|WikiProject Mathematics||(Rated Start-class, Low-importance)|
How to represent the answer?
If ai can only take the values 0 through 9, and the least i is zero, then how (for example) is the value 42 represented? mfc 19:18, 23 February 2006 (UTC)
- Well, it is 42.0000000....
- a_0=42, it does not have to be between 0 and 9, and 0<= a_i <= 9 for i=1, 2, .... Oleg Alexandrov (talk) 05:34, 24 February 2006 (UTC)
--I noticed the same thing until I looked closer and saw your correction. I think it might be clearest to rephrase it as: r == Sum from i=0:inf of ai / 10(i - k) where k is a non-negative integer and ai is an integer in [0,9]. As it is I think you might be assuming that a non-negative integer looks a certain way. —Preceding unsigned comment added by 220.127.116.11 (talk) 01:53, 28 January 2008 (UTC)
Renaming Decimal representation to Decimal expansion
- The page Decimal representation should more properly be called Decimal expansion, since that name is the common way of referring to the topic described on page Decimal representation, as in "The decimal expansion of π is 3.14159265358979323..."
- It can't be moved there because the page Decimal expansion exists. (Why isn't there a simple way of accomplishing the swapping a redirecting page and a redirected-to page not requiring the involvement of sysops?)
- After links to Decimal representation have been replaced by links to Decimal expansion, it would then make sense to redirect Decimal representation to Numeral system.
- Heh. I seem to like more decimal representation, rather than "expansion", but don't have a good reason for why. Other comments? Oleg Alexandrov (talk) 03:39, 21 March 2006 (UTC)
- The current state of the article is weird: decimal expansion redirects to decimal representation, but then the article uses only the phrase decimal expansion, and never defines the term decimal representation. It's a bit jarring. Anyway, I favor decimal representation slightly, but my feelings on the matter are quite weak. -lethe talk + 04:10, 21 March 2006 (UTC)
- I like "representation". What are you expanding, exactly? --Trovatore 04:51, 21 March 2006 (UTC)
- I vote for "representation" also, but here's one way in which "expansion" could make sense: especially in the math that most people know, numbers come in two forms, namely fractions and decimals. The fraction is a compact notation that "stands in" for a division, while the decimal is somehow a terminal form which expands the fraction notation. "Expand", as a phrase used informally in mathematics, tends to mean eliminating immanent operations, so to speak, and obtaining a canonical form for an expression. Thus it is with reducing fractions, which are not at all canonical for the purposes of arithmetic or comparison, to decimals, which are. The canonicity tends to rely upon increasingly redundant information, hence expansion.
- I don't like this, however, since it only works if you compare decimals to fractions. The modern perspective is more abstract than either, and employing decimals in describing elements of the real field is precisely representing them in terms of arithmetic on numerical quantities, namely integers. This is what represents them as numbers, rather than simply symbols. Not only this, but "representation" de-emphasizes the decimal system (unless prefixed with "decimal", of course) since one could have a "vingtesimal" representation, or a "binary" representation, and so on, all "representing" the same abstract number. "Expansion" makes it sound too much like the desired representation is waiting inside to spring out and make the number legitimate. Just my two cents. Ryan Reich 05:11, 21 March 2006 (UTC)
- I tick the box called "representation" instinctively, and the argument that an "expansion" is a better term for a natural operation that doesn't require an arbitrary choice of base is a good one. Elroch 15:51, 21 March 2006 (UTC)
- I prefer "representation". If we didn't have the more accessible article decimal, then I might have gone the other way. But given that this is aimed at a more sophisticated audience, I think "representation" conveys more accurately what this article is about. But really, I don't think it matters. Dmharvey 16:02, 21 March 2006 (UTC)
- Lambiam speaking. For me, it is not so much a matter of what I like, but what is most useful. The term "decimal expansion" is used or linked to in several articles: 19 (number), 1000 (number), 142857 (number), Ackermann function, Adriaan van Roomen, Cantor's diagonal argument, Computable number, Construction of real numbers, Contact (novel), Continued fraction, Countable set, Decimal, Digits of π, Feynman point, Fractal, Fransén-Robinson constant, Forcing (mathematics), Golden ratio, Irrational number, Largest known prime, List of paradoxes, List of YTMND fads, Logarithmic scale, Long division, Normal number, On-Line Encyclopedia of Integer Sequences, Periodic function, Pi, Positional notation, Prime constant, Proof that 22 over 7 exceeds π, Rational number, Regular number, Unique prime, Week, and Yasumasa Kanada. In almost all cases with precisely the meaning described in this article, exceptions being: Ackermann function (writing out A(4, 2) is indeed very expansive :) ), Largest known prime (ditto), and List of paradoxes. That is not a coincidence. Whether or not "decimal expansion" is the "right" term, it is the most commonly used term for this. And no, I did not go around changing all these links — although I may have changed one or two and I indeed adapted this article in preparation for the move, which I expected to be a walk-over. Secondly, to me "3055" is the decimal representation of the number whose hexadecimal representation is "BEF". The use of the term "decimal representation" in the preceding sentence is what I believe to be the primary one. It is used in this sense in the articles 1000 (number), 1729 (number), 53 (number), Cantor's diagonal argument, Colón (currency), Computer numbering formats, DRTE Computer, Intel BCD opcodes, Irrational number, Markup (business), Polymorphism (computer science), Sexy prime, Trigraph, and Unix time. It is used in the sense of decimal expansion representing a real number in Bounded complete poset, Bzip2, Continued fraction, Decimal, Enneagram, Methods of computing square roots, and Real number, and in both senses in Numeral system. (Disclaimer: I used Google search; the articles may have changed.) So in the majority of cases where "decimal representation" is used, it means something not described in this article, while in the overwhelming majority of cases they use "decimal expansion" to describe it. This is all a result of how the editors of these articles naturally used the terms, even with the bias towards using the terms that give you a meaningful link. I rest my case (unless someone dares to contradict me :) ). LambiamTalk 14:52, 22 March 2006 (UTC)
- Hey lambiam, judging by the strength of the opinions expressed above, I think either title is okay, so do what you like. But I'm not sure that people would agree with making decimal representation redirect to numeral system as you suggested above. Perhaps a "see also numeral system" entry on this article would work better. Dmharvey 15:28, 22 March 2006 (UTC)
infinite decimal representations
the article claims that "every real number except zero has two representations". Is this true? What are two decimal representations for 1/3? I am under the impression that this only holds for numbers that admit representations with finitely many nonzero terms. -lethe talk + 20:00, 22 March 2006 (UTC)
- This comment is in response to an edit by mfc, which was subsequently fixed by Dmharvey. So, nevermind. -lethe talk + 20:02, 22 March 2006 (UTC)
Merging two sections
I think the uniqueness is already covered above with the wording "normalized representation". Anyway, I would suggest merging the two paragraphs into one, because as it is now, the repetition of the 999-000 argument is most striking. Hylas 16:18, 24 March 2006 (UTC)
- I agree. The sections Multiple decimal representations and Infinite decimal representations need some kind of mergin. Oleg Alexandrov (talk) 17:28, 24 March 2006 (UTC)
- I thought the expansion .999… is chosen for mere technical reasons. It certainly seems artificial to say that .999… is an infinite expansion while 1.000… is not. Hylas 09:23, 25 March 2006 (UTC)
Recurring decimal representations - some logical considerations
I reworded this entry from: "This happens precisely when the number is a rational number." - with a "special case" when the entry ends in infinite zeros (or nines).
While initially perfectly coherent, the way this was said was logically incoherent. In this case I would ask what do you mean by a special case. If anything it might allow the integers and real numbers with finite decimal expansions, which are both rational numbers, to be included in your definition that might not otherwise have. But the fact that the rest of the entry was not consistent with that (a reference to "the decimal expansion" instead of "a decimal expansion" appeared earlier) leads me to conclude that that was not the intent. My correction should make the entry make more sense to anyone reading it.
I have two issues with the definition given here. The are, in my opinion, not that important. First, why are do we never point out that every real number has a decimal representation? Sure it is a bit easier to define for non-negative real numbers, and everything that is interesting to talk about happens in this case, but some comment should be made. Also the way we handle a0 isn't perfect. When we make the correspondence with the r=a0.a1a2a3… we are sort of implicitly implying everything should be written base 10. I think the definition would be better served if we summed over some finite number of negative indices. And later made a comment that for negative numbers it is typical to represent them by -a0.a1a2a3 where |r|=a0.a1a2a3… by our previous definition. Thenub314 (talk) 09:14, 20 January 2009 (UTC)
negative indices / integer part
Isn't 1024 = 1*10^3+0*10^2+2*10^1+4*10^0 also the decimal representation of a real which is an integer? Why is a0 allowed to be >9 in contrast to the other digits? Shouldn't the integer part also be split up in decimal digits? — MFH:Talk 15:52, 5 January 2014 (UTC)