INFINITY COMPUTER AND CALCULUS Yaroslav D. Sergeyev University of Calabria, Rende, Italy N.I. Lobachevski University of Nizhni Novgorod, Russia
e-mail: yaro@si.deis.unical.it http://wwwinfo.deis.unical.it/~yaro
Abstract Traditional computers execute arithmetical operations with finite numbers. The talk introduces a new positional system with infinite radix allowing one to write down finite, infinite, and infinitesimal numbers as particular cases of a unique framework (see [1,2]). The new numeral system gives possibility to introduce a new type of computer – Infinity Computer – able to operate not only with finite numbers but also with infinite and infinitesimal ones. The new approach both gives possibilities to execute calculations of a new type and simplifies fields of mathematics where usage of infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, etc.). Computational abilities of computers determine classes of applications that can be treated by the computers. In the following there are listed both operations that traditional computers are not able to perform and some of new areas of applications. It will become possible:
to substitute symbols +∞ and -∞ by spaces of positive and negative infinite numbers, to represent them in the memory of Infinity Computer and to execute arithmetical operations with all of them using this new type of computer, as we are used to do with usual finite numbers on traditional computers; to substitute qualitative descriptions of the type „a number tends to zero‟ by precise infinitesimal numbers, to represent them in the memory of Infinity Computer, and to execute arithmetical operations with them using Infinity Computer as we are used to do with usual finite numbers using traditional computers; to calculate divergent limits and series, providing as results explicitly written different infinite numbers, to be possibly used in further calculations on Infinity Computer; to calculate indeterminate forms (e.g., difference and division of divergent series) that can give as results different infinitesimal, finite or infinite numbers; to calculate on Infinity Computer sums of divergent series and improper integrals of various types that can give as results different explicitly written infinite numbers; to evaluate functions and their derivatives at infinitesimal, finite, and infinite points (infinite and infinitesimal values of functions and their derivatives can be also explicitly calculated); to study divergent processes at different infinite points; to extend definition of volume to objects having parts of different dimensions and to calculate these volumes in a unique framework using infinite and infinitesimal numbers; to introduce notions of lengths, areas, and volumes of fractal objects obtained after infinite numbers of steps and compatible with traditional lengths, areas, and volumes of non-fractal objects and to calculate them in a unique framework; to introduce notions of numbers of elements in infinite sets compatible with the notion used traditionally for finite sets and to calculate them in a unique framework (significantly evolving traditional approaches able to distinguish numerable sets from continuum only).
A demonstration of the Infinity Calculator using the Infinity Computer technology is given during the presentation.
Presentations Results described in the book and in the patent have been presented at the following congresses:
invited plenary lecture at the XXI Autumn Meeting of the Polish Information Processing Society, Katowice, Poland, December 5-9, 2005; invited plenary lecture at the International Workshop on Global Optimization, Almeria, Spain, September 18-22, 2005; tutorial at the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Blacksburg, Virginia, USA, August 15-17, 2005; invited plenary lecture, International Conference on Difference Equations, Special Functions and Applications, Munich, July 25-30, 2005; invited plenary lecture, International Conference “Numerical Analysis: the State of the Art”, Rende (CS), May, 19-21, 2005; invited semi-plenary lecture at the International conference on Selected Problems of Modern Mathematics, dedicated to the 200th anniversary of K.G. Jacobi and the 750th anniversary of the Koenigsberg foundation, Kaliningrad, April, 4-8, 2005; tutorial at the Workshop "Numerical Methods and Mathematical Software", Montecatini Terme (PT), January 31 - February 1, 2005; invited plenary lecture at the VI-th International Congress on Mathematical Modelling, Nizhni Novgorod, Russia, September, 21-26, 2004; opening invited lecture at the congress “Infinity in Mathematics, Physics, and Philosophy”, Pisa, Italy, 26.03.2004;
and at the universities of: Aberdeen, Bari, Bologna, Cagliari, Calabria, Cardiff, Florence, Kaliningrad, Nizhni Novgorod, Pisa, Rome “La Sapienza”, Salerno, St. Petersburg, Trieste, and also at the Italian Institute of Philosophical Studies, Naples, Computing Centre of the Russian Academy of Sciences, Bologna Academy of Sciences, Institute of Computer Science of the Academy of Sciences of the Czech Republic, Nizhni Novgorod Mathematical Society, Institute for Mathematical Modelling of the Russian Academy of Sciences, Institute for High Performance Computing and Networks and Institute of Applied Mathematics and Information Technology, both of the Italian National Research Council, and also at INTEL Research Labs in Nizhni Novgorod and St. Petersburg. Media citations
Interview to the largest Italian National business newspaper “Il Sole 24 Ore”, 27.08.2005; Interview to Italian National newspaper “Italia Oggi” reprinted also by another Italian National newspaper “MF” (Milan Finance), 12.07.2005. Interview to Italian National radio station Radio24, 23.06.2005. Cover story in the department Technology, Notable New Patents of the U.S.A. magazine of intellectual property and technology “IPfrontline”, 15.06.2005.
All these texts and some additional information can be downloaded from the internet page http://www.theinfinitycomputer.com References 1. Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, CS, 2003. 2. Ya.D. Sergeyev, Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them, patent pending, 08.03.04.