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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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    @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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    @BillDubuque : It is $p$recisely because they will not appreciate the di$f$ference 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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    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