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Computations with Grossone-Based Infinities

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Unconventional Computation and Natural Computation (UCNC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9252))

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Abstract

In this paper, a recent computational methodology is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA and EU) working numerically with infinite and infinitesimal numbers that can be written in a positional system with an infinite radix. On a number of examples dealing mainly with infinite sets and Turing machines with different infinite tapes it is shown that it becomes possible to execute a fine analysis of these mathematical objects. The accuracy of the obtained results is continuously compared with results obtained by traditional tools used to work with mathematical objects involving infinity.

Y.D. Sergeyev—This research was partially supported by the Russian Foundation for Basic Research, grant no. 15-01-06612.

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Notes

  1. 1.

    The last two numerals, \(\doteq \) and Ĩ, are probably less known. The former belongs to the Maya numeral system where one horizontal line indicates five and two lines one above the other indicate ten. Dots are added above the lines to represent additional units. For instance, \(\doteq \) means eleven in this numeral system. The latter symbol, Ĩ, belongs to the Cyrillic numeral system derived from the Cyrillic script. This numeral system was developed in the late \(X^{th}\) century and was used by South and East Slavic peoples. The system was used in Russia as late as the early \(XVIII^{th}\) century when it was replaced with Arabic numerals. To distinguish numbers from text, a titlo, \(\tilde{}\), is drawn over the symbols showing so that this is a numeral and, therefore, it represents a number and not just a character of text.

  2. 2.

    Notice that nowadays not only positive integers but also zero is frequently included in \(\mathbb {N}\). However, since zero has been invented significantly later than positive integers used for counting objects, zero is not include in \(\mathbb {N}\) in this text.

  3. 3.

    This is a difference with respect to non-standard analysis where infinities it works with do not belong to \(\mathbb {N}\).

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Author information

Authors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. Lobatchevsky State University, Nizhni Novgorod, Russia

    Yaroslav D. Sergeyev

  3. Institute of High Performance Computing and Networking, C.N.R., Naples, Italy

    Yaroslav D. Sergeyev

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  1. Yaroslav D. Sergeyev
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Correspondence to Yaroslav D. Sergeyev .

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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Cristian S. Calude

  2. University of Auckland, Auckland, New Zealand

    Michael J. Dinneen

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Sergeyev, Y.D. (2015). Computations with Grossone-Based Infinities. In: Calude, C., Dinneen, M. (eds) Unconventional Computation and Natural Computation. UCNC 2015. Lecture Notes in Computer Science(), vol 9252. Springer, Cham. https://doi.org/10.1007/978-3-319-21819-9_6

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