The main subject of the present paper is to define the four algebraic operations - additions, subtraction, multiplication and division in the set of the asymptotic numbers A [7] and to deduce the corresponding formulas for the components of the asymptotic number, representing the result as functions of the components of the arguments. The definitions of the operations, in fact, are introduced as a special case of the more general notion of a quasiclassical function - one special class of functions defined on A. The discussion of the algebraic and some other properties of the asymptotic numbers is put off for a next paper.

The set of asymptotic numbers, introduced by the same authors in [7], is a generalization of the system of real (complex) numbers, comprising infinitely small and infinitely large numbers [1], [2]. The reasons for introducing these numbers are connected with concrete problems of the quantum mechanics [5], [6], [8], but it seems to us that they are also interesting for themselves.

The definition of the asymptotic numbers and some of their properties are reminded in the introductory chapter, by which we achieve logical independence of [7].



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