Talk:Decimal representation

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Contents 1 How to represent the answer? 2 Renaming Decimal representation to Decimal expansion 3 infinite decimal representations 4 Merging two sections

[edit] How to represent the answer?

If a_i 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)

You are right, it was confusing. I tried to clarify it a bit. Oleg Alexandrov (talk) 05:38, 24 February 2006 (UTC)

[edit] 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• After links to Decimal representation have been replaced by links to Decimal expansion, it would then make sense to redirect Decimal representation to Numeral system.

Lambiam^Talk 21:20, 20 March 2006 (UTC)

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 guess the article uses "expaniòn" because it was changed in preparation of the move. I prefer "decimal expansion", seeing that the article only talks about infinite series sum_k a_k 10^(-k). But it's only a slight preference. -- Jitse Niesen (talk) 04:22, 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 :) ). Lambiam^Talk 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)

Yes, perhaps a redirect to Decimal is better, or a short informal article covering both integers and reals with appropriate "See further"s. But in any case, I can't do "what I like"; only a sysop can do this (see my post above and [1]). Lambiam^Talk 16:00, 22 March 2006 (UTC)

[edit] 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)

[edit] Merging two sections

(discussion started at my talk page. Oleg Alexandrov (talk) 17:28, 24 March 2006 (UTC))

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)

(In haste) "normalized representation" handled above can be finite. This section is about infinite expansions. Lambiam^Talk 22:19, 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)