The Importance of the Golden Number for Mathematics and Computer Science: Exploration of the Bergman’s System and the Stakhov's Ternary Mirror-symmetrical System (Numeral Systems with Irrational Bases)

The main goal of the article is to explore two unusual numeral systems, which alter radically our ideas on the positional numeral systems. We are talking on the numeral systems with irrational bases. The first of them is the binary (0,1) numeral system with the irrational base 152+Φ =(the golden ratio), proposed in 1957 by the 12-year American mathematician George Bergman, the second is the ternary mirror-symmetrical numeral system with the base 2352+Φ =, proposed by the author of the present article and published in 2002 in The Computer Journal (British Computer Society). Bergman’s system is the newest mathematical discovery in number theory and the greatest modern mathematical discovery in the field of positional numeral systems after Babylonian numeral system with the base 60, decimal and binary systems. Bergman’s system can be considered as a new definition of real numbers and is a source of new unusual properties of natural numbers. Bergman’s system generates the ternary mirror-symmetrical numeral system, having unique mathematical property of mirror symmetry, which can be used for effective detection of errors in all arithmetical operations. These numeral systems alter our ideasabout positional numeral systems and can affect on future development of mathematics and computer science. The ternary mirror-symmetrical numeral system is possibly the final stage in the long historical development of the concept of ternary numeral systems, because in the ternary mirror-symmetrical numeral system two scientific problems, the sign problem and representation of negative numbers and problem of error detection, based on the principle of mirror symmetry, are solving simultaneously. The famous American mathematician and expert in computer science Donald Knut evaluated highly the ternary mirror-symmetrical numeral system. The author is ready to offer consulting services for any electronic company with advanced technology, which can be interested in the technical implementation of the ternary mirror-symmetrical processors and computers on this basis.