Intervals and (non-)negative numbers

Abstract

Intervals have a double nature: they can be considered as compact sets ofВ real numbers (set-intervals) or as approximate numbers. A set-interval isВ presented as an ordered pair of two real numbers (interval end-points), whereasВ an approximate number is an ordered pairВ consisting of a real ``exact'' number and a nonnegative error bound.В Thus, differently to the case withВ set-intervals, where both endpoints are real numbers, when operating with approximate numbers,В one should know the algebraic properties of the arithmetic operationsВ over error bounds, that is over nonnegative numbers. ThisВ work is devoted to the algebraic study ofВ the arithmetic operations addition and multiplication by scalars for approximate numbers, resp. forВ errors bounds. Such a setting leads to so-called quasilinear spaces. We formulate andВ prove several new properties of such spaces, which areВ important from computational aspect. In particular, we focus our study on theВ operation ``distance between two nonnegative numbers''.В We show that this operation plays an important role in the study of theВ concept of linear independence of interval vectors, the latter being correctly defined.