>> the limiting case with the endless 9s is.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> I am not talking about 'limiting case'. Limit theory is full of inconsistency.
>>>>>>>>>>>>>> (Every one learned 'limit method' should have a sense of this. I do not what to dig into this shit deep)
>>>>>>>>>>>>>> We should be interested in the case that 0.999... equal to 1 or not, not the "limiting case".
>>>>>>>>>>>>> Maybe you should look at it again.
>>>>>>>>>>>>>
>>>>>>>>>>>>> If you aren't going to use the right definition of Limit, and the range
>>>>>>>>>>>>> of the Natural, Rational, and Real number, don't use those terms.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> And, here, right now, the density property in this thread.
>>>>>>>>>>>>>> Does not 'density property' mean to hold infinitely?
>>>>>>>>>>>>>> The problem is: The density property procedure can go on infinitely. Can not?
>>>>>>>>>>>>>>
>>>>>>>>>>>>> Not sure what you mean by 'infinitely' here, especially if you reject
>>>>>>>>>>>>> the concept of a limit. It can be done an unbounded number of times.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Remember, when we are talking about counting with Natural numbers, there
>>>>>>>>>>>>> is NO infinity. Infinity is just a limit we can approach.
>>>>>>>>>>>>
>>>>>>>>>>>> Right. "Infinity is just a limit we can approach". So the following:
>>>>>>>>>>>> 0.999... can never reach the limit 1.
>>>>>>>>>>> Wrong, by that logic we can't have a number like 0.9999.... or 0.3333...
>>>>>>>>>>> because they are only that value 'in the limit' when we get to the
>>>>>>>>>>> infinite number of digits.
>>>>>>>>>>
>>>>>>>>>> By what logic? I have no problem you can't have the number like 0.999... or
>>>>>>>>>> 0.333... in mind or practice. As said, you just keep fixating on LIMIT theory
>>>>>>>>>> and fabricating stories from the copy in brain to fool yourself again and again
>>>>>>>>>> (like PO?). But I won't say lying.
>>>>>>>>> Except that we KNOW that a number like 0.3333.... does exist in
>>>>>>>>> practice. So we need some notation to handle it, or do you just want to
>>>>>>>>> 'give up' and say that the only rationals that (in reduced form) have a
>>>>>>>>> denominator consisting only of powers of 2 and 5 exist in decimal form?
>>>>>>>>>
>>>>>>>>> If 0.3333.... doesn't exist, then does 1/3? (Its the same number) Or is
>>>>>>>>> 1/3 just not expressible as a decimal?
>>>>>>>>
>>>>>>>> If you insist changing the subject to the limit theory,
>>>>>>>> as said, I don't want to dig deep into the shit deep of limit theory.
>>>>>>>> If it is no problem to you, a possible reason is that you don't really
>>>>>>>> use it, understand it, never encountered the contradictory, you just reciting
>>>>>>>> the shallow memory imprint.
>>>>>>> Except that the MEANING of any INFINITE series, which is what the ...
>>>>>>> notation implies, is derived via limit theory (or related concepts).
>>>>>>>
>>>>>>> Just like I tell PO, if you won't follow the DEFINITIONS of the system
>>>>>>> you claim to talk about, you aren't talking about that system, but
>>>>>>> something else.
>>>>>>>
>>>>>>> The sets of Natural, Rational, and Real numbers do NOT have a member
>>>>>>> that represents 'Infinity', but only have it as an auxilary concept that
>>>>>>> corresponds to limits.
>>>>>> It is you not talking in the topic the thread "0.999...=1 or not and the
>>>>>> density p", and insist I claimed I was talking about your "an established
>>>>>> field of Mathematics" (what is that is also debatable).
>>>>>>
>>>>>>> Series APPROACH infinity.
>>>>>>
>>>>>> And, APPROACH means EQUAL (Pythagoreans' logic)
>>>>>> Why not limit theory admit this plainly straight?
>>>>>>>>
>>>>>>>> E.g. given an interval [0,1/3), question: does 1/3 in [0,1/3) or not?
>>>>>>>> (0.333... is an irrational number, no exact rational p/q form.)
>>>>>>> Nope. 0.33333..... is the EXACT value of 1/3.
>>>>>>>
>>>>>>> Of course 1/3 is NOT in the open interval that ends at 1/3, that is the
>>>>>>> DEFINITION of the open interval.
>>>>>> Typo, it should be: Given an interval [0,1/3), question: does 0.333... in [0,1/3) or not?
>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>> No finite number of 9s in 0.99999 will be equal to 1, but IN THE LIMIT,
>>>>>>>>>>> when we imagine that we reach that infinite end, it is.
>>>>>>>>>>
>>>>>>>>
>>>>>>>> Yes, BY IMAGINE we reach the infinite end, not by proof.
>>>>>>> No, by the proof using the property of limits, which is how these system
>>>>>>> DEFINE dealing with the infinite value.
>>>>>> You said "when we imagine that we reach that infinite end, it is."
>>>>>> lim(x->1) x=1
>>>>>> limit theory explicitly says x approaches 1 but never be exactly 1.
>>>>>> (but in the end, x is 1)
>>>>>>>>
>>>>>>>>>> Where is the evidence 0.999... WILL be equal to 1 (without breaking the density
>>>>>>>>>> property).
>>>>>>>>>> Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
>>>>>>>>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
>>>>>>>>>> yet still not exactly 1. Not a number? Or just too small(or too great)?
>>>>>>>>> You seem to be stuck on the finite. Yes, No FINITE listing of the digits
>>>>>>>>> of 0.999... will be equal to 1.
>>>>>>>>
>>>>>>>> You said: "...when we imagine that we reach that infinite end, it is."
>>>>>>>>
>>>>>>>>> The equality ONLY happens in the limit
>>>>>>>>> when we allow for there to be the INFINITE number of digits. (and no
>>>>>>>>> finite number is infinite).
>>>>>>>>
>>>>>>>> I have shown 0.999... can be INFINITE long, and yet not 1.
>>>>>>> Nope, you have shown that for an unbounded length, it is differnt, not
>>>>>>> for an INFINITE length.
>>>>>> What are you talking about? I used the word "INFINITELY" explicitly, and
>>>>>> it became "unbounded" in you eye.
>>>>>>
>>>>>> [quote] ...Assume the digit 9 could be as small as Plunk length, the number 0.999... can be
>>>>>> n*13.8 billion light years and beyond to another universe INFINITELY (eternal if you like),
>>>>>> yet still not exactly 1 ...
>>>>>>>>
>>>>>>>>>
>>>>>>>>> The 'proof' is in the definition of the limit.
>>>>>>>>
>>>>>>>> limit has no valid proof. It has 'definition' and 'explanation' and smoke.
>>>>>>>> But, definition is a lowest level of understanding --- limit don't understand
>>>>>>>> what 0.999.... is but must use it as non-1 at the beginning and use it as
>>>>>>>> exactly 1 latter.
>>>>>>> LIMIT is how infinite is defined to exist in the discussion.
>>>>>>>>
>>>>>>>>> Give me any positive real
>>>>>>>>> number, no matter how small, and I can find the finite number of 9's
>>>>>>>>> that will make that value, and all those pass it closer to 1 than that.
>>>>>>>>>
>>>>>>>>
>>>>>>>> And, the number you use to approach is still finite. Isn't it?
>>>>>>> Right, for every FINITE error, there is a FINITE length you need to
>>>>>>> acheive, which means that in the LIMIT, the error goes to ZERO when you
>>>>>>> include the INFINITE length.
>>>>>>
>>>>>> "Give me any positive real number, no matter how small, and I can find the finite number of 9's
>>>>>> that will make that value, and all those pass it closer to 1 than that."
>>>>>> Where can you find 1 in the sequence 0.999..., where every element in the sequence
>>>>>> has non-zero error.
>>>>>>>>
>>>>>>>>> This is the way the Reals, et all, handle infinite series.
>>>>>>>>
>>>>>>>> Your Real is Q plus numbers that have finite notation.
>>>>>>> Nope, Sqrt(2) has no finite decimal notation, but is a Real.
>>>>>>>
>>>>>>> 0.333... is a non-finite notation, but is in Q.
>>>>>> you need to prove 0.333...=1 (NOT BY your favorite DEFINITION)
>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Let f(n)= (2*n+1)!/((n!)^2*2^(3*k+1))
>>>>>>>>>>>> S= Σ(n=0,∞) f(n) = √2
>>>>>>>>>>>> S can never reach the limit √2 (albeit infinitely approaching, and, all the
>>>>>>>>>>>> instances of the sequence and the sum are rational).
>>>>>>>>>>> Wrong, No S(k) = Σ(n=0,k) f(n) will equal √2, but IN THE LIMIT, S does.
>>>>>>>>>>>>
>>>>>>>>>>>> This is the blind spot of Pythagoreans:
>>>>>>>>>>>> --- Infinitely approaching means equal. Number too small equals zero. ---
>>>>>>>>>>>>
>>>>>>>>>>>> Yes, we should remember, "Infinity is just a limit we can approach".
>>>>>>>>>>> And thus, 'In the limit', we reach it.
>>>>>>>>>>> Not for any finite step, but in the limit.
>>>>>>>>>>
>>>>>>>>>> What is in discussion is whether 0.999...=1 or not without breaking the density property.
>>>>>>>>>> Not the limit theory.
>>>>>>>>>>
>>>>>>>>>> --- Pythagoreans' Code ---
>>>>>>>>>> Infinitely approaching means equal. Number too small equals zero.
>>>>>>>>>>
>>>>>>>>> Except that on common definition of what the ... notation means is based
>>>>>>>>> on limit theory.
>>>>>>>>
>>>>>>>> Nope. I would say infinite series.
>>>>>>> Whose value is based on Limit Theory.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> We actually don't need limit theory to handle 0.9999.... as being e
>>>>>>>>> equal to 1.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Really? this showed you don't even really understand limit.
>>>>>>> Do you deny the property is true? Can you find a counter example?
>>>>>>>>
>>>>>>>>> There is the other property, that any repeating fraction 0.xyzxyzxyz...
>>>>>>>>> can be also expressed as a fraction of the unit xyz divided by the
>>>>>>>>> number of 9's of the repeat cycle (in this case xyz/999)
>>>>>>>>>
>>>>>>>>> For example: 1/7 = 0.142857 142857 .... = 142857 / 999999
>>>>>>>>>
>>>>>>>>> If there are some leading decimal digits that aren't part of the repeat,
>>>>>>>>> put those as a fraction over the right power of 10 and then add the same
>>>>>>>>> number of 0s after the 9's.
>>>>>>>>>
>>>>>>>>> By this property 0.333... = 3/9 = 1/3, and 0.9999.... = 9/9 = 1.
>>>>>>>>>
>>>>>>>>> (Note, this also works for ANY base >= 2)
>>>>>>>>
>>>>>>>> Show me the whole derivation/argument.
>>>>>>> Its been a while, let me look it up.
>>>>>
>>>>> Typo:
>>>>> you need to prove 0.333...=1/3 (NOT BY your favorite DEFINITION)
>>>> Just do the math by long division.
>>>>
>>>> If you think it isn't, do you think that not all rational number have
>>>> decimal representations when including repeating representations?
>>>>
>>>> 1/3 gives us 0 remainder 1, so 0 integal part.
>>>> Going down a digit we multiply the remainder by ten (the base) and
>>>> divide again.
>>>>
>>>> This gives us 10/3 = 3 remainder 1, so the tenths digit is 1.
>>>>
>>>> Since we previously had a 1 remained, we can just repeat this pattern:
>>>>
>>>> 1/3 = 0.3...
>>>
>>> You have just confirmed/proved the conversion of 1/3 to 0.333... always leave a '1'
>>> remainder behind? To be exact, 1/3= 0.333... + non_zero_remainder.
>>> So, the conclusion that "1/3 = 0.333..." exactly is wrong.
>> Wrong. That is the difference between any FINITE representation an the
>> infinite representation.
>>
>> You seem to be using a different definition of the number system then
>> assumed by convention, which means you need to state that or you are
>> just being deceptive.
>>
>> That is as bad as just insisting that 1 + 1 = 10 without giving a hint
>> that instead of using decimal numbers we are using binary.
>
> The allegation is too wild.
>
> 1/3= 0.333... + non_zero_remainder
> 1/3-0.333...= non_zero_remainder

On 12/16/21 11:24 AM, wij wrote:
> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
>> On 12/12/2021 2:48 AM, wij wrote:
>>> Example 1:
>>
>> WRONG
>>
>> Are Repeating Decimals Rational?
>> Repeating or recurring decimals are decimal representations of numbers
>> with infinitely repeating digits. Numbers with a repeating pattern of
>> decimals are rational because when you put them into fractional form,
>> both the numerator a and denominator b become non-fractional whole numbers.
>>
>> For example, when you use long division to divide 1 by 3, the resultant
>> quotient is 0.33333…. However, when put it into fractional form, it's
>> made of positive integers that don’t have decimal points:
>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>>
>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>> If A=1, what is the 3rd digit after the decimal point of A?
>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>
>>> Example 2:
>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
>>> always exists as the prerequisite of 'infinite' repeating)
>>>
>>> Note that A is DEFINED exactly 0.999..., not 1.
>>> Since the non-zero fractional pattern of the number (repeating decimal) is
>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
>>> decimal) is irrational.
>>>
>>> ----
>>> [Tip] More about "repeating decimals":
>>> A0= 0.9 9 9 9 ...
>>> A1= 0.99 99 99 99 ...
>>> A2= 0.9 99 999 9999 ...
>>> A3= 0.999 9 9999 9 99999 ...
>>> A4= lim(n->∞) 1-1/n
>>> A5= lim(n->∞) 1-2/n
>>> A6= lim(n->∞) 1-3/10^n
>>> A7= lim(n->∞) n/(n+1)
>>> ...
>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>
>>
>> --
>> Copyright 2021 Pete Olcott
>>
>> Talent hits a target no one else can hit;
>> Genius hits a target no one else can see.
>> Arthur Schopenhauer
>
> Then, try to explain the density property
>
> A(0)=0
> A(n)=(A(n-1)+1)/2
>
> When and how A(n)=1 and the density property still holds.

The issue is that you seem to think that you As above is a member of the
set A(n). But there is no requrement that it be so.

The limit of a sequenc4 does not need to be a member of that sequence.
In fact, you can have a sequence of Rational Numbers with the limit not
being a Rational Number but an Irrational Number, which of course CAN'T
be a member of that set.

On 12/16/21 11:09 AM, wij wrote:
> On Thursday, 16 December 2021 at 21:37:25 UTC+8, Andy Walker wrote:
>> On 16/12/2021 12:23, Malcolm McLean wrote:
>>> On Thursday, 16 December 2021 at 12:06:13 UTC, richar...@gmail.com wrote:
>>>> On 12/16/21 6:15 AM, wij wrote:
>> [Much snipped -- 500+ line articles are just /too long/.]
>>>>> Pythagoreans' Code:
>>>>> "Infinitely approaching means equal. Number too small equals zero.
>>>>> All 'real' numbers are in form of p/q."
>> Whatever else Wij may be thinking of, this is not the
>> "Pythagoreans' Code". The existence of irrational numbers was
>> [notoriously] known to the Pythagoreans themselves. The fact
>> that ℝ does not contain infinitesimals is usually attributed to
>> Archimedes and/or Eudoxus. Note that this is an /axiom/ of ℝ,
>> not a fact about numbers in general. "Infinitely approaching
>> means equal" is not a sensible statement. It's literally
>> nonsense in talking about ℝ;
>
> "Pythagoreans' Code" is a title like in news paper to draw attention.
> The contents is a slightly exaggerated statement.
>
>> if you, Wij, want to avoid the
>> use of limits, then you need to spell out more carefully what
>> you are actually talking about.
>
> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>
> This link has been shown many times. All the basics from bottom up are there,
> but I don't think people had really read it.
>
>>>>> People have no problem using 'real' number this way, lasted for >1500 years.
>> They did have problems. You can do some simple calculus
>> by intuition, largely as was done from Archimedes to Newton, but
>> that approach was always controversial and ran into real problems
>> in the 18thC. Solved by axiomatic constructions, including limits,
>> in the 19thC plus developments/generalisations in the 20thC.
>>>>> Modern Pythagoreans use just a little bit more different numbers and no idea
>>>>> they are the essentially the same as ancient Pythagoreans also have no problem
>>>>> using 'real' numbers.
>> I try to make all possible allowances when discussing with
>> non-native writers of English, but the above has defeated me. I
>> can't make any sense of it, even in what I understand to be Wij's
>> version of mathematics.
>
> [Revised passage] (If I understand what you mean. Sorry for the sloppiness)
> Modern Pythagoreans use just a few more different numbers(π,e,√,...) and yet,
> have no idea they are essentially the same as ancient Pythagoreans. They also
> have no problem using 'real' numbers.
>
> Thank all people tolerated my English.

But if they aren't following the definitions of modern mathematics as it
defines 'The Real Number System', the system they are describing may be
'a number system', but isn't 'The Real Number System'.

The Real Number System is the name of a SPECIFIC theory of numbers, with
SPECIFIC properties.

It is incorrect to call anything else by that name, just as people would
be wrong to call your brother (if you have one) YOU.

Because The Real Number System is so prevelent, if context doesn't make
it clear that you are NOT talking about it, it will generally be assumed
you are, or its related fields The Rational Numbers and the Natural Numbers,

IT sounds like the Pythagoreans just have an alternate idea of a similar
number system.

I will point out that alternate number system (and even in depth
discussion of The Real Number System) isn't really on topic for this
group, as the group is about Computation Theory which is something
different.

Sci.math (or maybe some other group) might be a better fit for that more
general sort of discussion.

On 12/16/21 11:24 AM, wij wrote:
> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
>> On 12/12/2021 2:48 AM, wij wrote:
>>> Example 1:
>>
>> WRONG
>>
>> Are Repeating Decimals Rational?
>> Repeating or recurring decimals are decimal representations of numbers
>> with infinitely repeating digits. Numbers with a repeating pattern of
>> decimals are rational because when you put them into fractional form,
>> both the numerator a and denominator b become non-fractional whole numbers.
>>
>> For example, when you use long division to divide 1 by 3, the resultant
>> quotient is 0.33333…. However, when put it into fractional form, it's
>> made of positive integers that don’t have decimal points:
>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>>
>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>> If A=1, what is the 3rd digit after the decimal point of A?
>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>
>>> Example 2:
>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
>>> always exists as the prerequisite of 'infinite' repeating)
>>>
>>> Note that A is DEFINED exactly 0.999..., not 1.
>>> Since the non-zero fractional pattern of the number (repeating decimal) is
>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
>>> decimal) is irrational.
>>>
>>> ----
>>> [Tip] More about "repeating decimals":
>>> A0= 0.9 9 9 9 ...
>>> A1= 0.99 99 99 99 ...
>>> A2= 0.9 99 999 9999 ...
>>> A3= 0.999 9 9999 9 99999 ...
>>> A4= lim(n->∞) 1-1/n
>>> A5= lim(n->∞) 1-2/n
>>> A6= lim(n->∞) 1-3/10^n
>>> A7= lim(n->∞) n/(n+1)
>>> ...
>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>
>>
>> --
>> Copyright 2021 Pete Olcott
>>
>> Talent hits a target no one else can hit;
>> Genius hits a target no one else can see.
>> Arthur Schopenhauer
>
> Then, try to explain the density property
>
> A(0)=0
> A(n)=(A(n-1)+1)/2
>
> When and how A(n)=1 and the density property still holds.

Well since no A(n) is 1, that is a non-sense statement.

A, (not A(n)) which you define as = 0.999... is equal to one, but none
of your series of A(n) is that value, and for every A(n) your A(n+1)
proves that there is a number between A(n) and 1.

The Limit as n goes to infinity of A(n) is 1, as is your A above, but
that doesn't mean that any particular A(n) will have that value.

In fact, if A was NOT 1, then we can't have the density property, as
there can be no decimal number between 0.9999... and 1 as it would need
some digit greater than 9, which doesn't exist.

That can actually be one way to show that 0.999... can't be different
than 1 without breaking the basic properties of The Real Number System.

Yes, there ARE alternate formulation of numbers with different
properties, and some of these may allow for these two representations to
be different numbers, but they are not 'The Real Number System'.

On 12/16/21 3:40 PM, wij wrote:
> On Friday, 17 December 2021 at 00:33:11 UTC+8, olcott wrote:
>> On 12/16/2021 10:24 AM, wij wrote:
>>> On Thursday, 16 December 2021 at 22:33:15 UTC+8, olcott wrote:
>>>> On 12/12/2021 2:48 AM, wij wrote:
>>>>> Example 1:
>>>>
>>>> WRONG
>>>>
>>>> Are Repeating Decimals Rational?
>>>> Repeating or recurring decimals are decimal representations of numbers
>>>> with infinitely repeating digits. Numbers with a repeating pattern of
>>>> decimals are rational because when you put them into fractional form,
>>>> both the numerator a and denominator b become non-fractional whole numbers.
>>>>
>>>> For example, when you use long division to divide 1 by 3, the resultant
>>>> quotient is 0.33333…. However, when put it into fractional form, it's
>>>> made of positive integers that don’t have decimal points:
>>>> https://tutorme.com/blog/post/are-repeating-decimals-rational/
>>>>
>>>>> Let A≡ Σ(n=1,∞) 9/10^n= 0.999...= 999.../1000...= (3*3*(11...1))/(5*2)^n
>>>>> If A=1, can we conclude that (3*3*(11..1)) equals to (5*2)^∞ ?
>>>>> If A=1, what is the 3rd digit after the decimal point of A?
>>>>> If A=1, density property (of ℚ and ℝ) is broken (false).
>>>>>
>>>>> Example 2:
>>>>> 1/3≈0.333... + x (the conversion never divides completely, non-zero remainder x
>>>>> always exists as the prerequisite of 'infinite' repeating)
>>>>>
>>>>> Note that A is DEFINED exactly 0.999..., not 1.
>>>>> Since the non-zero fractional pattern of the number (repeating decimal) is
>>>>> defined to repeat infinitely, no p,q∈ℕ such that A=p/q, therefor, A (repeating
>>>>> decimal) is irrational.
>>>>>
>>>>> ----
>>>>> [Tip] More about "repeating decimals":
>>>>> A0= 0.9 9 9 9 ...
>>>>> A1= 0.99 99 99 99 ...
>>>>> A2= 0.9 99 999 9999 ...
>>>>> A3= 0.999 9 9999 9 99999 ...
>>>>> A4= lim(n->∞) 1-1/n
>>>>> A5= lim(n->∞) 1-2/n
>>>>> A6= lim(n->∞) 1-3/10^n
>>>>> A7= lim(n->∞) n/(n+1)
>>>>> ...
>>>>> This is just tip of the iceberg, "0.999..." is an infinite set of numbers.
>>>>> Actually, card("0.999...") is greater than ℵ1,ℵ2,ℵ3..., and more:
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
>>>>
>>>>
>>>> --
>>>> Copyright 2021 Pete Olcott
>>>>
>>>> Talent hits a target no one else can hit;
>>>> Genius hits a target no one else can see.
>>>> Arthur Schopenhauer
>>>
>>> Then, try to explain the density property
>>>
>>> A(0)=0
>>> A(n)=(A(n-1)+1)/2
>>>
>>> When and how A(n)=1 and the density property still holds.
>> Any number that can be expressed as the ratio of two positive integers
>> is a rational number. That is all there is to it.
>> --
>> Copyright 2021 Pete Olcott
>>
>> Talent hits a target no one else can hit;
>> Genius hits a target no one else can see.
>> Arthur Schopenhauer
>
> When you use long division to divide 1 by 3, the resultant quotient is
> accumulated by a repeating '3' (the quotient is 0.333...) and a remainder '1',
> This fact forms the equality:
>
> 1/3= 0.333... + non_zero_remainder

But at the limit of infinite digits, the remainder becomes smaller than
any number and becomes zero, so isn't 'non-zero'.

>
> Is this correct? If so, from the equation above we can deduce
> 1/3 - 0.333...= non_zero_remainder

Except that at in infinite number of digits, the remainder is goes to
zero, as The Reals don't represent infinitesimal values.

>
> Since RHS is non-zero, therefore LHS is non-zero. So we can conclude
> 1/3 ≠ 0.333...
>
> Is there any flaw?

The fact that while for a FINITE number of digtis, you have a non-zero
result, but as you add more and more digits the value shrinks, and
shrinks fast enough that when we get to the infinite number of digits
(in the limit) it becomes 0.

Infinity isn't a 'Number' in 'The Real Number System', only a limit, and
when you invoke it, you invoke the limit property, and the limit of that
remainder is zero, and of 0.999... is 1.

Yes, if you EXTEND the system to add infinties and infintesimals, you
get a different set of answers, but then you are not talking about The
Real Number System.

wij <wyniijj@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...".

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.

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?

--
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 */

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]).

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

Note: From practical usecases, ∞ and |ℕ| are likely the same, i.e. ∞=|ℕ|
[Ref] For the development history that infinity is a number:
https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
------

"0.999..." can be built in many ways:
A1= Σ(n=1,∞) 9/10^n = 0.999...
A2= Σ(n=1,∞) 1/2^n = 0.999...
Borrow limit notation:
A3= lim(n->∞) 1-1/n = 0.999...
A4= lim(n->∞) 1-2/n = 0.999...
A5= lim(n->∞) 1-3/10^n = 0.999...
A6= lim(n->∞) n/(n+1) = 0.999...

"0.333..." (repeating decimal in general) can be understood as below:
1/3=0.3 +0.1
1/3=0.33 +0.01
1/3=0.333 +0.001
1/3=0.333... + 0.000...1
1/3=0.333... + 1/10^n // The remainder is only longer/more complex.
// Scale/magnitude is irrelevant in logic.

By "0.333...", I mean exactly, infinite repeating decimal 3.
So, yes, various definitions of 0.333... is not exactly equal to 1/3.

> 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.

> 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).

[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

> --
> 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 */

On Friday, 24 December 2021 at 08:21:47 UTC+8, wij wrote:
> 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]).
>
> Firstly, the introduction of the symbol '∞':
> '∞' (infinity)::=
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>
> Note: From practical usecases, ∞ and |ℕ| are likely the same, i.e. ∞=|ℕ|
> [Ref] For the development history that infinity is a number:
> https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
> ------
>
> "0.999..." can be built in many ways:
> A1= Σ(n=1,∞) 9/10^n = 0.999...
> A2= Σ(n=1,∞) 1/2^n = 0.999...
> Borrow limit notation:
> A3= lim(n->∞) 1-1/n = 0.999...
> A4= lim(n->∞) 1-2/n = 0.999...
> A5= lim(n->∞) 1-3/10^n = 0.999...
> A6= lim(n->∞) n/(n+1) = 0.999...
>
> "0.333..." (repeating decimal in general) can be understood as below:
> 1/3=0.3 +0.1
> 1/3=0.33 +0.01
> 1/3=0.333 +0.001
> 1/3=0.333... + 0.000...1
> 1/3=0.333... + 1/10^n // The remainder is only longer/more complex.
> // Scale/magnitude is irrelevant in logic.
>
> By "0.333...", I mean exactly, infinite repeating decimal 3.
> So, yes, various definitions of 0.333... is not exactly equal to 1/3.
> > 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.
> > 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).
>
> [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
> > --
> > 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 */
> "0.333..." (repeating decimal in general) can be understood as below:

Error correction:
1/3=0.3 +1/30
1/3=0.33 +1/300
1/3=0.333 +1/3000
....
1/3=0.333... + 1/(3*10^n) // The remainder is only longer/more complex.
// Scale/magnitude is irrelevant in logic.

On 12/23/21 7:21 PM, wij wrote:
> 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]).
>
> Firstly, the introduction of the symbol '∞':
> '∞' (infinity)::=
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

Which means you are not talking about The Real Number System, so the
term 'irrational' doesn't really apply.

The Reals only deal with FINITE values.

>
> Note: From practical usecases, ∞ and |ℕ| are likely the same, i.e. ∞=|ℕ|
> [Ref] For the development history that infinity is a number:
> https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
> ------
>
> "0.999..." can be built in many ways:
> A1= Σ(n=1,∞) 9/10^n = 0.999...
> A2= Σ(n=1,∞) 1/2^n = 0.999...
> Borrow limit notation:
> A3= lim(n->∞) 1-1/n = 0.999...
> A4= lim(n->∞) 1-2/n = 0.999...
> A5= lim(n->∞) 1-3/10^n = 0.999...
> A6= lim(n->∞) n/(n+1) = 0.999...
>
> "0.333..." (repeating decimal in general) can be understood as below:
> 1/3=0.3 +0.1
> 1/3=0.33 +0.01
> 1/3=0.333 +0.001
> 1/3=0.333... + 0.000...1
> 1/3=0.333... + 1/10^n // The remainder is only longer/more complex.
> // Scale/magnitude is irrelevant in logic.
>
> By "0.333...", I mean exactly, infinite repeating decimal 3.
> So, yes, various definitions of 0.333... is not exactly equal to 1/3.
>
>> 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.
>
>> 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).
>
> [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
>
> > --
>> 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 */

On Friday, 24 December 2021 at 08:40:15 UTC+8, richar...@gmail.com wrote:
> On 12/23/21 7:21 PM, wij wrote:
> > 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]).
> >
> > Firstly, the introduction of the symbol '∞':
> > '∞' (infinity)::=
> > 1. ∀n∈ℕ, n<∞
> > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> Which means you are not talking about The Real Number System, so the
> term 'irrational' doesn't really apply.
>
> 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
= 9/10 +9/100 +9/1000 +... = 1?

As noted before, the digit 9 can be billion light years long and beyond to another
universe INFINITELY (eternal), yet still not exactly 1.

> >
> > Note: From practical usecases, ∞ and |ℕ| are likely the same, i.e. ∞=|ℕ|
> > [Ref] For the development history that infinity is a number:
> > https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
> > ------
> >
> > "0.999..." can be built in many ways:
> > A1= Σ(n=1,∞) 9/10^n = 0.999...
> > A2= Σ(n=1,∞) 1/2^n = 0.999...
> > Borrow limit notation:
> > A3= lim(n->∞) 1-1/n = 0.999...
> > A4= lim(n->∞) 1-2/n = 0.999...
> > A5= lim(n->∞) 1-3/10^n = 0.999...
> > A6= lim(n->∞) n/(n+1) = 0.999...
> >
> > "0.333..." (repeating decimal in general) can be understood as below:
> > 1/3=0.3 +0.1
> > 1/3=0.33 +0.01
> > 1/3=0.333 +0.001
> > 1/3=0.333... + 0.000...1
> > 1/3=0.333... + 1/10^n // The remainder is only longer/more complex.
> > // Scale/magnitude is irrelevant in logic.
> >
> > By "0.333...", I mean exactly, infinite repeating decimal 3.
> > So, yes, various definitions of 0.333... is not exactly equal to 1/3.
> >
> >> 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.
> >
> >> 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).
> >
> > [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
> >
> > > --
> >> 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 */

On 12/23/21 7:58 PM, wij wrote:
> On Friday, 24 December 2021 at 08:40:15 UTC+8, richar...@gmail.com wrote:
>> On 12/23/21 7:21 PM, wij wrote:
>>> 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]).
>>>
>>> Firstly, the introduction of the symbol '∞':
>>> '∞' (infinity)::=
>>> 1. ∀n∈ℕ, n<∞
>>> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>> Which means you are not talking about The Real Number System, so the
>> term 'irrational' doesn't really apply.
>>
>> 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)?:

But A1 is not the value for ANY finite value of N, but the notation
means it is the LIMIT as n gets arbitraryly large.

By the definition of Limits, given any positive error bound e, you can
find an N such that for all elements of that sequnce after that that N
will have an error less that that bound.

For this case |A1-1| < e if N > -log10(e)

IF you reject the Limit property, you reject The Real Number System.

> A1= Σ(n=1,∞) 9/10^n
> = 9/10 +9/100 +9/1000 +... = 1?
>
> As noted before, the digit 9 can be billion light years long and beyond to another
> universe INFINITELY (eternal), yet still not exactly 1.

Right, but the LIMIT looks at how it goes in an unbounded number, at
which point there is no measurable distance betwenn 0.999... and 1 so
they are the same value.

>
>>>
>>> Note: From practical usecases, ∞ and |ℕ| are likely the same, i.e. ∞=|ℕ|
>>> [Ref] For the development history that infinity is a number:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/Infinity-en.txt/download
>>> ------
>>>
>>> "0.999..." can be built in many ways:
>>> A1= Σ(n=1,∞) 9/10^n = 0.999...
>>> A2= Σ(n=1,∞) 1/2^n = 0.999...
>>> Borrow limit notation:
>>> A3= lim(n->∞) 1-1/n = 0.999...
>>> A4= lim(n->∞) 1-2/n = 0.999...
>>> A5= lim(n->∞) 1-3/10^n = 0.999...
>>> A6= lim(n->∞) n/(n+1) = 0.999...
>>>
>>> "0.333..." (repeating decimal in general) can be understood as below:
>>> 1/3=0.3 +0.1
>>> 1/3=0.33 +0.01
>>> 1/3=0.333 +0.001
>>> 1/3=0.333... + 0.000...1
>>> 1/3=0.333... + 1/10^n // The remainder is only longer/more complex.
>>> // Scale/magnitude is irrelevant in logic.
>>>
>>> By "0.333...", I mean exactly, infinite repeating decimal 3.
>>> So, yes, various definitions of 0.333... is not exactly equal to 1/3.
>>>
>>>> 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.
>>>
>>>> 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).
>>>
>>> [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
>>>
>>>> --
>>>> 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 */

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

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.

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. 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.

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.

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

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