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

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... mals 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.