Intervals and (non-)negative numbers

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