Numerical computations and mathematical modelling with infinite and infinitesimal numbers

TL;DR

This paper introduces a novel computational methodology using the Infinity Computer to perform numerical calculations with finite, infinite, and infinitesimal numbers, enabling new mathematical modeling and solving complex problems.

Contribution

It presents a new numerical approach for working with infinite and infinitesimal quantities, expanding computational capabilities beyond traditional finite number systems.

Findings

Demonstrates numerical computations with divergent series and limits

Shows applications in probability theory and linear algebra

Calculates volumes of complex objects with different dimensions

Abstract

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers \textit{numerically}. This can be done on a new kind of a computer - the Infinity Computer - able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.