It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...) or it has the cardinality of the continuum, the ①-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in ①-based numbers. (shrink)