Regarding circle rotations and composites

Question

Let $S^1$ be the unit circle in the additive notation $[0,1)~mod~ 1$ and $0< \alpha<1$ be an irrational number. Let $\tau$ be the irrational rotation defined by $\alpha$ given by $\tau(x)=x+\alpha ~mod ~1$. For $n \in \mathbb{Z}$, I have seen that $\tau^n(x)=x+n\alpha ~mod ~1$ that is rotation by $n\alpha$. But for large $n$, $x+n\alpha$ can be very large right? What will be the inverse of $\tau$? is $\tau^{-1}(x)=x-\alpha ~mod ~(-1)$? I couldn't understand the rotation map in the additive notation of $S^1$. I wanted to prove that if $\beta \in$ Orbit($\alpha)$, then $0 \leq x <\tau^{-n}(\beta)<x'<\beta$ implies that $0<\tau^n(x)<\beta<\tau^n(x)\leq 1$