# When Is the Product of Intervals Also an Interval?

When Is the Product of Intervals Also an Interval? Interval arithmetic is based on the fact that for intervals on the real line, the element-wise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is an element-wise product or a sum of two intervals always an interval? In this paper, we analyze this problem in a general algebraic setting: we need the corresponding algebraic structures to have (related) addition, multiplication, and order; thus, we consider (consistently) ordered rings. We describe all consistently ordered rings for which the element-wise product and sum of two intervals are always intervals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# When Is the Product of Intervals Also an Interval?

, Volume 4 (2) – Oct 6, 2004
12 pages

/lp/springer_journal/when-is-the-product-of-intervals-also-an-interval-4F0DWVxYUS
Publisher
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/A:1009937210234
Publisher site
See Article on Publisher Site

### Abstract

Interval arithmetic is based on the fact that for intervals on the real line, the element-wise product of two intervals is also an interval. This property is not always true: e.g., it is not true if we consider intervals on the set of integers instead of intervals on the set of real numbers. When is an element-wise product or a sum of two intervals always an interval? In this paper, we analyze this problem in a general algebraic setting: we need the corresponding algebraic structures to have (related) addition, multiplication, and order; thus, we consider (consistently) ordered rings. We describe all consistently ordered rings for which the element-wise product and sum of two intervals are always intervals.

### Journal

Reliable ComputingSpringer Journals

Published: Oct 6, 2004

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