Numerical computations with infinite and infinitesimal numbers: Theory and applications

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this chapter. 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 not related to non-standard analysis theories. The Infinity Computer working with numbers of a new kind is described (its simulator has already been realized). The concept of accuracy of mathematical languages and its importance for a number of theoretical and practical issues regarding computations is discussed. Numerous examples dealing with divergent series, infinite sets, probability, limits, fractals, etc. are given.