Let C _sc and C _wc be classes of the semi-computable and weakly computable real numbers, respectively, which are discussed by Weihrauch and Zheng 12. In this paper we show that both C _sc and C _wc are not closed under the total computable real functions of finite length on some closed interval, although such functions map always a semi-computable real numbers to a weakly computable one. On the other hand, their closures under general total computable real functions are the same and are in fact an algebraic field. This field can also be characterized by the limits of computable sequences of rational numbers with some special converging properties.