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.

We analyze the performance of the unary arithmetical algorithm which computes a Moebius transformation in bimodular number systems which extend the binary signed system. We give statistical evidence that in some of these systems, the algorithm has linear average time complexity.

We analyze the time complexity of exact real arithmetical algorithms in Möbius number systems. Using the methods of Ergodic theory, we associate to any Möbius number system its transaction quotient {\bf T}\ge 1 and show that the norm of the state matrix after n transactions is of the order {\bf T}^n. We argue that the Bimodular Möbius number system introduced in Kůrka has transaction quotient less than 1.2, so that it computes the arithmetical operations faster than any standard positional system.