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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
Proof without words of the AM–GM inequality: PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. Visual proof that (x + y) 2 ≥ 4xy. Taking square roots and dividing by two gives the AM ...
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians [1] because of their importance in geometry and music.
Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b [note 1] Fréchet mean; Generalized mean; Inequality of arithmetic and geometric means; Sample mean and covariance; Standard deviation
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In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.. It is the most appropriate average for ratios and rates such as speeds, [1] [2] and is normally only used for positive arguments.
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umm, in the polya proof, the meaning of mu and rho are not given —Preceding unsigned comment added by 128.36.86.60 (talk • contribs) But they are, in the second paragraph, where it says: Let μ be the arithmetic mean, and let ρ be the geometric mean. Michael Hardy 22:07, 11 December 2006 (UTC)