A new look at infinitely large and infinitely small quantities: Methodological foundations and practical calculations with these numbers on a computer

This article describes a recently proposed methodology that allows one to work with infinitely large and infinitely small quantities on a computer. The approach uses a number of ideas that bring it closer to modern physics, in particular, the relativity of mathematical knowledge and its dependence on the tools used by mathematicians in their studies are discussed. It is shown that the emergence of new computational tools influences the way we perceive traditional mathematical objects, and also helps to discover new interesting objects and problems. It is discussed that many difficulties and paradoxes regarding infinity do not depend on its nature, but are the result of the weakness of the traditional numeral systems used to work with infinitely large and infinitely small quantities. A numeral system is proposed that not only allows one to work with these quantities analytically in a simpler and more intuitive way, but also makes possible practical calculations on the Infinity Computer, patented in a number of countries. Examples of measuring infinite sets with the accuracy of one element are given and it is shown that the new methodology avoids the appearance of some well-known paradoxes associated with infinity. Examples of solving a number of computational problems are given and some results of teaching the described methodology in Italy and Great Britain are discussed.

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.

Representation of grossone-based arithmetic in simulink for scientific computing

Numerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent.