Best $k$-digit rational bounds for irrational numbers: Pre- and super-computer era

Abstract

We present here the best $k$-digit rational bounds for a given irrational number, where the numerator has $k$ digits. Of the two bounds, either the upper bound or the lower bound, will be the best $k$-digit rational approximation for the given irrational number. The rational bounds derived from the corresponding $k$-digit decimal bounds are not often the best rational bounds for an irrational number. Such bounds not only allow a possible introduction of irrational numbers such as $\pi$, e, and log_e2 but also to compute error-bounds in an error-free computational problem. We have also focused on the importance of twenty-first century supercomputers with steadily increasing computing power–both sequential and parallel–in computing the best bounds as well as in determining error-bounds for a problem in error-free computational environment. We have also focused on the tremendous activities during/after pre-historic era on obtaining rational approximation/bounds of famous irrational numbers to justify the relevance and possible importance of this study in the current ultra-high speed computing age.