All the existing computers are able to execute arithmetical operations only 
with finite numbers. Operations with infinite and infinitesimal quantities 
could not be realized. The talk introduces a new positional system with 
infinite radix allowing us to write down finite, infinite, and infinitesimal 
numbers as particular cases of a unique framework. This new numeral system 
gives possibility to introduce a new type of 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 (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 infinite numbers, to represent them in the memory of 
     infinity computer and to execute arithmetical operations with them 
     using this computer of a new type as we are used to do with usual 
     finite numbers using traditional computers;
   * to substitute qualitative description of the type 'a number tends
     to zero' by precise infinitesimal numbers, to represent them in
     the memory of infinity computer and to execute mathematical
     operations with them using infinity computer as we are used to do
     with usual finite numbers using traditional computers;
   * to calculate limits (including indeterminate forms) as
     arithmetical expressions using infinity computer;
   * to calculate sums of divergent series and improper integrals of
     various types using infinity computer and to execute operations
     being indeterminate forms in traditional approaches, e.g.,
     difference and division of divergent series;
   * 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 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 this notion used traditionally for finite sets and
     to calculate them in a unique framework (and not only to
     distinguish numerable sets from continuum as it happens in
     traditional approaches).

References

  1. Ya.D. Sergeyev, Computer system  for storing infinite,
     infinitesimal, and finite quantities and executing arithmetical
     operations with them, Patent filed in 2004.
  2. Ya.D. Sergeyev, Arithmetic of infinity
     <http://si.deis.unical.it/%7Eyaro/arithmetic.html>, Edizioni
     Orizzonti Meridionali, 2003, ISBN 88-89064-01-3.