Difference between revisions of "Geometric mean"
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− | The ''' | + | The '''geometric mean''' of a collection of <math>n</math> [[positive]] [[real number]]s is the <math>n</math>th [[root]] of the product of the numbers. Note that if <math>n</math> is even, we take the positive <math>n</math>th root. It is analogous to the [[arithmetic mean]] (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers <math>b</math> and <math>c</math> is the number <math>a</math> such that <math>a + a = b + c</math>, while the geometric mean of the numbers <math>b</math> and <math>c</math> is the number <math>g</math> such that <math>g\cdot g = b\cdot c</math>. |
− | The Geometric Mean is a | + | == Examples == |
+ | The geometric mean of the numbers 6, 4, 1 and 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>. | ||
+ | |||
+ | The geometric mean features prominently in the [[Arithmetic Mean-Geometric Mean Inequality]]. | ||
+ | |||
+ | The geometric mean arises in [[geometry]] in the following situation: if <math>AB</math> is a [[chord]] of [[circle]] <math>O</math> with [[midpoint]] <math>M</math> and <math>M</math> divides the [[diameter]] passing through it into pieces of length <math>a</math> and <math>b</math> then the length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>. | ||
+ | |||
+ | <asy> | ||
+ | size(150); | ||
+ | pointfontsize=8; | ||
+ | pathfontsize=8; | ||
+ | pair A=(3,4),B=(3,-4),M=(3,0); | ||
+ | D((-5,0)--(5,0)); D(M--B); | ||
+ | MC("\sqrt{ab}",D(A--M,orange+linewidth(1)),W); | ||
+ | MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S); | ||
+ | MC("b",D((3,-0.3)--(5,-0.3),black,Arrows),S); | ||
+ | D(CR(D((0,0)),5)); | ||
+ | D("A",A,N); D("B",B);D("M",M,NE); | ||
+ | </asy> | ||
+ | |||
+ | == Practice Problems == | ||
+ | ===Introductory Problems=== | ||
+ | * [[1966 AHSME Problems/Problem 3]] | ||
+ | |||
+ | == See Also == | ||
+ | *[[Arithmetic Mean]] | ||
+ | *[[AM-GM]] |
Latest revision as of 21:04, 11 July 2008
The geometric mean of a collection of positive real numbers is the
th root of the product of the numbers. Note that if
is even, we take the positive
th root. It is analogous to the arithmetic mean (with addition replaced by multiplication) in the following sense: the arithmetic mean of two numbers
and
is the number
such that
, while the geometric mean of the numbers
and
is the number
such that
.
Examples
The geometric mean of the numbers 6, 4, 1 and 2 is .
The geometric mean features prominently in the Arithmetic Mean-Geometric Mean Inequality.
The geometric mean arises in geometry in the following situation: if is a chord of circle
with midpoint
and
divides the diameter passing through it into pieces of length
and
then the length of line segment
is the geometric mean of
and
.