\begin{definition}[Definition:Group Action of GL(2,Q) on Irrational Numbers]
Let $\GL {2, \Q}$ be the general linear group on the field of rational numbers.
Let $\R \setminus \Q$ be the set of irrational numbers.
The '''standard group action''' of $\GL {2, \Q}$ on $\R \setminus \Q$ is the group action:
:$\GL {2, \Q} \times \paren {\R \setminus \Q} \to \R \setminus \Q$:
:$\paren {\begin {pmatrix} a & b \\
c & d \end {pmatrix}, x} \mapsto \dfrac {a x + b} {c x + d}$
\end{definition}