$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$
Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to describe the cosets of $\R / \Q$.
I know from looking at the cardinality of the sets that because $\R$ is uncountable and $\Q$ is countable that $\R / \Q$ is uncountable. I am also thinking of $\R / \Q$ containing a "representative" of each irrational number. I am also aware that $\Q$ is dense in $\R$, so that each member of $\R$ is the limit of a sequence of numbers in $\Q$.
Both $\R$ and $\Q$ are ordered. But is there a natural order on $\R / \Q$? What else can we determine about $\R / \Q$?
Background: I am investigating functions satisfying
$f(a + b) = f(a)f(b)$ for all $a,b\in\R$.
If $f$ is required to be continuous then $f(x) = \exp(A x)$
but if $f$ is not required to be continuous then I think I can define
$f(x) = \exp(A_t x)$ where $x$ in $\Q$ where $t$ in some coset $\R / \Q$ and $t \Q = {t + q \text{ where } q\in \Q}$ and $A_t$ is different for each coset. This makes for quite an interesting function!!
Answer
By definition, $r\equiv s$ in the quotient iff $r-s\in \mathbb{Q}$. Said another way, given any real number $r$, the coset $\{r+q\mid q\in \mathbb{Q}\}$ is an element in the quotient.
This seems rather difficult to describe in terms of coset representatives! No obvious candidate for sensible coset representatives jumps to mind for me, because no matter what you pick, you can always strip one (or finitely many) decimal places away to get another rep.
For example $\pi\equiv 0.14159...\equiv0.04159\equiv 0.00159... $ etc.
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