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| [[Image:MathematicalMeans.svg|thumb|right|A geometric construction of the Quadratic mean and the Pythagorean means (of two numbers ''a'' and ''b''). Harmonic mean denoted by ''H'', Geometric by ''G'', Arithmetic by ''A'' and Quadratic mean (also known as [[Root mean square]]) denoted by ''Q''.]]
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| [[Image:Comparison_Pythagorean_means.svg|thumb|right|Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are [[asymptote]]s for the harmonic means.]]
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| In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (''A''), the [[geometric mean]] (''G''), and the [[harmonic mean]] (''H''). They are defined by:
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| * <math> A(x_1, \ldots, x_n) = \frac{1}{n}(x_1 + \cdots + x_n) </math>
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| * <math> G(x_1, \ldots, x_n) = \sqrt[n]{x_1 \cdots x_n} </math>
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| * <math> H(x_1, \ldots, x_n) = \frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}} </math>
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| Each mean has the following properties:
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| * Value preservation: <math> M(x,x, \ldots,x) = x </math>
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| * First order [[homogeneous function|homogeneity]]: <math> M(bx_1, \ldots, bx_n) = b M(x_1, \ldots, x_n) </math>
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| * Invariance under exchange: <math> M(\ldots, x_i, \ldots, x_j, \ldots ) = M(\ldots, x_j, \ldots, x_i, \ldots) </math> for any <math>i</math> and <math>j</math>.
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| * Averaging: <math> \min(x_1,\ldots,x_n) \leq M(x_1,\ldots,x_n) \leq \max(x_1,\ldots,x_n)</math>
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| These means were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians (Thomas Heath, History of Ancient Greek Mathematics) because of their importance in geometry and music.
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| There is an ordering to these means (if all of the <math> x_i </math> are positive), along with the [[quadratic mean]] <math>Q=\sqrt{\frac{x_1^2+x_2^2+ \cdots + x_n^2}{n}}</math>:
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| : <math> \min \leq H \leq G \leq A \leq Q \leq \max </math>
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| with equality holding if and only if the <math> x_i </math> are all equal. This is a generalization of the [[inequality of arithmetic and geometric means]] and a special case of an inequality for [[generalized mean]]s. This inequality sequence can be proved for the <math>n=2</math> case for the numbers ''a'' and ''b'' using a sequence of [[right triangle]]s (''x'', ''y'', ''z'') with [[hypotenuse]] ''z'' and the [[Pythagorean theorem]], which states that <math>x^2 + y^2 = z^2</math> and implies that <math>z > x</math> and <math>z > y</math>. The right triangles are<ref>Kung, Sidney H., "The Harmonic mean—geometric mean—arithmetic mean—root mean square inequality II," in Roger B. Nelsen, ''Proofs Without Words'', [[The Mathematical Association of America]], 1993, p. 54.</ref>
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| :<math>\left(\frac{b-a}{b+a}\sqrt{ab}, \frac{2ab}{a+b}, \sqrt{ab}\right) = \left(\frac{b-a}{b+a}\sqrt{ab}, H(a,b), G(a,b)\right),</math>
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| showing that <math>H(a,b) < G(a,b)</math>;
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| :<math>\left(\frac{b-a}{2}, \sqrt{ab}, \frac{a+b}{2}\right) = \left(\frac{b-a}{2}, G(a,b), A(a,b)\right),</math>
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| showing that <math>G(a,b) < A(a,b)</math>;
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| and
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| :<math>\left(\frac{b-a}{2}, \frac{a+b}{2}, \sqrt{\frac{a^2+b^2}{2}}\,\right) = \left(\frac{b-a}{2},A(a,b), Q(a,b)\right),</math>
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| showing that <math>A(a,b) < Q(a,b)</math>.
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| ==See also==
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| * [[Arithmetic-geometric mean]]
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| * [[Average]]
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| * [[Generalized mean]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{MathWorld|urlname=PythagoreanMeans|title=Pythagorean Means|author=Cantrell, David W.}}
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| * Nice comparison of Pythagorean means with emphasis on the [http://www.cse.unsw.edu.au/~teachadmin/info/harmonic3.html harmonic mean].
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| [[Category:Means]]
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