Difference between revisions of "Geometric mean"
m (needs an image; also, I just did an information dump -- could be greatly improved!) |
|||
| Line 7: | Line 7: | ||
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>. | 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> | ||
The geometric mean also arises in the following common [[word problem]]: if a driver travels half the distance of a trip at a speed of <math>a</math> miles per hour and the other half at a speed of <math>b</math> miles per hour, the average speed over the whole trip is the geometric mean of <math>a</math> and <math>b</math>. (If the driver spent half the ''time'' of the trip at each speed, we would instead get the arithmetic mean.) | The geometric mean also arises in the following common [[word problem]]: if a driver travels half the distance of a trip at a speed of <math>a</math> miles per hour and the other half at a speed of <math>b</math> miles per hour, the average speed over the whole trip is the geometric mean of <math>a</math> and <math>b</math>. (If the driver spent half the ''time'' of the trip at each speed, we would instead get the arithmetic mean.) | ||
Revision as of 18:49, 2 April 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 ![$\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}$](http://latex.artofproblemsolving.com/f/f/d/ffde0cff2f31a4ba62c966e51e6fd22fe1d905df.png)
.
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 .
![[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]](http://latex.artofproblemsolving.com/e/0/f/e0f7e268dd5d382995a16399cbc316344ad6f4af.png)
The geometric mean also arises in the following common word problem: if a driver travels half the distance of a trip at a speed of miles per hour and the other half at a speed of miles per hour, the average speed over the whole trip is the geometric mean of and . (If the driver spent half the time of the trip at each speed, we would instead get the arithmetic mean.)
