The main operations of the Fibonacci arithmetic Earlier when we considered the "algorithmic measurement theory" we introduced the concept of the It is very important to use an analogy between the classical binary number system and the However we begin from the simplest Fibonacci representation, which corresponds to the 1-Fibonacci code and uses the classical Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ...,
First of all we will consider a number of the distinctive and very important operations and concepts of the Fibonacci arithmetic. The
- The "convolution": 011 ® 100;
- The "devolution": 100 ® 011.
This means that if we have a possibility to fix some code combination 011 (or 100) in the Fibonacci code representation of natural number The
0111 = 10011010011 = 1010001010 (the "minimal" form).01Here all the "convolutions" carried out in the Fibonacci code representations are underlined. Here the right code representation is the "minimal" form. Its peculiarity consists of the fact that two binary 1's alongside do not meet. If we carry out in some Fibonacci code representation all possible operations of the "devolution" we will come to the "maximal" form. For example,
00 = 0101100 = 0101011 (the "maximal" form).100Here all the "devolutions" carried out in the Fibonacci code representations are underlined. Here the right code representation is the "maximal" form. Its peculiarity consists of the fact that two binary 0's alongside do not meet. Let's remind that the realization of the "convolution" and "devolution" in the Fibonacci code representation, which represents some natural number And now we demonstrate applications of these basic concepts of the Fibonacci arithmetic for realization of the simplest arithmetical operations, the Let's consider the functioning the Fibonacci "summing" counter:
We can see from the example that the functioning of the "summing" counter is reduced to the "convolution". Let's consider now the functioning the Fibonacci "subtracting" counter:
We can see from the example that the functioning of the "subtracting" counter is reduced to the "devolution".
The comparison procedure for the Fibonacci code combinations For example, to compare the Fibonacci code combinations
The comparison of the Note that first the Fibonacci arithmetic is stated in Stakhov's article "The redundancy binary positional number systems" published in the Journal of the Taganrog Radio University "Homogeneous Digital Computers and Integrated Structures" (Taganrog, 1974). Later this arithmetic was described in Stakhov's books "Introduction into Algorithmic Measurement Theory" (1977) and "Codes of Golden Proportion" (1984). And we will tell about Fibonacci's operations of addition, subtraction, multiplication and division at the next pages of our Museum. Follow us! |