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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 ...
Two thin lenses of focal length f 1 and f 2 in series is equivalent to two thin lenses of focal length f hm, their harmonic mean, in series. Expressed as optical power, two thin lenses of optical powers P 1 and P 2 in series is equivalent to two thin lenses of optical power P am, their arithmetic mean, in series.
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.
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.
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]
511 Gm (3.4 au) – average diameter of Mira, a pulsating red giant and the progenitor of the Mira variables. It is an asymptotic giant branch star. [182] 570 Gm (3.8 au) – length of the tail of Comet Hyakutake measured by Ulysses; the actual value could be much higher; 590 Gm (3.9 au) – diameter of the Pistol Star, a blue hypergiant star [183]
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