On the Significance of Digits in Interval Notation

On the Significance of Digits in Interval Notation We review the various interval notations as starting point for the analysis of the significance of the decimals used for interval bounds. As intervals are used to indicate that a number is known with a certain degree of uncertainty, we also review the rules used in physics to ensure the significance of the decimals used in recording a measurement. It turns out that according to these rules, commonly used interval notation carries too many decimals. To be able to investigate whether rules similar to those used in physics can be used to improve interval notation, we use information theory to determine the information content of the last decimals of the numerals used to denote the interval's bounds. We introduce the Law of One Tenth, stating that in many situations this content diminishes by a factor of ten on average for successive decimals. The law is especially useful in conjunction with a novel and little-used way of writing intervals that we call “factored notation.” The law implies that it is usually futile to write more than two or three decimals inside the brackets of factored notation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

On the Significance of Digits in Interval Notation

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2004 by Kluwer Academic Publishers
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/B:REOM.0000003996.60728.4e
Publisher site
See Article on Publisher Site

Abstract

We review the various interval notations as starting point for the analysis of the significance of the decimals used for interval bounds. As intervals are used to indicate that a number is known with a certain degree of uncertainty, we also review the rules used in physics to ensure the significance of the decimals used in recording a measurement. It turns out that according to these rules, commonly used interval notation carries too many decimals. To be able to investigate whether rules similar to those used in physics can be used to improve interval notation, we use information theory to determine the information content of the last decimals of the numerals used to denote the interval's bounds. We introduce the Law of One Tenth, stating that in many situations this content diminishes by a factor of ten on average for successive decimals. The law is especially useful in conjunction with a novel and little-used way of writing intervals that we call “factored notation.” The law implies that it is usually futile to write more than two or three decimals inside the brackets of factored notation.

Journal

Reliable ComputingSpringer Journals

Published: Oct 18, 2004

References

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