Numerical infinities and infinitesimals in a new supercomputing framework

Traditional computers are able to work numerically with finite numbers only. The Infinity Computer patented recently in USA and EU gets over this limitation. In fact, it is a computational device of a new kind able to work numerically not only with finite quantities but with infinities and infinitesimals, as well. The new supercomputing methodology is not related to non-standard analysis and does not use either Cantor's infinite cardinals or ordinals. It is founded on Euclid's Common Notion 5 saying 'The whole is greater than the part'. This postulate 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 numerals belonging to a positional numeral system with an infinite radix described by a specific ad hoc introduced axiom. Numerous examples of the usage of the introduced computational tools are given during the lecture. In particular, algorithms for solving optimization problems and ODEs are considered among the computational applications of the Infinity Computer. Numerical experiments executed on a software prototype of the Infinity Computer are discussed.