Notation of very large numbers

From: Philippe Verdy (verdy_p@wanadoo.fr)
Date: Sun Jan 22 2006 - 20:18:48 CST

  • Next message: Jukka K. Korpela: "Re: Notation of very large numbers"

    I fellaccidentally on this article about notations of very large numbers, for which the common positional system or the exponential notation fails to give compact and usable representation in text:

    http://mathworld.wolfram.com/Steinhaus-MoserNotation.html

    This is about the Steinhaus-Moser Notation, where:

    * the number "n in a triangle":
        denotes "n to the power of n" (representation is possible within existing Unicode plain-text),
        i.e. n^n;

    * the number "n in a square":
        denotes "n within n triangles" (impossible to represent within existing Unicode plain-text),
        for example "3 in a square" is "3 in 3 triangles",
        i.e the triangle of the triangle of the triangle of 3,
        or the triangle or the triangle of 27,
        or the triangle of 27^27, or (27^27)^(27^27) ;

    * the number "n in a circle" (or pentagon):
        denotes "n within n squares";

    * the number "n in a hexagon":
        denotes "n within n circles" (or pentagons);

    * etc (generalized to polygons with growing number of vertices)

    The number of exponentiation operations, i.e. its complexity to compute, grows extremely fast with a much order of magnitude than the value of n, such that it would be simply impossible to write all the digits making just the number 3 in a square, within this email.

    REFERENCES: Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999. ISBN 0486409147

    Such notation cannot be strictly represented as such within Unicode (but upper-layer mathemetical layout syntaxes may be used).

    However, for the generalization of the concept, polygons with large numbers of vertices would be rapidly impossible to differentiate, so may be it could be better represented with another notation like the following indiced functions:

    S_0(n) = n ;

    S_1(n) = n in a triangle = n * ...* n, with n factors
        = n ^ n ;

    S_2(n) = n in a square = S_1(...S_1(n)...), where the S_1 function is applied n times,
        = {S_1...S_1}(n) = M_{1, n}(n),
        using the LaTeX notation, and
        using classically for denoting the ordered composition of functions ;

    S_3(n) = n in a pentagon (or circle)
        = {S_2...S_2}(n) = M_{2, n}(n) ;
    ...
    S_{v+1}(n) = the number n, within n polygons of (v+3) vertices
       = {S_v...S_v}(n) = M_{v, n}(n)

    These functions may be defined recursively, using another bidimensional set M of functions with 2 indices, the first denoting the polygon vertices order, and the second indice the number of polygons with that number of vertices i.e. the number of recursive application of its associated polygon-function in the S set.



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