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Proceedings of GO 2005, pp. 3 – 4.Infinity computer and calculusYaroslav D. SergeyevFull Professor, Dipartimento di ElettronicaInformatica e SistemisticaUniversitá della CalabriaVia P.Bucci, Cubo 41C87036 Rende (CS), ItalyFull Professor, Lobatchevsky State University,Nizhni Novgorod, Russiayaro@si.deis.unical.itAll the existing computers are able to execute arithmetical operations only with finite numbers.Operations with infinite and infinitesimal quantities could not be realized. This tutorialintroduces a new positional system with infinite radix allowing us to write down finite,infinite, and infinitesimal numbers as particular cases of a unique framework. The new numeralsystem gives possibility to introduce a new type of computer able to operate not onlywith finite numbers but also with infinite and infinitesimal ones. The new approach bothgives possibilities to execute calculations of a new type and simplifies fields of mathematicswhere usage of infinity and/or infinitesimals is necessary (for example, divergent series,limits, derivatives, integrals, measure theory, probability theory, etc.).Particularly, the new approach and the infinity computer are able to do the following:to substitute symbols +infinity and -infinity by spaces of positive and negative infinitenumbers, to represent them in the computer memory and to execute arithmetical operationswith them as with normal finite numbers;to substitute qualitative description of the type "a number tends to zero" by precise infinitesimalnumbers, to represent them in the computer memory and to execute mathematicaloperations with them as with normal finite numbers;to introduce a new definition of continuity that is closer to the real world than the traditionalone;to calculate limits as arithmetical expressions;to calculate indeterminate forms in limits;to calculate sums of divergent series;to calculate improper integrals of various types;to calculate number of elements of infinite sets (and not only to distinguish numerablesets from continuum as it happens in the traditional approach);to evaluate functions and their derivatives at infinitesimal, finite, and infinite points(infinite and infinitesimal values of the functions and their derivatives can be also calculated);to study divergent processes at infinity;