On the continued fraction representation of computable real numbers

Abstract

The continued fraction representation of real numbers is compared with other types of representations of real numbers in the context of recursive analysis. The main result states that a modification of the natural continued fraction representation, based on the concept of principal convergents of real numbers, is polynomially equivalent to the left cut representation in the sense that, for any given real number x, the two representations of x are computable from each other in polynomial time. Following from earlier studies on the left cut representation of real numbers, this result verifies the intuition that there is no efficient algorithm for implementing addition of real numbers in the continued fraction form. On the other hand, when considering computable real functions, the continued fraction representation behaves differently from the left cut representation: a computable real function must be continuous if it is defined as a mapping on Cauchy sequences of real numbers; it must be left-continuous, but is not necessarily continuous, if defined as a mapping on left cuts; and it ist necessarily left- or right-continuous if defined as a mapping on continued fractions.