# Talk:Decimal representation

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## 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)
You are right, it was confusing. I tried to clarify it a bit. Oleg Alexandrov (talk) 05:38, 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 74.139.216.101 (talk) 01:53, 28 January 2008 (UTC)

## Renaming Decimal representation to Decimal expansion

LambiamTalk 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)
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]). LambiamTalk 16:00, 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

(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. LambiamTalk 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)

## 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.

## Annoying comment

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)