Finite Continued Fractions

Definition 2.1   A finite continued fraction is an expression

$$a_0 + \frac{1}{a_1+\frac{1}{a_2 + \cdots + \frac{1}{a_m},}}$$

where each $a_n$ is a rational number and $a_n>0$ for all $n\geq 1$. If the $a_n$ are integers, we say that the continued fraction is integral.

To get a feeling for continued fractions, observe that

$$[a_0] = a_0,$$

$$[a_0, a_1] = a_0 + \frac{1}{a_1} = \frac{a_0 a_1 + 1}{a_1},$$

$$[a_0, a_1, a_2] = a_0 + \frac{1}{a_1 + \frac{1}{a_2}} = \frac{a_0 a_1 a_2 + a_0 + a_2}{a_1 a_2 + 1}.$$

Also,

$$[a_0, a_1, \ldots ,a_{m-1}, a_m] = [a_0, a_1, \ldots, a_{m-2}, a_{m-1} + \frac{1}{a_m}]$$

$$= a_0 + \frac{1}{[a_1,\ldots, a_m]}$$

$$= [a_0, [a_1,\ldots, a_m]].$$