A new computational methodology for executing calculations with infinite andinfinitesimal quantities is described in this paper. It is based on theprinciple `The part is less than the whole' introduced by Ancient Greeks andapplied to all numbers (finite, infinite, and infinitesimal) and to all setsand processes (finite and infinite). It is shown that it becomes possible towrite down finite, infinite, and infinitesimal numbers by a finite number ofsymbols as particular cases of a unique framework. The new methodology hasallowed us to introduce the Infinity Computer working with such numbers (itssimulator has already been realized). Examples dealing with divergent series,infinite sets, and limits are given.
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