The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic

We consider two important generalizations of the golden proportion: golden p-proportions [Stakhov AP. Introduction into algorithmic measurement theory. Soviet Radio, Moscow, 1977 [in Russian]] and “metallic means” [Spinadel VW. La familia de números metálicos en Diseño. Primer Seminario Nacional de Gráfica Digital, Sesión de Morfología y Matemática, FADU, UBA, 11-13 Junio de 1997, vol. II, ISBN 950-25-0424-9 [in Spanish]; Spinadel VW. New smarandache sequences. In: Proceedings of the first international conference on smarandache type notions in number theory, 21-24 August 1997. Lupton: American Research Press; 1997, p. 81-116. ISBN 1-879585-58-8]. We develop a constructive approach to the theory of real numbers that is based on the number systems with irrational radices (Bergman's number system and Stakhov's codes of the golden p-proportions). It follows from this approach ternary mirror-symmetrical arithmetic that is the basis of new computer projects.