9

Continued fraction expansions and *β*-expansions

Beside *b*-ary expansions, there are many classical ways to represent a real number, including by means of its continued fraction expansion, or of its *β*-expansion. In the first section, we define the notion of normal continued fraction and construct explicitly a real number having a normal continued fraction. In the next section, we present several results on the continued fraction expansion of algebraic numbers. Section 9.3 is devoted to a short survey on *β*-expansions.

9.1 Normal continued fractions

The sequence of partial quotients of an irrational real number *ξ* in (0, 1) can be obtained by iterations of the Gauss map defined by (0) = 0 and (*x*) = {1/*x*} for *x* ∈ (0, 1). Namely, if [0; *a*_{1}, *a*_{2}

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