The Infinity Computer and Numerical Computations with Infinite and Infinitesimal Numbers
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