Been reading "The Theory of Groups: An Introduction (2nd
Edition)", a huge and advanced tome by Rotman.  In one of the
exercises, he casually mentions that Z[x], the polynomials with
integer coefficients, considered as a group under addition, is
isomorphic to the positive rationals considered as a group under
multiplication.  The exercise was to prove this by exhibiting such an
isomorphism explicitly, and I solved it thus:

a  
- = (p1^k1) * (p2^k2) * (p3^k3) * ...
b

where p1,p2,... are the prime numbers and k1,k2,... are integers
which, by the fundamental theorem of arithmetic, exist and are unique.
 (They are not necessarily nonnegative, as for example 1/2 = 2^(-1))

Therefore, given a positive rational q, we are also given k1,k2,k3,...
and we use these to construct
i(q) = k1*(x^0) + k2*(x^1) + ... + kn(x^(n-1)) + ...,
a function from the positive rationals to Z[x].
It's not hard to show that i is an isomorphism.

As some examples:
i(0)   = the zero polynomial
i(1/2) = -1 (the polynomial)
i(3/5) = x - x^2
i(3.14159265) = -8 + 2x - 7x^2 + x^3 + x^30 + x^991

This all seems incredibly interesting and profound.  It means, for
example, that we can sensibly refer to the "zeros", the "degree", the
"maxima", the "derivative", etc. of a positive rational, or of "the
value at x=5 of" the same.  Given a rational, we can figure it's
isomorphic image among Z[x], differentiate this and go back to Q to
obtain a new rational.. like so:
d/dx 3.14159265 = i^-1(2 - 14x + 3x^2 + 30x^29 + 991x^990)
                = (2^2)*(5^3)*(113^30)*(7841^991)/(5^14)

It seems there is no limit to the amount of new operations we can
invent using this isomorphism, the above are just a few examples.  We
could speak of "irreducible" rationals, "the minimum rational" of an
algebraic (ie, the rational image of its minimal polynomial), etc.
etc. etc.