Advancements in number representation for high-precision computing

Efficient representation of data is a fundamental prerequisite for addressing computational problems effectively using computers. The continual improvement in methods for representing numbers in computers serves as a critical step in expanding the scope and capabilities of computing systems. In this research, we conduct a comprehensive review of both fundamental and advanced techniques for representing numbers in computers. Additionally, we propose a novel model capable of representing rational numbers with absolute precision, catering to specific high precision applications. Specifically, we adopt fractional positional notation coupled with explicit codification of the periodic parts, thereby accommodating the entire rational number set without any loss of accuracy. We elucidate the properties and hardware representation of this proposed format and provide the results of extensive experiments to demonstrate its expressiveness and minimal codification error when compared to other real number representation formats. This research contributes to the advancement of numerical representation in computer systems, empowering them to handle complex computations with heightened accuracy, making them more reliable and versatile in a wide range of applications.