## 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...
10
33%
No, 1 ≠ 0.999...
14
47%
Other
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20%

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

Only a loxodrome caused by a transversal cutting a sphere at the same angle result in appearent straight lines, but that misses the point.

The point is, that nomenclature is to support the ideal shape, - which basically is an approximation to the level of descernibility of near straight pieces which make up a sphere, with the idea, that close to infinite discernibility such differentiations are apprehandable as straight. However, only just before the presumed point of infinity, can such straightness be conceivable.

The presumption becomes necessary, to assume an infinitely large sphere, with infinite radii. This becomes necessary to uphold the idea of an infinite universe.

otherwise infinity becomes just another nomenclature, without ground.
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:
James wrote:the most extreme member [of an infinite series]

Fault.

James S Saint wrote:EVERY member of that set had to have a "0" as its right digit because you multiplied every member by 10.

There's no difference between 1.0 and 1.00. As I argued earlier, there's an implied infinite string of zeros at the end of every real number that terminates.

10 times 1.0 is NOT 1.00

You are multiplying EACH member of the infinite set of elements. There are no decimals involved. When you add them all back together to form a single number again, you must maintain the exact same cardinality/size for the set or "count of decimal digits". And that requires that there be a "0" resulting from your multiplication that wasn't there before. If the 0 isn't there, then you didn't really multiply the entire number by 10.

Uccisore wrote:
James S Saint wrote:The "..." notation, specifically means that there is no end to be obtained. That means that it never, ever gets up to being exactly 1.0.

If that were true then that would confirm that 1.0 = .999... since the difference between them is 0.000... with 'no end to be obtained'. In order for the two numbers to be different, you'd have to propose that there is a '1' somewhere out there at the end of all those zeros, and you just said that in fact there is not.

Quite the opposite.
The difference is NOT expressed by:
1.0 - 0.99 = 0.00
1.0 - 0.999 = 0.000
1.0 - 0.9999 = 0.0000
.
.
.
Thus leading to an endless list of 0s.

The difference between them is expressed exactly as:
1.0 - 0.9 = 0.1
1.0 - 0.99 = 0.01
1.0 - 0.999 = 0.001
1.0 - 0.9999 = 0.0001
.
.
.
Note that EVERY term has a "1" at the end of the string of 0s regardless of how many 0s there are.
Last edited by James S Saint on Thu Jul 07, 2016 2:47 pm, edited 1 time in total.
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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### Re: Is 1 = 0.999... ? Really?

Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit
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:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.

1.1 times 10 equals 11, there's no extra zero. Similarly, multiplying every member of the infinite series by 10 gives 3/1, a term that wasn't there before. But the rest of the series is exactly the same series as before: the sum from n=1 to infinity of 3/(10^n). To say that that that series is actually two different series is problematic.

Countably-infinite and countably-infinite-minus-one are the same cardinality: each element can be uniquely mapped between two such sets. Again, this is the Hotel Paradox.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

James S Saint wrote:Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit

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

I think for those of us unschooled or unsophisticated in mathematics what makes this exchange fascinating is the fact that with regard to mathematics one would think that there is a way in which to frame an argument such that "all rational men and women are obligated to embrace it."

But apparently even here that is not really the case at all.

And that is still before the part where those who think they are sophisticated enough to provide the "objective truth" here, are able to reconfigure the "right answer" into a frame of mind that also includes a reference to the world that we interact in from day to day.

Or is that just me?

It all seems somewhat analogous to the mysteries that revolve around the world of the very, very small somehow reconfiguring into the world of the very, very large.

We don't know exactly how they are intertwined but we do know for certain that they are.

Aren't we?
Objectivists: Like shooting fish in a barrel!

He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382

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

Carleas wrote:
James S Saint wrote:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.

1.1 times 10 equals 11, there's no extra zero. Similarly, multiplying every member of the infinite series by 10 gives 3/1, a term that wasn't there before. But the rest of the series is exactly the same series as before: the sum from n=1 to infinity of 3/(10^n). To say that that that series is actually two different series is problematic.

Countably-infinite and countably-infinite-minus-one are the same cardinality: each element can be uniquely mapped between two such sets. Again, this is the Hotel Paradox.

You begin with an endless string of single digits (the digits that comprise the endless series of decimals; "0.999..." or "0.111..." or "0.333..."). You have no option but to add a "0" placeholder to each and every one.

Last chance, Carleas:
James S Saint wrote:Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit

jerkey wrote:
James S Saint wrote:Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit

b

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

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

James S Saint wrote:You begin with an endless string of single digits (the digits that comprise the endless series of decimals; "0.999..." or "0.111..." or "0.333..."). You have no option but to add a "0" placeholder to each and every one.

Why? You can just shift the decimal point one space to the right. And you don't add a zero to the end because there is no end.

James S Saint wrote:Last chance, Carleas:

In the context of mathematics, A.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:
James S Saint wrote:You begin with an endless string of single digits (the digits that comprise the endless series of decimals; "0.999..." or "0.111..." or "0.333..."). You have no option but to add a "0" placeholder to each and every one.

Why? You can just shift the decimal point one space to the right. And you don't add a zero to the end because there is no end.

Do you really want to try to go through the rigor to prove that? You will fail.
You multiply an infinite series of digits by multiplying each individual decimal digit and carrying (as with all arithmetic), then seeing where it logically leads as the process is presumed infinite.

Carleas wrote:
James S Saint wrote:Last chance, Carleas:

In the context of mathematics, A.

I don't trust that answer. What do you mean by "context of mathematics"? If you are referring to whatever the consensus of mathematicians might currently think, that is not good enough. Logical consistency is independent of what anyone might be currently believing concerning any subject.

So try again.
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:Do you really want to try to go through the rigor to prove that? You will fail.
You multiply an infinite series of digits by multiplying each individual decimal digit and carrying (as with all arithmetic), then seeing where it logically leads as the process is presumed infinite.

Sounds good:

10(.333...) = 10 (.3 + .03 + .003 + .0003 + ...)
10(.333...) =     3 + .3  + .03  + .003  + ...)
10(.333...) = 3 +(.3 + .03 + .003 + .0003 + ...)

Turns out that was easy as fuck, not sure why you are having so much difficulty with it.

James S Saint wrote:What do you mean by "context of mathematics"?

I mean, I'm not talking about art or poetry or philosophy of religion or whatever. I'm saying, in the context of mathematical thinking and reasoning, a logical contradiction shows that either a starting point is false, or the reasoning employed is flawed.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:
James S Saint wrote:Do you really want to try to go through the rigor to prove that? You will fail.
You multiply an infinite series of digits by multiplying each individual decimal digit and carrying (as with all arithmetic), then seeing where it logically leads as the process is presumed infinite.

Sounds good:

10(.333...) = 10 (.3 + .03 + .003 + .0003 + ...)
10(.333...) =     3 + .3  + .03  + .003  + ...)
10(.333...) = 3 +(.3 + .03 + .003 + .0003 + ...)

Turns out that was easy as fuck, not sure why you are having so much difficulty with it.

I don't have difficulty with it, but you will shortly.

Carleas wrote:
James S Saint wrote:What do you mean by "context of mathematics"?

I mean, I'm not talking about art or poetry or philosophy of religion or whatever. I'm saying, in the context of mathematical thinking and reasoning, a logical contradiction shows that either a starting point is false, or the reasoning employed is flawed.

Okay.

Next Question:
The quantity of digits to the right of the decimal in the number represented by "0.333..." is equal to the quantity of integers (since they are produced one at a time through long division and thus merely counted with integers). And for convenience we can call that a set of quantity "infA". InfA is the number of elements in that set.

A) True
B) False
or Forfeit.
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:
Carleas wrote:
James S Saint wrote:You are multiplying EACH member of the infinite set of elements. ... [T]hat requires that there be a "0" resulting from your multiplication that wasn't there before.

1.1 times 10 equals 11, there's no extra zero. Similarly, multiplying every member of the infinite series by 10 gives 3/1, a term that wasn't there before. But the rest of the series is exactly the same series as before: the sum from n=1 to infinity of 3/(10^n). To say that that that series is actually two different series is problematic.

Countably-infinite and countably-infinite-minus-one are the same cardinality: each element can be uniquely mapped between two such sets. Again, this is the Hotel Paradox.

You begin with an endless string of single digits (the digits that comprise the endless series of decimals; "0.999..." or "0.111..." or "0.333..."). You have no option but to add a "0" placeholder to each and every one.

Last chance, Carleas:
James S Saint wrote:Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit

jerkey wrote:
James S Saint wrote:Let's take this one step at a time.

To be correct, there can be no logical contradiction.

A) Agree
B) Disagree
or Forfeit

b

Only through an intuition of math/reason, can a reductive differentiation of two types of logic manifest a connection.
Difference through similarity-familiarity, and identity through reason if, not exclusive, then at some point may share a functional derivative. The difference at both absolute limits, are somehow limited without violating the definition of the limitless. They are at once limited and limitless at this point, and in spite of contradiction of nomenclature such contradiction resolves. Perhaps this is because the way the contradixtion effects and changes, as do quantum effects seem to change by the effected quantum particles.
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### Re: Is 1 = 0.999... ? Really?

Because of Carleas objection of not introducing non mathematical considerations, i withdraw my response to James quiery into logical contradixtion, but let stand as a valid conjecture holding that a purely mathematical demonstration will lead into that very contradiction.
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### Re: Is 1 = 0.999... ? Really?

jerkey wrote:Only through an intuition of math/reason, can a reductive differentiation of two types of logic manifest a connection.

The only two types of logic are:
1) Valid
2) Invalid
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:b]Next Question:[/b]
The quantity of digits to the right of the decimal in the number represented by "0.333..." is equal to the quantity of integers (since they are produced one at a time through long division and thus merely counted with integers). And for convenience we can call that a set of quantity "infA". InfA is the number of elements in that set.

A) True
B) False
or Forfeit.

Here, we have to be careful. InfA is not a number on the standard real line. It's true that the set of decimal places of .333... is of the same cardinality as the set of integers. But calling that cardinality a "quantity" or a "number" are being a little too loose with language, especially since it looks like you mean to build a rigorous syllogism. I applaud that effort, but if that's the goal, we should be wary of accidentally introducing ambiguities.

Does that answer the question, or if not, is there a way to reword the question to address my concerns? Or perhaps my concerns are misplaced?
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Next Question:
The quantity of digits to the right of the decimal in the number represented by "0.333..." is equal to the quantity of integers (since they are produced one at a time through long division and thus merely counted with integers). And for convenience we can call that a set of quantity "infA". InfA is the number of elements in that set.
"quantity of digits to right of the decimal" ??

What "quantity"??

The "quantity" is infinite. And infinite isn't a number. Therefore, there is no quantity in a meaningful numeric sense.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:
James S Saint wrote:b]Next Question:[/b]
The quantity of digits to the right of the decimal in the number represented by "0.333..." is equal to the quantity of integers (since they are produced one at a time through long division and thus merely counted with integers). And for convenience we can call that a set of quantity "infA". InfA is the number of elements in that set.

A) True
B) False
or Forfeit.

Here, we have to be careful. InfA is not a number on the standard real line. It's true that the set of decimal places of .333... is of the same cardinality as the set of integers. But calling that cardinality a "quantity" or a "number" are being a little too loose with language, especially since it looks like you mean to build a rigorous syllogism. I applaud that effort, but if that's the goal, we should be wary of accidentally introducing ambiguities.

Does that answer the question, or if not, is there a way to reword the question to address my concerns? Or perhaps my concerns are misplaced?

We have previously agreed that infinites can be of different sizes. One particular size can be ("may validly be ") set as a standard with which to compare others. Obviously if there are different sizes, there are differences between those sizes to be potentially measured.

I am giving one particular infinite size a name, "infA", to be the size of the set of all integers. InfA does not have to be called a "number" or a "real number" or a "quantity" or anything other than a standard size (aka "infA elements") of the infinite set of integers.

So we have agreed on the 2nd question of the set of digits to the right of the decimal being of size infA. Or that there are "infA elements in the set".

Correct?
Phyllo?
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 am giving one particular infinite size a name, "infA", to be the size of the set of all integers. InfA does not have to be called a "number" or a "real number" or a "quantity" or anything other than a standard size (aka "infA elements") of the infinite set of integers.
Therein lies your fault. You treat something that is not a number as a number. Which makes you think that there is a zero at a particular position on the end. In fact, there is no end.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Yeah, I'm with Phyllo here too. Your reluctance to talk about it as a cardinality rather than a size suggests that you mean something different. Is that right? My concern is that when you call it a 'size', there will be a temptation to make an equation like

InfA + 1 > InfA

which is not the case if InfA is the cardinality of the integers (and 1 is the cardinality of a set with 1 element).

And again, I mean to be pedantic here, because I expect the syllogism you are building to be sensitive to the details. Cardinality is a specific mathematical concept; 'size' is ambiguous. The set of integers has the same cardinality as the set of decimal places of .333... If you mean something different by size, you need to be explicit.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
I am giving one particular infinite size a name, "infA", to be the size of the set of all integers. InfA does not have to be called a "number" or a "real number" or a "quantity" or anything other than a standard size (aka "infA elements") of the infinite set of integers.
Therein lies your fault. You treat something that is not a number as a number. Which makes you think that there is a zero at a particular position on the end. In fact, there is no end.

Do you believe that there are different sizes of infinite?

A) Yes
B) No
or Forfeit

Of course, I am merely saying that I am naming one of the sizes.

Carleas wrote:Yeah, I'm with Phyllo here too. Your reluctance to talk about it as a cardinality rather than a size suggests that you mean something different. Is that right? My concern is that when you call it a 'size', there will be a temptation to make an equation like

InfA + 1 > InfA

which is not the case if InfA is the cardinality of the integers (and 1 is the cardinality of a set with 1 element).

And again, I mean to be pedantic here, because I expect the syllogism you are building to be sensitive to the details. Cardinality is a specific mathematical concept; 'size' is ambiguous. The set of integers has the same cardinality as the set of decimal places of .333... If you mean something different by size, you need to be explicit.

ˌkärdəˈnalədē/
noun
Mathematics
noun: cardinality; plural noun: cardinalities

the number of elements in a set or other grouping, as a property of that grouping.
Merriam-Webster wrote:Definition of cardinality
plural cardinalities

: the number of elements in a given mathematical set

That is exactly what I mean by "cardinality" and by "size" - "the number of elements".

So what do YOU mean by those words?
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?

jerkey wrote:Only a loxodrome caused by a transversal cutting a sphere at the same angle result in appearent straight lines, but that misses the point.

The point is, that nomenclature is to support the ideal shape, - which basically is an approximation to the level of descernibility of near straight pieces which make up a sphere, with the idea, that close to infinite discernibility such differentiations are apprehandable as straight. However, only just before the presumed point of infinity, can such straightness be conceivable.

The presumption becomes necessary, to assume an infinitely large sphere, with infinite radii. This becomes necessary to uphold the idea of an infinite universe.

otherwise infinity becomes just another nomenclature, without ground.

Using terminology I've never heard before does not increase the validity of your claims, jerkey.

You often do this in many of your posts, hoping to "confuse" the opposition, with esotericism, as they say.
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### Re: Is 1 = 0.999... ? Really?

James S Saint wrote:That is exactly what I mean by "cardinality" and by "size" - "the number of elements".

So what do YOU mean by those words?

You cite to non-technical sources, so we should expect those definitions to be a bit blunt for many purposes, and they are for this discussion: Infinity isn't a number, so the cardinality of an infinite set can't simply be the "number of elements" (though it is for finite sets). And infinite sets and their cardinalities have weird properties that make treating them as numbers problematic; for example,

InfA + 1 = InfA (where InfA is the cardinality of the integers, and 1 is the cardinality of a set with one element).

By contrast, if 1 and 3 are the cardinalities of sets with 1 and 3 elements, then

1 + 3 = 4

where the + symbol in relation to cardinalities is the disjoint union operation (if it is a regular union, then 1 + 3 <= 4)

If all of this is accepted, and you are just using "size" to mean "cardinality", then proceed. Otherwise, could you clarify how they are different?
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Do you believe that there are different sizes of infinite?

A) Yes
B) No
or Forfeit

Of course, I am merely saying that I am naming one of the sizes.
B) No

Certainly not in the sense that it is being used in general math equations.
An infinite series, or sum, or integral does not specify some particular infinity as opposed to another infinity.

The reason seems intuitively obvious - the very concept of 'infinity' does away with all size concerns. What applies for the "lowest" infinity also applies to all "higher" infinities. IOW, the results for all infinities are exactly the same.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Perhaps a better term would be "extension" for size. Then there may be better correspondence with "Cardinal"
Meno_
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### Re: Is 1 = 0.999... ? Really?

Why are you all able to talk about this endlessly, but cannot think of a simple solution for raytracing?
trogdor

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