Set of double complex numbers as a special extension of the field of complex numbers

Abstract

Aim. Definition of a special algebraic system based on the field of complex numbers with its addition and multiplication operations and their properties, concepts of norm and argument of new numbers, forms of their representation, special space, as an interpretation of this system.

Methodology. Analysis of algebraic number theory, complex and hypercomplex numbers, theory of vector spaces, theory of functions of a complex variable.

Results. On the basis of complex numbers, an algebraic system of a new type is defined, the operations of which are also invertible, as operations on the field of complex numbers, the properties of operations are defined and proved. This system differs from known systems of hypercomplex numbers and is a special extension of the field of complex numbers. A special space is defined that interprets this system, the concepts of the module and argument of new numbers, the form of their presentation. The possibility of solving in a new system of a certain type of equations insoluble on the set of complex numbers has been proven.

Research implications. The new algebraic system is an element of further development of number theory and the theory of vector spaces, expands the direction of research in other areas (solvability of algebraic equations, function theory, etc.). The results are also applied. In particular, the solvability in the new algebra of previously insoluble complex equations can be used in practical problems, the ability to graphically represent functions of a complex variable can be used in mathematical modeling of spatial objects.