\begin{definition}[Definition:Lagrange Number]
Let $\xi$ be an irrational number.
Consider a real number $L_n$ such that there are infinitely many relatively prime integers $p, q \in \Z$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {L_n q^2}$
We define the '''Lagrange number''' of $\xi$ to be $\map L \xi = \map \sup L$ over all $L$ satisfying the inequality above.
{{expand|According to MathWorld there are two kinds of Lagrange number. Recommend that we compare various sources for this. Also recommend that we embed this definition into some kind of mathematical context.}}
\end{definition}