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.