## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
41%
No, 1 ≠ 0.999...
16
50%
Other
3
9%

### Re: Is 1 = 0.999... ? Really?

Actually I clear something up with my perspective here...

I think James and Carleas are making the same mistake, James from a theistic view and carleas not ...

That infinites converge!!!

Ponder it!!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

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

In anticipation of your response, I had updated my post to include "number or pattern" before you posted that. It just depends on which kind of list you are making whether it is a single number pattern or a more complicated pattern. Either way, the pattern doesn't change such as to become so small that a part of the pattern must become different.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
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Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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".

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.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

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".

No...
Second Proof that the two "numbers" are not equal, the "set definition proof".
The definition of the set is:
0.1
0.01
0.001
0.0001
.
.
.
That is an infinite set of non-zero items.
Do you agree that every single item in that set has a 1 at its end and thus non-zero?

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.

That's not true.

The above defined set is formed by the following set:
1.0 - 0.9 = 0.1
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
.
.
.
That is the same infinite set of non-zero items.
"1.0 - 0.999..." is a member of that set.

The First Proof was from the "summation definition" of "0.999...":
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
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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
Except that the 'proof' is wrong because 0.9+0.09+0.009+ ...

is a convergent series which obviously converges to 1.

https://en.wikipedia.org/wiki/Convergent_series

(The convergence of a particular series solves Zeno' Dichotomy Paradox. https://en.wikipedia.org/wiki/Zeno's_pa ... my_paradox )
phyllo
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### Re: Is 1 = 0.999... ? Really?

At some point, the infinitesimal reaches a critical point, where it transforms infinity into finity. That's got to be it. But it has to be a very high n to the X power.
Meno_
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### Re: Is 1 = 0.999... ? Really?

In any subset pattern, the functional anti derivative collapses to zero. There .999999999999->×--=1. ?
Meno_
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### Re: Is 1 = 0.999... ? Really?

The null space in an eigenvector function can be summed in linear systems, by linear algebra. - is proof of this.

I am only showing it this way, so as to desribe the irresistible truth in Plato's Meno-s slave,as a way to reach the anti derivative. Kind of like showing the wisdom of zeno's Paradox, and it' solution being solvable backwards.

Plato, it seems left it to the future to come up with an explanation of this idea, or of ideas in general.
Last edited by Meno_ on Wed Jun 29, 2016 4:24 pm, edited 2 times in total.
Meno_
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
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
Except that the 'proof' is wrong because 0.9+0.09+0.009+ ...

is a convergent series which obviously converges to 1.

https://en.wikipedia.org/wiki/Convergent_series

A "converging series" specifies a LIMIT for a summation. The LIMIT is equal to xxx. You can have two obviously different summation series with the same limit. Note that the summation must REACH INFINITY before it can equal its limit. And there is no "at infinity" to be reached.

And even a higher truth is that even if the summation could actually reach infinity, it still would not equal its limit because its definition forbids it from EVER summing up the last tiny bit.

phyllo wrote:(The convergence of a particular series solves Zeno' Dichotomy Paradox. https://en.wikipedia.org/wiki/Zeno's_pa ... my_paradox )

That is irrelevant and Zeno's can be solved easier than that via relative infinitesimals, "calculus".

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.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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?
Last edited by Arminius on Wed Jun 29, 2016 4:55 pm, edited 2 times in total.

Arminius
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### Re: Is 1 = 0.999... ? Really?

I don't see anything astrange aboutbit, but perhaps i am missing something.
Meno_
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### Re: Is 1 = 0.999... ? Really?

But wait, it happened again
Meno_
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### Re: Is 1 = 0.999... ? Really?

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.

Yes. Therefore all this operations have to do with the infinitesimal calculus (inveted by Gottfried Wilhelm Leibniz). So James is right with his answer.
Last edited by Arminius on Wed Jun 29, 2016 5:11 pm, edited 1 time in total.

Arminius
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### Re: Is 1 = 0.999... ? Really?

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?

Did you just make 2 edits to that post??
Or perhaps it is that your continued posts are being counted as edits due to their rapidity.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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!

Arminius
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Joined: Sat Mar 08, 2014 10:51 pm
Location: Saltus Teutoburgiensis

### Re: Is 1 = 0.999... ? Really?

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.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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. A really, really long finite string has a last digit, an infinite string does not. A really,really long finite string would indeed leave a "last little bit" left over, but we aren't talking about a really, really long finite string, we're talking about an infinite string. That distinction is crucially important.
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Carleas
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Posts: 6107
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Location: Washington DC, USA

### Re: Is 1 = 0.999... ? Really?

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.

And it was about 5:30 pm when I read that Jerkey edited his post.

I have never seen that on ILP before.

In addition: the server does not "know" when the next poster is going to post.

The rule is that it is not possible to read that you edited your post when you did it before the next poster posted.
Last edited by Arminius on Wed Jun 29, 2016 5:51 pm, edited 1 time in total.

Arminius
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### Re: Is 1 = 0.999... ? Really?

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.
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

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.

Arminius
ILP Legend

Posts: 5732
Joined: Sat Mar 08, 2014 10:51 pm
Location: Saltus Teutoburgiensis

### Re: Is 1 = 0.999... ? Really?

So could not the two impossibilities somehow have some sort of congruence?
Meno_
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Joined: Tue Dec 08, 2015 2:39 am
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### Re: Is 1 = 0.999... ? Really?

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.

No. That is what YOU are doing when you claim that there is a 0 "at the end once it gets to infinity".

I am saying that because there is no end, the 9s must still be going - ALWAYS (true by definition). And thus never, ever becoming the string of 0s required for "1.000...".

jerkey wrote:So could not the two impossibilities somehow have some sort of congruence?

What two?
Clarify, Verify, Instill, and Reinforce the Perception of Hopes and Threats unto Anentropic Harmony
Else
From THIS age of sleep, Homo-sapien shall never awake.

The Wise gather together to help one another in EVERY aspect of living.

You are always more insecure than you think, just not by what you think.
The only absolute certainty is formed by the absolute lack of alternatives.
It is not merely "do what works", but "to accomplish what purpose in what time frame at what cost".
As long as the authority is secretive, the population will be subjugated.

Amid the lack of certainty, put faith in the wiser to believe.
Devil's Motto: Make it look good, safe, innocent, and wise.. until it is too late to choose otherwise.

The Real God ≡ The reason/cause for the Universe being what it is = "The situation cannot be what it is and also remain as it is".
.
James S Saint
ILP Legend

Posts: 25976
Joined: Sun Apr 18, 2010 8:05 pm

### Re: Is 1 = 0.999... ? Really?

I was expecting that,James let me go back and recite from Your responses. Get back in a second
Meno_
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Location: Mysterium Tremendum

### Re: Is 1 = 0.999... ? Really?

In Your response ending in 'that clould never happen' -5 of Your responses back in the forum, and the one two back, ending with 'impossible' - referring to machine intelligence.

It could never happen - is equivocal to - impossible
Meno_
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### Re: Is 1 = 0.999... ? Really?

As per patterns, congruence and possibility.
Meno_
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