Talk:Conway base 13 function

13

Is the exact number 13 important? Or could this have been done with 14, 17, 23? Crasshopper (talk) 16:10, 30 September 2011 (UTC)

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. David Radcliffe (talk) 04:39, 4 November 2011 (UTC)

This could be done with anything greater than 13. — Preceding unsigned comment added by 158.144.68.25 (talk) 17:09, 19 January 2017 (UTC)

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'... --CiaPan (talk) 19:29, 19 January 2017 (UTC)

No, it is base 5 function, "0"=0, "1"=1, "+"=2, "-"=3, "."=4 — Preceding unsigned comment added by 115.82.177.54 (talk) 22:28, 2 May 2018 (UTC)

That's correct, it would be base 5. --CiaPan (talk) 20:24, 3 May 2018 (UTC)

Reference

Needs a better reference (94.212.42.237 (talk) 19:21, 5 October 2011 (UTC)).

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 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. Julian Gilbey (talk) 22:56, 31 July 2013 (UTC)

Digits

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. — Preceding unsigned comment added by 131.162.130.215 (talk) 15:07, 25 October 2011 (UTC)

I agree, and I've changed the article to match this. 128.86.179.86 (talk) 17:37, 25 January 2012 (UTC)

Error correction

"Indeed, $f$ takes on the value of every real number on any closed interval $[a,b]$ where $b>a$." This is false. For example, if $a=.++$ (base 13) and $b=.+++$, then $f$ is constantly 0 on $[a,b]$.

I edited the page accordingly, but am still not sure if the article %100 correct.

Quinn (talk) 03:34, 26 December 2011 (UTC)

(In the new notation, this example reads a=0.ACC, b=0.ACCC.)

The article was correct before, so I changed it back. I think you missed the fact that the expansion only has to end in the required form; I've now highlighted that in the definition. For example, the interval you've given includes 0.ACC0 up to 0.ACC1, so you can encode any decimal in this region e.g. f(0.AAC0B123A456...)=123.456.... Let me know if you want more details (I haven't completely figured out the algorithm for picking the analogue to 0.ACC0 for arbitrary a and b, but I don't think it's too hard). 128.86.179.86 (talk) 18:02, 25 January 2012 (UTC)

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 > 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 > 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.) 69.195.54.191 (talk) 18:34, 8 April 2012 (UTC)

Reverted recent change

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. User:Jdgilbey (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.

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 then interpreting the symbols literally, is not least confusing! There is even this long discussion on mathematics StackExchange 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.

This has swept up some reference-related edits that were also negative: a conversation is not a valid Wikipedia 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 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. Quietbritishjim (talk) 13:39, 2 January 2014 (UTC)

Response

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 ([1]) or the old version of this Wikipedia page, in that the base-13 "digits" are underlined in my version, whereas the Wikipedia 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 actual 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 I'm not quite sure how one would take a vote on the matter.

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. Julian Gilbey (talk) 14:40, 2 January 2014 (UTC)

Reverting the changes back to the version I had, with a few improvements, since no objection was received within almost two months. Julian Gilbey (talk) 15:36, 21 February 2014 (UTC)

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. 134.206.80.210 (talk) 13:15, 28 June 2016 (UTC)

Why base 13?

We can also use 10 symbols: 0~9, and regard "7" as "+", "8" as "-", "9" as ".", Thus:

f(823.9471415926535023...) (in base 10) = +1415.26535023... (in base 7) — Preceding unsigned comment added by 49.214.90.24 (talk) 13:01, 24 August 2015 (UTC)

New, better source? (op strikes back)

http://www.uccs.edu/Documents/goman/Converse%20of%20IVT.pdf

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 a base-13 function, proving stuff about it, and then getting more general/abstract. Check it out!

and thanks xo --Odinegative (talk) 09:06, 8 May 2017 (UTC)

Hi Odinegative, you can see the paper has already been added to the article on 1 Sept 2015 (in the Special:Diff/678988765 edit). --CiaPan (talk) 11:00, 8 May 2017 (UTC)

Almost everything maps to 0?

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" C/– C/. . This doesn't invalidate the conclusions. I wonder if

1. I'm mistaken
2. I'm right, and those who wrote the article haven't noticed
3. I'm right, but those who wrote the article didn't mention it because it's irrelevant and could be confusing. Maproom (talk) 21:58, 26 July 2023 (UTC)

To put it another way, for readers familiar with regular expressions: we search the tridecimal expression of the number for a match with the regex
/[AB]?\d*C\d+$/
But there generally is no $, so we get the result 0. Maproom (talk) 06:47, 27 July 2023 (UTC)

@Maproom: I'm not sure Almost all is corrrectly used here. The subset mapped to non-zeros has the same cardinality of continuum as its complement, and both are dense on $\mathbb {R}$. So none is 'negligible', which would be required for 'almost all' or 'almost no'. --CiaPan (talk) 19:14, 27 July 2023 (UTC)

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? Maproom (talk) 19:24, 27 July 2023 (UTC)

I've now read Lebesgue measure, and belief that "Almost all" is correct. In any interval, the set of reals that the function maps to a non-zero value has measure 0. But that claim is WP:OR. I'll see if I can find a source. Maproom (talk) 08:33, 6 August 2023 (UTC)

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? Maproom (talk) 08:52, 6 August 2023 (UTC)

References

missing article

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). Kontribuanto (talk) 06:24, 3 March 2024 (UTC)