TOENS: A Practical Approach to Handling Uncertainty in Numerical Computations

Yueshui Lin

panzhihua university; linyueshui@pzhu.edu.cn

Abstract

This paper introduces the Third-Order Exact Number System (TOENS), a practical framework for working with numbers in situations where uncertainty is inevitable. Unlike traditional number systems that pretend values are exact, TOENS openly acknowledges and quantifies uncertainty through two key components: type indicators (like ·, *, ~, ?) and precision levels (0-4095). This approach helps in applications ranging from quantum computing to structural monitoring, where we need to understand how errors accumulate.  Our tests show TOENS can significantly improve accuracy while being computationally efficient. The core insight is that in real-world problems, it’s often more valuable to

understand the limits of our knowledge than to pretend we have perfect precision.

Keywords: uncertainty quantification; numerical precision; error propagation; quantum computing;

on or fluid dynamics, tiny differences in starting conditions can lead to completely different outcomes.