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Difference between three quantities

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.

where θ is a primitive root of polynomial x 3 − 1. In Refs. [5], [4], this formula of difference have been used in order to formulate a notion of ternary commutator. Apparently, this definition belong to the field of complex numbers and possesses with the following feature In the present paper we suggest a definition of the notion of the difference between three and more quantities making use of a feature of the Vandermonde determinant. Denote by [a, b, c] difference between three quantities a, b, c. With respect to additional quantity d this definition of the difference is decomposed as follows All quantities belong to the field of real numbers. 1 Difference between three quantities Let us start with the fraction of type 1 where x i , i = 1, 2, 3 are roots of the cubic polynomial The following expansion for that fraction holds true where by V we denoted the Vandermonde's determinant of matrix of order (3 × 3) Collecting the terms in the right-hand side of (1.3) we obtain From this equality we come to the conclusion, that This equation displays an interesting feature of Vandermonde's determinant, which to our knowledge still has not been revealed [6]. We suggest to use formula (1.6) as a formula of difference between three amounts. The interval of this formal distance is bounded by three points x 1 , x 2 , x 3 and it is divided into three parts by using only one point, x, among them. Thus, the following formula is an analogue of the interval (x 1 − x 2 ), and the property given by formula (1.6) is an analogue of the following property of the interval between two points The Proof of formula (1.6).
Formula (1.6) is a consequence of one of the properties of Vandermonde's determinant. We have to prove that Consider the following matrix and calculate determinant of this matrix in two ways. Firstly, let us calculate the determinant on making use of the method of expansion with respect to lines of the matrix. In this way we find that The last determinant is equal to zero. In this way we get Secondly, let us expand the determinant with respect to columns. For that purpose it is convenient to use the following notation of the Vandermonde determinant: In this notation Then, the following expansion holds true Notice, the first term is the Vandermond's determinant. Therefore, Hence, End of proof.
2 Difference between n ≥ 2 quantities Since we have found the concept of difference between three quantities this generalization to the case of n ≥ 3 quantities is straightforward. Consider n-th order Vandermonde's matrix (2.1) The determinant of this matrix is given by well-known Vandermonde's formula: Consider the following auxiliary matrix Now, let us prove that the determinant of this matrix equal to 2V . Firstly, expand this determinant with respect to l-th line. The result is given by sum of two determinants Det(AV (x)) = Det(AV 1 (x, x l k )) + Det(AV 2 (x, x l )), (2.4) where we denoted and, The determinant of the second matrix is trivial because there l-th line is proportional to the first one. Continue to expand the first determinant AV 1 (x, x l ) with respect to other lines. At the final step of this process the determinant of the auxiliary matrix is reduced to the following form which obviously equal to 2V , Det(AV (x)) = 2V.
(2.6) Now let us calculate the determinant Det(AV (x)) by expanding with respect to columns. For the sake of convenience denote the Vandermonde's determinant (2.1) as follows (2.7) Correspondingly, the determinant of the auxiliary matrix will be written in the form The expansion process with resect to columns of this determinant is worked out as follows. [ The second term in right-hand side is equal to Continue to expand the first of the sum, The last term is equal to zero. The first one is represented as follows At the final step of this process we come to the following equation This formula implies one of the important features of the Vandermond's determinant. That is the formula which we suggest to use as a definition of the difference between n quantities.
Ternary algebraic operations and cubic relations have been considered, although quite sporadically, by several authors already in the XIX-th century, e.g. by A. Cayley ([7]) and J.J. Sylvester ( [8]. The development of Cayley's ideas, which contained a cubic generalization of matrices and their determinants, can be found in a recent book by M. Kapranov, I.M. Gelfand and A. Zelevinskii ( [9]). A discussion of the next step in generality, the so called n − ary algebras, can be found in ( [10]).
The difference between two quantities has direct geometrical interpretation as a distance between two points on a straight line. Let O, A, B  This definition of the difference we suggest use in geometry in order to found a concept of ternary distance between three points. A generalization to the case of n ≥ 3 points is straightforward.