Continued fraction and classification of real numbers.

Question

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.

Answer 1

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:

$$e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,\ldots]$$

Answer 2

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.