We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infinite product of matrices, with non-negative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S1, is expressed as the image of the base interval [0, ∞] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first so-called sign matrix determines an interval on which the real number lies.