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The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean ...
In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of those two segments equals the altitude.
Suppose AC = x 1 and BC = x 2. Construct perpendiculars to [AB] at D and C respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean.
The geometric mean of two positive numbers is never greater than the arithmetic mean. [3] So the geometric means are an increasing sequence g 0 ≤ g 1 ≤ g 2 ≤ ...; the arithmetic means are a decreasing sequence a 0 ≥ a 1 ≥ a 2 ≥ ...; and g n ≤ M(x, y) ≤ a n for any n. These are strict inequalities if x ≠ y.
The power mean could be generalized further to the generalized f-mean: (, …,) = (= ()) This covers the geometric mean without using a limit with f(x) = log(x). The power mean is obtained for f(x) = x p. Properties of these means are studied in de Carvalho (2016).
Mazur's torsion theorem (algebraic geometry) Mean value theorem ; Measurable Riemann mapping theorem (conformal mapping) Mellin inversion theorem (complex analysis) Menelaus's theorem ; Menger's theorem (graph theory) Mercer's theorem (functional analysis) Mermin–Wagner theorem ; Mertens's theorems (number theory)
It can be obtained from the mean value theorem by choosing () = . S 2 ( x , y ) {\displaystyle S_{2}(x,y)} is the arithmetic mean . S 3 ( x , y ) = Q M ( x , y , G M ( x , y ) ) {\displaystyle S_{3}(x,y)=QM(x,y,GM(x,y))} is a connection to the quadratic mean and the geometric mean .
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