The Infinity Computer and Numerical Calculus

Yaroslav D. Sergeyev

Dipartimento di Elettronica, Informatica e Sistemistica, Università della Calabria, 87030 Rende (CS), Italy and N.I. Lobatchevsky State University, Nizhni Novgorod, Russia

There exist different ways to generalize traditional arithmetic for finite numbers to the case of infinite and infinitesimal numbers (see Belaga [Version 2007], Benci and Di Nasso [2003], Cantor [1955], Cohen [1966], Conway and Guy [1996], Hardy [1910], Kanamori [1996], Mayberry [2001], Robinson [1996] and references given therein). However, arithmetics developed for infinite numbers are quite different with respect to the finite arithmetic we are used to deal with. Moreover, very often they leave undetermined many operations where infinite numbers take part (for example, ∞−∞, ∞/∞, sum of infinitely many items, etc.) or use representation of infinite numbers based on infinite sequences of finite numbers. These crucial difficulties did not allow people to create computers working with infinite and infinitesimal quantities.

In this survey lecture, we describe a new methodology (see Sergeyev [2003], Sergeyev [2004a], Sergeyev [2004b], Sergeyev [2005], Sergeyev [2006b], Sergeyev [2006a], Sergeyev [2007], Sergeyev [2008c], Sergeyev [2008a], Sergeyev [2008b], Sergeyev [25 July 2008]) having a strong numerical character for treating infinite and infinitesimal quantities based on the principle ‘the part is less than the whole’. In order to understand how it is possible to affront the problem of infinity in this new way let us consider a study published in Science by Peter Gordon (see Gordon [2004]) where he describes a primitive tribe living in Amazonia - Pirahã - that uses a very simple numeral system¹ for counting: one, two, many. For Pirahã, all quantities bigger than two are just ‘many’ and such operations as 2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak numeral system Pirahã are not able to see, for instance, numbers 3, 4, 5, and 6, to execute arithmetical operations with them, and, in general, to say anything about these numbers because in their language there are neither words nor concepts for that. Moreover, the weakness of their numeral system leads to such results as

‘many’+ 1= ‘many’, ‘many’ + 2 = ‘many’,

which are very familiar to us in the context of views on infinity used in the traditional calculus

∞ + 1= ∞, ∞ + 2 = ∞.

This observation leads us to the following idea: Probably our difficulty in working with infinity is not connected to the nature of infinity but is a result of inadequate numeral systems used to express infinite numbers.

The lecture 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 the survey Sergeyev [2008b] and also Sergeyev [2003], Sergeyev [2004b], Sergeyev [2005], Sergeyev [2006a], Sergeyev [2007]). The new numeral system gives possibility to introduce a new type of a computer – the Infinity Computer – able to operate not only with finite numbers but also with infinite and infinitesimal ones (the European Patent Office has expressed its positive opinion with respect to the patent Sergeyev [2004a]).

The new approach has a strong applied character and is not related to the non-standard analysis. It both gives possibilities to execute calculations of a new type and simplifies fields of Mathematics where usage of the infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, etc.).

Applications that can be treated by computers are determined by their computational abilities. In the following there are listed both operations that the Infinity Computer can execute and traditional computers are not able to perform and some of new areas of applications. It becomes possible:

0cm 0cm 0cm 0cm to introduce notions of numbers of elements for infinite sets in a way compatible with the notion used traditionally for finite sets significantly evolving traditional approaches able to distinguish numerable sets from continuum only (the principle ‘The part is less than the whole’ is used for this purpose);
to substitute symbols +∞ and −∞ by spaces of positive and negative infinite numbers, to represent them in the memory of the Infinity Computer and to execute arithmetical operations with all of them using this new type of a 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 the Infinity Computer, and to execute arithmetical operations with them using the 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 the 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 the 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 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.

The Infinity Calculator using the Infinity Computer technology is presented during the talk. Additional information can be downloaded from the page http://www.theInfinitycomputer.com

References

Belaga [Version 2007]: E. Belaga. Mathematical infinity, its inventors, discoveres, detractors, defenders, masters, victims, users, and spectators. Version 2007.
Benci and Di Nasso [2003]: V. Benci and M. Di Nasso. Numerosities of labeled sets: a new way of counting. Advances in Mathematics, 173: 50–67, 2003.
Cantor [1955]: G. Cantor. Contributions to the founding of the theory of transfinite numbers. Dover Publications, New York, 1955.
Cohen [1966]: P. Cohen. Set Theory and the Continuum Hypothesis. Benjamin, New York, 1966.
Conway and Guy [1996]: J. Conway and R. Guy. The Book of Numbers. Springer-Verlag, New York, 1996.
Gordon [2004]: P. Gordon. Numerical cognition without words: Evidence from Amazonia. Science, 306 (15 October): 496–499, 2004.
Hardy [1910]: G. Hardy. Orders of infinity. Cambridge University Press, Cambridge, 1910.
Kanamori [1996]: A. Kanamori. The mathematical development of set theory from Cantor to Cohen. Bull. Symbolic Logic, 2 (1): 1–71, 1996.
Mayberry [2001]: J. Mayberry. The Foundations of Mathematics in the Theory of Sets. Cambridge University Press, Cambridge, 2001.
Robinson [1996]: A. Robinson. Non-standard Analysis. Princeton Univ. Press, Princeton, 1996.
Sergeyev [25 July 2008]: Y. Sergeyev. Numerical computations and mathematical modelling with infinite and infinitesimal numbers. Journal of Applied Mathematics and Computing, 25 July 2008. published online.
Sergeyev [2006a]: Y. Sergeyev. Misuriamo l’infinito. Periodico di Matematiche, 6(2): 11–26, 2006a.
Sergeyev [2005]: Y. Sergeyev. A few remarks on philosophical foundations of a new applied approach to Infinity. Scheria, 26-27: 63–72, 2005.
Sergeyev [2006b]: Y. Sergeyev. Mathematical foundations of the Infinity Computer. Annales UMCS Informatica AI, 4: 20–33, 2006b.
Sergeyev [2003]: Y. Sergeyev. Arithmetic of Infinity. Edizioni Orizzonti Meridionali, CS, 2003.
Sergeyev [2004a]: Y. Sergeyev. Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. patent application 08.03.04, 2004a.
Sergeyev [2008a]: Y. Sergeyev. Modelling season changes in the infinite processes of growth of biological systems. Transactions on Applied Mathematics and Nonlinear Models, page (to appear), 2008a.
Sergeyev [2007]: Y. Sergeyev. Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers. Chaos, Solitons & Fractals, 33(1): 50–75, 2007.
Sergeyev [2008b]: Y. Sergeyev. A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica, 19(4): 567–596, 2008b.
Sergeyev [2008c]: Y. Sergeyev. Measuring fractals by infinite and infinitesimal numbers. Mathematical Methods, Physical Methods & Simulation Science and Technology, 1(1): 217–237, 2008c.
Sergeyev [2004b]: Y. Sergeyev. http://www.theinfinitycomputer.com. 2004b.

1: We remind that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘5’, ‘five’, and ‘V’ are different numerals, but they all represent the same number.

This document was translated from L^AT_EX by H^EV^EA.