"Fibonacci Arithmetic, Coding and Computers"
For the students who learn the computer theory and design the course "Fibonacci Arithmetic, Coding and Computers" is rather useful and interesting.
Program of the course:
- Number systems. Basic concepts and definitions. Positional and non-positional notations. A little of a history. Radix of notation. Requirements to notations. Canonical positional notations. Binary notations and binary arithmetic. Special binary notations. Conversion algorithms of numbers from one notation in other. Optimal radix for notations. Symmetrical notations. Ternary symmetrical notation and ternary arithmetic. Nega-positional notations and arithmetics. Notations with the complex radix. Factorial notation. Notation in residual classes.
- Golden Section and Fibonacci Numbers. Golden Section problem. Algebraic and geometric properties of the golden section. Fibonacci and Lucas numbers. Generalized Fionacci numbers. Generalized golden sections. Fibonacci Q-matrix. Generalized Fibonacci matrices.
- Algorithmic measurement theory. Fibonacci's "weighing problem". Asymmetry principle of measurement. Classical measurement algorithms. "Binary" measurement algorithm and its generalization. Measurement algorithms based on Pascal Triangle. Fibonacci measurement algorithms.
- Fibonacci codes and arithmetic. Fibonacci representation. Zeckendorf representation. Redundancy of Fibonacci codes. Fibonacci addition and subtraction. Fibonacci multiplication and division. Basic micro operations. Noise-tolerant Fibonacci arithmetic. Concept of Fibonacci computer. Fibonacci self-synchronization codes.
- Bergman's notation. Mathematical properties of the "Tau-system". Multiplicity of number representation. Z-property of natural numbers. F- è L-codes. "Golden" arithmetic. Notations based on generalized golden proportions. New definition of a number.
- Ternary mirror-symmetrical notation. Number conversion from the "Tau-system to the ternary "golden" representation. Ternary F- and L-codes. Property of "mirror symmetry". Representation of negative numbers. Range of number representation. Redundancy of the ternary mirror-symmetrical representation. Ternary mirror-symmetrical addition and subtraction. "Swing"-phenomenon. Ternary mirror-symmetrical multiplication and division. Mirror-symmetrical representation with the floating point. Ternary flip-flap-flop. Ternary mirror-symmetrical adder.
- "Golden" analog-to-digit and digit-to-analog converters. Resistive divisor for the "Tau-system". Resistive divisor for the ternary mirror-symmetrical number system. Digit-to-analog converters for number systems with irrational radices. Self-correcting analog-to-digit converters.
- Coding theory based on Fibonacci matrices. Coding based on the Q-matrix. Error detection. Basic check correlation for Fibonacci coding. Error correction. Fibonacci coding based on the generalized Fibonacci matrices. Application to cryptography.