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Thread: Very interesting average - I'm thinking of three numbers whose average is (pi) π.

  1. #1
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    Very interesting average - I'm thinking of three numbers whose average is (pi) π.

    from https://www.quora.com/Im-thinking-of...tany?srid=oNrJ

    The three numbers are 2,3,4 and the “average” you are thinking of is a generalized mean:
    Mp= [(2p+3p+4p)/3] ^ (1/p) = π
    .
    The arithmetic mean is M1=[2+3+4] / 3=3

    but you would never think of such a boring average.

    The geometric mean is M0=limq0Mq = [2*3*4]^(1/3) 2.8845
    .
    The quadratic mean is M2 = [(4+9+16)/3] ^ (1/2) 3.1091
    .
    The cubic mean is M3 = [(8+27+64)/3] ^ (1/3) 3.2075
    .
    It turns out that the generalized mean is a continuous monotonic increasing function of p
    so, by the intermediate value theorem, there is some 2<p<3 such that Mp=π.

    Clearly the average about which you were thinking ¨

    Of course the boring old arithmetic mean of any finite set of Rational numbers is Rational. It is therefore never π
    because that value is known to be Irrational, in fact Transcendental.

    More evidence that you weren’t thinking of this average or, indeed, any generalized mean with an Integer value for p.
    Last edited by Subhotosh Khan; 12-02-2017 at 09:02 AM.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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