Moderator: Flannel Jesus
Ecmandu wrote:James S Saint wrote:There is no "technique" involved, Ec..
It is merely a question of the definition of the terms involved.
"..." means that the preceding number continues eternally without change.
Well, decimals that are non zero regress, that is change... What happens with those, is that we find a pattern in regression, infinite regression to be exact.
But to say they don't change is to say they don't continue... Just to articulate the issue
James wrote:Again, I am asking how any of a set of numbers that have a 1 at the end loses the 1.
James wrote:What possible logic could say that despite being defined as only numbers that have a 1 at their end, one of that exact same set actually doesn't have a 1 at the end.
Carleas wrote:James wrote:Again, I am asking how any of a set of numbers that have a 1 at the end loses the 1.
Again, I'm not saying that anything loses a 1. Rather, finite strings just aren't relevant here, because we're talking about an infinite string that by definition cannot have any digit "at the end".
Do you agree that every single item in that set has a 1 at its end and thus non-zero?Second Proof that the two "numbers" are not equal, the "set definition proof".
The definition of the set is:That is an infinite set of non-zero items.
0.1
0.01
0.001
0.0001
.
.
.
Carleas wrote:James wrote:What possible logic could say that despite being defined as only numbers that have a 1 at their end, one of that exact same set actually doesn't have a 1 at the end.
The difference between .999... and 1 is not a member of that set. That is the number we're talking about.
The above defined set is formed by the following set:That is the same infinite set of non-zero items.
1.0 - 0.9 = 0.1
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
.
.
.
"1.0 - 0.999..." is a member of that set.
James S Saint wrote:The series 0.999... is the result of an operation:
0.999... = 0.9 + 0.009 + 0.0009 + 0.00009 + ... =wherein Σ is the accumulated sum at each stage
That series of operations has no opportunity to become different no matter how long it is carried out. That is why it is "endless" or "infinite". The result of every operation is that there is required to still be a small percentage between the accumulated result of the operation and 1.0. As long as the operation adds only 90%, there will always be a 10% not added into the accumulated sum. Thus the difference between the sum and 1.0 can never become zero. QED
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.
Except that the 'proof' is wrong because 0.9+0.09+0.009+ ...James S Saint wrote:
The series 0.999... is the result of an operation:
0.999... = 0.9 + 0.009 + 0.0009 + 0.00009 + ... =
wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.
That series of operations has no opportunity to become different no matter how long it is carried out. That is why it is "endless" or "infinite". The result of every operation is that there is required to still be a small percentage between the accumulated result of the operation and 1.0. As long as the operation adds only 90%, there will always be a 10% not added into the accumulated sum. Thus the difference between the sum and 1.0 can never become zero. QED
phyllo wrote:Except that the 'proof' is wrong because 0.9+0.09+0.009+ ...James S Saint wrote:
The series 0.999... is the result of an operation:
0.999... = 0.9 + 0.009 + 0.0009 + 0.00009 + ... =
wherein Σ is the accumulated sum at each stage
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
90% (1 - Σ) +
.
.
.
That series of operations has no opportunity to become different no matter how long it is carried out. That is why it is "endless" or "infinite". The result of every operation is that there is required to still be a small percentage between the accumulated result of the operation and 1.0. As long as the operation adds only 90%, there will always be a 10% not added into the accumulated sum. Thus the difference between the sum and 1.0 can never become zero. QED
is a convergent series which obviously converges to 1.
https://en.wikipedia.org/wiki/Convergent_series
phyllo wrote:(The convergence of a particular series solves Zeno' Dichotomy Paradox. https://en.wikipedia.org/wiki/Zeno's_pa ... my_paradox )
jerkey wrote:At some point, the infinitesimal reaches a critical point, where it transforms infinity into finity. That's got to be it.
James S Saint wrote:jerkey wrote:At some point, the infinitesimal reaches a critical point, where it transforms infinity into finity. That's got to be it.
That is what must happen in order to make 1.0 = 0.999..., but the problem is that such can never happen.
Arminius wrote:Strange! Before James and I posted one could read the following text below Jerkey's last post, although he was the last one who posted (before James and me):
"Last edited by jerkey on Wed Jun 29, 2016 5:24 pm, edited 2 times in total."
How is that posible? Is it a wonder?
Arminius wrote:Look at the time that is mentioned below Jerkey's post:
"Last edited by jerkey on Wed Jun 29, 2016 5:24 pm, edited 2 times in total."
You posted after it. Look at the time that is mentioned above your post:
"by James S Saint » Wed Jun 29, 2016 5:48 pm".
Strange!
James S Saint wrote:Arminius wrote:Look at the time that is mentioned below Jerkey's post:
"Last edited by jerkey on Wed Jun 29, 2016 5:24 pm, edited 2 times in total."
You posted after it. Look at the time that is mentioned above your post:
"by James S Saint » Wed Jun 29, 2016 5:48 pm".
Strange!
Yeah 24 minutes seems like a lot of settle time required by a server.
Arminius wrote:In Addition: the serrver does not "know" when the next poster is going to post.
James S Saint wrote:Arminius wrote:In addition: the serrver does not "know" when the next poster is going to post.
Realize that as they make machines more and more clever, machines begin to be able to do what humans would think impossible.
Carleas wrote:James, you're treating an infinite string as though it's just a really, really long finite string (thus leaving a "last little bit", presumably after the "last digit" at the "end" of the "endless" string...). That isn't true.
jerkey wrote:So could not the two impossibilities somehow have some sort of congruence?
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