# Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

*EMS Surveys in Mathematical Sciences*4 (2):219–320 (2017)

**Abstract**

In this survey, a recent computational methodology paying a special attention to the separation
of mathematical objects from numeral systems involved in their representation is described.
It has been introduced with the intention to allow one to work with infinities and infinitesimals
numerically in a unique computational framework in all the situations requiring these notions. The
methodology does not contradict Cantor’s and non-standard analysis views and is based on the
Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite,
infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a
computational device called the Infinity Computer (patented in USA and EU) working numerically
(recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite
and infinitesimal numbers that can be written in a positional numeral system with an infinite radix.
It is argued that numeral systems involved in computations limit our capabilities to compute and lead
to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility
to use the same numeral system for measuring infinite sets, working with divergent series, probability,
fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally
different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical
objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher
accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not
related to their nature but is a consequence of the weakness of traditional numeral systems used to
study them. It is shown that the introduced methodology and numeral system change our perception
of the mathematical objects studied in the two problems.

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**Revision history**

Archival date: 2018-12-17

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References found in this work BETA

Exact and Approximate Arithmetic in an Amazonian Indigene Group.Pierre Pica, Cathy Lemer, Véronique Izard & Stanislas Dehaene - 2004 -

*Science*306 (5695):499-503.Non-Standard Analysis.Robinson, A.

Satan, Saint Peter and Saint Petersburg: Decision Theory and Discontinuity at Infinity.Bartha, Paul; Barker, John & Hájek, Alan

Tasks and Supertasks.Thomson, James

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Citations of this work BETA

Independence of the Grossone-Based Infinity Methodology From Non-Standard Analysis and Comments Upon Logical Fallacies in Some Texts Asserting the Opposite.Sergeyev, Yaroslav

Single-Tape and Multi-Tape Turing Machines Through the Lens of the Grossone Methodology.Sergeyev, Yaroslav & Garro, Alfredo

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2018-12-17

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