D
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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-o...are-the-numbers/answer/Alan-Bustany?srid=oNrJ
The three numbers are [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]4[/FONT] and the “average” you are thinking of is a generalized mean:
[FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Math])/3[/FONT][FONT=MathJax_Main]] ^ (1/p) [/FONT][FONT=MathJax_Main]= [/FONT][FONT=MathJax_Math]π[/FONT]
.
The arithmetic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4] / [/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]3
[/FONT]
but you would never think of such a boring average.
The geometric mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math]q[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]q [/FONT][FONT=MathJax_Main]= [[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]*[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]*[/FONT][FONT=MathJax_Main]4]^(1/3) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]2.8845[/FONT]
.
The quadratic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]2 [/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]16)/3] ^ (1/2) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]3.1091[/FONT]
.
The cubic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]3 [/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]27[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]64)/3] ^ (1/[/FONT][FONT=MathJax_Main]3) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]3.2075[/FONT]
.
It turns out that the generalized mean is a continuous monotonic increasing function of [FONT=MathJax_Math]p[/FONT]
so, by the intermediate value theorem, there is some [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Main]3[/FONT] such that [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]π[/FONT].
Clearly the average about which you were thinking [FONT=MathJax_AMS]⌣[/FONT][FONT=MathJax_Main]¨
[/FONT]Of course the boring old arithmetic mean of any finite set of Rational numbers is Rational. It is therefore never [FONT=MathJax_Math]π[/FONT]
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 [FONT=MathJax_Math]p[/FONT].
from https://www.quora.com/Im-thinking-o...are-the-numbers/answer/Alan-Bustany?srid=oNrJ
The three numbers are [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]4[/FONT] and the “average” you are thinking of is a generalized mean:
[FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Math])/3[/FONT][FONT=MathJax_Main]] ^ (1/p) [/FONT][FONT=MathJax_Main]= [/FONT][FONT=MathJax_Math]π[/FONT]
.
The arithmetic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4] / [/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]3
[/FONT]
but you would never think of such a boring average.
The geometric mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]lim[/FONT][FONT=MathJax_Math]q[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]q [/FONT][FONT=MathJax_Main]= [[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]*[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]*[/FONT][FONT=MathJax_Main]4]^(1/3) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]2.8845[/FONT]
.
The quadratic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]2 [/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]16)/3] ^ (1/2) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]3.1091[/FONT]
.
The cubic mean is [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Main]3 [/FONT][FONT=MathJax_Main]= [([/FONT][FONT=MathJax_Main]8[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]27[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]64)/3] ^ (1/[/FONT][FONT=MathJax_Main]3) [/FONT][FONT=MathJax_Main]≈ [/FONT][FONT=MathJax_Main]3.2075[/FONT]
.
It turns out that the generalized mean is a continuous monotonic increasing function of [FONT=MathJax_Math]p[/FONT]
so, by the intermediate value theorem, there is some [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]<[/FONT][FONT=MathJax_Main]3[/FONT] such that [FONT=MathJax_Math]M[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]π[/FONT].
Clearly the average about which you were thinking [FONT=MathJax_AMS]⌣[/FONT][FONT=MathJax_Main]¨
[/FONT]Of course the boring old arithmetic mean of any finite set of Rational numbers is Rational. It is therefore never [FONT=MathJax_Math]π[/FONT]
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 [FONT=MathJax_Math]p[/FONT].
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