The Normed Algebra of Binary Numbers

The richness of the theory of functions of a complex variable, the effectiveness of its methods have always served as a stimulus and a source of ideas when constructing a theory of the function of a hypercomplex variable. It should be noted that hypercomplex number systems are an extension of the field of complex numbers. Modern hypercomplex studies can be divided into algebraic and analytical; the latter are often called hypercomplex analysis in the broad sense. Hypercomplex systems are an effective tool in mathematical modeling that allows representing complex multidimensional data and operations on them in a more compact form. In this paper, a new system of hypercomplex numbers is constructed by establishing an isomorphism to a specific matrix algebra. The matrix algebra is studied using classical methods of matrix theory and endowed with a norm, enabling the construction of analysis elements within it using matrix analysis techniques. The obtained results for matrices are transferred to elements of the isomorphic algebra of finite rank, namely, hypercomplex numbers called «binary numbers». This allows endowing the algebra of binary numbers with a topological structure and laying the foundations for analysis within it, including the construction of functions of a binary variable. The article introduces the concept of convergent sequences of binary numbers, and through them, convergent binary series. The novelty of our approach is that the set of hypercomplex numbers is considered as an algebra of rank 2 over the field R and through the found matrix representation, the n-th power of a binary number, and therefore the sum of certain power series, is determined in algebraic form. Functions of a binary variable are defined by sums of corresponding power series. And as a result, a general formula for constructing elementary functions of a binary variable is derived. Hypercomplex numbers are already used to describe quantum states, model dynamical systems, in computer graphics and cryptography. And the further development of hypercomplex analysis and the construction of new numerical systems leads to the expansion of the range of their applications in modeling real states and processes.