Although arithmetic coders produce optimum compression, corresponding to the entropy of the source data string, when based on the exact probabilities of occurrence of the symbols constituting the data string, in fact prior implementations of arithmetic coding procedures have tended to introduce approximations on account of the difficulty in determining the exact probabilities. Such approximations reduce the efficiency of the arithmetic coding operation and result in an output data string being generated which has more symbols than the theoretical minimum, or entropy. Moreover, further approximations have been introduced in order to eliminate the multiplication operation, which is required for determining the width of each successive sub-interval.