The exact real binary arithmetical algorithm is an on-line algorithm which computes the sum, product or ratio of two real numbers to arbitrary precision. The algorithm works in general Moebius number systems which represent real numbers by infinite products of Moebius transformations.

We consider a number system of binary continued fractions in which this algorithm is computed faster than in the binary signed system. Moreover, the number system of binary continued fractions circumvents the problem of nonredundancy and slow convergence of continued fractions.