Bergman's number system

However at the dawn of the computer era the new original discovery in the number system theory was made. This one was directed on the overcoming the second essential lack of the binary number system, the problem of the "zero" redundancy. In 1957 the American mathematician George Bergman introduced the positional number system of the following kind:


where A is some real number and ai is the i-th digit binary numeral, 0 or 1, i = 0, ±1, ±2, ±3); ti is the weight of the i-th digit, t is the base of the number system (1).

At the first glance, there dose not exist any difference between the formulas (1) and the formulas for the traditional positional number representation but it is only at the first glance. The principal distinction between the number system (1) and the traditional positional number systems consists of the fact that Bergman used the irrational number called the "golden ratio" as the base of the number system (1). That is why Bergman called it the "number system with an irrational base" or "Tau System". Although Bergman's paper contained the result of great importance for the number system theory, however in that period it simply was not noted either by mathematicians or engineers. And Bergman assessed his mathematical discovery very modestly:

"I do not know of any useful application for systems such as this, except as a mental exercise and pastime, though it may be of some service in algebraic number theory".

However the development of computers cut across Bergman's pessimistic opinion about practical application of his number system. In contrast to the classical binary number system Bergman's number system possesses the "natural" redundancy, which can be used effectively for computer control. In the 70th and 80th of the 20th century the scientific and engineering developments based on the redundant Bergman's number system were realized in the former Soviet Union. These developments showed exceptional effectiveness of Bergman's number system for the design of the self-correcting analog-to-digit converters (ADC) and the noise-tolerant processors. At the preceding pages of our Museum we considered so-called Z-property of natural numbers, which arise at their representation in Bergman's number system. This property and others ("minimal" and "maximal" forms etc.) can be used for control of computer information on all levels of its organization.

There arises a question: could one design the new number system joining the advantages of Bergman's number system with the advantages of the ternary symmetrical number system? Just this question became the topic of Prof. Stakhov's investigations in the "Libyan period" of his life. We will tell at the next pages of our Museum what resulted from this. Follow us!