Continued Fraction Expansion via Gauss Map

Theorem
Let $T : \closedint 0 1 \to \closedint 0 1$ be the Gauss map.

Let $x \in \closedint 0 1 \setminus \Q$.

Then $x$ has the simple infinite continued fraction:

where:
 * $\map {a_n} x := \floor {\dfrac 1 {\map {T^{n - 1} } x} }$
 * $\floor \cdot$ denotes the floor.