  We show a correspondence between simple continued fraction expansions of
irrational numbers and irreducible permutative representations of the Cuntz
algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown
that the equivalence of real numbers with respect to modular transformations is
equivalent to the unitary equivalence of representations. Furthermore, we show
that quadratic irrationals are related to irreducible permutative
representations of ${\cal O}_{\infty}$ with a cycle.
