Mathematical foundations of the infinity computer

All the existing computers are able to do arithmetical operations only with finite numerals. Operations with infinite and infinitesimal quantities could not be realized. The paper describes a new positional system with infinite radix allowing us to write down finite, infinite, and infinitesimal numbers as particular cases of a unique framework. The new approach both gives possibilities to do calculations of a new type and simplifies fields of mathematics where usage of infinity and/or infinitesimals is required. Usage of the numeral system described in the paper gives possibility to introduce a new type of computer – Infinity Computer – able to operate not only with finite numbers but also with infinite and infinitesimal ones.