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Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of constructing the field of fractions of an integral domain.

One case where one (typically??) does not know of such a thing in advance is the field of "convolution quotients"---the field of fractions of a ring of functions of a real variable in which the "multiplication" is convolution.

But convolution quotients will not be appreciated by students who just finished a first-semester calculus course last week. Is there some example one could mention to such students where they wouldn't think they already know what is meant by division of the objects in question?

Later edit suggested by answers and comments posted so far: I had in mind two or three purposes. One was that I wanted to mention this topic a bit obliquely in something the students are to read, and that had to be really terse, so I can't do anything really involved. Less than an hour after I posted the question, this ended up being a parenthetical comment on the course web site that said: "(for example, why is it that one can `divide' one divergent series by another?)". Here I had in mind the ring of formal power series suggested by Chris Eagle, but of course I needed to ruthlessly avoid mentioning power series.

A second purpose concerned possible future uses. Not only in courses: if we get some good examples here, I'd like to add them to Wikipedia's article titled "field of fractions".

A possible third purpose was just the satisfaction of knowing more than one decent example (since the only one mentioned above that's "decent" in the relevant sense is convolution quotients).

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    Maybe formal power series and formal Laurent series?2012-04-26
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    COnstruct the coordinate ring of some curve, define the ring of germs at a point, and then show that you can get it by localizing.2012-04-26
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    @MarianoSuárez-Alvarez : I think your example suffers from the same difficulty that afflicts convolution quotients.2012-04-26
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    +1 Nice pedagogical question. Do the examples need to be *total* quotient fields or can the be proper subrings thereof, such as localizations? It might help to make the title more specific, e.g. replace "certain" by "unanticipated" or "examples... new to novices", or somesuch.2012-04-26
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    At some point thy will have to come to terms with the fact that the notions they are being exposed to are interesting for reasons that thy have not previously been exposed to.2012-04-26
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    @MarianoSuárez-Alvarez : +1 for your last remark. That is actually what I mentioned in the thing I just posted on the course web site. I said (paraphrasing) there are reasons to prefer to avoid fractions when proving a certain statement.... and mentioned that sometimes it's not clear in advance that it's possible to divide the objects in question; hence one can't use fractions.2012-04-26
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    @Mariano: I remember one of my undergraduates teachers said once: "You know you are getting closer to finishing your degree when, instead of hearing a lot of "And this is important for reasons you will see in ..." you hear a lot of "And this is important, for reasons you saw in ...."2012-04-27
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    @ChrisEagle : If you make your comment into an answer with emphasis on the fact that this allows division of _divergent_ series (so that it's not just an example where they _already_ "know" what division is) then I'll "accept" it.2012-05-08

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Ask them how many intermediate rings $\mathbb Z\subset R \subset \mathbb Q$ there are and whether they can classify them all.
Amaze them by telling them that there is a continuum ($=2^{\aleph_0}$) of them and that you can classify all of them explicitly: they are indexed by the subsets $P\subset \lbrace 2,3,5,7,\ldots \rbrace $ of the primes and they are the $S_P^{-1}\mathbb Z=\mathbb Z[\frac {1}{p}\mid p\in P]$ (where $S_P$ is the multiplicative monoid generated by $P$)

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    Well, this answer obviously won't serve the immediate purpose. Students of the kind I have in mind are from _this_ Reality, not some other. (I.e. computer science students whose professional interest is in things other than math. And they really haven't had anything beyond first-semester calculus before this. Really. I mean: really.) It's an interesting answer, but for the immediate purpose it can't be taken literally.2012-04-26
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    ....However, some version of this answer may well serve the less immediate purposes of the question.2012-04-26
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    @Michael But these examples are neither *fields* nor examples of elements that students don't already know how to divide. Could you *please* clarify if either of these points are essential (as they seem to be in you query).2012-04-26
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    @BillDubuque : Well, definitely it's better to use examples of things where it's not obvious that division is possible.2012-04-26
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    @Michael Note that *formal* rational "functions" are not really quotients of *functions*, so this example may already meet your requirements, i.e. they really are "formal" quotients. However, your students might not yet be ready to appreciate the distinction between formal vs. functional polynomials (and their fractions).2012-04-26
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    @BillDubuque : It is precisely because they will not appreciate the difference between "formal" quotients of _integers_ and actual rational numbers that I wanted a different sort of example.2012-04-27
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Here's a mean trick you can pull if they really did just finish first semester calculus. Introduce the ring $$ \mathbb{R}\left[\frac{d}{dx}\right] $$ which is of course just a polynomial ring, but don't say this. Show that it "acts'' on the space of smooth functions in the manner suggested by the notation.

Then ask a student what the field of fractions should be, suggestively saying "and what will the inverse of $\frac{d}{dx}$ be?"

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    Well...... There's such a thing as $\dfrac{D}{e^D-1}$ (where of course $D=d/dx$, and it doesn't even involve a "constant of integration!, and in fact $\dfrac{D}{e^D - 1} x^n = B_n(x)$ $=\text{the $n$th-degree Bernoulli polynomial}$. But I still have some qualms about using this.2012-04-26
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    Alternatively you could have the polynomial ring act on smooth functions mod constants, in which case the action really does extend to the field of fractions.2012-04-26