Difference between three quantities

TL;DR

This paper explores the concept of difference among three or more quantities, linking it to properties of the Vandermonde determinant and proposing a new understanding of multi-quantity differences.

Contribution

It introduces a novel definition of difference for three or more quantities and connects it to the Vandermonde determinant's properties.

Findings

The difference between three quantities can be decomposed similarly to binary differences.

The proposed difference satisfies a property analogous to the binary case when considering an additional quantity.

This difference is related to a feature of the Vandermonde determinant.

Abstract

The notion of difference between two quantities plays a basic role in mathematics, consequently in all branches of human activity where the mathematics is applied. However the long stand question is: what is the difference between three (or more) quantities? The binary operation [a,b]=(a-b) possesses the following principal feature: with respect to the third quantity (c) this operation is decomposed into a sum of the same operations between (a) and (c), and (c)and (b), i.e., [a,b]=[a,c]+[c,b]. Denote by [a,b,c] difference between three quantities (a,b,c). With respect to additional quantity (d) this definition of the difference has to possess with the following property [a,b,c]=[d,b,c]+[a,d,c]+[a,b,d]. We prove that this property of difference between three (or n>2) quantities is satisfied by one of the features of Vandermonde determinant.