707:) or the old version of this Knowledge page, in that the base-13 "digits" are underlined in my version, whereas the Knowledge version of 03:31, 26 December 2011‎ were not (and possibly hence the confusion). Conway's idea was simply that if you have an underlined trailing decimal number, that's the value of the function. I have to say that personally I found the "A", "B", "C" version far less obvious, though others may disagree. (I could actually improve my version by writing 3.1415... before "= pi" in the example.) Have you found anyone complaining about not understanding the version I produced based on Conway's
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The main argument for using .,+,- for the extra digits seems to be "that's what Conway used", but this is not an argument at all. Unless this convention has become widespread (it hasn't) we're free to use whatever notation is least confusing. As the comments above note, using . + and - as digits, and
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Hey all, original creator of the article here, returning now after reading the first section of this cool paper. I'm not much of an editor (at all), but created this page back in college and have checked in on its growth from time to time. Very fun! This paper does a very complete job of defining
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I put in a reference to a conversation with Conway. I haven't found a written source, unfortunately, but in the absence of this, surely this is better than nothing? I'm also unclear how it is less reliable (especially the explanation of the reason for creating it) than a lecture about it, which is
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notation (using the underlining) as opposed to the pre-your-edits version? (The stackexchange discussion was about the old version.) I propose reverting your edits in a week or two unless you can give a good reason - I certainly think the underlined version makes the key idea much clearer, though
684:
This has swept up some reference-related edits that were also negative: a conversation is not a valid
Knowledge reference. Neither is a lecture really, but at least it implies some level of review (people in the audience could in principle object!), and I really don't want this article deleted. So
745:
I, for one, was confused by Conway's notation. There is no argument for using Conway's notation besides tradition, whereas the standard notation for tridecimal numbers is, at least, standard. I believe the explanation is clearer now.
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Separately, I understand your comment about the conversation not being a valid reference. I'm not sure how one could come up with a better reference - I could ask Conway to explain this in a lecture, I suppose.
948:
True enough. "Almost all" was wrong. But if you apply the function to a real that wasn't chosen with the Conway 13 function in mind, your probability of getting a nonzero result is less than 0.0000001, right?
214:
It could be also done with bases less than 13. If you start with real numbers' representation in binary instead of decimal (plus '+' and '–' signs) you could easily construct an analogue 'base 4 function'...
989:
I've found this. But
Mathoverflow is a blog, in that anyone can post whatever they like there. Is a response at Mathoverflow, with 16 upvotes, consisting of a short and clear proof, a reliable source?
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0 so that a_(n+m) < C, then any number with expansion a_1, ... , a_(n+m) + 1, ... lies in the interval. (We can always find such an m since we're avoiding expansions with C repeating.)
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There ought to be an article on the 'Bergfeldt function', and the article on 'Darboux's theorem' should be updated to point to it (rather than to the Conway base 13 function).
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The only importance of base 13 is that this particular encoding of real numbers uses 13 symbols. If you used an encoding with two symbols then it would be a base 2 function.
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Disclosure: It was me who made changes as 128.86.179.86 (see comments above), in particular the change from using 0,...,9,.,+,- as tridecimal digits to 0,...,9,A,B,C.
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673:(who I will notify of this discussion) changed back to using .,+,- for the extra digits, and I have now reverted this change. That's what this discussion is about.
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My edited version, which you've just undone, was what Conway showed me. It is fundamentally different from the website referred to in the stackexchange article (
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that can be summarised as "the notation is this article is confusing, can someone explain it to me". So I'm afraid I have just reverted the changes wholesale.
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currently the only given reference. So I'll put it back in the general references for the time being, but won't cite it as an in-article reference.
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until a better reference is found I suggest that that reference be left in place, along with a notice at the top calling for better references.
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My understanding of the Conway base 13 function is that it maps almost all real numbers to zero, because the base-13 expansion has no "last"
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a_n, if the differ by at least two than any number with expansion a_1, ..., a_n + 1, ... lies in the interval. Otherwise, take some m : -->
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is corrrectly used here. The subset mapped to non-zeros has the same cardinality of continuum as its complement, and both are dense on
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Reverting the changes back to the version I had, with a few improvements, since no objection was received within almost two months.
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Algorithm: starting from the left, find the first digit in the base 13 expansion where a and b differ, say, digit n. Then b_n : -->
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in the required form; I've now highlighted that in the definition. For example, the interval you've given includes
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The article was correct before, so I changed it back. I think you missed the fact that the expansion only has to
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I'm right, but those who wrote the article didn't mention it because it's irrelevant and could be confusing.
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It is more conventional, and would be less confusing, to use the digits 0,1,2,3,4,5,6,7,8,9,A,B,C.
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a base-13 function, proving stuff about it, and then getting more general/abstract. Check it out!
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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We can also use 10 symbols: 0~9, and regard "7" as "+", "8" as "-", "9" as ".", Thus:
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I edited the page accordingly, but am still not sure if the article %100 correct.
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Is the exact number 13 important? Or could this have been done with 14, 17, 23?
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then interpreting the symbols literally, is not least confusing! There is even
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f(823.9471415926535023...) (in base 10) = +1415.26535023... (in base 7)
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takes on the value of every real number on any closed interval
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No, it is base 5 function, "0"=0, "1"=1, "+"=2, "-"=3, "."=4
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C/. . This doesn't invalidate the conclusions. I wonder if
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I'm not quite sure how one would take a vote on the matter.
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http://www.uccs.edu/Documents/goman/Converse%20of%20IVT.pdf
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I'm right, and those who wrote the article haven't noticed
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But there generally is no $ , so we get the result 0.
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I agree, and I've changed the article to match this.
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1025:"Is Conway's base-13 function measurable?"
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583:(In the new notation, this example reads
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262:That's correct, it would be base 5. --
791:New, better source? (op strikes back)
627:, but I don't think it's too hard).
95:This article is within the scope of
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975:. I'll see if I can find a source.
38:It is of interest to the following
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444:." This is false. For example, if
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1081:Low-priority mathematics articles
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115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
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135:This article has been rated as
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172:16:10, 30 September 2011 (UTC)
1:
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109:and see a list of open tasks.
927:{\displaystyle \mathbb {R} }
843:Almost everything maps to 0?
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353:17:37, 25 January 2012 (UTC)
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225:19:29, 19 January 2017 (UTC)
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187:04:39, 4 November 2011 (UTC)
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882:/?\d*C\d+$ /
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164:Crasshopper
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103:mathematics
59:Mathematics
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969:Almost all
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363:"Indeed,
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264:CiaPan
217:CiaPan
36:scale.
973:WP:OR
570:Quinn
427:: -->
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
