5 ms·

There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and

by pacala 9y ago

There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p

alangpierce 9y ago
Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
- igravious 9y ago
  I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...
  - umanwizard 9y ago
    0.999... does not represent a function or a process or anything like that. It represents a fixed number: the limit of sum from k=1 to n of 9/10^k , as n approaches infinity. You're demonstrating why so many people get confused by the 0.999... notation: they (and you) are thinking of it as the sequence (0.9, 0.99, 0.999, ...) ; but actually it doesn't represent this sequence, or a process for producing this sequence; rather, it represents the limit of this sequence, which is a different type of object: a real number. > Mathematicians are using the symbol … ambiguously Sure, it can mean other things in other contexts (like in my description above of a sequence), but we're talking about what it means in the case 0.j_1 j_2 j_3 ... where the j_k's are base 10 digits. In that case it has a well-defined, non-ambiguous meaning: \lim_{n\to\infty}\sum_{k=1}^n \frac{j_k}{10^k} . > If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. OP is right. I have a degree in math and this is what mathematicians mean by 0.999... > What I'm saying is that the expression could be taken to be one of any number of things Yeah we could have had the notation mean something else. But we didn't. We have other notations for the things you're talking about. Who cares? > That you say the convention is that it most assuredly is 4 is not something I'm buying into. What would it take to make you believe that this is what mathematicians mean by this notation? I promise with all my heart that this is true. Don't take my word for it. Here's some random analysis book. Flip to page 31 (51/686 in the numbering of the PDF): http://www.matematica.net/portal/e-books/Apostol%20-%20CALCULUS%20-%20VOLUME%201%20-%20One-Variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra.pdf http://www.matematica.net/portal/e-books/Apostol%20-%20CALCU... > (I'd nearly go a step further and assert that it ought not be defined this way). Plenty of stuff in math has pedagogically dubious notation, it's not really novel or philosophically interesting to claim you dislike some piece of notation or think it could have been better. It's all just convention. > Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. No mathematician is saying that. The sequence (0.9, 0.99, 0.999...) quite clearly never reaches 1, and any mathematician will agree with that. But its limit is 1. > Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. Well, things don't have to "exist" for us to be able to study them in mathematics, so what's the point of claims like this?
- kutkloon7 9y ago
  Or define what = means. In fact, most university courses make this same mistake.
  - umanwizard 9y ago
    Yeah, this is an abuse of notation (it should be written as set membership, not equality), but the "=" in "0.99... = 1" is really equality, so I don't think this is a good example when talking about that.
    - kutkloon7 9y ago
      True. My point is that equality is never defined well.
      - umanwizard 9y ago
        Never? Equality always means the same thing in every case I can think of, except big-O notation. You picked the one example I'm aware of in all of mathematics where it means something different.
        kutkloon7 9y ago
        I am not arguing that it is ill-defined. I am arguing that no-one takes the effort to rigorously define it.
cgmg 9y ago
> There is no such thing as 0.9... Yes, there is. Ask yourself: What's the decimal expansion of 1/3, or pi?
- wyldfire 9y ago
  I'm not a math pro but I don't think those are good evidence to refute that claim. I have no idea whether the claim is valid or not, though I vaguely recall this discussion from school. From your examples -- one's a ratio and the other is irrational. Almost as if you suggest "The set of real numbers exists, ergo 0.999999... exists"?
  - umanwizard 9y ago
    "0.999... = 1" isn't really a "claim", it's a definition. Well, it's technically a claim, but one that is obviously true once we agree on what 0.999... means. Anyone who doesn't agree that 0.999... = 1 has a different definition of 0.999... from the one used in mainstream math notation.
  - yorwba 9y ago
    The battle over the question whether or not something exists in the mathematical sense has essentially been won by the formalists, who will allow any definition to claim the existence of something so long as it is consistent with the rest of mathematics. Then, given that 0.99999... exists, what useful properties could it possibly have, if it is to be treated as a number. 0.99999... = 0.9999... * 1 = 0.99999... * 9/9 = 0.99999... * (10 - 1)/9 = (0.99999... * 10 - 0.99999... * 1)/9 = (9.99999... - 0.99999...)/9 = 9.00000.../9 = 9/9 = 1 So if 0.99999... exists and has the same arithmetic properties as decimal expansions of rational numbers, its value must be 1. Now whether you think it should have these properties is an entirely different question, but most mathematicians seem to like it this way.
  - cgmg 9y ago
    No, I'm simply saying those are other examples of numbers with infinite decimal expansions. In fact, most numbers don't have a finite decimal expansion. And the claim is correct. It is the statement that the limit of the partial sums of the inverse powers of 9 is 1, which can be rigorously proven with the epsilon-delta definition of a limit.
umanwizard 9y ago
There is such a thing as 0.9... . This notation is defined as meaning the limit as n approaches infinity of the sum from k=1 to n of 9/10^k . A sequence doesn't have to reach its limit!! For example, the limit of the sequence 1/2, 1/4, 1/8, ... is 0, even though the sequence never reaches 0. The limit of the sequence 0.9, 0.99, 0.999, ... is 1 (and this is BY DEFINITION what 0.999... means).
- pacala 9y ago
  In this Universe, no n has been observed to approach infinity.
  - marcosdumay 9y ago
    In this Universe, numbers do no exist, operations on quantities are meaningless, and things are not something. Or, you can decide to give some arbitrary meaning to all that stuff, and get some incredibly useful results. It's your choice.
  - umanwizard 9y ago
    Math isn't directly about the concrete universe; it's an abstraction.

by pacala 9y ago

alangpierce 9y ago

Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.

igravious 9y ago
I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...
- umanwizard 9y ago
  0.999... does not represent a function or a process or anything like that. It represents a fixed number: the limit of sum from k=1 to n of 9/10^k , as n approaches infinity. You're demonstrating why so many people get confused by the 0.999... notation: they (and you) are thinking of it as the sequence (0.9, 0.99, 0.999, ...) ; but actually it doesn't represent this sequence, or a process for producing this sequence; rather, it represents the limit of this sequence, which is a different type of object: a real number. > Mathematicians are using the symbol … ambiguously Sure, it can mean other things in other contexts (like in my description above of a sequence), but we're talking about what it means in the case 0.j_1 j_2 j_3 ... where the j_k's are base 10 digits. In that case it has a well-defined, non-ambiguous meaning: \lim_{n\to\infty}\sum_{k=1}^n \frac{j_k}{10^k} . > If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. OP is right. I have a degree in math and this is what mathematicians mean by 0.999... > What I'm saying is that the expression could be taken to be one of any number of things Yeah we could have had the notation mean something else. But we didn't. We have other notations for the things you're talking about. Who cares? > That you say the convention is that it most assuredly is 4 is not something I'm buying into. What would it take to make you believe that this is what mathematicians mean by this notation? I promise with all my heart that this is true. Don't take my word for it. Here's some random analysis book. Flip to page 31 (51/686 in the numbering of the PDF): http://www.matematica.net/portal/e-books/Apostol%20-%20CALCULUS%20-%20VOLUME%201%20-%20One-Variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra.pdf http://www.matematica.net/portal/e-books/Apostol%20-%20CALCU... > (I'd nearly go a step further and assert that it ought not be defined this way). Plenty of stuff in math has pedagogically dubious notation, it's not really novel or philosophically interesting to claim you dislike some piece of notation or think it could have been better. It's all just convention. > Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. No mathematician is saying that. The sequence (0.9, 0.99, 0.999...) quite clearly never reaches 1, and any mathematician will agree with that. But its limit is 1. > Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. Well, things don't have to "exist" for us to be able to study them in mathematics, so what's the point of claims like this?
kutkloon7 9y ago
Or define what = means. In fact, most university courses make this same mistake.
- umanwizard 9y ago
  Yeah, this is an abuse of notation (it should be written as set membership, not equality), but the "=" in "0.99... = 1" is really equality, so I don't think this is a good example when talking about that.
  - kutkloon7 9y ago
    True. My point is that equality is never defined well.
    - umanwizard 9y ago
      Never? Equality always means the same thing in every case I can think of, except big-O notation. You picked the one example I'm aware of in all of mathematics where it means something different.
      - kutkloon7 9y ago
        I am not arguing that it is ill-defined. I am arguing that no-one takes the effort to rigorously define it.

cgmg 9y ago

> There is no such thing as 0.9... Yes, there is. Ask yourself: What's the decimal expansion of 1/3, or pi?

wyldfire 9y ago
I'm not a math pro but I don't think those are good evidence to refute that claim. I have no idea whether the claim is valid or not, though I vaguely recall this discussion from school. From your examples -- one's a ratio and the other is irrational. Almost as if you suggest "The set of real numbers exists, ergo 0.999999... exists"?
- umanwizard 9y ago
  "0.999... = 1" isn't really a "claim", it's a definition. Well, it's technically a claim, but one that is obviously true once we agree on what 0.999... means. Anyone who doesn't agree that 0.999... = 1 has a different definition of 0.999... from the one used in mainstream math notation.
- yorwba 9y ago
  The battle over the question whether or not something exists in the mathematical sense has essentially been won by the formalists, who will allow any definition to claim the existence of something so long as it is consistent with the rest of mathematics. Then, given that 0.99999... exists, what useful properties could it possibly have, if it is to be treated as a number. 0.99999... = 0.9999... * 1 = 0.99999... * 9/9 = 0.99999... * (10 - 1)/9 = (0.99999... * 10 - 0.99999... * 1)/9 = (9.99999... - 0.99999...)/9 = 9.00000.../9 = 9/9 = 1 So if 0.99999... exists and has the same arithmetic properties as decimal expansions of rational numbers, its value must be 1. Now whether you think it should have these properties is an entirely different question, but most mathematicians seem to like it this way.
- cgmg 9y ago
  No, I'm simply saying those are other examples of numbers with infinite decimal expansions. In fact, most numbers don't have a finite decimal expansion. And the claim is correct. It is the statement that the limit of the partial sums of the inverse powers of 9 is 1, which can be rigorously proven with the epsilon-delta definition of a limit.

umanwizard 9y ago

There is such a thing as 0.9... . This notation is defined as meaning the limit as n approaches infinity of the sum from k=1 to n of 9/10^k . A sequence doesn't have to reach its limit!! For example, the limit of the sequence 1/2, 1/4, 1/8, ... is 0, even though the sequence never reaches 0. The limit of the sequence 0.9, 0.99, 0.999, ... is 1 (and this is BY DEFINITION what 0.999... means).

pacala 9y ago
In this Universe, no n has been observed to approach infinity.
- marcosdumay 9y ago
  In this Universe, numbers do no exist, operations on quantities are meaningless, and things are not something. Or, you can decide to give some arbitrary meaning to all that stuff, and get some incredibly useful results. It's your choice.
- umanwizard 9y ago
  Math isn't directly about the concrete universe; it's an abstraction.