Two complement representations and a sign-magnitude one are introduced which allow for handling negative numbers using only binary coefficients in Fibonacci base expansions. These are developed for practical implementation in Fibonacci computers.

background

Complement representations in the Fibonacci computer

Fibonacci p-codes and Codes of the “Golden” p-proportions: New Informational and Arithmetical Foundations of Computer Science and Digital Metrology for Mission-Critical Applications

For computers whose main components are on a single chip, as microprocessors, the errors, instead of being randomly scattered, are more apt to be "localized" in the form of bursts in the chip Burst-errors, also known as non-uniform errors, or errors to consecutive digits, are becoming an important and practical problem in these new computers and their resulting arithmetic coding theory With technology moving toward making integrated circuit (IC) chips of groups of bits (the bytes) the building block units, burst-error-correcting codes are useful indeed
In order to produce better burst-error-correcting codes, it is felt that not only the present day hardware needs to be improved but also a better number base is needed to be developed in which the computer does its calculation, as well as a superior coding theory
The Fibonacci numbers are proposed as a new number base system for computers, which could lead to more error-free operations The motivation to use these numbers comes from a property of considerable importance: the existence of a non-adjacent-form Fibonacci representation of integers
This study is divided into four major parts: (1) section one formulates the Fibonacci number system (2) section two shows the implementation of the Fibonacci Computer's Arithmetic Unit using existing binary logic (3) section three studies the two basic Fibonacci arithmetic codes: the Fibonacci AN codes and the Fibonacci biresidue codes (4) section four gives the theory of the Fibonacci burst-error-correcting AN codes
It is the intention of this study to demonstrate that the Fibonacci computer's architecture produces more superior arithmetic burst-error-correcting codes than their binary system's counterparts This "burst-error-correcting" property, not seen in the binary system, makes Fibonacci computer theoretically interesting as well as practically important

Complement representations in the Fibonacci computer

Citations

Fibonacci p-codes and Codes of the “Golden” p-proportions: New Informational and Arithmetical Foundations of Computer Science and Digital Metrology for Mission-Critical Applications

A class of arithmetic burst-error-correcting codes for the fibonacci computer

References

Note on Complete Sequences of Integers

A class of arithmetic burst-error-correcting codes for the fibonacci computer

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