Talk:Non-integer base of numeration

Base 0.1
The idea of base 0.1 is not valid. Any base system must use numerals that are less than the base under consideration. The article makes the assertion that using numerals 0 through 9 is okay for a base that is smaller than 1. There is no other source I can find for this numeral system. HumphreyW (talk) 12:06, 22 August 2009 (UTC)
 * It has been a week and no one else seems to be concerned about it either way so I am going to remove the unsourced text from the article. HumphreyW (talk) 02:49, 29 August 2009 (UTC)
 * There's no golden rule that says that you can only uses numerals that are less than the base being used; in general, the rule is that you have to use all of the numerals needed before another column can start to be filled. With base 0.1 it takes 10 1's before another column starts to fill (the 10's) and it takes 10 0.1's before a 1 can jump into the 1's column, so the base has to use 10 symbols. With base 0.125 it takes 8 1's before a 1 goes into the 8's column so there are 8 symbols used, and in general we can say that if the base is less than 1 then with base 1/x x numerals are used. Robo37 (talk) 13:40, 20 September 2009 (UTC)
 * I think it is better to stick with the convention that most sources adopt, that &beta; > 1 and that the coefficients in the expansion are nonnegative integers less than &beta;. It would be very difficult to structure a coherent article otherwise.  Other conventions, if they can be sourced, perhaps belong in a dedicated section towards the end.  Le Docteur (talk) 00:52, 12 November 2009 (UTC)
 * I agree. Furthermore, it seems like the formulation that Robo is proposing would be equivalent to base 1/&beta; except written backwards. Cheers, — sligocki (talk) 15:52, 12 November 2009 (UTC)
 * Roman numerals are another example of using numerals that are greater then the base. They are base 10, (because there is always a symbol for 10x but not 5x or any other number.) but use numerals as large as 1000.  Algr (talk) 08:26, 24 November 2009 (UTC)
 * Roman numerals are not a positional number system. So there is no notion of a base in Roman numerals. HumphreyW (talk) 10:12, 24 November 2009 (UTC)
 * I disagree. Roman numerals are a rather quirky system, but if unary is considered a base, then being positional is not necessary. Algr (talk) 09:46, 25 November 2009 (UTC)
 * I think it's arguable whether "positional number system" should be defined so that it includes unary, in the same sense that it's arguable whether the natural numbers should include 0, or the prime numbers should include 1 (in the latter case there is a clear standard and the alternative wording "positive irreducible numbers" for the alternative). In a completely different direction it's also arguable whether the Roman numbers are positional number system: Position does play a role, as IX is not the same as XI, and the number 10 does have special function. Hans Adler 09:58, 25 November 2009 (UTC)

Comment I never read anything about the beta convention, so I don't know whether there was any special reason for its constraints, but taken out of context they sound nutty to me. I don't see any practical way to use 0 as a radix, and it patently is not worth using 1 or -1 as a radix (possible of course, but clumsy), but beyond those, any integral power (including negative integral) power of an integer can in principle be used simply and with full precision as a base. This includes in particular their reciprocals. So can the negatives of those radices be used, and very useful they could be in principle. (No negatives needed, for example!) So can variable bases. Nor is there any reason why the digits should be kept larger or smaller than the radix. Knuth showed how to use the root of -2 (or was it -4?) as a radix, and though I don't believe it came to anything in practice, it was a very neat idea, which is of course what one expects from Knuth!!! Oh, and PS: using 0.1 as a radix, or -1/16 is almost exactly as easy as using 10 or -16. Don't whine, try it! And in case anyone thinks that that little rant exhausts the subject, let alone the foregoing thread, or the article as it stands, don't kid yourself! JonRichfield (talk) 12:29, 19 April 2014 (UTC)

√2
The section on Base √2 seems highly questionable to me. While it's true that one representation of 1911 is 101010001010100010101 base √2, the expansion, in the sense given in the article, is 1000100010000001000100.0001... . If any representations are to be allowed then why not just take the number itself as the expansion in a single digit? According to the references given, the expansions of integers by bases that are not Pisot–Vijayaraghavan numbers are not guaranteed to terminate or even be periodic. So I would suggest that since the expansion base √2 of most integers produces an uninteresting sequence of pseudo-random digits, it's not notable in this context. The Base π and most of the Base e sections also seem ORish and suspect for similar reasons, but the Hayes article does give some justification for a brief mention of Base e.--RDBury (talk) 00:29, 22 November 2009 (UTC)

Applications section
I added a link to Beta encoder as an application. A section on applications would be nice. History2007 (talk) 17:02, 17 September 2010 (UTC)

Base pi radius to area
Shouldn't the area of a circle with radius 100_pi be (10^4)_pi instead of (10^5)_pi? No matter how I slice it, 10^5 doesn't seem to make a logical progression. 2600:6C54:4E00:2E0:1864:10D6:F9B3:AFBF (talk) 02:13, 5 July 2017 (UTC)


 * Write it in base ten. r = 100_pi = pi^2. Area = pi * r^2 = pi * pi^4 = pi^5 = 100000_pi 76.99.216.210 (talk) 01:23, 5 August 2018 (UTC)

Clarification needed
The sentence in the intro “Every real number has at least one (possibly infinite) β-expansion” is unclear. Does it mean that for every β and every real number x there exists a β-expansion of x? Or just that for every real x there exists some β such that x has a β-expansion? The latter is trivial (pick β=x) so I’m assuming the former, which probably requires further exposition. 76.99.216.210 (talk) 01:14, 5 August 2018 (UTC)

Radix point
This article uses the term "decimal point" regardless of the base. The correct term is "radix point" if one does not want or need to specify a base. Binary, octal, hexadecimal, etc. points are particular cases. Even "point" could be wrong as some locations use "comma". George Rodney Maruri Game (talk) 03:14, 3 March 2023 (UTC)