Diophantine approximation deals with approximations of real numbers by rational numbers. The classical finite-difference approximations for numerical differentiation are ill-conditioned. The simplest method is to use finite difference approximations. The EMU operates by calculating polynomial approximations of these functions. This motivates the study and application of numerical approximations of integrals. This requires more choices, such as numerical approximations or the use of heuristics. Not all mathematical advances relating to were aimed at increasing the accuracy of approximations. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. In fact could be computed from these rational approximations. There are also techniques for iteratively improving linear approximations (Matsui 1994). Thus there is no choice but to use the linearized approximations. The method used is differential reinforcement of successive approximations. There are many other smooth, analytic approximations to the step function. Mesh cells are used as discrete local approximations of the larger domain. In some cases large sample approximations for the variance are used.