Symbol for rational/irrational part of a number

Question

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?

Answer 1

You are talking in the realm of e.g. quadratic rings like $\mathbb{Q}(\sqrt{d})$. Often $d$ is negative (Gaussian integers, for instance), and (even when it isn't) you might as well use the notations $\Re(z)$ and $\Im(z)$. But make double sure your audience knows what you are talking about.

Note that if you want to talk about cubic or higher rings, you get more "basis vectors," and you'd need to extend the notation somehow.

Answer 2

Not that I know of. Rational and irrational numbers are all elements of the same set complex numbers.

Answer 3

I think the closest thing you can get is the floor function. Where the "rational" or integer part of $x$ would be the largest integer less than or equal to $x$. But this doesn't really guarantee that what is left over will be irrational.

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Edit

Thinking about it a bit more I think the following is at least well defined for real values of $x$,

$$ RationalPart(x) = \begin{cases} x \qquad x \in \mathbb{Q} \\ \lfloor x \rfloor \qquad x\notin \mathbb{Q} \end{cases} $$

Of course this doesn't have any of the nice properties like linearity that $Im$ and $Re$ have.

Answer 4

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.