The Infinity Computer and numerical computations with infinite and infinitesimal numbers Yaroslav D. Sergeyev University of Calabria, Rende, Italy and N.I. Lobatchevsky University of Nizhni Novgorod, Russia Via P. Bucci, Cubo 42-C 87036 Rende (CS), Italy
[email protected] Keywords: Infinities and infinitesimals, Infinity Computer, numerical computations A new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers (see [7,9,15]) on a new type of a computer – the Infinity Computer – is introduced (see EU, USA, and Russian patents [8]). A calculator using the Infinity Computer technology is presented during the talk. The new approach is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks that is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework (different from that of the non-standard Analysis). The new methodology (see surveys [9,15]) evolves ideas of Cantor and Levi-Civita in a more applied way and (among other things) introduces infinite integers that possess both cardinal and ordinal properties as usual finite numbers (its relations with traditional approaches are discussed in [4,5,9,15]). Note that foundations of the Set Theory dealing with infinity have been developed starting from the end of the XIX-th century until more or less the first decades of the XX-th century. Foundations of the classical Analysis dealing both with infinity and infinitesimal quantities have been developed even earlier, more than 200 years ago, with the goal to develop mathematical tools allowing one to solve problems arising in the real world in that remote time. As a result, these parts of Mathematics reflect ideas that people had about Physics (including definitions of the notions continuous and discrete) more than 200 years ago. Thus, these mathematical tools do not include numerous achievements of Physics of the XX-th century. Even the brilliant non-standard Analysis of Robinson made 1
in the middle of the XX-th century has been also directed to a reformulation of the classical Analysis (i.e., Analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of Analysis that would incorporate new achievements of Physics. The point of view on infinite and infinitesimal quantities presented in this talk uses strongly two methodological ideas borrowed from the modern Physics: relativity and interrelations holding between the object of an observation and the tool used for this observation. The latter is directly related to connections between different numeral systems used to describe mathematical objects and the objects themselves. The new computational methodology gives the possibility both to execute numerical (not symbolic) computations of a new type and simplifies fields of Mathematics where the usage of the infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, fractals, etc., see [1-3,6,7,9-20]). In particular, a number of results related to the First Hilbert Problem and Turing machines are established (see [13,14]) and a new concept of continuity better reflecting the modern view of physicists on the world around us is introduced (see [12]). Numerous examples and applications are given: approximation, automatic differentiation, cellular automata, differential equations, linear and non-linear optimization, fractals, percolation, processes of growth in biological systems, etc., see [1-3,6,9-12,17-19]. A lot of additional information on the new methodology (theoretical and applied papers of several authors, the list of awards, reviews appeared in MIT Technology Review, International Journal of Unconventional Computing, etc.) can be downloaded from the page http://www.theinfinitycomputer.com References: [1] L. D’Alotto, Cellular automata using infinite computations, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8077–8082. [2] S. De Cosmis, R. De Leone, The use of Grossone in Mathematical Programming and Operations Research, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8029–8038. [3] D.I. Iudin, Ya.D. Sergeyev, M. Hayakawa, Interpretation of percolation in terms of infinity computations, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8099–8111. [4] G. Lolli, Infinitesimals and infinites in the history of Mathematics: A brief survey, Applied Mathematics and Computation, 218 (2012), No. 16, 2
pp. 7979–7988. [5] M. Margenstern, Using Grossone to count the number of elements of infinite sets and the connection with bijections, p-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), No. 3, pp. 196–204. [6] M. Margenstern, An application of Grossone to the study of a family of tilings of the hyperbolic plane, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8005–8018. [7] Ya.D. Sergeyev, Arithmetic of Infinity, Edizioni Orizzonti Meridionali, CS, 2003. [8] Ya.D. Sergeyev, Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them, EU patent 1728149, issued 03.06.2009; RF patent 2395111, issued 20.07.2010; USA patent 7,860,914 issued 28.12.2010. [9] Ya.D. Sergeyev, A new applied approach for executing computations with infinite and infinitesimal quantities, Informatica, 19 (2008), No. 4, pp. 567–596. [10] Ya.D. Sergeyev, Numerical computations and mathematical modelling with infinite and infinitesimal numbers, Journal of Applied Mathematics and Computing, 29 (2009), pp. 177–195. [11] Ya.D. Sergeyev, Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge, Chaos, Solitons & Fractals, 42 (2009), pp. 3042–3046. [12] Ya.D. Sergeyev, Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains, Nonlinear Analysis Series A: Theory, Methods & Applications, 71 (2009), No. 12, pp. e1688–e1707. [13] Ya.D. Sergeyev, Counting systems and the First Hilbert problem, Nonlinear Analysis Series A: Theory, Methods & Applications, 72(2010), No. 3-4, pp. 1701–1708. [14] Ya.D. Sergeyev, A. Garro, Observability of Turing Machines: a refinement of the theory of computation, Informatica, 21 (2010), No. 3, pp. 425–454.
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[15] Ya.D. Sergeyev, Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals, Rendiconti del Seminario Matematico dell’Universit e del Politecnico di Torino, 68 (2010), No. 2, pp. 95–113. [16] Ya.D. Sergeyev, On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function, p-Adic Numbers, Ultrametric Analysis and Applications, 3 (2011), No. 2, pp. 129–148. [17] Ya.D. Sergeyev, Higher order numerical differentiation on the Infinity Computer, Optimization Letters, 5 (2011), pp. 575–585. [18] Ya.D. Sergeyev, Using blinking fractals for mathematical modeling of processes of growth in biological systems, Informatica, 22 (2011), No. 4, pp. 559–576. [19] M.C. Vita, S. De Bartolo, C. Fallico, M. Veltri, Usage of infinitesimals in the Menger’s Sponge model of porosity, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8187–8196. [20] A.A. Zhigljavsky, Computing sums of conditionally convergent and divergent series using the concept of grossone, Applied Mathematics and Computation, 218 (2012), No. 16, pp. 8064–8076.
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