A new applied approach for executing computations with infinite and infinitesimal quantities

TL;DR

This paper introduces a novel computational framework and Infinity Computer for performing calculations with finite, infinite, and infinitesimal quantities using a unified symbolic approach, enabling new analysis of divergent series and limits.

Contribution

It presents a new methodology and a dedicated computer for calculations involving infinite and infinitesimal numbers within a unified framework.

Findings

Finite, infinite, and infinitesimal numbers can be represented with a finite number of symbols.

The Infinity Computer can perform calculations with these numbers, demonstrated through examples.

Applications include divergent series, infinite sets, and limits.

Abstract

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. 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. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.