## 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...
9
33%
No, 1 ≠ 0.999...
14
52%
Other
4
15%

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

.999... is not a quantity or fixed value. It is an endless series of diminishing decimal values. The ellipsis "..." means "infinitely" - "never ending". The decimals begin but never end - open ended.

1 is an obvious quantity and fixed value with a beginning and an obvious end.
So, sometimes when you divide two fixed values(1/3, 2/3, 1/9, etc) , the result is not a fixed value. It's "something else".

Yet oddly enough, the rules of mathematics seem to deal with it adequately. It doesn't break any theorems. Math doesn't come to a grinding halt at these particular operations.
phyllo
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
.999... is not a quantity or fixed value. It is an endless series of diminishing decimal values. The ellipsis "..." means "infinitely" - "never ending". The decimals begin but never end - open ended.

1 is an obvious quantity and fixed value with a beginning and an obvious end.
So, sometimes when you divide two fixed values(1/3, 2/3, 1/9, etc) , the result is not a fixed value. It's "something else".

Yet oddly enough, the rules of mathematics seem to deal with it adequately. It doesn't break any theorems. Math doesn't come to a grinding halt at these particular operations.

What is so shocking about that?

The distinction between and infinitely diminishing series sum and a fixed quantity is irrelevant to science or any practical concern. The logic of the distinction doesn't interfere with any normal mathematical operations. The existence of the entire hyperreal system of reasoning changes nothing concerning practical issues. It is entirely an issue of keeping your mind, language, and logic coherent. It is a pedantic concern, not a life threatening concern, much like arguing whether there is life in distance galaxies.
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?

What is so shocking about that?
According to you, one can't even write this :

1/3 = 0.333...

Because it's not true. It's not equal. 0.333... isn't even a number.

Sounds like a major problem.
The logic of the distinction doesn't interfere with any normal mathematical operations.
Well, either 0.999... =1 or it doesn't (as Iambig might say) and that has some effect on math results. You can't just wave it off. You can't wave off a fundamental problem with division (1/3 =/= 0.333...). Where is the rigor then?
It is entirely an issue of keeping your mind, language, and logic coherent.
Coherent???
You just whacked a bunch of coherence in math, if your analysis is correct. You just broke a bunch of math properties.
phyllo
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
What is so shocking about that?
According to you, one can't even write this :

1/3 = 0.333...

Because it's not true. It's not equal. 0.333... isn't even a number.

Being pedantic, you are correct. The exact truth is that 1/3 => 0.333... {leads to, not equates to}

But no one has serious practical reason to care. And an argument from pragmatism is fallacious. Truth doesn't depend on how practical the conclusion might be for society (except to the priesthood).

phyllo wrote:
The logic of the distinction doesn't interfere with any normal mathematical operations.
Well, either 0.999... =1 or it doesn't (as Iambig might say) and that has some effect on math results. You can't just wave it off. You can't wave off a fundamental problem with division (1/3 =/= 0.333...). Where is the rigor then?
It is entirely an issue of keeping your mind, language, and logic coherent.
Coherent???
You just whacked a bunch of coherence in math, if your analysis is correct. You just broke a bunch of math properties.

The only incoherence being corrected in common math leads to the inclusion of the proven hyperreal system. And the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities. Such can be merely stated in a foot note. Nothing very serious changes anywhere.

The provable incoherences in Relativity Ontology and especially Quantum Physics are far more serious.
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?

The exact truth is that 1/3 => 0.333... {leads to, not equates to}
It's unclear what "leads to" means. You need to define it.
And the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities.
Then they are not numbers or quantities in the hyperreal system either.
The only incoherence being corrected in common math leads to the inclusion of the proven hyperreal system.
You have 3 possible results in the hyperreal system:
1) 0.333... is not a number
2) 0.333... is a number and it's exactly equal to 1/3 (so 0.333... is a fixed value)
3) 0.333... = 1/3 + a fixed infinitesimal ( so 0.333... is again a fixed value)

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

phyllo wrote:
The exact truth is that 1/3 => 0.333... {leads to, not equates to}
It's unclear what "leads to" means. You need to define it.

Yeah right. And I suppose that you don't know what the word "range" means either.

phyllo wrote:
And the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities.
Then they are not numbers or quantities in the hyperreal system either.

The hyperreals are about degrees of infinite, not quantities.

phyllo wrote:
The only incoherence being corrected in common math leads to the inclusion of the proven hyperreal system.
You have 3 possible results in the hyperreal system:
1) 0.333... is not a number - is a type and degree of infinity
2) 0.333... is a number and it's exactly equal to 1/3 (so 0.333... is a fixed value) - incoherent logic
3) 0.333... = 1/3 + a fixed infinitesimal ( so 0.333... is again a fixed value) - that would be compatible with hyperreals as the addition would drop the value from being a degree of infinity into being a set quantity. But the exact degree of infinitesimal would also have to be defined, not stated as merely "+ infinitesimal" ( "+ 1/H" would work).

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?

Yeah right. And I suppose that you don't know what the word "range" means either.
I suppose that neither do you.
The hyperreals are about degrees of infinite, not quantities.
Of course they are about quantities - infinities and infinitesimals become usable quantities within the system. That's the whole point, that infinity and infinitesimal are no longer hand-wavy concepts.
is a type and degree of infinity
Degree of infinity? What??
incoherent logic
That's what Wtf says that the transfer principle produces. You haven't disproved his assertion.
that would be compatible with hyperreals as the addition would drop the value from being a degree of infinity into being a set quantity. But the exact degree of infinitesimal would also have to be defined, not stated as merely "+ infinitesimal" ( "+ 1/H" would work).
"degree of infinity" , "degree of infinitesimal" - What is that??
phyllo
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### Re: Is 1 = 0.999... ? Really?

James S Saint wrote:
fergusrush wrote:There is no controversy here: the equation 0.999999.....= 1 merely demonstrates that the two "numbers" are different representations of the same thing.

Yes, we have all read Wiki and heard from the priesthood. We know what we are supposed to believe to be the holy truth from above that our small minds are just too inept to conceive. But interestingly throughout history, in almost every case, the priesthood has turned out to be, if not entirely wrong, at least distorted and in need of correction .. which eventually comes after centuries of debates similar to this wherein the outsider is continually accused of being just too inept to see the greater wisdom of the holy enlightened priesthood.

Frankly, I prefer pure logic. And it seems that pure logic dictates that 1 and 0.999... are entirely different things, for several reasons.

fergusrush wrote:The notion of the sum of an infinite series is sound;

True.
fergusrush wrote: the problem is that we have difficulty wrapping our minds around the concept that it NEVER ends.

Some people do.
fergusrush wrote: We tend to "feel" that the decimal will always be smaller if only to some very minute amount but, of course, that would only be true if the decimal truncated at some far distant point.

That seems to be contradictory. Please believe that nothing that I am arguing has anything to do with "feelings" (other than attitudes towards certain obnoxious debaters).

The infinite series summations dictate that there is ALWAYS a smaller amount to be added (that is what "infinite" means) in the attempt to resolve the ratio of 1/3 into decimal form and the function that leads to 0.999.... It has nothing to do with what we poor mentally backward, decrepit souls "feel".

I would submit that those following their "feelings" rather than their rationality are those who out of hand, without a single rational argument, accept the holy preaching that 1 = 0.999... simply because "THEY say...". They are the ones thinking in terms of a truncation that isn't there, because only with a truncation can the series ever reach 1.0.

By "feel" I was referring to intuition, that sense that something is a particular way because similar things are that particular way. The infinite series is merely the mathematical equivalent to the word "infinity" itself: the word allows us to name that which has not and cannot be achieved in the material universe but surely exists (yes, I know this leads to another conversation entirely!), while the series allows us to measure that which is endless. Mathematics is the art of measurement, after all.
The infinite series is not smaller because "there is ALWAYS a smaller amount to be added" because THAT is not what the infinite series IS. That kind of thinking implies a truncation: one has looked NOT at the series as a whole but only up to a point...and found it wanting. Just as we say "infinity" to name something endless, the series must be seen as a single name for the process.
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### Re: Is 1 = 0.999... ? Really?

If something is infinite, then it's not a whole.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

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

Just because the two propositions are not differentiable, does not mean they are not whole, meaning by logic they fall into each other. Wheather that kind of union consist of some interpretation of what 'whole' means, is a matter of definition.
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### Re: Is 1 = 0.999... ? Really?

Whole = a set of elements that may or may not be a subset of some other set of elements

Part = a set of elements that is a subset of some other set of elements or simply an element that belongs to some set

Sets are by definition finite.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

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

The set of all real numbers is infinite.
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### Re: Is 1 = 0.999... ? Really?

There is no such a thing as "infinite set". That's just poetry.

What there are are algorithms -- not sets -- that have no exit point. This means algorithms that do not tell you when to stop following their instructions.

The algorithm that tells you how to list real numbers is one such algorithm.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

Magnus Anderson
Philosopher

Posts: 3716
Joined: Mon Mar 17, 2014 7:26 pm

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

fergusrush wrote:By "feel" I was referring to intuition, that sense that something is a particular way because similar things are that particular way.

Yes, I understood what you meant. And my reply still stands.

fergusrush wrote:The infinite series is merely the mathematical equivalent to the word "infinity" itself: the word allows us to name that which has not and cannot be achieved in the material universe but surely exists (yes, I know this leads to another conversation entirely!), while the series allows us to measure that which is endless.

But do you not realize that "infinity" does not exist period, not merely in the physical, but even the concept is an error (produced by those who rely too much on the intuition). That which is infinite has no end. The concept of "infinity" is the concept of "The greatest point of that which has no greatest point" or "the end of endlessness" - an oxymoron.

in·fin·i·ty
inˈfinədē/
noun
1.
the state or quality of being infinite.
"the infinity of space"
synonyms: endlessness, infinitude, infiniteness, boundlessness, limitlessness; More
2.
MATHEMATICS
a number greater than any assignable quantity or countable number (symbol ∞).

The "state of being infinite" is called "infinite", not called "infinity". But they had to make up something because people keep using the word as if the noun actually made sense. The word "infinite" is the adjective, not "infinity".

And then in math, obviously there is no number greater than any number that can be applied to an infinite series. That is the oxymoron in the common usage.

fergusrush wrote:Mathematics is the art of measurement, after all.

More or less true. Mathematics is logic as applied to quantities. Science is more the art of measurement.

fergusrush wrote:The infinite series is not smaller because "there is ALWAYS a smaller amount to be added" because THAT is not what the infinite series IS. That kind of thinking implies a truncation: one has looked NOT at the series as a whole but only up to a point...and found it wanting. Just as we say "infinity" to name something endless, the series must be seen as a single name for the process.

This is the part that is interesting and again, seemingly self-contradictory.

"The infinite series is not smaller because "there is ALWAYS a smaller amount to be added" because THAT is not what the infinite series IS."
It's hard to be certain, but I think you're stating that an infinite series is not a series that always has more to add. That would be an error in definition on your part. The ellipsis, "..." specifically means "infinitely/endlessly extended", having no end or destination. The word "infinity" implies an final destination.

"That kind of thinking implies a truncation:"
That is the contrary part. Thinking that something is endless, is not implying a truncation at all, quite the reverse. Thinking that infinity exists, even as a valid concept, is implying a truncation. It implies that there is a point "at infinity", yet there is no "at infinity". Those who think that 1 = 0.999... are thinking in terms of "once the series reaches infinity, it will be equivalent to 1.0". But of course, there is no "infinity" to be reached. They "truncate" the endless process.

"one has looked NOT at the series as a whole but only up to a point...and found it wanting."
One has looked at what the series states and found that there is no end and thus "found it wanting". The more simple minded imagines an end "at infinity" and thus sees it as a completed process. It is a common error of imagination.

"Just as we say "infinity" to name something endless,"
That part would be largely true, but more like "we say 'infinity' to name the destination of something endless". And that of course, is logically senseless, but intuitively implies the greatest possible even though there is no "greatest possible".

"the series must be seen as a single name for the process."
An ENDLESS process, yes.
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... ? Real

The point is that an infinite set may be closed, as between .9 and 1, where the terms .9+.09+.009 are the members of the set. That this series can be interpreted as an infinitely divisible set, (by definition), has no linguistic barrier, that declares that such a proposal is logically unsound. Between logic and language there is this unassailable affinity.

On the other hand, the same set is bounded , by 0, and infinity, an infinity which again is a conceptual bind, vis: where the bind is the antimony, or the contradiction within it's own meaning structure.

For instance, the infinite is contradicted by the finite, there can not be a definitionally infinite, without the positing of the finite structures abiding within the structure of language it'self.

Now comes the usual critique as to the reality of the logic of language, as it corresponds to reality. Does it?

Does language create reality, or conversely does reality create language? Or, does it really matter, or even make sense to pose this question in such terms?

Linguists would say that it is language and logic which need to correspond to reality, and I don't think they say anything else, or whether it matters at all which takes precedence.

Why? Because of the spational -temporal relativity, but that is a conclusion, and sorrily, I have to argue backwards, and if you do, you have to come to view spational relationships as formative. I really do not want to get into this , though, because of a belief in the intuitive basis of math, is as suspect as it is esoteric, almost borderline occult.
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### Re: Is 1 = 0.999... ? Really?

Every algorithm is defined by some set of rules which means that the range of possible outcomes is limited (as you say, closed.) Otherwise, it wouldn't be much of an algorithm, right? It would be an absence of rules, and thus an absence of algorithm, which can be poetically expressed as an all-permitting pseudo-algorithm that says "do whatever you want".
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

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

If an algorithm, which is basically a set of restrictions, is such that it has a definite number of permitted values, then it can be represented using a set. However, if an algorithm is such that it does not have a definite number of permitted values, then it cannot be represented using a set, only poetically using the idea of "infinite set" which is just a reflection of our desire for something that isn't there.

Nonetheless, even though the number of permitted values is indefinite, we can still, somehow, determine which among such algorithms has the greatest number of permitted values. The question is how is that possible if we do not have the corresponding sets?

The answer is because we are not comparing sets of permitted values (which are non-existent) but sets of restrictions that define these algorithms (which are existent.)

The algorithm that has the least number of restrictions -- the least restrictive algorithm -- is said to have the greatest number of permitted values.

If you have a rule that says "do anything except for lying" and a rule that says "do anything except for lying and stealing" the former rule is clearly less restrictive and therefore more permissive which means it has a greater number of permitted values (in this case, values would be behaviors.)

0.999... is strictly speaking an algorithm without an exit point that generates a number. You need to force quit it to get the final number (the algorithm does not produce one) or the algorithm will proceed generating the number by making it closer to 1 but never 1.

How can an algorithm be a number?
Especially one that does not have an exit point?
Unless you plug it into some function (such as limitOf) that returns a number. But then, it would be f(algorithm)=number and not algorithm=number.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

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

Sorry I only have my Samsung phone to write with. But generally yes you can talk or think in terms of an infinite set ,non poetically, a set of numbers is a set whether it is finite or not.

Now we could refer to James in saying that infinity as a concept or a set of numbers does not exist, but the only boundary which separates a finite set of numbers from an infinite one, is the idea behind putting a limit on the number of a decimal, after which there is no functionally derived significance.

But the problem with that is, that such a value Is incalculable.

Therefore, the calculus of differentials can not solve that problem, inasmuch as Outside the boundary of the problem.

It is not a matter of the existence of infinity, because, the tools by which it can be shown to 'exist' have no such capacity, except to the degree there is a function to which there is a derivative.

Therefore, it becomes a matter of an intuitive
process, the idea of which goes back to Meno,
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### Re: Is 1 = 0.999... ? Really?

The first step would be to establish whether the concept of infinite makes any sense. Without this step, no discussion can take place.

My contention is that there is NO such a thing. Noone ever observed anything that is without an end. The act of observation requires that what is observed has an end.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

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

Logic dictates that the concept of "infinite" is valid in both the physical and the mental.
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?

What does the concept of infinite refer to?
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

Magnus Anderson
Philosopher

Posts: 3716
Joined: Mon Mar 17, 2014 7:26 pm

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

Magnus Anderson wrote:What does the concept of infinite refer to?

Endlessness, such as space extending without bound or the number system having no greatest number.

... or a repeating decimal that never reaches its resolution limit.
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?

How do you observe endlessness?
How do you observe an absence of an end?
How do you observe absence in general?
Finally, what is an end?
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

Magnus Anderson
Philosopher

Posts: 3716
Joined: Mon Mar 17, 2014 7:26 pm

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

Basically, absence cannot be observed. Absence is an unmet expectation. When we expect to observe X but end up observing Y, we say that X is absent and Y is present.

Endlessness, being an absence of end, is an unmet expectation of an end. It's not something you observe. It's something you expect to observe but do not observe.
I got a philosophy degree, I'm not upset that I can't find work as a philosopher. It was my decision, and I knew that it wasn't a money making degree, so I get money elsewhere.
-- Mr. Reasonable

Magnus Anderson
Philosopher

Posts: 3716
Joined: Mon Mar 17, 2014 7:26 pm

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

An ENDLESS process, yes.
There is no endless process.

There are no tireless minions endlessly calculating one digit after another.

Both processes and algorithms require sequential steps and therefore time to complete. Since there is no time involved, it's impossible for there to be any sort of process or algorithm.
phyllo
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