Hypercomplex Numbers

Key words: complex numbers, triples, quaternions, multiplication of generalized complex numbers, torus, algorithm The article "Hypercomplex Numbers" is devoted to the study of the generalization of complex numbers. First, complex numbers are generalized to quaternions, and then they are generalized to hypercomplex numbers of any length (n+1). Here, the definition of the product of these numbers is of great importance, for which it is necessary to define all products of the form i_k i_m, where k and m independently take natural values from 1 to n, and the values of those products depend on the cases k<m and k>m, which are thoroughly considered in the work. The concepts of right and left quotients are distinguished. A general algorithm has been developed for defining a similar product of hypercomplex numbers of any length. To implement this algorithm, a program has been written in the C++ programming language, which obtains a table of products i_k i_m for hypercomplex numbers of length n≥3. An algorithm for quickly filling the table of products has been proposed, which is based on the principle of considering the table as a grid and applying the circular "modulo n" operation along the diagonals of the table. Hypercomplex numbers can find applications in coding systems. They are used in geometry, physics, Finsler spaces, fractals, etc. [3].