On the optimality of ternary arithmetic for compactness and hardware design

TL;DR

This paper rigorously analyzes the efficiency of ternary arithmetic, demonstrating mathematically that base 'e' is optimal and ternary is the closest integer, thus supporting its use for compact and cost-effective hardware design.

Contribution

It provides a formal mathematical proof that the natural logarithm base 'e' is optimal for numeric encoding, with ternary being the best integer approximation for hardware efficiency.

Findings

e is proven as the optimal radix for numeric systems

Ternary arithmetic closely approximates the optimal base

Supports ternary as efficient for hardware implementation

Abstract

In this paper, the optimality of ternary arithmetic is investigated under strict mathematical formulation. The arithmetic systems are presented in generic form, as the means to encode numeric values, and the choice of radix is asserted as the main parameter to assess the efficiency of the representation, in terms of information compactness and estimated implementation cost in hardware. Using proper formulations for the optimization task, the universal constant 'e' (base of natural logarithms) is proven as the most efficient radix and ternary is asserted as the closest integer choice.