An algebraic approach to the set of intervals (a new approach of arithmetic of intervals)

TL;DR

This paper introduces a novel algebraic framework for intervals, modeling them as a normed vector space and a four-dimensional algebra, enabling new operations like divisibility and Euclidean division, with applications in differential calculus.

Contribution

It presents a new algebraic approach to interval arithmetic by modeling intervals as a normed vector space and a four-dimensional algebra, facilitating advanced operations.

Findings

Intervals form a normed vector space.

A four-dimensional algebra models interval multiplication.

New notions of divisibility and Euclidean division are introduced.

Abstract

In this paper we present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some cases an euclidian division. We introduce differential calculus and give some applications.