Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 12F / a / Solution

The Liouville approximation theorem says that for a real algebraic number $\xi$ of degree $d\ge 2$, there is a constant $C(\xi )>0$ such thatfor all integers $p$ and $q\ge 1$ with $p/q\ne \xi$. In particular a quadratic irrational cannot have arbitrarily accurate rational approximations of order $o(q^{-2})$.