  A new computational methodology for executing calculations with infinite and
infinitesimal quantities is described in this paper. It is based on the
principle `The part is less than the whole' introduced by Ancient Greeks and
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. The new methodology has
allowed us to introduce the Infinity Computer working with such numbers (its
simulator has already been realized). Examples dealing with divergent series,
infinite sets, and limits are given.
