Processing of the residuals of numbers in real and complex numerical domains

The chapter discusses the procedures for the formation and use of real residuals of real numbers on a real module, as well as complex and real residues of an integer complex number on a complex module. The chapter focuses on the processing of complex and real residuals of an integer complex number by a complex module. This procedure is based on using the results of the first fundamental Gauss theorem. The chapter of the proposed procedure provides examples of determining deductions in a complex numerical domain. On the basis of the considered procedure, an algorithm was developed for determining the real deduction of an integral complex number using a complex module in accordance with which the device was synthesized for its technical implementation. The device received a patent of Ukraine for the invention, which confirms the novelty and practical value of research results. The results obtained in the chapter are advisable to be used when implementing tasks and algorithms in real and complex numerical domains. In particular, the use of real numbers for cryptographic applications was considered.