Continuous fractions, complexity of rational nubmbers and languages with infinite alphabet

In this paper continuous fractions are treated as description language of rational numbers. Alphabet of the language is natural numbers $N$ — infinite set. So standard definition of complexity (a la A. N. Kolmogorov) is not applicable. New approach is needed. Here you find one natural way to define complexity in the case of infinite countable languages. Besides, some known results about continuous fractions and some news, drawing from this new approach are stated. There are new tasks related with the complexity distributions and efficiency of a description language.