Best k-digit rational approximation of irrational numbers: Pre-computer versus computer era

摘要

The best k-digit rational approximation for a given irrational number, where the numerator has k digits, is presented. Such an approximation allows better error-free computation in the real-world scenario than that permitted by the k-digit decimal approximation of the irrational number. It is pointed out that such a best k-digit rational approximation, though exponential, has been made possible due to current high-speed computation performed by a digital computer. To appreciate the importance of such an approximation, we have also focused on the rational approximation – not always the best one – of some of the famous irrational numbers such as the π and the exponential function e by the ingenuity of super-mathematicians in several parts of the world during the pre-computer era spanning over centuries.