Real numbers are usually represented by finite strings of digits belonging to some digit set. However, finite strings of digits can only represent a limited subset of the real numbers exactly because many real numbers have too many significant digits or are too large or too small. In the literature, there are broadly three frameworks for exact real computer arithmetic: infinite sequences of linear maps, continued fraction expansions and infinite compositions of linear fractional transformations. We introduce here a new, feasible and incremental representation of the extended real numbers, based on the composition of linear fractional transformations with either all non-negative or all non-positive integer coefficients.