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Mathematics Without Boundaries pp 181–220Cite as

Which Numbers Simplify Your Problem?

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Abstract

An extension of the ring of scalar quantities, from the usual field of real numbers to a non-Archimedean, sometimes permits to simplify some problems which, at a first sight, may seem not correlated with infinitesimal and infinite numbers. We present four simple cases, each one at the level of possibility for the creativity of a motivated student. The ring of Fermat reals and its applications to physics and differential geometry, the ring of Colombeau generalized numbers and its applications to the foundations of generalized functions, the Levi-Civita field and the derivation of complicated computer functions and the Surreals numbers as a universal non-Archimedean ring. The definition of each one of these rings is strongly motivated at elementary level and some open problems and ideas are introduced in the first two cases.

The author has been supported by FWF grants M1247-N13 and P25116-N25.

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Notes

  1. 1.

    The notation with square brackets \(x[q]\) permits to avoid confusion when one consider functions defined on the LCF \(\mathcal{R}\).

  2. 2.

    Really, the same field of numbers has been predate by [14] (in Spanish) and [36] (in German).

  3. 3.

    That is \(L_{x}<R_{x}\) and \(L_{y}<R_{y}\). Let us note that using a notation like \(x=\{L_{x}\,|\, R_{x}\}\) we do not mean that a number \(x\in{\bf No}\) uniquely determines the subsets L x and R x .

  4. 4.

    Of course, at this stage of development and using this not-strictly formal point of view, our use of the notion of “simplicity” is only informal and it is natural to ask for a more formal definition, considering, moreover, its uniqueness. This will be done in the next section.

  5. 5.

    From this point of view the name “surreal numbers” is less meaningful than the original Conway’s “numbers” without any adjective.

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  1. Fakultät für Mathematik, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Paolo Giordano

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  1. Paolo Giordano
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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Giordano, P. (2014). Which Numbers Simplify Your Problem?. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_7

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