0.9999... = 1

[Tenek]

Because the latest argument, which I suppose I have to be credit for going way beyond what the usual ones do, is something I'm unable to come up with a decent response to. So here goes:

-0.9999... = 1 - 10^-infinity. Ignoring the infintesimal part would be like saying that 0.9 = 1 - 0.1 = 1. It's 1 minus some number that is smaller than any real number, which apparently leads to 0.9999... not being a real number. Since I'm a bit short on math required to define reals, anyone up for showing that 0.9... is real?

[Tom Sawyer]

Regardless of how small the value is, it's still a value.

It may be that for all intents and purposes, it's impossible to actually ever measure the difference between the two numbers in reality and you could never find anything that corresponds to this difference, but if you're just talking about mathematical proofs instead of real world applications, then a difference is a difference and it doesn't matter if that difference is one infinitieth or forty-seven.

[Preno]

Not again.

0.999... = \lim_{n\to\infty} \sum_{i=1}^n 9 \cdot 10^{-i} = \lim_{n\to\infty} (1 - 10^{-n}) = 1 - \lim_{n\to\infty} 10^{-n} = 1 - 0 = 1

That's all there is to say to it, other than "look up the definitions".

[IanC]

1/3 = 0.3'
3* (1/3)=0.9'
3/3=1
0.9' = 1

There is no difference, they are the same number represented in two different ways.

Another proof:
0.9' / 10 = 0.09'
0.9' - 0.09' = 0.9
0.9' - (0.9'/10) = 0.9
0.9' * (9/10) = 0.9
0.9' = 1

That last trick can be done for any recurring number. If it's one recurring digit, divide by 10, if two divide by 100, etc.

I'll let someone more versed in maths talk about whether you can divide by infinity or not.

[TNorthover]

That's all there is to say to it, other than "look up the definitions".

Just beat me to it, with virtually identical LaTeX too.

[Simen]

There is no smallest real number.

[Vicious Love]

There is no smallest real number.

Sure there is. It's half of whatever number you're thinking of. Or double, if it's negative.

[Tenek]

Well, I'm not terribly surprised at the answers. What I'm missing is exactly how to prove that 0.999... is a real number.

Construction_of_real_numbers (Yes, I incidentally see the note about 0.99... = 1 on this page. Still don't get it though.)

[Tenek]

There is no smallest real number.

No, and there's no largest integer either, but there's still |N| which is bigger than all of them.

[Simen]

Sure there is. It's half of whatever number you're thinking of. Or double, if it's negative.
That's kind of the point. However long you go, you can go longer.

No, and there's no largest integer either, but there's still |N| which is bigger than all of them.
Um, what? What's this mysterious |N| that is larger than every integer? It wouldn't happen to be infinity, now would it?

[Tenek]

That's kind of the point. However long you go, you can go longer.


Um, what? What's this mysterious |N| that is larger than every integer? It wouldn't happen to be infinity, now would it?

Yes, I didn't feel like scribbling down Hebrew.

[ZouPrime]

It may be that for all intents and purposes, it's impossible to actually ever measure the difference between the two numbers in reality and you could never find anything that corresponds to this difference, but if you're just talking about mathematical proofs instead of real world applications, then a difference is a difference and it doesn't matter if that difference is one infinitieth or forty-seven.
When asking a mathematical question, it is assumed that people are talking about mathematics. "Real world" doesn't mean anything here; the only proper definition of numbers (especially structures like 0.999…) is the one provided by mathematicians.

[IanC]

Well, I'm not terribly surprised at the answers. What I'm missing is exactly how to prove that 0.999... is a real number.
Well, it's 1. 1 is a real number.

Unless the arrow never hits the tortoise, that is. Get experimenting, and we can have a bbq with kebabs later to talk about the results.

[cpollett]

Well, I'm not terribly surprised at the answers. What I'm missing is exactly how to prove that 0.999... is a real number.

Construction_of_real_numbers (Yes, I incidentally see the note about 0.99... = 1 on this page. Still don't get it though.)

I think of the definitions on the Wikipedia, the one based on Cauchy Sequences is probably the most common; the second most common is using Dedekind cuts. Unfortunately, the Wiki article leaves out a lot of useful information for first time readers. The basic idea with the Cauchy sequence approach is that you start with the rationals, that is, the fractions -- numbers of the form p/q with p and q integers. i.e., 2/3, 3/8, etc. One then considers sequences over this set which converge to some limit.

For instance,
3/10, 33/100, 333/1000,...

4/10, 34/100, 334/1000,...

44/1000, 4004/10000, 400004/1000000,...
are sequences which converge. The first two converge to the same limit (1/3), the second converges to a different one (4/10). More formally the meaning of converge to a limit is that if you give me a rational such as 1/10000, I can find a position in my sequence such that all numbers further out in the sequence differ by less than this number (in this example 1/10000). Converging to the same limit means that if you given me two sequences, I can find some position (say the hundreth or the thousandth number in the sequence) after which the difference in value between elements in the two sequences is less than the rational you gave me.

Once we have this set of sequences mentioned above, we say a real number is the largest subset of this set all of whose members converge
to the same limit. That is, we identify sequences which have the same limit. Identifying sequences might seem strange at first but we do a similar thing with the rationals. For instance we identify rationals like 3/4, 6/8, etc.

Hope this helps.

[His Noodly Appendage]

Call your 0.999... X.

As X approaches one, so one minus X approaches zero.

When the number of digits reaches infinity, one minus X equals zero.

Ask: One minus what equals zero?

Why, one, of course.

And there is your answer.

[whichphilosophy]

Because the latest argument, which I suppose I have to be credit for going way beyond what the usual ones do, is something I'm unable to come up with a decent response to. So here goes:

-0.9999... = 1 - 10^-infinity. Ignoring the infintesimal part would be like saying that 0.9 = 1 - 0.1 = 1. It's 1 minus some number that is smaller than any real number, which apparently leads to 0.9999... not being a real number. Since I'm a bit short on math required to define reals, anyone up for showing that 0.9... is real?


US$1,000,000 X 0.9999 = 999,900, only US$100 short hence just below the full value of one unit (of US1,000,000).

Therefore 0.9999 is not 1 but pretty close.

[ZouPrime]

US$1,000,000 X 0.9999 = 999,900, only US$100 short hence just below the full value of one unit (of US1,000,000).

Therefore 0.9999 is not 1 but pretty close.

Except that 0.999 is not the same as 0.999…

The … is very important.

[Tenek]

I think of the definitions on the Wikipedia, the one based on Cauchy Sequences is probably the most common; the second most common is using Dedekind cuts. Unfortunately, the Wiki article leaves out a lot of useful information for first time readers. The basic idea with the Cauchy sequence approach is that you start with the rationals, that is, the fractions -- numbers of the form p/q with p and q integers. i.e., 2/3, 3/8, etc. One then considers sequences over this set which converge to some limit.

For instance,
3/10, 33/100, 333/1000,...

4/10, 34/100, 334/1000,...

44/1000, 4004/10000, 400004/1000000,...
are sequences which converge. The first two converge to the same limit (1/3), the second converges to a different one (4/10). More formally the meaning of converge to a limit is that if you give me a rational such as 1/10000, I can find a position in my sequence such that all numbers further out in the sequence differ by less than this number (in this example 1/10000). Converging to the same limit means that if you given me two sequences, I can find some position (say the hundreth or the thousandth number in the sequence) after which the difference in value between elements in the two sequences is less than the rational you gave me.

Once we have this set of sequences mentioned above, we say a real number is the largest subset of this set all of whose members converge
to the same limit. That is, we identify sequences which have the same limit. Identifying sequences might seem strange at first but we do a similar thing with the rationals. For instance we identify rationals like 3/4, 6/8, etc.

Hope this helps.

So you're looking for the largest subset of, in this case, the set of all sequences that converge to 1/3? Wouldn't they all be infinite anyways?

[whichphilosophy]

Except that 0.999 is not the same as 0.999…

The … is very important.

I don't understand. 0.999 is the same value as 0.999. The figure talked about is 0.9999 which I think you meant.

What is ... ?

[Tenek]

I don't understand. 0.999 is the same value as 0.999. The figure talked about is 0.9999 which I think you meant.

What is ... ?

... is how lazy people indicate "repeats forever", i.e. 0.999... = 0.9999... = 0.(infinite # of 9's)

[Tom Sawyer]

... is how lazy people indicate "repeats forever", i.e. 0.999... = 0.9999... = 0.(infinite # of 9's)

I don't think it's laziness, but more that by the time they'd finished typing in an infinite number of 9's, the conversation would have passed them by.

[ZouPrime]

... is how lazy people indicate "repeats forever", i.e. 0.999... = 0.9999... = 0.(infinite # of 9's)

"..." is not the standard mathematical definition to write periodicity, but giving the limited writing tools provided by the board software, it's good enough.

[Simen]

"..." is not the standard mathematical definition to write periodicity, but giving the limited writing tools provided by the board software, it's good enough.

Out of curiosity, what is the standard notation?

[perfessor]

Out of curiosity, what is the standard notation?
Standard notation is 0.9 with a bar over the 9.

1/7 = 0.142857 with a bar over those six digits, which repeat in sequence. This is better than 0.142857... which is unclear.

0.999... is a representation of a number. That number is 1. The problem is that our finite earthling minds always want to stop somewhere along the line and ask, "Are we there yet? No? Then it's not equal to 1 yet." As was pointed out before, this is the difference between mathematics, where infinity has meaning; and everyday life, where it doesn't.

[ZouPrime]

Out of curiosity, what is the standard notation?
IIRC, it's a "bar" above the periodicity. Something like:

..._
0.9

means the 9 is repeated infinitely

(Note that I'm using . instead of spaces above the numbers because the board deletes leading spaces)


Representing periodic numbers that way is useful, because sometimes only a portion of the decimal portion is repeated. For example:

......____
0.419728

Would mean 0.41972897289728...

Hope this is clear enough...

[lenrek]

After I read post #3 from Preno, I don't understand why there is a need to discuss further?

[perfessor]

After I read post #3 from Preno, I don't understand why there is a need to discuss further?
I've done a bit of math teaching, and it never hurts to explain a concept in several different ways. You can't always tell in advance which proof will "take" with a particular student.

Down in MD right now, there's a thread about the cardinality of sets. Boy, talk about a drunkard's walk! That conversation is veering all over the place.

[Tenek]

After I read post #3 from Preno, I don't understand why there is a need to discuss further?

Well if you'd read any of the other posts you might understand that the usual litany of proofs are not what interested me in the first place. Sue me for giving context, I guess.

[Jesse]

-0.9999... = 1 - 10^-infinity. Ignoring the infintesimal part would be like saying that 0.9 = 1 - 0.1 = 1. It's 1 minus some number that is smaller than any real number, which apparently leads to 0.9999... not being a real number. Since I'm a bit short on math required to define reals, anyone up for showing that 0.9... is real? In standard analysis there are no such thing as "infinitesimals", so in the context of the ordinary real numbers that argument is just wrong, 0.999... and 1 are just different ways of writing the same real number. In Nonstandard analysis (http://en.wikipedia.org/wiki/Non-standard_analysis) you can have hyperreal numbers (http://en.wikipedia.org/wiki/Hyperreal_number) which include infinitesimals, but I have my doubts that "10^-infinity" is a valid infinitesimal, you'd have to check with someone well-versed in hyperreal numbers to learn how infinitesimals are constructed.

edit: according to "Hurkyl" and "matt grime" on this physicsforums.com thread (http://www.physicsforums.com/showthread.php?t=106212&page=2), 0.999... is equal to 1 in the hyperreals too. According to a poster dealing with various false arguments here (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1083464351), any number expressed in decimal notation is a real number by definition, the hyperreals would not be expressed this way: 11) 0.999... is a hyperreal number, not real

This sounds apealling, but only because only us severe math wonks have ever even heard of hyperreals.

The idea is that hyperreal numbers extend the reals, including in "infinitely small" numbers. That's it! we say: 1 - 0.999... is one of these infinitely small numbers, rather than 0!

Sorry, but it ain't so. By definition 0.999... = limit[subn] {0.999...([smiley=n.gif] 9s)}. Each element of the sequence is a real number. By the definition of limits, the limit of a sequence of real numbers is a real number. So 0.999... is real, not hyperreal.

No matter what larger set of numbers we decide to work in, decimal notation is defined for real numbers, and it is in the real numbers alone that this matter is settled.

[Morgana]

No, and there's no largest integer either, but there's still |N| which is bigger than all of them.
But |N| is not an integer.

[cpollett]

So you're looking for the largest subset of, in this case, the set of all sequences that converge to 1/3? Wouldn't they all be infinite anyways?

What do you think is infinite? Do you think the sequence is infinite?
A sequence is just some process where if you give a natural number n the process gives you back the n th element of the sequence. Depending on how constructive you want to be processes are very finite concrete objects.

Are you worried about the largest subset of sequences that converge to a given limit being infinite? This subset is defined by some kind of algorithm (maybe effective, maybe not) which can determine given one of the processes for sequences whether that sequence has the desired limit.

Usually when people talk of a real that pick one representative sequence out of the set and leave it to the reader to close under equivalent sequences. Are you worried about human abilities to do this?

[Brian_Boru]

IIRC, it's a "bar" above the periodicity. Something like:

..._
0.9

means the 9 is repeated infinitely

(Note that I'm using . instead of spaces above the numbers because the board deletes leading spaces)

It's easy to do in Latex: "0.\overline {9}" and replace the quotes with the latex vBCode 0.\overline {9}, although the ellipses is pretty common.

4fsc = 4(3.5795\overline {45} {\rm{ }}MHz) = 14.31\overline {81} {\rm{ }}MHz

[Tenek]

What do you think is infinite? Do you think the sequence is infinite?
A sequence is just some process where if you give a natural number n the process gives you back the n th element of the sequence. Depending on how constructive you want to be processes are very finite concrete objects.

Are you worried about the largest subset of sequences that converge to a given limit being infinite? This subset is defined by some kind of algorithm (maybe effective, maybe not) which can determine given one of the processes for sequences whether that sequence has the desired limit.

Usually when people talk of a real that pick one representative sequence out of the set and leave it to the reader to close under equivalent sequences. Are you worried about human abilities to do this?

No, but the sequence would have an infinite number of elements.

Also you said

a real number is the largest subset of this set all of whose members converge

I'm not sure I get pi as a set of anything. Or would that just be the value to which all the sequences in your set converge?

[Jesse]

No, but the sequence would have an infinite number of elements. But the value of the real is the limit as the number of terms approaches infinity. If L is the limit of the sequence, that just means that for any arbitrarily small number r you choose, you can always find some integer N such that after you add N terms of the sequence, the sum is larger than L-r but smaller than L (and no matter what number of terms you add, it never goes above L). If your sequence is 9/10 + 9/100 + 9/1000 + ...., the limit must be 1, there's no other number that satisfies these requirements.

[Tenek]

But the value of the real is the limit as the number of terms approaches infinity. If L is the limit of the sequence, that just means that for any arbitrarily small number r you choose, you can always find some integer N such that after you add N terms of the sequence, the sum is larger than L-r but smaller than L (and no matter what number of terms you add, it never goes above L). If your sequence is 9/10 + 9/100 + 9/1000 + ...., the limit must be 1, there's no other number that satisfies these requirements.

So can we define a real number as the limit of a sequence of rationals?

[Jesse]

So can we define a real number as the limit of a sequence of rationals? I think that could be one way to think of them conceptually, although it may not be totally rigorous since the definition of a limit depends on that "arbitrarily small real number r". But the decimal expansion is definitely understood in terms of this sort of limit.

[His Noodly Appendage]

Or we could just say that zero is a number of infinitely small magnitude...

[AdamWho]

Another of at least 3 threads on this subject.

http://www.iidb.org/vbb/showthread.php?t=184617

[cpollett]

No, but the sequence would have an infinite number of elements.


A sequence is a function from the natural numbers to some other set.
We are right now interested in sequences of rationals.
I prefer to think of a sequence of rationals as some kind of fixed finite program which when you give me a natural number such 530, it computes a while and then spits our a rational such a 88/999. Here I am leaning towards something like a computable real. In my previous posts, I was trying to use words like the program could be non-effective so that I can maintain a finitistic perspective but still agree with the usual definition of real.


Also you said

I'm not sure I get pi as a set of anything. Or would that just be the value to which all the sequences in your set converge?

Yes.

[Sven]

Not again.

Exactly what I thought. Threads like this one are the only ones which are worse than the ones on global warming.

[ZouPrime]

Exactly what I thought. Threads like this one are the only ones which are worse than the ones on global warming.
I see your "global warming" and raise you with the "monty hall problem".

[Tears In The Rain]

x = .99999...

10x = 9.99999...

9x = 9.99999... - .999999... = 9

x = 1

[His Noodly Appendage]

*golf clap*

[Blui]

You must realise that it depends on what number system you are using, for the REAL number system, it is indeed 1, for there is no value between 1 and 1-infinitely small.

The Hyper-Real system does indeed not have it as 1.

[J-D]

Because the latest argument, which I suppose I have to be credit for going way beyond what the usual ones do, is something I'm unable to come up with a decent response to. So here goes:

-0.9999... = 1 - 10^-infinity.Bzzzt! Stop right there.

No it doesn't.

The expression 10^-infinity is undefined.

[Jesse]

You must realise that it depends on what number system you are using, for the REAL number system, it is indeed 1, for there is no value between 1 and 1-infinitely small.

The Hyper-Real system does indeed not have it as 1. I think you're incorrect about the hyperreals, see my post #29 above.

[invisible trousers]

so you have a giant conveyor belt and a jet...

[Blui]

I think you're incorrect about the hyperreals, see my post #29 above.
Well we are just going off what other people are arent we?


I hope I didnt make too many mistakes. For the hyperreals , 0.9... <> 1. First I clarify and set up. If A is an *R statement then R --> *R (if true in R then R* for A and *R statement) , the converse, *R --> R is false. Thus simply stating that something is true in R makes it always true in *R is not enough. There is only a gurantee that if A is a properly defined statement then there will be a behaviour in *R that "functions" as expected. But it need not be the only one. An internal statement in *R is not necessarily true in all subsets of *R, especially depending on your language strength. Also, *N is an internal subset of *R but not N. For example, x + y = 3 is only true in *R when we have (x,y) = (1-e, 2 + e) otherwise, x + y <> 3 even though st(x + y)=3. The behaviour of infinitesimals and finites hint that there exists gaps in any cuts we make in *R (if we began with sets containing only finite hyperreals for example).

Note though, that A an R statement --> R <--> *R. But then *R is not Dedekind Complete, it is not possible to define cuts such that there will always be a number in a gap. It not possible to extend the completeness axiom over to *R. The proof for 0.999... = 1 depedends on an ability to define cuts at its heart and Dedekind completeness. I will not go into that but will argue intuitively.

There exists a hyperreal number e that is infinitely near to any finite number r. I use ~= to mean infinitely approximately (infiniproixmitely) close to. Since there exists e infinitely near to all r then there exists r ~= r - e ~= r + e. The real, r = st(r), the standard part of the hypernumber r. Suppose we have two finite hyperreals r ~= 0.99...9 and s ~=1 then you cannot prove that it converges or whatever because there exists hyperreals r,s,t members of some set S where r < t < s with t not in the interval. Which is undefineable anyway. There is no intreval between the two numbers since such a thing is undefined. You can still however, use hyperreals to study convergance and limits and all that lovely stuff since we would actually be operating on reals but using hyperreals in place of infinities, infinitesimals and stuff. Just as rigorous but much more efficient than epsilon-delta. limits.

st(0.9...) = st(s) = 1 ~= 1 - e < 1+e ~= 1 ~= 1 + e.

This hyperreals vs limits remind me of the old fight between quaternions and vectors and grassman's algebras. limits map to quaternions and hypers to Grassman.

-http://www.physicsforums.com/showthread.php?t=106212&page=3


I'll guess ill wait till common consensus amongst mathematicians arrives.

[Jesse]

Well we are just going off what other people are arent we?


-http://www.physicsforums.com/showthread.php?t=106212&page=3


I'll guess ill wait till common consensus amongst mathematicians arrives. It's pretty clear that the guy you quoted is not a mathematician, he was making fuzzy verbal arguments like this: Since there exist non zero infinitesimals we can only say that the series gets aproximately close to 1 since we are approximately close to 0 (there exists 0 < e ~= 0). But in fact the whole endeavoure is meaningless since the cuts that created the entire decimals should not have been possible! Since we have e, the gaps created should have been empty. There is no concept of inifinite precision, just a line riddled with gaps as we go through all c in our original set. In fact, the proof is not possible since a meaningful upper bound cannot be found in *R due to the existance of infinitesimals. And this: There is only a gurantee that if A is a properly defined statement then there will be a behaviour in *R that "functions" as expected. ... The behaviour of infinitesimals and finites hint that there exists gaps in any cuts we make in *R (if we began with sets containing only finite hyperreals for example). I don't need to know anything about hyperreal mathematics myself to see that this person is not making the types of arguments a mathematician would, his arguments are all handwavey. The others responding to him did not make these sorts of handwavey arguments (see Hurkyl's response to the post you quoted in post #47 for example), and they all seemed agreed that he didn't know what he was talking about. This is not to say it's not possible they have made a mistake, just that I'm pretty certain the guy you quote is not worth paying attention to. Either way, I'm sure the question has a clear-cut answer, mathematics is not an area where you have to "wait till common consensus arrives."

[Blui]

It's pretty clear that the guy you quoted is not a mathematician, he was making fuzzy verbal arguments like this: And this: I don't need to know anything about hyperreal mathematics myself to see that this person is not making the types of arguments a mathematician would, his arguments are all handwavey. The others responding to him did not make these sorts of handwavey arguments (see Hurkyl's response to the post you quoted in post #47 for example), and they all seemed agreed that he didn't know what he was talking about. This is not to say it's not possible they have made a mistake, just that I'm pretty certain the guy you quote is not worth paying attention to. Either way, I'm sure the question has a clear-cut answer, mathematics is not an area where you have to "wait till common consensus arrives."
But that is the only way you and i can even remotely come to a rational conclusion, for even you allude to this with the fact you rely on 'the others' discrediting him.

Neither you nor i seem to be good enough mathematicians to argue about the hyperreals, so we must rely on expert opinion.

[JohannGoodflag]

I'm sure the question has a clear-cut answer, mathematics is not an area where you have to "wait till common consensus arrives."

Well, that depends on your philosophy of mathematics. Realists of various stripes think that these things really refer to some collection of facts about the real world, formalists say that it all depends on your definitions (and so ultimately on what you care to investigate and how you do it) some intuitionists would object to the notion of an infinite sum required to expres 0.9-repeating at all.

There will always be cranks out there who claim to have better definitions (or what they call "the real" definitions for these things, which of course differ from the ones used by the vast majority of working mathematicians, physicists, engineers, etc.) Some of these may be amusing mathematical systems to play with, and may occasionally be illustrative of some interesting idea or another. These variants may even be useful from time to time. But that doesn't make them "the real" or "the more real" definitions or nature of things; and if you're waiting for a time when everyone who has anything to say about 0.9-repeating to agree with one another, you will never stop waiting, on account of people such as these. (Of course, this is true of just about any topic, from mathematics to international policy.)

To use an analogy, a French immigrant to the United States can't protest that "fuck" is not in any way obscence, simply because the french homonym "phoque" refers to an acquatic mammal native to the arctic ocean, and that this is "the real meaning" of the word. The french meaning of "phoque" in French is irrelevant in the United States, except amoung comparably small groups of people who speak to each other in french. The fact that there isn't perfect unanimity within the United States of what the sound "fuck" can refer to is also irrelevant: unless you know you are addressing a group with a background significantly different from the average, the pertinent meaning of the word "fuck" in the United States is as a provocative expletive which occasionally refers to a sexual act of some sort.

Similarly, the absence of perfect agreement between all parties who ever write anything about repeating fractions does not entail anything. (Notice that the disagreement between these parties usually only occurs for representations of numbers such as 0.9-repeating; you'll find no such debate about 0.142857-repeating, for instance --- which should be a warning sign that one of the sides is not arguing from a well-defined system.) Over 99% of mathematicians, physicsts, chemists, and engineers --- in short, those professions for which this debate has any real importance --- follow a single convention, and this suffices in practise to make a consensus, and you needn't wait for one to form.

The realists, the formalists, and the people who don't care and just do their work, almost unanimously agree on a common definition of the real numbers, and of infinite summation, as being either "the correct" or "the most interesting" definitions available. What is therefore pertinent is that

the "real numbers" are by consensus given one of about three or so equivalent definitions (Dedekind's definition, in terms of what are now called 'Dedekind cuts'; Cantor's definition, in terms of equivalence classes of Cauchy sequences over the rationals; and a third definition in terms of sequences of nested intervals of the real line),
there is a well-defined criterion given, by consensus, given for when a sequence which converges to zero has a well-defined infinite sum, and what the sum evaluates to, which is necessary to even attribute any value of 0.9-repeating at all;
the consequence of these definitions is that 0.9-repeating is equal to 1.

The proofs that show this are all the old familiar ones, both simple and complicated, that get brought up and ignored in discussions such as this one.

For my part, I'm a formalist, but a utilitarian formalist. You can adopt whatever bizarre set of definitions and axioms you like, but that has no bearing to what is common practise or, more importantly, to what is useful. In my opinion, the system in common usage, where 0.9-repeating is identical to 1, is a system in which useful ideas can be expressed concisely and used easily; and any reasonably simple-to-express system in which 0.9-repeating differs from 1 is likely to make doing anything of value in either pure or applied mathematics a headache.

[J-D]

Well, that depends on your philosophy of mathematics. Realists of various stripes think that these things really refer to some collection of facts about the real world, formalists say that it all depends on your definitions (and so ultimately on what you care to investigate and how you do it) some intuitionists would object to the notion of an infinite sum required to expres 0.9-repeating at all.

There will always be cranks out there who claim to have better definitions (or what they call "the real" definitions for these things, which of course differ from the ones used by the vast majority of working mathematicians, physicists, engineers, etc.) Some of these may be amusing mathematical systems to play with, and may occasionally be illustrative of some interesting idea or another. These variants may even be useful from time to time. But that doesn't make them "the real" or "the more real" definitions or nature of things; and if you're waiting for a time when everyone who has anything to say about 0.9-repeating to agree with one another, you will never stop waiting, on account of people such as these. (Of course, this is true of just about any topic, from mathematics to international policy.)

To use an analogy, a French immigrant to the United States can't protest that "fuck" is not in any way obscence, simply because the french homonym "phoque" refers to an acquatic mammal native to the arctic ocean, and that this is "the real meaning" of the word. The french meaning of "phoque" in French is irrelevant in the United States, except amoung comparably small groups of people who speak to each other in french. The fact that there isn't perfect unanimity within the United States of what the sound "fuck" can refer to is also irrelevant: unless you know you are addressing a group with a background significantly different from the average, the pertinent meaning of the word "fuck" in the United States is as a provocative expletive which occasionally refers to a sexual act of some sort.

Similarly, the absence of perfect agreement between all parties who ever write anything about repeating fractions does not entail anything. (Notice that the disagreement between these parties usually only occurs for representations of numbers such as 0.9-repeating; you'll find no such debate about 0.142857-repeating, for instance --- which should be a warning sign that one of the sides is not arguing from a well-defined system.) Over 99% of mathematicians, physicsts, chemists, and engineers --- in short, those professions for which this debate has any real importance --- follow a single convention, and this suffices in practise to make a consensus, and you needn't wait for one to form.

The realists, the formalists, and the people who don't care and just do their work, almost unanimously agree on a common definition of the real numbers, and of infinite summation, as being either "the correct" or "the most interesting" definitions available. What is therefore pertinent is that

the "real numbers" are by consensus given one of about three or so equivalent definitions (Dedekind's definition, in terms of what are now called 'Dedekind cuts'; Cantor's definition, in terms of equivalence classes of Cauchy sequences over the rationals; and a third definition in terms of sequences of nested intervals of the real line),
there is a well-defined criterion given, by consensus, given for when a sequence which converges to zero has a well-defined infinite sum, and what the sum evaluates to, which is necessary to even attribute any value of 0.9-repeating at all;
the consequence of these definitions is that 0.9-repeating is equal to 1.

The proofs that show this are all the old familiar ones, both simple and complicated, that get brought up and ignored in discussions such as this one.

For my part, I'm a formalist, but a utilitarian formalist. You can adopt whatever bizarre set of definitions and axioms you like, but that has no bearing to what is common practise or, more importantly, to what is useful. In my opinion, the system in common usage, where 0.9-repeating is identical to 1, is a system in which useful ideas can be expressed concisely and used easily; and any reasonably simple-to-express system in which 0.9-repeating differs from 1 is likely to make doing anything of value in either pure or applied mathematics a headache.As far as I understand this, I think I agree with it, and I would sum it up as follows:

According to the usual mathematical definition of 0.9 recurring, it is equal to 1. In order to say that 0.9 recurring is not equal to 1, you need to use different definitions. People who argue that 0.9 recurring is not equal to 1 generally don't appreciate this point, and don't even understand what the usual mathematical definitions are, and it is this that invalidates their arguments. No argument that 0.9 recurring is not equal to 1 can be a useful contribution to the discussion if it does not begin with a definition of how 0.9 recurring is being defined.

[Jesse]

But that is the only way you and i can even remotely come to a rational conclusion, for even you allude to this with the fact you rely on 'the others' discrediting him. I don't understand, what is the only way? I agree that if we are not informed about the details of non-standard analysis and the hyperreals we have to rely on expert opinion, but this isn't saying we have to "wait till common consensus arrives", because I'm confident that there already is perfect consensus among actual experts in this area about what non-standard analysis would say about the question (as JohannGoodflag says, one might question whether the axioms of non-standard analysis were 'true' in some non-axiomatic sense, but that's not what we're talking about here, we just want to know whether the set of axioms that is used when talking about 'hyperreals' can be used to show 0.999... is different from 1 or the same as 1.) Neither you nor i seem to be good enough mathematicians to argue about the hyperreals, so we must rely on expert opinion. Right, and I gave reasons why I feel very confident that the guy you quoted is not an expert. Of course it's possible the other people I quoted are also not experts but are just better at sounding like they know what they're talking about, but I would bet not, and in any case I stand by my previous statement that "This is not to say it's not possible they have made a mistake, just that I'm pretty certain the guy you quote is not worth paying attention to."

[Amedeo]

Well, I'm not terribly surprised at the answers. What I'm missing is exactly how to prove that 0.999... is a real number.

Construction_of_real_numbers (Yes, I incidentally see the note about 0.99... = 1 on this page. Still don't get it though.)
I don't have a formal proof, but I find this clear:

Divide a line-segment into three parts. They are such that

1/3 + 2/3 = 1

1/3 = O.333333333... or 0.3

2/3 = 0.666666666... or 0.6
Their sum =
====0.999999999... or 0.9

So, 0.9 = 1

[jeffevnz]

I am amazed at how this issue never seems to go away. I don't get why this thread is still going, and I don't know why a new one on the same topic gets started on an almost monthly basis. The question has been settled, quite definitively, by several posts in this thread alone.


I'll try to make this as clear as possible:

The symbol, "0.9999..." represents the limit of a sequence, namely the infinite sequence of numbers that goes like this: 0.9, 0.99, 0.999, 0.9999, etc.

The limit of a sequence is where the sequence is "headed", in the long run. What's it moving towards? Is it approaching some value? We don't care whether it ever gets there. Maybe it does, maybe it doesn't. We just want to know whether it's headed somewhere in particular, and, if so, where it's headed.

The sequence {0.9, 0.99, 0.999, 0.9999, etc.} is clearly headed towards 1. (If you need me to prove this, I'll be glad to. The bottom line is that it gets closer to 1 each time, it's not going to overshoot, and it's not going to back away.) To state the answer formally, the limit of the sequence is 1. Or, in shorthand,

0.9999... = 1

[TNorthover]

I'll guess ill wait till common consensus amongst mathematicians arrives.

I'll weigh in with Jesse. The hyperreals embed the reals, and conventionally use decimal notation for that embedding, so 0.999... = 1. The fact that there are extra infinitesimals lurking around is irrelevant. God knows what the bloke you quoted was on about, but it looked like nonsense to me. In particular
Suppose we have two finite hyperreals r ~= 0.99...9 and s ~=1 then you cannot prove that it converges or whatever because there exists hyperreals r,s,t members of some set S where r < t < s with t not in the interval. Which is undefineable anyway. There is no intreval between the two numbers since such a thing is undefined.
is high grade bullshit.

[Preno]

As for the real / hyperreal thing:
Another angle is to look at the hyperdecimal numbers. If we have a transfinite, but hyperfinite number of 9's (that is, the number of 9's is a transfinite hypernatural number) in 0.999.....9, then this does, in fact, denote a number infinitessimally close, but unequal, to 1. However, this is a terminating hyperdecimal. The nonterminating hyperdecimal 0.999... is, in fact, equal to 1.
(from here (http://www.physicsforums.com/showthread.php?t=124382))Basically, 0.999... is a real number and any statement about real numbers is true in hyperreals (just like any statement about natural numbers is true even if we consider them as reals). In order to be the notation of a hyperreal, it would apparently have to be 0.999...9.

[Chamaeleon]

It should be obvious that 0.999... is not 1, and that the difference between the two is the God the gaps. :)

[Amedeo]

I am amazed at how this issue never seems to go away. I don't get why this thread is still going, and I don't know why a new one on the same topic gets started on an almost monthly basis. The question has been settled, quite definitively, by several posts in this thread alone.


I'll try to make this as clear as possible:

The symbol, "0.9999..." represents the limit of a sequence, namely the infinite sequence of numbers that goes like this: 0.9, 0.99, 0.999, 0.9999, etc.

The limit of a sequence is where the sequence is "headed", in the long run. What's it moving towards? Is it approaching some value? We don't care whether it ever gets there. Maybe it does, maybe it doesn't. We just want to know whether it's headed somewhere in particular, and, if so, where it's headed.

The sequence {0.9, 0.99, 0.999, 0.9999, etc.} is clearly headed towards 1. (If you need me to prove this, I'll be glad to. The bottom line is that it gets closer to 1 each time, it's not going to overshoot, and it's not going to back away.) To state the answer formally, the limit of the sequence is 1. Or, in shorthand,

0.9999... = 1

As I pointed out in my above post, I myself have no problem understanding that 0.9 = 1. Nevertheless, what is mind-boggling is the idea that a finite line [which is designated as 1] can be the sum of an infinite number of partitions.... which are always of some size or other. It seems absurd that a finite line consists of an infinity of tiny segments, whether they are equal or decreasing in size.

The problem is that arithmentical numbers do NOT designate lines or anything else; they designate individuals or portions of individuals. Thus, if 1 is the standard volume of a sound, 0.9 sounds means that the sum of these portions,

9/10 + 9/100 + 9/1000 +.... = the 1 volume.

Each portion is a conceptual or possible portion, since we do not know whether there can exist such p[ortions as one trillionth of a sound-volume or not; what 0.9 means is that if volumes come in such portions as the decimals state, it would take an infinity of them to have one sound of standard size. (This was just an example, since we do not know if the addition of two sounds adds up to a third sound in the artithmetical way.)

The sum of these three segments, 1/2 + 1/3 + 1/9, does not add up to 1; therefore, 1 cannot be partitioned into those three segments. Any finite number of portions of the 9/10^n type does not add up to 1; therefore, any line cannot be partitioned into a finite number of thus defined portions. This means that any finite number of partitionings does not cover the whole line. An infinite number of partitionings is one that is never completed; 0.9 never attains completion [1]. Thus "1" ir rightly called a limit-number.

But then we keep on falling into the same mind-boggling issue, if we think of the sequence of the decimals as static constituents of a line. If we are moving over a line or road, and if for every 1/100 of the line we decrease the speed to one half of the previous one, the road will be fully covered in a finite sequence of speed portions:
d/t, d/2t, d/4t, etc.

We can choose the speed pattern to be,

d/t= 9/10, d/t'= 9/100, d/t''= 9/1000, etc.

So, there is decelaration according to the decimal pattern, but after a number of speed portions, the end of the line will be reached; the deceleration is not such that there is an infinite sequence of portions (which would mean that the end of the road is never reached).

The Zeno problem of being unable to reach the end of the line, if, for instance, you have to keep on cutting the speed by a half, is that for every unit if time, half of the previous distance had to be covered. Obviously the sum of these distances,

1/2 + 1/4 + 1/8 + 1/16 etc.

will not add to one. On the contrary, walking over same distances at a slower and slower pace will get you there -- precisely because the line is finite.

[jeffevnz]

Amadeo - my post wasn't necessarily directed at you. In fact, I thought your previous argument was somewhat convincing. But, to be honest with you, I didn't follow your latest one. There are a few problems that jump out at me.

1) You're using terms that aren't standard, or using standard terms in a nonstandard way, like "individual" and "portion" in this sentence:
The problem is that arithmentical numbers do NOT designate lines or anything else; they designate individuals or portions of individuals.

The problem with doing that is that standard terms have precise definitions that everyone knows. If you don't have totally clear definitions of your terms, then you can't make an airtight argument with them. (Google the definition of "interval", for example, to get an idea of what a rigorous definition looks like.) Now it may happen that you know what goes into a rigorous definition, and that you have such definitions in mind for your words. However, if there are already standard terms with the same definitions, then all you're doing by making up new words is creating unnecessary confusion. So try to avoid doing that unless you're sure the idea hasn't already been named.


2) Frankly, I have no idea what any part of this phrase means:

Each portion is a conceptual or possible portion, since we do not know whether there can exist such p[ortions as one trillionth of a sound-volume or not;

3) Speed and acceleration have nothing to do with any of this. Get speed and acceleration out of your head. They're red herrings. I have a feeling that's what's tripping you up. Whether or not Zeno finishes the race has nothing to do with how fast he goes. Similarly, whether the sequence {0.9, 0.99, 0.999....} approaches 1 has nothing to do with how quickly it approaches 1.

Here's how to think about it. Zeno runs, as fast or as slow as he wants. For simplicity, let's say the race is a mile long. At some point he has finished half the race, so the total distance he has run is 1/2 mile. Later on, he finishes half of what's remaining, so his total distance is 3/4 mile. Then he runs half of what's remaining, bringing his total distance to 7/8 mile....

The total distance he has run is approaching some value. What is it?

[JohannGoodflag]

[...] what is mind-boggling is the idea that a finite line [which is designated as 1] can be the sum of an infinite number of partitions.... which are always of some size or other. It seems absurd that a finite line consists of an infinity of tiny segments, whether they are equal or decreasing in size.

Well, you can consider the same thing with halves and quarters and eights, etc, a la Zeno. Take any line, and divide it in half. Take the latter part, and divide it in half, and then divide the latter part of those two pieces in half, etc. In a finite number of time, you will only obtain a finite number of pieces; the insight is that those partitions can be considered to "be there" whether or not you actually carry out the dividing, and there is an infinite descent of potential divisions.

There are three reactions to this, of varying reasonableness. (This isn't meant to be provocative, or to demand that you choose; it is meant only to exhibit the spectrum of seemingly intellectually sound responses that I can concieve of.)

Assert that a line segment cannot be divided into infinitely many segments because space or matter isn't continuous: this is to assert a sort of material realist position with respect to mathematics, as an even more hardcore version of finitism, essentially claiming that there is no mathematical truth except that which we can directly model in controllable processes in the observable universe. (This would also entail having to come up with what criterion you would accept as sufficient to justify a mathematical entity; for example, if you only accept that math which you can "observe" in some sense, would you consider numbers which you cannot yourself count, such as 10^23, "real"?)
Assert that it is unreasonable to consider the infinitely many potential divisions as if they were real: that is, to adopt an intuitionist/finitist position, and from that position reject the meaningfulness of infinite sums.
Accept the infinite set of line segments, but not accept the concept of summing over this infinite set: that is, assert that the limit of the partial sums, while a well-defined number, doesn't correspond to a reasonable extension of summation. Basically, this is just refusing to call it a sum, but a recognition of the mechanics behind it. It would be a consistent but sterile position --- unless you can find a really interesting alternative interpretation for the limit of the partial sums (aside from being a sum of an infinite set, that is).
Adjust your intuition to accomodate the possibility of an infinite sum of length being at least bounded from above by a finite length; that is, to decide that Zeno was worried about nothing. --- This position is actually necessary to consider the number 0.9 a "finite" quantity (that is, bounded on either side by some integers) of any sort, as the definition of 0.9 is perfectly analogous to the above.

[jeffevnz]

Amadeo -

After re-reading your post, I can sort of understand why you're talking about speed and acclereration, but rest assured that they don't matter here.

Zeno's race has an infinite number of legs to it, but the distance and/or time involved in each one shrinks toward zero. I think you're seeing all these infinities and zeros floating around and you're worried that we're doing something invalid like dividing by zero or multipying by infinity at some point. What happens to his speed as he runs through an infinite number of infinitesimally shrinking legs of the race? If that's the concern, I can kinda see where you're coming from.

Here's why that's not a problem. Zeno runs through an infinite number of legs of the race, but each successive leg is shrinking at such a rate that the total distance he runs is something finite, like a mile. Similarly, each of the infinite number of legs takes some time, but that amount of time is shrinking at such a rate that the total amount of time is also something finite, like 10 minutes. So total time and total distance are finite. Therefore his average speed is also finite, and there's no reason his accleration and speed can't be finite at every point along the race.

Zeno's just an ordinary guy running a race. It's a plain old finite distance like 1 mile, and he takes a plain old finite (and pathetic) amount of time like 10 minutes. He averages 6 mph. All these infinites and 0's are artifacts of the math we use to analyze the race. They have no impact on the race itself.

[Corona688]

-0.9999... = 1 - 10^-infinity. Ignoring the infintesimal part would be like saying that 0.9 = 1 - 0.1 = 1. You yourself said it's infintesimal -- arbitrarily small.

The smallest arbitrarily small number is zero.It's 1 minus some number that is smaller than any real number No, it's 1 minus some number that is infinitely small.

There happens to be an infinitely small number.

It's called zero.which apparently leads to 0.9999... not being a real number. Incorrect. Since 0.99... is equivalent to 1, obviously it is a real number.Since I'm a bit short on math required to define reals, anyone up for showing that 0.9... is real? Sure.

1-0.999999 = 1/inf
1/inf = 0
therefore 1-0.99999 = 0
therefore 1 = 0.999999

Since 1 is a real number, 0.999999 is also a real number.

Let me guess... "no it's not." Sigh.

[Corona688]

I'll let someone more versed in maths talk about whether you can divide by infinity or not. You can. Any finite number divided by infinity is zero.

[seebs]

Er, uhm.

It's real because it's not imaginary. Transcendentals are real too; the fact that you can't nail down the exact value of pi doesn't make it not a real number.

It's obviously real. The question is whether it's 1 or not, and the answer is "of course it is".

[Astinus]

I've heard that this sort of thread is common at the WoW forums. I would've thought this place would be able to get this.

.9999... is a real number. In fact, it's even a rational number, as it can be represented as a fraction.

1/9 = .1111111....
2/9 = .222222...
3/9 = 1/3 = .3333333...
4/9 = .4444444....
5/9 = .55555...
6/9 = 2/3 = .666666...
7/9 = .77777....
8/9 = .8888888....
9/9 = .999999... = 1

Simple proof: between any two distinct, real numbers, there is an infinite number of other distinct, real numbers. For example, between 1 and 2, there is 1.1, 1.2, 1.20001, etc. Between 1 and 1.01, there is 1.001, 1.0000000001, etc. There is not one number between .9999... and 1, because they are equal.

[jeffevnz]

Simple proof: between any two distinct, real numbers, there is an infinite number of other distinct, real numbers. For example, between 1 and 2, there is 1.1, 1.2, 1.20001, etc. Between 1 and 1.01, there is 1.001, 1.0000000001, etc. There is not one number between .9999... and 1, because they are equal.
Ooh nice.

[orac]

Here's the thing I'm (possibly) missing.

Back in the Jurassic period when I rode a dinosaur to school and was taught to do long division by hand, there was a process known as "carrying the one". I've heard of the "new math" which has a whole different way to do division, but I'm not personally familier with it.

If I did "one divided by three" by hand I still wouldn't be finished - and at each digit, I'ld be carrying my "1" on to the next digit. Then if I took the result and multiplied it by 3, I'ld carry my 1 all the way back, and the result of "one divided by three then multiplied by three" would end up as "one". In the phrase "zero point three repeating" I see no acknowledgement of that extra bit.

I'll admit that mathematicians often have weird notation and that the whole "zero point three repeating" may have an implicit acknowledgement of the "1" that got carried

But all the "proofs" that "0.9 repeating is equal to 1" seem to be based on a process that involves assuming that "one divided by three" really is "a zero, a period, and an infinitely long line of threes" without any reference to the process of arriving at that number, or the problem of dealing with the carried "one". Noone ever multiplies one-third by three while mentioning that little extra detail.

Am I missing something? Are we just debating obscure mathematical notation, getting conufsed by terminology, or are people actually making flawed arguments to "prove" invalid conclusions?

As far as I can see, if you do the division and multiplication by hand and don't "forget" about a detail that doesn't often get mentioned in these debates then there's no meaningful way to claim that "zero point nine repeating" and "one" are the same thing.

[Astinus]

orac - I don't even understand what you're concerned about. 1/3 = .33333.... That's a fact of mathematics. It doesn't matter if you "recognize" how you got there or not.

[mac_philo]

I find it funny that "common sense" arguments against evolution get nothing but scorn and ridicule here, yet "common sense" arguments against mathematical and logical truths get a lot of play.

[J-D]

Here's the thing I'm (possibly) missing.

Back in the Jurassic period when I rode a dinosaur to school and was taught to do long division by hand, there was a process known as "carrying the one". I've heard of the "new math" which has a whole different way to do division, but I'm not personally familier with it.

If I did "one divided by three" by hand I still wouldn't be finished - and at each digit, I'ld be carrying my "1" on to the next digit. Then if I took the result and multiplied it by 3, I'ld carry my 1 all the way back, and the result of "one divided by three then multiplied by three" would end up as "one". In the phrase "zero point three repeating" I see no acknowledgement of that extra bit.

I'll admit that mathematicians often have weird notation and that the whole "zero point three repeating" may have an implicit acknowledgement of the "1" that got carried

But all the "proofs" that "0.9 repeating is equal to 1" seem to be based on a process that involves assuming that "one divided by three" really is "a zero, a period, and an infinitely long line of threes" without any reference to the process of arriving at that number, or the problem of dealing with the carried "one". Noone ever multiplies one-third by three while mentioning that little extra detail.

Am I missing something? Are we just debating obscure mathematical notation, getting conufsed by terminology, or are people actually making flawed arguments to "prove" invalid conclusions?

As far as I can see, if you do the division and multiplication by hand and don't "forget" about a detail that doesn't often get mentioned in these debates then there's no meaningful way to claim that "zero point nine repeating" and "one" are the same thing.Yes, you are missing something.

The definition of 'nought point three recurring' is not 'what you get when you carry out by hand a long division of one by three'. That's what you're missing.

[orac]

Yes, you are missing something.

The definition of 'nought point three recurring' is not 'what you get when you carry out by hand a long division of one by three'. That's what you're missing.

Well, then, what is the definition of "nought point three recurring" because it has been used as "what you get when you divide one by three" followed closely by "hey, looky, if you multiple 3 by 3 you get 9, so if you divide one by three and then multiply all those "threes" by three again you get nought point nine recurring".

(My point was, pretty much, that people use "nought point three recurring" as a synonym for "one divided by three" and I'm not convinced you can actually do that unless you always remember to use the same definition for the entire argument. If 1/3 is defined a particular way, any operations on the "written form" have to be consistent with the actual definition rather than a written approximation.)

Maybe I'm still dense, though.

1 / 3 = X
X * 3 = 1

I always understood that the rules of math mean that the value of 'X' in those two equations is what we call "one third" and that they follow quite simply.

While there is the "nought point three recurring" notation that apaprantly is "defined" as a long-winded way of writing "1/3", what noone seems to be explaining is where they get a value of 'X' that is the same in both equations and yet gives a result other than '1' in the second equation.

These debates aren't "hey look, the number 3 in numerals is the same as the number III in ye olde fashion roman numerals - just written differently and I can prove it". Time and again people claim to have "proven" that two different numbers are also the same number.

I'm still not convinced.

If some mathematician defined "squibble" as "the result of one divided by three" then we'ld have an exciting new term, but "three times squibble" would still be "one".

[NZSkep]

orac - I don't even understand what you're concerned about. 1/3 = .33333.... That's a fact of mathematics. It doesn't matter if you "recognize" how you got there or not.

But if you don’t accept that 0.999…. is exactly equal to 1 then you are also unlikely to accept that 0.333…. is exactly equal to 1/3 (for the same reasons)so using the latter to prove the former would be pointless.

[Robertson]

But if you don’t accept that 0.999…. is exactly equal to 1 then you are also unlikely to accept that 0.333…. is exactly equal to 1/3 (for the same reasons)so using the latter to prove the former would be pointless.

This is true. I think there are a couple things confusing people with this issue. One is the difference between numbers and representations of numbers. The number that is one third of one can be represented an infinite number of ways. Consider in base 9 it would be 0.3, not repeating. But it's still one-third. So why the fuss when 1 is represented a different way?

The other problem is imagining repeating decimals as a process. It's natural because that's how we calculate. One digit at a time. But the number we're calculating doesn't really change with each step in the calculation. We are arriving at the number, or discovering it. All those 3's in 0.333... are "already there"; we just discover each one as we do long division. The process is just our calculation. Numbers exist without any entity performing a process.

[J-D]

Well, then, what is the definition of "nought point three recurring"A string of digits following the decimal point in which, for every positive integer N, the Nth digit is three. Using this definition, your problems do not arise.

[DigitalDruid]

As far as I can see, if you do the division and multiplication by hand and don't "forget" about a detail that doesn't often get mentioned in these debates then there's no meaningful way to claim that "zero point nine repeating" and "one" are the same thing.
If you do not accept that 0.9.... is not exactly the same as 1, then you have to agree that a dropped ball never hits the floor.
Suppose we drop a ball from certain height (call it one unit of distance) on to a floor. We reckon the distnace traveled by the ball towards the floor this way:
It first travels 9/10 of the distance to the floor.
So, distance traveled so far = 0.9 units

Next it travels 9/10 (=0.9) of the remaining 0.1 units of distnace, which is 0.9x0.1 = 0.09 unit.
So, distance traveled so far = (0.9 + 0.09) units

Continuing in this manner, the ball travels
0.9 + 0.09 + 0.009 + 0.0009+ .......... units and so on. Which is the same as 0.999....... forever repeating.

This process never seems to end, because there is always 9/10 the of the remaining distance to go. So we expect the ball to take forever to reach the floor. But we are mistaken. The ball always hits the floor in finite time, which is Nature's way of proving that 0.999..... unit = 1 unit . In this case, Nature takes a finite amount of time to complete an infinite sequence of steps, each taking a calculably finite (however small) amount of time. It is our assumption that the sum of any infinite sequence of numbers can not be a finite number. But it is an assumption, and however intuitive, it is still an assumption. For a convergent infinite series that assumption is not true, as demonstrated by the dropped ball.

[TNorthover]

Well, then, what is the definition of "nought point three recurring" because it has been used as "what you get when you divide one by three"

It's a theorem that 0.3... = 1/3, not a definition. The definition would probably run something like "the real number x, such that for any e > 0, there exists an N such that x differs from 0.3...3 (with n 3s)by less than e for any n > N" -- or more informally, the real number that you can get as close as you like to by truncating 0.3... far enough along the string.

You then prove that this number is in fact 1/3.

[Blui]

If you do not accept that 0.9.... is not exactly the same as 1, then you have to agree that a dropped ball never hits the floor.
Suppose we drop a ball from certain height (call it one unit of distance) on to a floor. We reckon the distnace traveled by the ball towards the floor this way:
It first travels 9/10 of the distance to the floor.
So, distance traveled so far = 0.9 units

Next it travels 9/10 (=0.9) of the remaining 0.1 units of distnace, which is 0.9x0.1 = 0.09 unit.
So, distance traveled so far = (0.9 + 0.09) units

Continuing in this manner, the ball travels
0.9 + 0.09 + 0.009 + 0.0009+ .......... units and so on. Which is the same as 0.999....... forever repeating.

This process never seems to end, because there is always 9/10 the of the remaining distance to go. So we expect the ball to take forever to reach the floor. But we are mistaken. The ball always hits the floor in finite time, which is Nature's way of proving that 0.999..... unit = 1 unit . In this case, Nature takes a finite amount of time to complete an infinite sequence of steps, each taking a calculably finite (however small) amount of time. It is our assumption that the sum of any infinite sequence of numbers can not be a finite number. But it is an assumption, and however intuitive, it is still an assumption. For a convergent infinite series that assumption is not true, as demonstrated by the dropped ball.
I don't think this is quite right, you are assuming a growing infinite is equivalent to a narrowing infinite.

[Astinus]

It may not be exactly relational, but it is a sound analogy. This is why knowledge of limits is good for understanding this.

The actual Calculus of this problem is earlier in this thread, but it's more complicated, so here's a simpler example: lim x-> infinity (1/x). In that limit, x gets larger and larger (towards infinity), so what is the limit of (1/x)? The answer is 0. As x gets larger and larger, 1/x will get closer to 0, but the limit actually is 0.

It may be counterintuitive, but numbers already exist. We can discover and work them out one number at a time, but they're already there. 0.99999... is not "an extension 9's at the end to get it closer and closer to 1," it's "an infinite string of 9's after the decimal." That infinite string is equal to 1.

Just as the ball goes 9/10 of the way in increments but eventually hits the ground, that infinite string of 9's after the decimal point reaches 1.

[ghetto astronaut]

I've read all the threads on this topic, but I'm still not convinced that 0.999... equals 1. I don't think I'm stupid, and if I really think someone makes a solid case, I'll believe it. That just hasn't happened yet.

a) I don't consider 'infinity' to be a valid concept when it comes to numbers. It's useful theoretically, but not literally. I don't think 'infinity' anything can exist other than in theory. Everything is finite, no matter how large.

b) I think that 0.999... = 1, as well as any other infinitely-repeating number, is just an approximation. Any fraction like that [like 0.333... being 1/3] is an approximation. It's perfectly acceptable in a real-world, pragmatic kind of way, but it doesn't seem literally right. It seems that there isn't an answer we can come to that is literally right - but that still doesn't mean the approximation is the literal truth. Someone mentioned the god of the gaps here, and that's what it reminds me of too.

c) I accept that there is no definable number between 0.999... and 1. But there still has to be one, even if we're incapable of pinning it down. Sort of like how oxygen is matter, even though we can't see, taste, hear or feel it. It's still there.

There could be reasonable rebuttals to these points, but I haven't seen them specifically tackled. Basically what I want to know is, is it being argued that 0.999... is, in a pragmatic way, equal to 1, or literally? One seems reasonable to me and the other doesn't. Just because we're not capable of quantifying the difference doesn't mean that there isn't one, or at least it seems that way.

[TNorthover]

There could be reasonable rebuttals to these points, but I haven't seen them specifically tackled. Basically what I want to know is, is it being argued that 0.999... is, in a pragmatic way, equal to 1, or literally? One seems reasonable to me and the other doesn't. Just because we're not capable of quantifying the difference doesn't mean that there isn't one, or at least it seems that way.

It's literal, and a matter of definitions. We have to decide what we mean by any set of symbols. If you had 0.99, it would be easy: 9/10 + 9/100. But with the recurring decimal, you can't just add up an infinite number of terms like that. Mathematicians looked at this problem, and eventually settled on something called limits which carefully avoid direct mention of infinity and skirt around the subject to keep everything consistent. The alternative was to declare any nonterminating decimal meaningless, and so have no simple notation for most numbers. It turned out that with this definition, the number is equal to 1.

Well, that's not strictly true, they could have come up with some method that made it not equal to 1, but they'd lose all kinds of nice properties that they wanted (largely arithmetic working as expected for nonterminating decimals). Regardless, they chose as they did, and arguing it now is like arguing 1+2 isn't 2.

[Sven]

I've read all the threads on this topic, but I'm still not convinced that 0.999... equals 1. I don't think I'm stupid, and if I really think someone makes a solid case, I'll believe it. That just hasn't happened yet.

a) I don't consider 'infinity' to be a valid concept when it comes to numbers. It's useful theoretically, but not literally. I don't think 'infinity' anything can exist other than in theory. Everything is finite, no matter how large.
Umm, mathematics is theory. Just as 0.999... and 1 are theory.

c) I accept that there is no definable number between 0.999... and 1. But there still has to be one, even if we're incapable of pinning it down. No. Between any two real numbers, there are infinitely many real numbers. One number in between is simply not possible.

Sort of like how oxygen is matter, even though we can't see, taste, hear or feel it. It's still there. Sorry, this analogy is really shit. Cool oxygen down to -200 °C and you have no problem to see it. Light a candle - you immediately see if oxygen is there or not. Etc. etc. etc.

There could be reasonable rebuttals to these points, but I haven't seen them specifically tackled. Basically what I want to know is, is it being argued that 0.999... is, in a pragmatic way, equal to 1, or literally? Literally. Numbers simply don't exist in a pragmatic way (well, except for the natural numbers, perhaps), they are theoretical concepts.

One seems reasonable to me and the other doesn't. Just because we're not capable of quantifying the difference doesn't mean that there isn't one, or at least it seems that way. This is mathematics, not physics. We can prove that there is no difference, it's not that we can not find a difference.

[Sven]

Regardless, they chose as they did, and arguing it now is like arguing 1+2 isn't 2.
I see no problem in arguing that 1 + 2 is not 2 ... :-p

[Chamaeleon]

Someone mentioned the god of the gaps here, and that's what it reminds me of too.
I'm going to use my non-recurring 3 post count to say that my gap of the gods post was entirely satirical, and that I believe this gap exist to the same extent I think a god of the gaps gets you from one species to another. In other words, I don't. I cannot understand why it's a problem accepting the equivalence, nor do I understand the need for so many threads to be started on the topic. If the multitude of reasons for the equivalence already given doesn't convince someone, odds are no future reason will persuade that person either.

[JohannGoodflag]

a) I don't consider 'infinity' to be a valid concept when it comes to numbers. It's useful theoretically, but not literally. I don't think 'infinity' anything can exist other than in theory. Everything is finite, no matter how large.

We're talking about mathematics: that is theory par excellence.

If you don't accept the 'existence' of infinitely many numbers, then we have bigger differences in understanding that just whether it makes sense for 0.9 = 1 to hold.

Ditto if you don't accept that it is reasonable to contemplate a set of infinitely many numbers. Considering such a set is again entirely 'theoretical'; the numbers of that set are just a class of concepts, upon which you are performing operations as a body, in much the same way you perform mental operations on an infinite class of possible things when you say 'one plus one equals two'. You take an adjective (the word 'one') that refers to the size of a collection, and you treat it as an object for the purposes of mathematical contemplation.

If you don't accept the meaningfulness of infinite sets, then you are a finitist, in which case you shouldn't accept the meaningfulness of the number 0.9 in the first place.

If you accept the meaningfulness of infinite sets of numbers, but not of infinite sums of them, then your position is consistent, if a bit sterile. But I can give a reason why you should not only accept infinite sums of numbers, but in particular see that it can yield a finite result.

Consider a set such as
D = {9 * 10^(-n) : n a positive integer} = {0.9, 0.09, 0.009, ... }.
These are all positive, which means you can arrive at a reasonable idea of what it would mean to sum up this set, even if you don't think this would come to a real number. Intuitively, if there was a meanginful sum of all of D, it should certainly be at least as large as the sum of any finite subset of D. (We will call a sum of any finite subset of D a partial sum of D.)

A reasonably conservative thing to say is that if D has a sum, then it should be the smallest real* number which is greater than or equal to any partial sum of D. That is, if we let S be the set of partial sums of D, the sum of D would be the smallest number which is greater than or equal to all of the elements of S (if such a number exists).
* By "real" number, we are referring to an element of the set called "the real numbers", and are not making any ontological claims about this number.

If summing the elements of D were to give an infinitely large result, then for any large number N, we should be able to give a partial sum which is larger than N, which would clearly show that summing all of the elements of D gives an infinitely large result. But then I demand that this rule go both ways: that if all of the partial sums of D are less than some number N, it follows that no amount of summing of elements of D will give me a result larger than N, in which case the sum of the whole of S cannot be larger than N either. That is: if a sum is going to become larger than a given number N, it must do so after only finitely many terms added.

This has been a bit hand-wavy: but the bounding argument is the essence of how we treat infinite sums (of sets of positive numbers; the argument for sets of both positive and negative numbers is similar, but slightly more detailed; but we won't need that to treat the number 0.9).

I will again note that if you don't accept that a sum of an infinite set can yield a finite result, then you shouldn't accept the meaningfulness of the number 0.9 in the first place.

Finally, we apply the definition for the sum of an infinite set that we gave above, to see if we get any finite result in the case of the set D. For the set D given above, the set S of partial sums contain such numbers as:

0.9, 0.09, 0.009, 0.0009, ...
0.99, 0.909, 0.9009, 0.90009, ...
0.999, 0.9909, 0.99009, 0.990009, ...

In particular, for any number x which is less than 1, there is an element of S (0.99...9, for some large enough but finite sequence of 9s) which is larger than x. So if the infinite set D has a sum, it is at least as large as 1. But it is also easy to show that all of the elements of S are less than 1, so the sum of the elements of D is finite (because none of the partial sums exceed 1) and cannot be larger than 1 (for the same reason).

Thus, the sum of the elements of D is at least as large as 1, but no larger; it must then be equal to 1.

If you disagree with this argument, it seems to me that you must disagree with at least one of the the ideas above: that there are infinitely many numbers, that you can have an infinite set, or that the above definition of the sum of an infinite set of real numbers is reasonable.

[JohannGoodflag]

b) I think that 0.999... = 1, as well as any other infinitely-repeating number, is just an approximation. Any fraction like that [like 0.333... being 1/3] is an approximation. It's perfectly acceptable in a real-world, pragmatic kind of way, but it doesn't seem literally right. It seems that there isn't an answer we can come to that is literally right - but that still doesn't mean the approximation is the literal truth. Someone mentioned the god of the gaps here, and that's what it reminds me of too.

The infinitely-repeating number is not an approximation. Any finite truncation of it is, though. Again, if you insist that it is an approximation, then you must disagree with one of the several points of my previous post, and I would be interested in knowing which one.

c) I accept that there is no definable number between 0.999... and 1. But there still has to be one, even if we're incapable of pinning it down. Sort of like how oxygen is matter, even though we can't see, taste, hear or feel it. It's still there.

Why must there be one? What is the basis of your claim? Again, if you insist this, you must disagree with one of the points of my previous post.

This isn't a matter of mathematicians striving to understand something that we find in the real world and falling short: it is a matter of mathematicians defining what we mean by a sum of infinitely many numbers --- which is necessary for the number 0.9 to mean anything at all --- and then persuing the consequences of that definition.

[ghetto astronaut]

It's literal, and a matter of definitions. We have to decide what we mean by any set of symbols. If you had 0.99, it would be easy: 9/10 + 9/100. But with the recurring decimal, you can't just add up an infinite number of terms like that. Mathematicians looked at this problem, and eventually settled on something called limits which carefully avoid direct mention of infinity and skirt around the subject to keep everything consistent. The alternative was to declare any nonterminating decimal meaningless, and so have no simple notation for most numbers. It turned out that with this definition, the number is equal to 1.

Well, that's not strictly true, they could have come up with some method that made it not equal to 1, but they'd lose all kinds of nice properties that they wanted (largely arithmetic working as expected for nonterminating decimals). Regardless, they chose as they did, and arguing it now is like arguing 1+2 isn't 2.

So does that mean that if the question was literally referring to infinity, I would be correct in saying that 0.999... doesn't equal 1? I could concede to your point if it's not really infinity.

[ZouPrime]

I cannot understand why it's a problem accepting the equivalence, nor do I understand the need for so many threads to be started on the topic. If the multitude of reasons for the equivalence already given doesn't convince someone, odds are no future reason will persuade that person either.
The problem is that people see mathematics the same way they see other sciences; i.e. a model that can be refined by observations. And since everyone can make observations, everyone can feel like they have a good case to refine the model. But mathematics don't work that way: they are a construct. They make sense because they are based on precise definitions of each terms. Concept like "real number" and "infinity" have very precise and permanent definitions. If you change these definitions, you're not talking about mathematics anymore.

[ghetto astronaut]

If you don't accept the meaningfulness of infinite sets, then you are a finitist, in which case you shouldn't accept the meaningfulness of the number 0.9 in the first place.

Well that's just it, I don't literally accept it. I understand why we use it, and I know it's the best we can do, but I don't think it literally exists. I don't see how something can go on forever - it just doesn't make sense.

Consider a set such as
D = {9 * 10^(-n) : n a positive integer} = {0.9, 0.09, 0.009, ... }.
These are all positive, which means you can arrive at a reasonable idea of what it would mean to sum up this set, even if you don't think this would come to a real number. Intuitively, if there was a meanginful sum of all of D, it should certainly be at least as large as the sum of any finite subset of D. (We will call a sum of any finite subset of D a partial sum of D.)

This makes sense to me. My issue is just that it's not 100% precise, even though it's clearly good enough. I take people saying it's "literally" true as meaning that it's absolutely true, even if that means being out of our reach. "Essentially true" /= "true," basically.

[ghetto astronaut]

The infinitely-repeating number is not an approximation. Any finite truncation of it is, though. Again, if you insist that it is an approximation, then you must disagree with one of the several points of my previous post, and I would be interested in knowing which one.



Why must there be one? What is the basis of your claim? Again, if you insist this, you must disagree with one of the points of my previous post.

This isn't a matter of mathematicians striving to understand something that we find in the real world and falling short: it is a matter of mathematicians defining what we mean by a sum of infinitely many numbers --- which is necessary for the number 0.9 to mean anything at all --- and then persuing the consequences of that definition.

If it wasn't an approximation, then it wouldn't go on forever - we would be able to actually define the number without resorting to say "it goes on forever." Those two things contradict.

However now I'm confused, because I can also see your point that truncating it would also be an approximation. :huh:

What I mean is, how can something both be "infinitely large" yet have its size actually defined? I don't understand how these two things can be true at once.

[Sven]

Well that's just it, I don't literally accept it. I understand why we use it, and I know it's the best we can do, but I don't think it literally exists. I don't see how something can go on forever - it just doesn't make sense.
What you miss is that no number (apart from the natural ones) really exist.
And 0.999... does not go on forever! You are still looking at this the wrong way - all digits are already there as soon as you write 0.999.... down, it's not a temporal proces with one digit "appearing" at each time intervall.


This makes sense to me. My issue is just that it's not 100% precise, even though it's clearly good enough. I take people saying it's "literally" true as meaning that it's absolutely true, even if that means being out of our reach. "Essentially true" /= "true," basically.
"not precise" does not exist in mathematics. Things are either wrong or right.

[Sven]

If it wasn't an approximation, then it wouldn't go on forever - we would be able to actually define the number without resorting to say "it goes on forever." Those two things contradict.
Point is: We can define the number. It's 1. No contradiction here.
"it goes on forever" is just a way to describe it in non-formal language; the statement is not correct if you use the language of math.

What I mean is, how can something both be "infinitely large" yet have its size actually defined? I don't understand how these two things can be true at once.
What exactly is infinitely large here? :confused:

[ghetto astronaut]

Point is: We can define the number. It's 1. No contradiction here.
"it goes on forever" is just a way to describe it in non-formal language; the statement is not correct if you use the language of math.

What exactly is infinitely large here? :confused:

If we can define the number, then why does it go on forever? And if you're saying it doesn't go on forever, then why is it 0.999... ? What are those dots supposed to represent if not an infinite string of nines?

I don't know math language, so can you tell me what the difference is?

[Chamaeleon]

The problem is that people see mathematics the same way they see other sciences; i.e. a model that can be refined by observations. And since everyone can make observations, everyone can feel like they have a good case to refine the model. But mathematics don't work that way: they are a construct. They make sense because they are based on precise definitions of each terms. Concept like "real number" and "infinity" have very precise and permanent definitions. If you change these definitions, you're not talking about mathematics anymore.
I guess my question was a rethorical one. Sorry. :) I'm in complete agreement with your statements here.

[ZouPrime]

I guess my question was a rethorical one. Sorry. :) I'm in complete agreement with your statements here.
Yes, I know. Sorry for the use of "you", I was talking in the general sense.

[Astinus]

ghetto astronaut: did you see the calculus proof? This has been mathematically proven. There is no actual debate.

Also:
c) I accept that there is no definable number between 0.999... and 1. But there still has to be one, even if we're incapable of pinning it down. Sort of like how oxygen is matter, even though we can't see, taste, hear or feel it. It's still there.

Between any two distinct, real numbers there's not just ONE "definiable number." There's an INFINITE amount of them. There isn't between .9999... and 1 because they are merely different representations of the same number.

[ghetto astronaut]

Between any two distinct, real numbers there's not just ONE "definiable number." There's an INFINITE amount of them. There isn't between .9999... and 1 because they are merely different representations of the same number.

Why are there two different representations, then? Why would 0.999... exist if it was obviously 1? And does it have to be infinite? Is 0.9999999999999999999999999999999999 mathematically considered to be 1 also? Where is the line? Does it only work with infinite numbers?

[Mike Rosoft]

No, it's not an approximation.

0.9 is defined to mean the sum of elements 9*10^-n, for n=1 .. infinity. (Note: for no element is n actually equal to infinity; it just means that the sequence goes through all natural numbers and never ends.)

The infinite sum is defined as a limit of the series of partial sums (s_1=9*10^-1=0.9, s_2=s1+9*10^-2=0.99, s_3=s2+9*10^-3=0.999, etc.) We can see that s_n=1-10^-n. By the above-given definition, the infinite sum is equal to the following limit: lim (1-10^n) for n->infinity.

The limit of an infinite series is defined as the unique number L (if there is one) for which the following holds: Given any (arbitrarily small) positive real number e, there exists a number n (depending on e) and for every m>n, s_m falls into the interval (L-e,L+e). [n and m are positive integers here.]

For the series (1-10^n) the number L indeed exists, and is exactly equal to one. In other words, 0.9 is exactly equal to 1.


Mike Rosoft

[ghetto astronaut]

No, it's not an approximation.

0.9 is defined to mean the sum of elements 9*10^-n, for n=1 .. infinity. (Note: for no element is n actually equal to infinity; it just means that the sequence goes through all natural numbers and never ends.)

:confused:

[JohannGoodflag]

If it wasn't an approximation, then it wouldn't go on forever - we would be able to actually define the number without resorting to say "it goes on forever." Those two things contradict.

We can specify it exactly. There is nothing wrong with infinitely repeating sequences of digits: for instance, we have a formula to determine what each and every one of those digits are. We have complete knowledge of that number; it is completely specified, and therefore not "merely an approximation" of something.

If you would prefer a finitely long representation of 0.9, how about this:
\sum_{j \in \mathbb N} 9 \cdot 10^{-j-1}
This uses a finite number of symbols; it's then a matter of defining what they mean. Intuitively, we refer to an infinite sum: but as so many people are fond of pointing out, we can't actually carry out an infinite number of summations in a finite amount of time, so we resort to a definition which we can evaluate in a finite amount of time. This is basically what I was outlining in the longer post above.

For that matter, how about this finitely long expression for 0.9?
0.9
This notation certainly uses a finite number of symbols, and is part of a well-accepted shorthand for summations of the sort above, which can be given in terms of periodic (i.e. infinitely repeating) functions and negative powers of 10 --- that is, for repeating decimals. For the binary system, we have a similar notation for periodic functions and negative powers of 2; for duodecimal, for periodic functions and negative powers of 12; and so on.

Why should something which has a completely specified, but infinitely long, expansion necessarily be an approximation? So long as we have some finite means of expressing it --- which we clearly do --- we can then perform calculations with it just as any other number.

However now I'm confused, because I can also see your point that truncating it would also be an approximation. :huh:

What I mean is, how can something both be "infinitely large" yet have its size actually defined? I don't understand how these two things can be true at once.

The reason is because it is not actually "infinitely large" --- there are many ways of expressing it in a finite number of symbols, which we can make meaningful by rules for how to perform computations.

We refer to something which has an infinitely long representation in one representation system; but we specify it exactly using a different representation system. This can be as simple as putting a bar or underline to specify "this portion repeats infinitely often in a decimal expansion", or it can be something more elaborate, like the summation notation above. All that matters is that we can specify it finitely by some means, and be able to compute with it by well-defined rules, where these rules produce consistent results across different systems of notation.

[y_feldblum]

-0.9999... = 1 - 10^-infinity. Ignoring the infintesimal part would be like saying that 0.9 = 1 - 0.1 = 1. It's 1 minus some number that is smaller than any real number, which apparently leads to 0.9999... not being a real number. Since I'm a bit short on math required to define reals, anyone up for showing that 0.9... is real?

Separate the expression representing a number from the number itself. The means of reference is not the thing itself. The mere fact that we have the capability of representing one number via two different means ... does not mean that the one number is at the same time two different numbers.

10^-infinity = lim(n->infinity) 10^-n = 0, by the definition of "infinity". No, infinity is not a real number, and no, 10^-infinity is not some really small number other than 0 ("infinitesimal"). 10^-infinity is, by definition, 0.

[TNorthover]

I see no problem in arguing that 1 + 2 is not 2 ... :-p

Yes, but you've always been a hidebound reactionary ;)

[Astinus]

Why are there two different representations, then? Why would 0.999... exist if it was obviously 1? And does it have to be infinite? Is 0.9999999999999999999999999999999999 mathematically considered to be 1 also? Where is the line? Does it only work with infinite numbers?

So why does 1/1, 2/2 or 3/3 also exist if 1 already exists? Actually, for a better example: does 6/8 not equal 3/4, even though they're just different representations of the same number?

.9999... and 2/2 are just different representations for the same number.

[TNorthover]

So does that mean that if the question was literally referring to infinity, I would be correct in saying that 0.999... doesn't equal 1? I could concede to your point if it's not really infinity.

If a sheep was a cow, would it moo? More seriously, that gets into exactly what you'd count as a real infinity. There's always going to be infinite sets lurking, but that's true for 1,2,3,... before we get anywhere near 0.999..., I'd say closest 0.999... gets to a real infinity is that it says that the 1st digit is 9, 2nd digit is 9 and so on.

To answer your actual question, you'd have to say exactly what you meant by changing it to literally refer to infinity.

Why are there two different representations, then?

Just how it works out, really. You're probably completely used to it for fractions: 1 = 2/2 = 3/3..., but it's just a few edge cases for decimals, so it's not so intuitive to us.

[Smullyan-esque]

Well that's just it, I don't literally accept it. I understand why we use it, and I know it's the best we can do, but I don't think it literally exists. I don't see how something can go on forever - it just doesn't make sense.

:devil1:

So, what's the largest number, then?
:blush:

[Smullyan-esque]

Just how it works out, really. You're probably completely used to it for fractions: 1 = 2/2 = 3/3..., but it's just a few edge cases for decimals, so it's not so intuitive to us.

Here's an idea I'm sure you will appreciate, TNorthover:

EVERY terminating decimal has two different ways of being represented. I remember seeing this for the first time in Cantor's diagonal argument.

For example: 3.73 could also be written as 3.729

The only decimals that come in only one form are the infinite decimals (which includes the irrationals and repeating rationals).

Since every terminating decimal comes in more than one form, most folks wouldn't call that "just a few edge cases".

.
..
...
....
.....
......

Except, there are LOTS more infinite decimals than terminating ones. If you picked a number at random, the chance of picking a terminating decimal is zero, so, from a mathematician's point of view, your original quote was correct! :D

[ghetto astronaut]

We can specify it exactly. There is nothing wrong with infinitely repeating sequences of digits: for instance, we have a formula to determine what each and every one of those digits are. We have complete knowledge of that number; it is completely specified, and therefore not "merely an approximation" of something.

Hm, interesting. For some reason I have an easier time understanding that something could infinitely repeat like 0.252525252525... than 0.9... The 0.9... one seems like it keeps trying to get at a smaller and smaller number but never succeeding (hence it going to infinity). Is that what you mean? That you already know the entire sequence, even if you can't write out the whole thing?

[Corona688]

Hm, interesting. For some reason I have an easier time understanding that something could infinitely repeat like 0.252525252525... than 0.9... The 0.9... one seems like it keeps trying to get at a smaller and smaller number but never succeeding (hence it going to infinity). I think I can see why it would puzzle you less. 0.2525... isn't an edge case. It's not about to turn into 0.26 because, no mater how many more ...25...'s you add, there is already a finite difference provably larger than any digits that come after it.

But the difference between 0.999... and 1 is proven to be infinitely small. There is such a thing as an infinitely small number. We call it "Zero".Is that what you mean? That you already know the entire sequence, even if you can't write out the whole thing? Precisely. We don't have to write down the whole sequence, just enough information to derive the whole sequence from.

[ghetto astronaut]

I think I can see why it would puzzle you less. 0.2525... isn't an edge case. It's not about to turn into 0.26 because, no mater how many more ...25...'s you add, there is already a finite difference provably larger than any digits that come after it.

But the difference between 0.999... and 1 is proven to be infinitely small. There is such a thing as an infinitely small number. We call it "Zero". Precisely. We don't have to write down the whole sequence, just enough information to derive the whole sequence from.

So are 0.252525... and 0.999... exactly the same concept? Or are they two different things? If it's the former, then I think I just got misled by the particular example - it makes it seem like it's saying something it isn't.

[Astinus]

If it wasn't an approximation, then it wouldn't go on forever - we would be able to actually define the number without resorting to say "it goes on forever." Those two things contradict.

However now I'm confused, because I can also see your point that truncating it would also be an approximation. :huh:

What I mean is, how can something both be "infinitely large" yet have its size actually defined? I don't understand how these two things can be true at once.

Well that's just it, I don't literally accept it. I understand why we use it, and I know it's the best we can do, but I don't think it literally exists. I don't see how something can go on forever - it just doesn't make sense.

Have you heard of pi? It goes on forever. It's irrational, though, so it can't be represented as a fraction.

And any rational number, such as a repeating decimal, can be defined by using a fraction. .9999... = 9/9 = 1. Those three representations are for the same number. It's not an approximation.

Another example: a line has an infinite number of points, but the line can be defined.

[ghetto astronaut]

Have you heard of pi? It goes on forever. It's irrational, though, so it can't be represented as a fraction.

And any rational number, such as a repeating decimal, can be defined by using a fraction. .9999... = 9/9 = 1. Those three representations are for the same number. It's not an approximation.

Another example: a line has an infinite number of points, but the line can be defined.

Is a decimal just an inadequate way of expressing that number, then? The point of confusion for me is the concept that 0.999... is supposed to at once be infinite, yet also clearly definable and literal.

By inadequate I mean less than perfect. Obviously it works perfectly well in everyday life.

[JohannGoodflag]

Hm, interesting. For some reason I have an easier time understanding that something could infinitely repeat like 0.252525252525... than 0.9... The 0.9... one seems like it keeps trying to get at a smaller and smaller number but never succeeding (hence it going to infinity). Is that what you mean? That you already know the entire sequence, even if you can't write out the whole thing?

Yes, the analogy is exact. Keep in mind that in the case of 0.25, if you write out more and more digits of it, you are also reaching closer and closer to some number --- in this case, 25/99. If you only specify a finite number of digits, you don't actually achieve 25/99; you achieve something which is somehwat smaller than it. Only if you actually indicate that it repeats infinitely (but according to a finite pattern) do you actually obtain precisely 25/99; but by the same token, you achieve precisely 25/99, not an approximation to it.

Not being able to write out the whole of a number in some particular representation is not a handicap to the number being well-defined. For instance, to the best of our knowledge, there isn't even enough matter in the universe to write out the number
googolplex = 10^(googol) = 10^(10^100);
having said that, googolplex is a perfectly respectable positive integer, with nothing really mysterious about it. Still, even if there were enough matter in the universe to write it out, it is almost certain that I, and all of my descendants, and the civilization in which they lived, would all cease to exist before it had been written out in full. But unless you're an ultrafinitist --- roughly speaking, someone who doesn't accept the existence of numbers that can be represented by piles of actual objects presented in front of them --- it's hard to dispute the usefulness of a number system in which a googolplex is a well-defined number. Not because it is crucial that we be able to use the number googolplex, but because anything reasonably useful will inevitably lead to the existence of such a number.

[JohannGoodflag]

Is a decimal just an inadequate way of expressing that number, then? The point of confusion for me is the concept that 0.999... is supposed to at once be infinite, yet also clearly definable and literal.

By inadequate I mean less than perfect. Obviously it works perfectly well in everyday life.

Almost, but not quite. It is quite fair to say that the decimal system is inadequate to represent the expansion 0.9. But it does have another perfectly reasonable representation of that same number: namely, 1. (That's the whole thrust of this topic, of course.)

Of course, decimal expansions are inadequate to represent numbers such as 0.3, 0.25, or pi with a finite number of symbols. It just so happens that 0.9 refers to a number which also has a finite representation in decimal expansions.

ETA: I should note that I consider the 'repeating' bar as an extension of decimal expansion notation, but not an actual part of decimal expnasion notation per se. It's all in how you choose your definitions: I'm just saying this to explain that the collection symbols "0.25" isn't a part of what I call decimal expansion notation, which is why I don't consider "0.25" a finite expression of 25/99 as a decimal expansion.

[ghetto astronaut]

Yes, the analogy is exact. Keep in mind that in the case of 0.25, if you write out more and more digits of it, you are also reaching closer and closer to some number --- in this case, 25/99. If you only specify a finite number of digits, you don't actually achieve 25/99; you achieve something which is somehwat smaller than it. Only if you actually indicate that it repeats infinitely (but according to a finite pattern) do you actually obtain precisely 25/99; but by the same token, you achieve precisely 25/99, not an approximation to it.

Not being able to write out the whole of a number in some particular representation is not a handicap to the number being well-defined.

Well, you have now cleared up my confusion. I was interpreting the representation as the actual thing, which led me to believe that because the representation could never be expressed in full, then the whole thing must be imprecise.

Someone's mind changed on the internet! :eek:

[Astinus]

Is a decimal just an inadequate way of expressing that number, then? The point of confusion for me is the concept that 0.999... is supposed to at once be infinite, yet also clearly definable and literal.

By inadequate I mean less than perfect. Obviously it works perfectly well in everyday life.

It's not inadequate. It may be inefficient or confusing, but it's the same number. We use "1" as the main foundation for that concept, but that is arbitrary. 1, 2/2, (1)1^(1/1), .9999... , et cetera are all simply different represenations of the same concept. You seem to have had difficulty with this because you think of the representation 1 as THE representation and concept both, but it's not; it's just one representation of a mathematical concept.

EDIT:
I was interpreting the representation as the actual thing, which led me to believe that because the representation could never be expressed in full, then the whole thing must be imprecise.

So of course, after I post it, I see you said almost the exact same thing. :p

[Corona688]

Why are there two different representations, then? Why would 0.999... exist if it was obviously 1? Because people refuse to accept that it's 1. When people see 1/1, 2/2, 3/3, and 4/4 they simplify them to 1. When people see 0.999... they go to iidb and tell people why it's not true because they feel so in their heart.And does it have to be infinite? Yes.

[untermensche]

Because people refuse to accept that it's 1. When people see 1/1, 2/2, 3/3, and 4/4 they simplify them to 1. When people see 0.999... they go to iidb and tell people why it's not true because they feel so in their heart. Yes.
The idea that a point on a line can be represented by an infinitely repeating digit is not intuitive.