A fractional weighted number system, based on Hensel’sp-adic number system, is proposed for constructing a unique code (called Hensel’s code) for rational numbers in a certain range. In this system, every rational number has an exact representation. The four basic arithmetic algorithms that use the code for the rational operands, proceed in one direction, giving rise to an exact result having the same code-wordlength as the two operands. In particular, the division algorithm is deterministic (free from trial and error). As a result, arithmetic can be carried out exactly and much faster, using the same hardware meant forp-ary systems.

Finite segmentp-adic number systems with applications to exact computation

Some rational as well as some irrational numbers, among all real numbers in mathematics, are very special and have fascinated many human minds. Associated with these numbers are not only the fascinating history but also remarkable physical phenomena observed by critical minds of scientists, artists, architects, engineers, naturalists and spiritualists. The rational number 2^n and the irrational number @p- a transcendental number, for example, have very special places in computer science and in mathematics, respectively. Some of the other famous numbers are the Hilbert number 2^2~2.66514414269023, the Liouville number ~0.1100010000000000000000010000 which has a 1 in the 1st, 2nd, 6th, 24th, 120th etc. places and 0s elsewhere, the Euler-Mascheroni constant @c=lim"n"->"~(@?"k"="1^n1k-lnn)~0.57721566490153, and the numbers i^i=e^-^@p^/^2~0.207879576350762,@p^e~22.4591577183611 (believed (not proved) to be a transcendental number) and e^@p~23.1406926327793. Presented here is yet another exceedingly delightful, extensively explored irrational algebraic number (1+5)/2~1.61803398874989 called the golden ratio @f and its widespread occurrence in mathematics, specifically geometry, computational science, biology, artistic creations, architecture, nature and beyond. Specifically, digits-even randomly or systematically chosen consecutive digits or consecutive blocks of digits-of golden ratio may be used as a source of uniformly distributed random numbers. Unlike any of the several quasi- and pseudo-random number generators using various methods, we need to use no method here; only we have to pick up the consecutive/nonconsecutive blocks of digits from the stored golden ratio and hence it would be a fastest means of obtaining random numbers. This idea of getting random sequences possibly opens up a new efficient way of solving numerous optimization problems including the NP-hard travelling salesman problem by polynomial-time heuristics such as ant system approaches, genetic algorithms, simulated annealing and other randomized algorithms. Also, whether these random numbers sieved out of the golden ratio are quasi- (more uniformly distributed) or pseudo-random numbers may be studied including its scope among other random number generators. Presented here is the golden ratio along with its computation up to a desired number of digits using the single Matlab command vpa. Also described are its occurrences in sciences in very many ways and a fixed-point iteration scheme besides other methods for its computation. Demonstrated are the uniform pseudo-random distribution of its digits and its capability to perform the Monte Carlo integrations using systematically its consecutive blocks of digits. Mentioned are some of the interesting happenings/occurrences in nature, art and architecture in which the golden ratio has been discovered in an exact/approximate form. The article is our way of viewing this amazing number, the golden ratio, and depicting its beauty. Included are several Matlab programs for the reader with Matlab facilities. These will enable him/her to have a deeper insight into its character in the background of our aesthetic sense and its extraordinary tendency to pop up in diverse situations through quick computation.

Golden ratio in science, as random sequence source, its computation and beyond

Negabinary is a component of the positional number system. A complete set of negabinary arithmetic operations are presented, including the basic addition/subtraction logic, the two-step carry-free addition/subtraction algorithm based on negabinary signed-digit (NSD) representation, parallel multiplication, and the fast conversion from NSD to the normal negabinary in the carry-look-ahead mode. All the arithmetic operations can be performed with binary logic. By programming the binary reference bits, addition and subtraction can be realized in parallel with the same binary logic functions. This offers a technique to perform space-variant arithmetic-logic functions with space-invariant instructions. Multiplication can be performed in the tree structure and it is simpler than the modified signed-digit (MSD) counterpart. The parallelism of the algorithms is very suitable for optical implementation. Correspondingly, a general-purpose optical logic system using an electron trapping device is suggested. Various complex logic functions can be performed by programming the illumination of the data arrays without additional temporal latency of the intermediate results. The system can be compact. These properties make the proposed negabinary arithmetic-logic system a strong candidate for future applications in digital optical computing with the development of smart pixel arrays. (C) 1999 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(99)00803-X].

Parallel optical negabinary signed-digit computing: algorithm and optical implementation

In the digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.

Negative base encoding in optical linear algebra processors

Algorithms are described for the basic arithmetic operations in a negative base. These algorithms are simpler, faster, and more general than those proposed by Sankar et al.

Arithmetic Algorithms in a Negative Base

Algorithms are described for the basic arithmetic operations and square rooting in a negative base. A new operation called polarization that reverses the sign of a number facilitates subtraction, using addition. Some special features of the negative-base arithmetic are also mentioned.

Described here is a deterministic division algorithm in a negative-base number system; here, the divisor is mapped into a suitable range by premultiplication, so that the choice of the quotient digit is deterministic.

Deterministic Division Algorithm in a Negative Base

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Arithmetic Algorithms in a Negative Base

Deterministic Division Algorithm in a Negative Base