Should the set of irrational numbers be divided into further subsets? The real numbers are divided into two subcategories; the rational numbers, those that can be written as a ratio, and the irrational ones, those that can't.
But we know, that of the irrational numbers, many, although they cannot be written as a ratio or fraction, can still be written as a continued fraction. So my question is: Do there exist, or should there exist, further subdivisions of irrational numbers, into those that can be written as a regular continued fraction (whose nominators and denominators follow a regular pattern as opposed to being random), and those that can't? By "existing", I refer to the set having a specific letter assigned to it, and being officially recognized as a set of numbers.
 A: Mahler's classification of irrationals is based on how well the irrational can be approximated by rationals, and this is very closely related to how fast the partial quotients grow in the simple continued fraction for the irrational. Type "Mahler's classification" into the web for links to papers where the details are given. 
Wikipedia is a good place to start. 
A: There are many further divisions and Gerry mentions some.  However, one very common further subdivision is into algebraic and transcendental.
Algebraic numbers are roots of a polynomial with rational or integer coefficients so they include $\sqrt 2$ but also some numbers which cannot be written with radicals.  
Transcendental numbers are those which are not algebraic.  Famous examples are $e$ and $\pi$.  Once a number has been proved to be irrational, it is common to ask whether it is transcendental.  Usually, proving this is much harder.  
The algebraic numbers are sometimes denoted by $\mathbb{A}$.
The algebraic numbers are countable so, since the transcendental numbers include all of the other real numbers, they must be uncountable.  So in a sense, most irrational numbers are transcendental.
Algebraic may be interpreted as including complex numbers; you could say real algebraic if you wish to exclude those. Actually, you may need to specify algebraic irrational as rational numbers are the roots of polynomials with rational coefficients (rather trivially).
