TITLE: Visualizing quotient groups: $\mathbb{R/Q}$
QUESTION [23 upvotes]: I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not topological) quotient is topologically equivalent to a circle.
But then, what does $\mathbb{R}/\mathbb{Q}$ look like?

REPLY [3 votes]: The quotient group R/Q is similar to R/Z in some respects, but is quite different and, I think, impossible to visualize in the way R/Z is.  First note that if p is a rational number, then it's equivalence class (i.e. coset generated by p) in R/Q, denoted [p] is equal to [0].  That is, all rationals collapse to the single coset Q.  Now, note that if r is an irrational number, we can write it as r=n+s, where s is an irrational number in the interval (0,1) and n is an integer.  That means that r-s=nis a rational number, which in turn means r and s are in the same equivalence class, i.e. [r] = [s].  That means the elements of R/Q look like {0} U {a set of irrationals in the interval (0,1)}.  But what set of irrationals, exactly?  Not all of them; for example take the decimal portion of PI (=0.14159...) and add 0.5 to it to get 0.64159....  Both are irrational numbers yet their difference is ½, so they generate the same coset, in other words, they are collapsed to the same element of R/Q.  On the other hand, it is known that sqrt(2), sqrt(3) and sqrt(2)-sqrt(3) are all irrational numbers.  That means the cosets [sqrt(2)] and [sqrt(3)] are distinct elements of R/Q.  So, some irrationals collapse to the same element in R/Q but not all do.  So the question becomes:  is there a way to choose or describe a set of irrational numbers that represent the distinct non-zero cosets of R/Q?  The Axiom of Choice implies that, yes, one can choose a set of irrational numbers in the interval (0,1) that form a distinct complete set of cosets for R/Q.  The catch, however, is that the Axiom of Choice gives no recipe for how to choose or describe such a set of representatives.  What we can say is that two non-zero elements of R/Q, call them [r] and [s], are equivalent if and only if their decimal representations differ by only a finite number of digits. So a non-zero coset consists of all the irrational numbers in (0,1) that differ from each other by only a finite number of digits.  It would seem that surely there must be a way to methodically pick from each coset a "canonical" representative.  A likely candidate might be to pick the smallest member in each coset, but of course that fails because there is no smallest member in each non-zero coset; same for largest. What one would ideally like is a choice function f:R/Q -> (0,1) such that for any two cosets C and D, f(C+D) = f( C)+f(D) mod(0,1). As far as I know, no such choice function has been described and, indeed I do not know if it is even possible to define such a function in the standard ZF language.  Note that the AC does not imply that such a function as f, above, exists.  It only says that a choice function exists, but says nothing about how it will behave arithmetically as described above.  
(*)Note:  The statement above that two cosets [r] and [s] are equal iff r and s differ by a finite number of digits is almost right, but ignores the possibility that r-s might be a repeating decimal, like 1/9.  If we permit ourselves to use the repeating decimal symbology of a bar over the repeating segment of decimals, then the original statement stands true.