On the Significance of Digits in Interval Notation
M. H. van Emden
TL;DR
This work analyzes the significance of digits in interval bounds through an information theoretic lens under the set interpretation of interval arithmetic. It shows that factored tail notation can greatly improve readability by reducing repeated decimals with negligible information loss, and it introduces inflation as a tool to trade precision for concise representation. Key insights include the Rule of One Tenth, which describes how information gained per digit decays, and the practical guideline of using two decimals inside interval brackets for a good balance of clarity and fidelity. The findings provide actionable guidance for producing readable interval outputs without compromising the integrity of the represented uncertainty.
Abstract
To analyse the significance of the digits used for interval bounds, we clarify the philosophical presuppositions of various interval notations. We use information theory to determine the information content of the last digit of the numeral used to denote the interval's bounds. This leads to the notion of efficiency of a decimal digit: the actual value as percentage of the maximal value of its information content. By taking this efficiency into account, many presentations of intervals can be made more readable at the expense of negligible loss of information.