In this contribution is deduced a generalisation of the 2-dimensional complex number system. The construction of a hyperbolic basis is one of the main topics in this paper. By the aid of this basis the authors succeed in a nice description of an \(n\)-dimensional direct product ring of reals. Problems which appear with zero divisors where explained in detail. Elementary functions similar to the ``elliptic'' case were introduced. Finally, analogues to the Cauchy-Riemann equations were studied. The notation of a hyperbolic conformal mapping is given and some examples strike the meaning of these considerations. Basic results were obtained in the paper: \textit{A. Duranona Vedia} and \textit{J. C. Vignaux}: On the theory of functions of a hyperbolic complex variable (in Spain) Univ. Nac. La Plata (Argentina), Publ. Fac. Ci. Fisicomat. Contr. 104, 139-183 (1935; JFM 62.1122.03).