Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which Fowler calls anthyphairesis. Theaetetus is said to develop a classification of quadratic irrationals in this setting. Given positive $x$ and $y$, with $x/y$ being a square root of a rational number, $x+y$ is called a binomial, $x-y$ an apotome (if $x>y$), and $\sqrt{xy}$ a medial (if it is irrational). There are some interesting results described in book X of Euclid's Elements: the classes are mutually exclusive, for any binomial or apotome the split into $x$ and $y$ is unique, etc. Further classes are considered, apotome of a medial, binomial of a medial, and so on.
Theaetetus seems to deal with positive elements of $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots)$, but I can not think of any algebraic or number theoretic classification scheme that would produce his classes. Are they related to something in modern algebra?