0.999...
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- Notified: Melchoir, Dedhert.Jr, JayBeeEll, Hawkeye7, WikiProject Mathematics; original nominator not notified as they have been inactive for over 6 years
This is the 1.999...nd FAR of this article. The 0.999...st was back in 2010 and FA status was retained. I placed a FAR notice on the article talk back in January and Dedhert.Jr, JayBeeEll, and Hawkeye7 have done some work on it, but the article has whole sections without citation, amounting to OR; while doing basic math isn't OR, there should be citations to the proofs since they're described as known proofs. Other editors have identified SYNTH in the article during the FAR listing. Hopefully more eyes on this can get it back to FA quality. voorts ( talk/ contributions) 20:55, 11 April 2024 (UTC)
- I was the only one to identify SYNTH during the previous discussion, and I
removed it back in February. There is no whole section without citation; "Elementary proof" has only a single citation for a longish stretch of text, so that can be improved, but overall the situation is not bad at all.
XOR'easter (
talk) 16:15, 12 April 2024 (UTC)
- I did some work on the "Elementary proof" section, and I think now it's in better shape.
XOR'easter (
talk) 14:11, 13 April 2024 (UTC)
- Looks much better. Thanks. voorts ( talk/ contributions) 22:54, 13 April 2024 (UTC)
- RIgorous proof has no citations. There are also statements that I think need citations, such as:
- The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it. Particularly since it's being described as "simple" (by whom? should that be in wikivoice?).
The first paragraph of "Proofs from the construction of the real numbers" has no citations.In 1802, H. Goodwin published an observation ... There's no citation to Goodwin here.
- There's also still the issue of deprecated citations being used in "Algebraic arguments"; {{
sfn}}s were added, but they still need page numbers.
voorts (
talk/
contributions) 22:53, 13 April 2024 (UTC)
- I would try to fix these issues, but I have no math background and wouldn't feel comfortable in case I miscite something.
voorts (
talk/
contributions) 22:54, 13 April 2024 (UTC)
- Added a Goodwyn cite. Feel free to cross that one off! Tito Omburo ( talk) 14:34, 14 April 2024 (UTC)
- The only {{ sfn}}s in "Algebraic arguments" that I'm seeing without page numbers are to journal articles, which don't need them. (I mean, journal articles are short enough that giving a specific page number within them provides basically no value beyond giving the page in the journal where they begin. Standard practice around here is to omit the excess detail, as far as I know.) I've added citations to the opening paragraph of "Proofs from the construction of the real numbers". XOR'easter ( talk) 16:57, 14 April 2024 (UTC)
- I would try to fix these issues, but I have no math background and wouldn't feel comfortable in case I miscite something.
voorts (
talk/
contributions) 22:54, 13 April 2024 (UTC)
- I did some work on the "Elementary proof" section, and I think now it's in better shape.
XOR'easter (
talk) 14:11, 13 April 2024 (UTC)
- I've been perennially disappointed that this article fails to make contact with modern mathematics. Points that could be made, but don't seem to be:
- If one works in base-10, then any fraction whose denominator is power of 2 and 5 will have exactly two distinct expansions: for example, 1/8 has 0.125000... and 0.1249999...
- Similar phenomena happen for base-N for any integer N. (there are always 2 choices)
- For base-N with N not an integer, there may be 1,2, a countable number or uncountable number of such expansions. When there's only one, it is called univoke. For base-phi, with phi the golden mean, there are a countable number of equivalent expansions: basically, you can repeat a finite number of times, and then switch over, or not, at that point. This continues to be a topic of modern research; I read a paper published in 2010 or 2015 that explored this.
- Whenever there are such "gaps" (two distinct reps) those two endpoints can be joined, ... or not. Joining them gives the de Rham curves, which are fractal curves.
- Most or almost all or all fractal watsizz are due exactly to there being two or more non-unique expansions. Whether it's "all" or just "almost all" remains a topic of academic debate. There are "classification theorems" that try to sort out all of the cases; they're called "non-classification theorems" where there's an uncountable number of alternative expansions.
- Something like this applies to chaotic dynamical systems. But there's argument about that.
- Some philosophers have used this in arguments about free will: basically: "a hah, I can choose 0.999 ... or I can choose 1.000... and nature (or neurons, or physics or whatever) will automatically amplify this difference to finite size in finite time and this is how/why one has free will" Not that I beleive this argument, but it is out there, in the wild.
- I do not have references fro most of the above, only for some (below). Failing to mention any of the above just misses an excellent teaching opportunity, to bridge some old ideas to modern, cutting-edge math. This is not some idea that sits in a heremetically-sealed vacuum; it continues to excite mathematicians and philosophers (and students) and should be presented as such.
- Here's some references:
- Nikita Sidorov, “Almost every number has a continuum of β-expansions.”, The American Mathematical Monthly, 110, 2003, pp. 838–842, URL http://www.maths.manchester.ac.uk/nikita/amm.pdf.
- Martijn de Vries and Vilmos Komornik, “Unique Expansions of Real Numbers”, ArXiv, arXiv:math/0609708, 2006, URL https://www.esi.ac.at/static/esiprpr/esi1810.pdf.
- Karma Dajani, et al., “The natural extension of the beta-transformation”, Acta Math Hungary, 73, 1996, pp. 97–109, URL https://www.researchgate.net/publication/2257842.
- Vaughn Climenhaga and Daniel J. Thompson, “Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors”, Israel Journal of Mathematics, 2010, URL https://arxiv.org/abs/1011.2780.
- P. Erdős and V. Komornik, “Developments in non-integer bases”, Acta Math Hungar, 79, 1998, pp. 57–83.
- Nikita Sidorov, “Universal β-expansions”, Arxiv, 2002, URL https://arxiv.org/abs/math/0209247v1.
- Daniel J. Thompson, “Irregular sets and conditional variational principles in dynamical systems”, , 2010, URL https://people.math.osu.edu/thompson.2455/thesis_thompson.pdf.
- Hmm. Actually, it seems I have 20 more of these. Above is a random sampling. Some touch more directly, some touch less directly on the subject matter. I have no references for the philosophy claims. 67.198.37.16 ( talk) 20:09, 12 April 2024 (UTC)
- Oh, ahh, huh, Perhaps I have to partly retract. Closer review indicates the article does touch on some of this. I suppose I have ADHD and didn't notice on first reading. 67.198.37.16 ( talk) 20:41, 12 April 2024 (UTC)
- "A different definition involves what Terry Tao refers to as ultralimit." Why is Terry Tao mentioned at all here? Would one say that Terry Tao is what Martin Hairer refers to as a professor? Gumshoe2 ( talk) 17:14, 14 April 2024 (UTC)
FYI, I'm not that active on Wikipedia these days, but let me know if there are particular questions about any old edits of mine. For example, if there's a cited reference that is hard for others to access, and we need the page number or the context of a quotation, I could look it up in my notes. Melchoir ( talk) 18:22, 17 April 2024 (UTC)
- Melchoir, thanks. XOR'easter ( talk) 19:14, 17 April 2024 (UTC)
0.999...
0.999... ( | talk | history | protect | delete | links | watch | logs | views)
| Toolbox |
|---|
- Notified: Melchoir, Dedhert.Jr, JayBeeEll, Hawkeye7, WikiProject Mathematics; original nominator not notified as they have been inactive for over 6 years
This is the 1.999...nd FAR of this article. The 0.999...st was back in 2010 and FA status was retained. I placed a FAR notice on the article talk back in January and Dedhert.Jr, JayBeeEll, and Hawkeye7 have done some work on it, but the article has whole sections without citation, amounting to OR; while doing basic math isn't OR, there should be citations to the proofs since they're described as known proofs. Other editors have identified SYNTH in the article during the FAR listing. Hopefully more eyes on this can get it back to FA quality. voorts ( talk/ contributions) 20:55, 11 April 2024 (UTC)
- I was the only one to identify SYNTH during the previous discussion, and I
removed it back in February. There is no whole section without citation; "Elementary proof" has only a single citation for a longish stretch of text, so that can be improved, but overall the situation is not bad at all.
XOR'easter (
talk) 16:15, 12 April 2024 (UTC)
- I did some work on the "Elementary proof" section, and I think now it's in better shape.
XOR'easter (
talk) 14:11, 13 April 2024 (UTC)
- Looks much better. Thanks. voorts ( talk/ contributions) 22:54, 13 April 2024 (UTC)
- RIgorous proof has no citations. There are also statements that I think need citations, such as:
- The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it. Particularly since it's being described as "simple" (by whom? should that be in wikivoice?).
The first paragraph of "Proofs from the construction of the real numbers" has no citations.In 1802, H. Goodwin published an observation ... There's no citation to Goodwin here.
- There's also still the issue of deprecated citations being used in "Algebraic arguments"; {{
sfn}}s were added, but they still need page numbers.
voorts (
talk/
contributions) 22:53, 13 April 2024 (UTC)
- I would try to fix these issues, but I have no math background and wouldn't feel comfortable in case I miscite something.
voorts (
talk/
contributions) 22:54, 13 April 2024 (UTC)
- Added a Goodwyn cite. Feel free to cross that one off! Tito Omburo ( talk) 14:34, 14 April 2024 (UTC)
- The only {{ sfn}}s in "Algebraic arguments" that I'm seeing without page numbers are to journal articles, which don't need them. (I mean, journal articles are short enough that giving a specific page number within them provides basically no value beyond giving the page in the journal where they begin. Standard practice around here is to omit the excess detail, as far as I know.) I've added citations to the opening paragraph of "Proofs from the construction of the real numbers". XOR'easter ( talk) 16:57, 14 April 2024 (UTC)
- I would try to fix these issues, but I have no math background and wouldn't feel comfortable in case I miscite something.
voorts (
talk/
contributions) 22:54, 13 April 2024 (UTC)
- I did some work on the "Elementary proof" section, and I think now it's in better shape.
XOR'easter (
talk) 14:11, 13 April 2024 (UTC)
- I've been perennially disappointed that this article fails to make contact with modern mathematics. Points that could be made, but don't seem to be:
- If one works in base-10, then any fraction whose denominator is power of 2 and 5 will have exactly two distinct expansions: for example, 1/8 has 0.125000... and 0.1249999...
- Similar phenomena happen for base-N for any integer N. (there are always 2 choices)
- For base-N with N not an integer, there may be 1,2, a countable number or uncountable number of such expansions. When there's only one, it is called univoke. For base-phi, with phi the golden mean, there are a countable number of equivalent expansions: basically, you can repeat a finite number of times, and then switch over, or not, at that point. This continues to be a topic of modern research; I read a paper published in 2010 or 2015 that explored this.
- Whenever there are such "gaps" (two distinct reps) those two endpoints can be joined, ... or not. Joining them gives the de Rham curves, which are fractal curves.
- Most or almost all or all fractal watsizz are due exactly to there being two or more non-unique expansions. Whether it's "all" or just "almost all" remains a topic of academic debate. There are "classification theorems" that try to sort out all of the cases; they're called "non-classification theorems" where there's an uncountable number of alternative expansions.
- Something like this applies to chaotic dynamical systems. But there's argument about that.
- Some philosophers have used this in arguments about free will: basically: "a hah, I can choose 0.999 ... or I can choose 1.000... and nature (or neurons, or physics or whatever) will automatically amplify this difference to finite size in finite time and this is how/why one has free will" Not that I beleive this argument, but it is out there, in the wild.
- I do not have references fro most of the above, only for some (below). Failing to mention any of the above just misses an excellent teaching opportunity, to bridge some old ideas to modern, cutting-edge math. This is not some idea that sits in a heremetically-sealed vacuum; it continues to excite mathematicians and philosophers (and students) and should be presented as such.
- Here's some references:
- Nikita Sidorov, “Almost every number has a continuum of β-expansions.”, The American Mathematical Monthly, 110, 2003, pp. 838–842, URL http://www.maths.manchester.ac.uk/nikita/amm.pdf.
- Martijn de Vries and Vilmos Komornik, “Unique Expansions of Real Numbers”, ArXiv, arXiv:math/0609708, 2006, URL https://www.esi.ac.at/static/esiprpr/esi1810.pdf.
- Karma Dajani, et al., “The natural extension of the beta-transformation”, Acta Math Hungary, 73, 1996, pp. 97–109, URL https://www.researchgate.net/publication/2257842.
- Vaughn Climenhaga and Daniel J. Thompson, “Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors”, Israel Journal of Mathematics, 2010, URL https://arxiv.org/abs/1011.2780.
- P. Erdős and V. Komornik, “Developments in non-integer bases”, Acta Math Hungar, 79, 1998, pp. 57–83.
- Nikita Sidorov, “Universal β-expansions”, Arxiv, 2002, URL https://arxiv.org/abs/math/0209247v1.
- Daniel J. Thompson, “Irregular sets and conditional variational principles in dynamical systems”, , 2010, URL https://people.math.osu.edu/thompson.2455/thesis_thompson.pdf.
- Hmm. Actually, it seems I have 20 more of these. Above is a random sampling. Some touch more directly, some touch less directly on the subject matter. I have no references for the philosophy claims. 67.198.37.16 ( talk) 20:09, 12 April 2024 (UTC)
- Oh, ahh, huh, Perhaps I have to partly retract. Closer review indicates the article does touch on some of this. I suppose I have ADHD and didn't notice on first reading. 67.198.37.16 ( talk) 20:41, 12 April 2024 (UTC)
- "A different definition involves what Terry Tao refers to as ultralimit." Why is Terry Tao mentioned at all here? Would one say that Terry Tao is what Martin Hairer refers to as a professor? Gumshoe2 ( talk) 17:14, 14 April 2024 (UTC)
FYI, I'm not that active on Wikipedia these days, but let me know if there are particular questions about any old edits of mine. For example, if there's a cited reference that is hard for others to access, and we need the page number or the context of a quotation, I could look it up in my notes. Melchoir ( talk) 18:22, 17 April 2024 (UTC)
- Melchoir, thanks. XOR'easter ( talk) 19:14, 17 April 2024 (UTC)