This paper will explore the concept of Arithmetic Dimensionality, D_A, for real numbers, whose analog is the fractal dimensionality of geometric constructs, D_G. Beginning with a short review of how the dimension of a geometric object is calculated, it will be shown how this same concept can be applied to real numbers, and that the current understanding of numbers having a geometric dimension of 0 is not accurate. Instead it is show,, that although the point on a number line, like the x-axis is 0 dimensional, the numbers themselves may possess a fractal dimension. It is beyond to scope of this paper to prove the exact Geometric Fractal Dimension of transcendental numbers (like π and e) or irrational numbers (like √2). The goal however is to show (1) there exists a D_A and D_G for real numbers, (2) that D_A→D_G as the number of decimal digits considered n→∞ and (3) where D is a generic stand in for both D_A and D_G, it exists in Information Space or Sub-Hilbert Space of Geometric Dimension such that 0≤D<1.