It is assumed that the reader is acquainted with the first two parts of the present paper, [1] and [2], in which there was developed informally the theory of a certain class of hypothetical computing devices, the Q-machines. In the present part of the paper we develop a way of describing Q-programs and Q-computations; then, following the theorem in [2], we obtain some further connections between the arithmetical nature of a number x and the computational schémas for generating sequences of rational numbers converging to x; and finally, by means of the usual device of Goedel numbering we begin to examine some properties of programs which operate on programs.