  In this notes we make a comparison between the arithmetic properties of
irrational numbers and their dynamical properties under the Gauss map. We show
some equivalences between different classifications of irrational numbers such
as the Diophantine classes and numbers admitting approximations by rational
numbers at a given 'speed'. We also show that irrational numbers with finite
upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.
