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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.

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:

$A(x_{1},\ldots ,x_{n})={\frac {1}{n}}(x_{1}+\cdots +x_{n})$
$G(x_{1},\ldots ,x_{n})={\sqrt[{n}]{x_{1}\cdots x_{n}}}$
$H(x_{1},\ldots ,x_{n})={\frac {n}{{\frac {1}{x_{1}}}+\cdots +{\frac {1}{x_{n}}}}}$

Each mean satisfies a set of common axiums:

Value preservation: $M(x,x,\ldots ,x)=x$
First order homogeneity: $M(bx_{1},\ldots ,bx_{n})=bM(x_{1},\ldots ,x_{n})$
Invariance under exchange: $M(\ldots x_{i},\ldots x_{j},\ldots )=M(\ldots x_{j},\ldots x_{i},\ldots )$ for any i and j.
Averaging: $\min(x_{1},\ldots ,x_{n})\leq M(x_{1},\ldots ,x_{n})\leq \max(x_{1},\ldots ,x_{n})$

There is an ordering to these means (if all of the $x_{i}$ are positive), along with the quadratic mean $Q={\sqrt {\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{n}}}$ :

H(x_{1},\ldots ,x_{n})\leq G(x_{1},\ldots ,x_{n})\leq A(x_{1},\ldots ,x_{n})\leq Q(x_{1},\ldots ,x_{n})

with equality holding if and only if the $x_{i}$ are all equal. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. This inequality sequence can be proved for the n=2 case for the numbers a and b using a sequence of right triangles (x, y, z) with hypotenuse z and the Pythagorean theorem, which states that x² + y² = z² and implies that z>x and z>y. The right triangles are^[1]

({\frac {b-a}{b+a}}{\sqrt {ab}},{\frac {2ab}{a+b}},{\sqrt {ab}})=({\frac {b-a}{b+a}}{\sqrt {ab}},H(a,b),G(a,b)),

showing that H(a,b) < G(a,b);

({\frac {b-a}{2}},{\sqrt {ab}},{\frac {a+b}{2}})=({\frac {b-a}{2}},G(a,b),A(a,b)),

showing that G(a,b) < A(a,b);

and

({\frac {b-a}{2}},{\frac {a+b}{2}},{\sqrt {\frac {a^{2}+b^{2}}{2}}})=({\frac {b-a}{2}},A(a,b),Q(a,b)),

showing that A(a,b) < Q(a,b).

References

^ 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.

External links

Pythagorean means on MathWorld

[1] 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.

[1]

@@ Line 6: / Line 6: @@
 * <math> H(x_1, \ldots, x_n) = \frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}} </math>
-Each of these means satisfies the properties:
+Each mean satisfies a set of common axiums:
+* Value preservation: <math> M(x,x, \ldots,x) = x </math>
-* <math> M(x,x, \ldots,x) = x </math>
+* First order homogeneity: <math> M(bx_1, \ldots, bx_n) = b M(x_1, \ldots, x_n) </math>
-* <math> M(bx_1, \ldots, bx_n) = b M(x_1, \ldots, x_n) </math>
+* Invariance under exchange: <math> M(\ldots x_i, \ldots x_j, \ldots ) = M(\ldots x_j, \ldots x_i, \ldots) </math> for any i and j.
+* Averaging: <math> \min(x_1,\ldots,x_n) \leq M(x_1,\ldots,x_n) \leq \max(x_1,\ldots,x_n)</math>
 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>:
-: <math> Q(x_1,\ldots,x_n) \geq A(x_1,\ldots,x_n) \geq G(x_1,\ldots,x_n) \geq H(x_1,\ldots,x_n) </math>
+: <math> H(x_1,\ldots,x_n) \leq G(x_1,\ldots,x_n) \leq A(x_1,\ldots,x_n) \leq Q(x_1,\ldots,x_n) </math>
 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 ''n''=2 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 ''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup> and implies that ''z''>''x'' and ''z''>''y''.  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>

Pythagorean means: Difference between revisions

Revision as of 23:11, 1 April 2011

See also

References

External links

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