# On the Arithmetic of Errors

• Volume: 4, Issue: 4, page 447-462
• ISSN: 1312-6555

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## Abstract

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An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008 and NATO project ICS.EAP.CLG 983334.

## How to cite

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Markov, Svetoslav, and Hayes, Nathan. "On the Arithmetic of Errors." Serdica Journal of Computing 4.4 (2010): 447-462. <http://eudml.org/doc/11399>.

@article{Markov2010,
abstract = {An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008 and NATO project ICS.EAP.CLG 983334.},
author = {Markov, Svetoslav, Hayes, Nathan},
journal = {Serdica Journal of Computing},
keywords = {Computer Arithmetic; Error Analysis; Interval Arithmetic; Approximate Numbers; Algebra of Errors; Quasilinear Spaces; inner subtraction; inner addition; interval analysis; generalized Hukuhara difference; fuzzy set theory; computer arithmetic; error analysis; interval arithmetic; approximate numbers; algebra of errors; quasilinear spaces},
language = {eng},
number = {4},
pages = {447-462},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On the Arithmetic of Errors},
url = {http://eudml.org/doc/11399},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Markov, Svetoslav
AU - Hayes, Nathan
TI - On the Arithmetic of Errors
JO - Serdica Journal of Computing
PY - 2010
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 4
IS - 4
SP - 447
EP - 462
AB - An approximate number is an ordered pair consisting of a (real) number and an error bound, briefly error, which is a (real) non-negative number. To compute with approximate numbers the arithmetic operations on errors should be well-known. To model computations with errors one should suitably define and study arithmetic operations and order relations over the set of non-negative numbers. In this work we discuss the algebraic properties of non-negative numbers starting from familiar properties of real numbers. We focus on certain operations of errors which seem not to have been sufficiently studied algebraically. In this work we restrict ourselves to arithmetic operations for errors related to addition and multiplication by scalars. We pay special attention to subtractability-like properties of errors and the induced “distance-like” operation. This operation is implicitly used under different names in several contemporary fields of applied mathematics (inner subtraction and inner addition in interval analysis, generalized Hukuhara difference in fuzzy set theory, etc.) Here we present some new results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008 and NATO project ICS.EAP.CLG 983334.
LA - eng
KW - Computer Arithmetic; Error Analysis; Interval Arithmetic; Approximate Numbers; Algebra of Errors; Quasilinear Spaces; inner subtraction; inner addition; interval analysis; generalized Hukuhara difference; fuzzy set theory; computer arithmetic; error analysis; interval arithmetic; approximate numbers; algebra of errors; quasilinear spaces
UR - http://eudml.org/doc/11399
ER -

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