Symbol for rational/irrational part of a number Just as $\Im(z)$ and  $\Re(z)$ denote the imaginary and real parts of $z$, respectively, do there exist symbols for the rational and irrational parts of a real number?

I had a similar issue and came across this question. Here is my own solution and I hope it still helps.
The real part and imaginary part always make sense because $\mathbb C$ is a degree 2 field extension of $\mathbb R$. Viewing $\mathbb C$ as a vector space over $\mathbb R$ we have a canonical basis $\{1,i\}$. Real part is the coefficient of $1$ while imaginary part is the coefficient of $i$.
Thus, for a field extension $K$ of $\mathbb Q$ of finite degree, we can make the notion of "rational part" meaningful by fixing a basis $B=\{1,e_1,e_2,\dots\}$, and define the coefficient of $1$ to be the "rational part". As such definition is sensitive to the basis $B$, I recommend the notation to be something like $\mathfrak Q_B(x)$.
For example in $\mathbb Q [\sqrt 2]$, we can take the basis $B_1=\{1,\sqrt 2\}$ or $B_2=\{1,126+\sqrt2\}$. With respect to different bases,
$\mathfrak Q_{B_1}(3+\sqrt 2)=3$, $\mathfrak Q_{B_2}(3+\sqrt 2)=-123$. But as long as we are using the same basis, comparing the rational part $\mathfrak Q_B(x)$ always makes sense.
