Continued fraction and classification of real numbers. I would be grateful if anyone can tell if there are any methods to classify real numbers using continued fraction. eg: Suppose $[a_0;a_1,a_2,\ldots,a_n]$  is the representation of some real number $\alpha$ then
1) if $n$ is finite then it represents a rational number.
2) if $n$ is infinite then it is an irrational number.
3) if $\alpha$ is a quadratic irrational then the pattern repeats indefinitely with a period of some number $t$. What we can say about the converse?
4) for 2) case  how we can distinguish between algebraic and transcendental numbers?
Thanks for your time.
 A: If $n$ is finite:
$n\in\mathbb{Q}$
This is because a finite continued fraction is the result of a finite amount of additions and divisions of integers.
If $n$ is infinite and periodic
$n$ is an irrational root of $ax^2+bx+c=0$ where $a,b,c\in\mathbb{Q}$
If $n$ is infinite and non-periodic
$n$ is not expressible as a root of a second degree or lower polynomial.
Algebraic vs transcendental do not show any differences as far as I figured out.
Transcendental numbers can show patterns, for example:
$$e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,\ldots]$$
A: I believe there is no presently known "reasonably explicit" description (i.e. an $n$-term recursion formulation for some positive integer $n)$ of continued fraction expansions that can be used to characterize even just the cubic irrationalities, although a recent paper by Nadir Murru---On the Hermite problem for cubic irrationalities---appears to give such a description for cubic irrationalities when multidimensional continued fraction expansions are used.
I've known about this for maybe a year, but I have not yet looked into what this all means, so hopefully an expert can provide more details and correct any inaccuracies I may have made.
