On Tuesday, 14 December 2021 at 20:02:31 UTC+8, richar...@gmail.com wrote:
> On 12/14/21 5:06 AM, wij wrote:
> > On Tuesday, 14 December 2021 at 08:32:06 UTC+8, richar...@gmail.com wrote:
> >> On 12/13/21 10:53 AM, wij wrote:
> >>> On Monday, 13 December 2021 at 23:30:04 UTC+8, Mike Terry wrote:
> >>>> On 13/12/2021 12:27, 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
> >>>>>
> >>>> So we were right!
> >>>>
> >>>> So if A=1, how does this break the density property?
> >>>
> >>> When there ever exists a time a n such that A(n)=1, "A(n)<A(n+1)<1"
> >>> then, the density property does not hold.
> >>> This leads to 0.999...≠1.
> >> But the question wasn't if A(n) was 1, the question was if A, the limit
> >> as n goes to infinity was 1. The limit of a sequence is normally NOT an
> >> element of the sequence.
> >
> > The question was whether A(n) equals 1 or not when n approaches infinity and the
> > related density property.
> > You were still stuck in using 'limit theory' to miss the issue.
> No, *A* was the INFINITE sum (not the finite partial sum on the way),
> which can only be evaluate IN THE LIMIT, since infinity isn't a number
> (in Natural, Rationals, or Reals).

If the infinity notation is in infinite series or limit notation is exempted !!!
lim(n->∞) f(n) // fine, no problem (a number?)
Σ(n=0,∞) f(n) // ditto
"0<|x-∞|<δ" // Even is allowed in ε-δ argument

> Now, if you want to talk about a different field, lke the Surreals, then
> the rules change.

It seem that the right answer to you depends on what the text
book one read and can recite. What I want to talk about is whether 0.999...=1
or not without breaking the density property.

--- Pythagoreans' Code ---
Infinitely approaching means equal. Number too small equals zero.

> >
> >> Note, as pointed below, 'infinity' is not a value in N, Q, or R.
> >>>
> >>>>>> 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
> >>>> Correct.
> >>>>
> >>>>>
> >>>>> When A(n)=1? Infinity?
> >>>>
> >>>> A(n) < 1 for all n. (n = oo is not in the range of n)
> >>>>
> >>>>
> >>>> Mike.
> >>>
> >>> Correct.
> >>>

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.

> 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.

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)?

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.

On 12/14/21 12:06 PM, wij wrote:
> On Tuesday, 14 December 2021 at 20:02:31 UTC+8, richar...@gmail.com wrote:
>> On 12/14/21 5:06 AM, wij wrote:
>>> On Tuesday, 14 December 2021 at 08:32:06 UTC+8, richar...@gmail.com wrote:
>>>> On 12/13/21 10:53 AM, wij wrote:
>>>>> On Monday, 13 December 2021 at 23:30:04 UTC+8, Mike Terry wrote:
>>>>>> On 13/12/2021 12:27, 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
>>>>>>>
>>>>>> So we were right!
>>>>>>
>>>>>> So if A=1, how does this break the density property?
>>>>>
>>>>> When there ever exists a time a n such that A(n)=1, "A(n)<A(n+1)<1"
>>>>> then, the density property does not hold.
>>>>> This leads to 0.999...≠1.
>>>> But the question wasn't if A(n) was 1, the question was if A, the limit
>>>> as n goes to infinity was 1. The limit of a sequence is normally NOT an
>>>> element of the sequence.
>>>
>>> The question was whether A(n) equals 1 or not when n approaches infinity and the
>>> related density property.
>>> You were still stuck in using 'limit theory' to miss the issue.
>> No, *A* was the INFINITE sum (not the finite partial sum on the way),
>> which can only be evaluate IN THE LIMIT, since infinity isn't a number
>> (in Natural, Rationals, or Reals).
>
> If the infinity notation is in infinite series or limit notation is exempted !!!
> lim(n->∞) f(n) // fine, no problem (a number?)
> Σ(n=0,∞) f(n) // ditto
> "0<|x-∞|<δ" // Even is allowed in ε-δ argument
>
>> Now, if you want to talk about a different field, lke the Surreals, then
>> the rules change.
>
> It seem that the right answer to you depends on what the text
> book one read and can recite. What I want to talk about is whether 0.999...=1
> or not without breaking the density property.

And, as I have said, from the basic properties of what 0.999... MEANS,
it becomes identical to the number 1, IF we are talking about the domain
of Reals or Rationals.

Remember, if you want to claim you are talking about an established
field of Mathematics, you have to accept the definitios of that field,
otherwise you aren't really talking about that field, and being like PO
and showing you lack of understanding.

>
> --- Pythagoreans' Code ---
> Infinitely approaching means equal. Number too small equals zero.
>
>>>
>>>> Note, as pointed below, 'infinity' is not a value in N, Q, or R.
>>>>>
>>>>>>>> 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
>>>>>> Correct.
>>>>>>
>>>>>>>
>>>>>>> When A(n)=1? Infinity?
>>>>>>
>>>>>> A(n) < 1 for all n. (n = oo is not in the range of n)
>>>>>>
>>>>>>
>>>>>> Mike.
>>>>>
>>>>> Correct.
>>>>>

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?

On Wednesday, 15 December 2021 at 10:02:46 UTC+8, richar...@gmail.com wrote:
> On 12/14/21 12:06 PM, wij wrote:
> > On Tuesday, 14 December 2021 at 20:02:31 UTC+8, richar...@gmail.com wrote:
> >> On 12/14/21 5:06 AM, wij wrote:
> >>> On Tuesday, 14 December 2021 at 08:32:06 UTC+8, richar...@gmail.com wrote:
> >>>> On 12/13/21 10:53 AM, wij wrote:
> >>>>> On Monday, 13 December 2021 at 23:30:04 UTC+8, Mike Terry wrote:
> >>>>>> On 13/12/2021 12:27, 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
> >>>>>>>
> >>>>>> So we were right!
> >>>>>>
> >>>>>> So if A=1, how does this break the density property?
> >>>>>
> >>>>> When there ever exists a time a n such that A(n)=1, "A(n)<A(n+1)<1"
> >>>>> then, the density property does not hold.
> >>>>> This leads to 0.999...≠1.
> >>>> But the question wasn't if A(n) was 1, the question was if A, the limit
> >>>> as n goes to infinity was 1. The limit of a sequence is normally NOT an
> >>>> element of the sequence.
> >>>
> >>> The question was whether A(n) equals 1 or not when n approaches infinity and the
> >>> related density property.
> >>> You were still stuck in using 'limit theory' to miss the issue.
> >> No, *A* was the INFINITE sum (not the finite partial sum on the way),
> >> which can only be evaluate IN THE LIMIT, since infinity isn't a number
> >> (in Natural, Rationals, or Reals).
> >
> > If the infinity notation is in infinite series or limit notation is exempted !!!
> > lim(n->∞) f(n) // fine, no problem (a number?)
> > Σ(n=0,∞) f(n) // ditto
> > "0<|x-∞|<δ" // Even is allowed in ε-δ argument
> >
> >> Now, if you want to talk about a different field, lke the Surreals, then
> >> the rules change.
> >
> > It seem that the right answer to you depends on what the text
> > book one read and can recite. What I want to talk about is whether 0.999...=1
> > or not without breaking the density property.
> And, as I have said, from the basic properties of what 0.999... MEANS,
> it becomes identical to the number 1, IF we are talking about the domain
> of Reals or Rationals.
>
>
> Remember, if you want to claim you are talking about an established
> field of Mathematics, you have to accept the definitios of that field,
> otherwise you aren't really talking about that field, and being like PO
> and showing you lack of understanding.
> >
> > --- Pythagoreans' Code ---
> > Infinitely approaching means equal. Number too small equals zero.
> >
> >>>
> >>>> Note, as pointed below, 'infinity' is not a value in N, Q, or R.
> >>>>>
> >>>>>>>> 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
> >>>>>> Correct.
> >>>>>>
> >>>>>>>
> >>>>>>> When A(n)=1? Infinity?
> >>>>>>
> >>>>>> A(n) < 1 for all n. (n = oo is not in the range of n)
> >>>>>>
> >>>>>>
> >>>>>> Mike.
> >>>>>
> >>>>> Correct.
> >>>>>

As I said: Definition is a lowest level of understanding.
If you are contented within it, be it.

On 12/15/21 7:23 AM, wij wrote:

> As I said: Definition is a lowest level of understanding.
> If you are contented within it, be it.

Definition is the basis of defining what you are talking about.

Change the definition, you change the system.

This is actually FINE, just don't say you are working in the original
system. PERIOD.

Yes, there ARE systems which include infinity (in many different forms)
as members of them. These are just not the Natural, Rational, or Real
Numbers.

If you want to talk about THOSE OTHER systems, fine, just admit what you
are talking about. Those systems exist and behave differently than those
base systems.

Every time you extend a system, like by adding some from of infinity,
you add some new things that can be expressed, but some of the basic
rules that used to hold, no longer hold.

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.

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.
>
> 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.)
>
>>>
>>>> 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.
>
>>> 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.
>
>>
>> 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.
>
>> 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?
>
>> This is the way the Reals, et all, handle infinite series.
>
> Your Real is Q plus numbers that have finite notation.
>
>>>
>>>>>
>>>>> 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.
>
>>
>> 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.
>
>> 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.

On Wednesday, 15 December 2021 at 13:14:54 UTC, richar...@gmail.com wrote:
> On 12/15/21 7:23 AM, wij wrote:
>
> > As I said: Definition is a lowest level of understanding.
> > If you are contented within it, be it.
> Definition is the basis of defining what you are talking about.
>
> Change the definition, you change the system.
>
>
> This is actually FINE, just don't say you are working in the original
> system. PERIOD.
>
> Yes, there ARE systems which include infinity (in many different forms)
> as members of them. These are just not the Natural, Rational, or Real
> Numbers.
>
They're not called the "Natural, Rational, or Real numbers". By convention. You
can argue that "natural" would be a more logical name for numbers in some
other number system. But the word is taken, so to avoid confusion, you must
use "natural" to refer to positive integers, and another word to refer to, say, all reals
plus +/- infinity and infinitesimal. You could call it the "idealised computer arithmetic
set" for example.

Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:

> ... But the word is taken, so to avoid confusion, you must use
> "natural" to refer to positive integers, and another word to refer to,
> say, all reals plus +/- infinity and infinitesimal. You could call it
> the "idealised computer arithmetic set" for example.

That set has a name. Casually, it's called the "extended reals", but
where there might be confusion, it's the affine (or protective)
extension or R.

--
Ben.

On 15/12/2021 16:43, Ben Bacarisse wrote:
> Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:
>
>> ... But the word is taken, so to avoid confusion, you must use
>> "natural" to refer to positive integers, and another word to refer to,
>> say, all reals plus +/- infinity and infinitesimal. You could call it
>> the "idealised computer arithmetic set" for example.
>
> That set has a name. Casually, it's called the "extended reals", but
> where there might be confusion, it's the affine (or protective)
> extension or R.
>

But there are no infinitesimals in the extended reals as they're
normally defined (e.g. within measure theory or functional analysis).
Just the normal reals with added points at +- infinity.

Mike.

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.

Click here to read the complete article

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.

On Wednesday, 15 December 2021 at 21:30:13 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.
> >
> > 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.)
> >
> >>>
> >>>> 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.
> >
> >>> 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.
> >
> >>
> >> 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.
> >
> >> 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?
> >
> >> This is the way the Reals, et all, handle infinite series.
> >
> > Your Real is Q plus numbers that have finite notation.
> >
> >>>
> >>>>>
> >>>>> 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.
> >
> >>
> >> 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.
> >
> >> 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.
> Here is a simple version of it.
>
> given the repeating decimal 0.xyz xyz.... = F
>
> Since it repeats with a period of 3 digits, let us also look at F * 10^3
>
> That value will be xyz.xyzxyz...
>
> Subtracting F from 1000*F we get 999*F and we also get xyz.xyz xyz... -
> 0.xyz xyz... which is just xyz
>
> thus we get that 999*F = xyz, or F = xyz/999
>
>
> This is basically the same method used to solve many infinite sums.
>
> Yes, it needs some polish to be formal with boiler plate to handle other
> bases and arbitrary repeat lengths, but that is the core of the proof.

Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:

> On 15/12/2021 16:43, Ben Bacarisse wrote:
>> Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:
>>
>>> ... But the word is taken, so to avoid confusion, you must use
>>> "natural" to refer to positive integers, and another word to refer to,
>>> say, all reals plus +/- infinity and infinitesimal. You could call it
>>> the "idealised computer arithmetic set" for example.
>
>> That set has a name. Casually, it's called the "extended reals", but
>> where there might be confusion, it's the affine (or protective)
>> extension or R.
>
> But there are no infinitesimals in the extended reals as they're
> normally defined (e.g. within measure theory or functional analysis).
> Just the normal reals with added points at +- infinity.

I did not spot that in what MM said. I was translating what I thought
"idealised computer arithmetic set" meant, and missed the fact that MM
had included infinitesimals. Not sure what they have to do with
computer arithmetic, idealised of otherwise.

--
Ben.

wij <wyniijj@gmail.com> writes:

> you need to prove 0.333...=1/3 (NOT BY your favorite DEFINITION)

What a silly request! 2+2=4 is easily proved using my favourite
definition of these symbols. 2+2=10 by my second favourite definition.
Likewise, 0.333...=1 using that same definition.

--
Ben.

On Wednesday, 15 December 2021 at 23:08:45 UTC, Ben Bacarisse wrote:
> Mike Terry <news.dead.p...@darjeeling.plus.com> writes:
>
> > On 15/12/2021 16:43, Ben Bacarisse wrote:
> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>
> >>> ... But the word is taken, so to avoid confusion, you must use
> >>> "natural" to refer to positive integers, and another word to refer to,
> >>> say, all reals plus +/- infinity and infinitesimal. You could call it
> >>> the "idealised computer arithmetic set" for example.
> >
> >> That set has a name. Casually, it's called the "extended reals", but
> >> where there might be confusion, it's the affine (or protective)
> >> extension or R.
> >
> > But there are no infinitesimals in the extended reals as they're
> > normally defined (e.g. within measure theory or functional analysis).
> > Just the normal reals with added points at +- infinity.
> I did not spot that in what MM said. I was translating what I thought
> "idealised computer arithmetic set" meant, and missed the fact that MM
> had included infinitesimals. Not sure what they have to do with
> computer arithmetic, idealised of otherwise.
>
I forgot that IEEE doesn't have a value for "infinitesimal". You need a
concept of infinitesimals to propose wij's argument that 0.999... is
a bit smaller than 1.000...

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)

On 12/15/21 5:38 PM, wij wrote:
> On Wednesday, 15 December 2021 at 21:30:13 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.
>>>
>>> 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.)
>>>
>>>>>
>>>>>> 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.
>>>
>>>>> 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.
>>>
>>>>
>>>> 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.
>>>
>>>> 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?
>>>
>>>> This is the way the Reals, et all, handle infinite series.
>>>
>>> Your Real is Q plus numbers that have finite notation.
>>>
>>>>>
>>>>>>>
>>>>>>> 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.
>>>
>>>>
>>>> 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.
>>>
>>>> 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.
>> Here is a simple version of it.
>>
>> given the repeating decimal 0.xyz xyz.... = F
>>
>> Since it repeats with a period of 3 digits, let us also look at F * 10^3
>>
>> That value will be xyz.xyzxyz...
>>
>> Subtracting F from 1000*F we get 999*F and we also get xyz.xyz xyz... -
>> 0.xyz xyz... which is just xyz
>>
>> thus we get that 999*F = xyz, or F = xyz/999
>>
>>
>> This is basically the same method used to solve many infinite sums.
>>
>> Yes, it needs some polish to be formal with boiler plate to handle other
>> bases and arbitrary repeat lengths, but that is the core of the proof.
>
> I tidied up and interpreted your statement (see fit):
>
> (1) 0.xyz xyz... = F
>
> (2) xyz.xyzxyz... = 1000*F // mul 1000 to both side, valid
>
> (3) xyz.xyzxyz... -F = 999*F // sub F from both sides, valid
>
> (4) xyz+0.xyzxyz... -F = 999*F // break the 1st term, valid
>
> (5) xyz+(0.xyzxyz... - F)= 999*F // rearangement, valid (a bit fishy, misguiding)
>
> (6) xyz = 999*F // oop, a different "0.xyzxyz..." =F is implicitly
> // assumed. The above "0.xyzxyz..." is not the
> // original one. Invalid deduction.
> (7) xyz/999 = F
>
> For (6) to be valid, the premise 1000*0.xzyxzy...-xyz= 0.xyzxyz... has to be asserted.
> Since there is no such premise, (6) is invalid.
> To see this clearly, try deduce backward from (6) to (5).

On Thursday, 16 December 2021 at 11:43:34 UTC+8, richar...@gmail.com wrote:
> On 12/15/21 5:38 PM, wij wrote:
> > On Wednesday, 15 December 2021 at 21:30:13 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.
> >>>
> >>> 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.)
> >>>
> >>>>>
> >>>>>> 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.
> >>>
> >>>>> 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.
> >>>
> >>>>
> >>>> 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.
> >>>
> >>>> 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?
> >>>
> >>>> This is the way the Reals, et all, handle infinite series.
> >>>
> >>> Your Real is Q plus numbers that have finite notation.
> >>>
> >>>>>
> >>>>>>>
> >>>>>>> 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.
> >>>
> >>>>
> >>>> 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.
> >>>
> >>>> 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.
> >> Here is a simple version of it.
> >>
> >> given the repeating decimal 0.xyz xyz.... = F
> >>
> >> Since it repeats with a period of 3 digits, let us also look at F * 10^3
> >>
> >> That value will be xyz.xyzxyz...
> >>
> >> Subtracting F from 1000*F we get 999*F and we also get xyz.xyz xyz... -
> >> 0.xyz xyz... which is just xyz
> >>
> >> thus we get that 999*F = xyz, or F = xyz/999
> >>
> >>
> >> This is basically the same method used to solve many infinite sums.
> >>
> >> Yes, it needs some polish to be formal with boiler plate to handle other
> >> bases and arbitrary repeat lengths, but that is the core of the proof.
> >
> > I tidied up and interpreted your statement (see fit):
> >
> > (1) 0.xyz xyz... = F
> >
> > (2) xyz.xyzxyz... = 1000*F // mul 1000 to both side, valid
> >
> > (3) xyz.xyzxyz... -F = 999*F // sub F from both sides, valid
> >
> > (4) xyz+0.xyzxyz... -F = 999*F // break the 1st term, valid
> >
> > (5) xyz+(0.xyzxyz... - F)= 999*F // rearangement, valid (a bit fishy, misguiding)
> >
> > (6) xyz = 999*F // oop, a different "0.xyzxyz..." =F is implicitly
> > // assumed. The above "0.xyzxyz..." is not the
> > // original one. Invalid deduction.
> > (7) xyz/999 = F
> >
> > For (6) to be valid, the premise 1000*0.xzyxzy...-xyz= 0.xyzxyz... has to be asserted.
> > Since there is no such premise, (6) is invalid.
> > To see this clearly, try deduce backward from (6) to (5).
> Why do youy say it isn't a basic premise.
>
> 0.xyzxyz... * 1000 by DEFINITION of decimal notaiton is xyz.xyzxyz...

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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...

On Thursday, 16 December 2021 at 11:01:08 UTC, wij wrote:
> On Thursday, 16 December 2021 at 11:43:34 UTC+8, richar...@gmail.com wrote:
> >
> > 0.xyzxyz... * 1000 by DEFINITION of decimal notaiton is xyz.xyzxyz...
> By what definition?
>
We say that a property of decimal notation is that muliplying or dividing by
a power of ten shifts the decimal point left or right. That's consistent with
the other things we say about that notation.
>
> > Multiply a decimal number by a power of ten is EXACTLY the same as
> > shifting the decimal point. This follows from the basic definition of
> > what the decimal notation means.
> "0.xyzxyz... * 1000 = xyz.xyzxyz..." (still not clear) is the premise
> (different place different name, e.g. you use it as 'definition') you must
> give out to interpret the meaning of ".xyz...".
>
Now you can say that "the property that multiplying by a power of ten
shifts the decimal point left or right dies not hold when an ellipsis is used."
However now you've got to work out all the implications of that. For
example, does it hold for 1.000...? If so, why is that special case? If not,
why does the shift property reappear when we remove the ellipsis?

You can declare any rules that you want. But it's hard to come up with a set of
rules which are both consistent, and interesting. It's not really desireable that
we have two decimal representations for 1.0 and it's perfectly reasonable to
try to add, change, or remove a rule about decimal representation to get rid
of that. However it's not easy to avoid making a much worse mess in the process.

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.

On 12/16/21 6:01 AM, wij wrote:
> On Thursday, 16 December 2021 at 11:43:34 UTC+8, richar...@gmail.com wrote:
>> On 12/15/21 5:38 PM, wij wrote:
>>> On Wednesday, 15 December 2021 at 21:30:13 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.
>>>>>
>>>>> 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.)
>>>>>
>>>>>>>
>>>>>>>> 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.
>>>>>
>>>>>>> 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.
>>>>>
>>>>>>
>>>>>> 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.
>>>>>
>>>>>> 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?
>>>>>
>>>>>> This is the way the Reals, et all, handle infinite series.
>>>>>
>>>>> Your Real is Q plus numbers that have finite notation.
>>>>>
>>>>>>>
>>>>>>>>>
>>>>>>>>> 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.
>>>>>
>>>>>>
>>>>>> 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.
>>>>>
>>>>>> 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.
>>>> Here is a simple version of it.
>>>>
>>>> given the repeating decimal 0.xyz xyz.... = F
>>>>
>>>> Since it repeats with a period of 3 digits, let us also look at F * 10^3
>>>>
>>>> That value will be xyz.xyzxyz...
>>>>
>>>> Subtracting F from 1000*F we get 999*F and we also get xyz.xyz xyz... -
>>>> 0.xyz xyz... which is just xyz
>>>>
>>>> thus we get that 999*F = xyz, or F = xyz/999
>>>>
>>>>
>>>> This is basically the same method used to solve many infinite sums.
>>>>
>>>> Yes, it needs some polish to be formal with boiler plate to handle other
>>>> bases and arbitrary repeat lengths, but that is the core of the proof.
>>>
>>> I tidied up and interpreted your statement (see fit):
>>>
>>> (1) 0.xyz xyz... = F
>>>
>>> (2) xyz.xyzxyz... = 1000*F // mul 1000 to both side, valid
>>>
>>> (3) xyz.xyzxyz... -F = 999*F // sub F from both sides, valid
>>>
>>> (4) xyz+0.xyzxyz... -F = 999*F // break the 1st term, valid
>>>
>>> (5) xyz+(0.xyzxyz... - F)= 999*F // rearangement, valid (a bit fishy, misguiding)
>>>
>>> (6) xyz = 999*F // oop, a different "0.xyzxyz..." =F is implicitly
>>> // assumed. The above "0.xyzxyz..." is not the
>>> // original one. Invalid deduction.
>>> (7) xyz/999 = F
>>>
>>> For (6) to be valid, the premise 1000*0.xzyxzy...-xyz= 0.xyzxyz... has to be asserted.
>>> Since there is no such premise, (6) is invalid.
>>> To see this clearly, try deduce backward from (6) to (5).
>> Why do youy say it isn't a basic premise.
>>
>> 0.xyzxyz... * 1000 by DEFINITION of decimal notaiton is xyz.xyzxyz...
>
> By what definition?