Why is the principal quantum number irrational? My textbook says that for a $d$-electron, the orbital angular momentum is: $$ \frac{\sqrt{6}h}{2\pi} $$
But since:
$$ mvr = \frac{nh}{2\pi}$$
$$ n \in N$$
where $N$ is the set of natural numbers, how is the value of $n$ an irrational number?
So, first, as I said in my comment, you confused the orbital angular momentum quantum number $l$ with the principal quantum number $n$. Both can take only integer values (positive and non-negative respectively), but for totally unrelated reasons.

Secondly, the orbital angular momentum itself is related to its quantum number $l$ by the following equation,
$$
    \hat{L}^2 \Psi = l (l + 1) \hbar^2 \Psi \, ,
$$
where $\hbar^2{l(l+1)}$ is the square of the magnitude of an electron orbital angular momentum. Thus, while the orbital angular momentum quantum number $l$ is always a non-negative integer, the magnitude of an electron orbital angular momentum $\sqrt{l(l+1)} \hbar$ can take irrational values.

For instance, for a d-electron, $l=2$, and thus, the magnitude of its orbital angular momentum is $\sqrt{2(2+1)} \hbar = \sqrt{6} \hbar = \sqrt{6} h / 2 \pi$.