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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 appli cations 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 computation ally 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.