In this paper, we establish a number system in $R^n$ which arises from a Haar wavelet basis in connection with decompositions of certain Cuntz algebra representations on $L^2$( $R^n$). Number systems in $R^n$ are also of independent interest [9]. We study radix-representations of $chi$ $in$ $R^n$: $chi$:$alpha$$_{ι}$ $alpha$$_{ι-1}$…$alpha$$_1$$alpha$$_{0}$ ㆍ$alpha$$_{-1}$ $alpha$$_{-2}$ … as $chi$= $M^{ι}$$alpha$$_{ι}$ $alpha$＋…M$alpha$$_1$＋$alpha$$_{0}$ ＋ $M^{-1}$ $alpha$$_{-1}$ ＋ $M^{-2}$ $alpha$$_{-2}$ ＋… where each $alpha$$_{k}$ $in$ D, and D is some specified digit set. Our analysis uses iteration techniques of a number-theoretic flavor. The view-point is a dual one which we term fractals in the large vs. fractals in the small,illustrating the number theory of integral lattice points vs. fractions.s vs. fractions.