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arithmetic, branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with Numbers for practical purposes, e.g., computation of areas, proportions, costs, and the like. The four fundamental operations of this study are addition, subtraction, multiplication, and division. In advanced study the concept of number is greatly generalized to include not only complex numbers, but also quaternions, tensors, and abstract entities with no other meaning than that they obey certain laws (see algebra). The division of arithmetic into the practical and the theoretical dates back to classical Greek times, when the term *logistic* referred to elementary arithmetic and the term *arithmetic* was reserved for the theory.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

In mathematics, the arithmetic-geometric mean (AGM) of two positive real numbers x and y is defined as follows:## Example

To find the arithmetic-geometric mean of a_{0} = 24 and g_{0} = 6, first calculate their arithmetic mean and geometric mean, thus:

First compute the arithmetic mean of x and y and call it a_{1}. Next compute the geometric mean of x and y and call it g_{1}; this is the square root of the product xy:

- $a\_1\; =\; frac\{x+y\}\{2\}$

- $g\_1\; =\; sqrt\{xy\}.$

Then iterate this operation with a_{1} taking the place of x and g_{1} taking the place of y. In this way, two sequences (a_{n}) and (g_{n}) are defined:

- $a\_\{n+1\}\; =\; frac\{a\_n\; +\; g\_n\}\{2\}$

- $g\_\{n+1\}\; =\; sqrt\{a\_n\; g\_n\}.$

These two sequences converge to the same number, which is the arithmetic-geometric mean of x and y; it is denoted by M(x, y), or sometimes by agm(x, y).

- $a\_1=frac\{24+6\}\{2\}=15,$

- $g\_1=sqrt\{24\; times\; 6\}=12,$

and then iterate as follows:

- $a\_2=frac\{15+12\}\{2\}=13.5,$

- $g\_2=sqrt\{15\; times\; 12\}=13.41640786500dots$ etc.

The first four iterations give the following values:

n a _{n}g _{n}0 24 6 1 15 12 2 13.5 13.41640786500... 3 13.45820393250... 13.45813903099... 4 13.45817148175... 13.45817148171... The arithmetic-geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately 13.45817148173.

## Properties

M(x, y) is a number between the geometric and arithmetic mean of x and y; in particular it is between x and y.If r > 0, then M(rx, ry) = r M(x, y).

There is a closed form expression for M(x,y):

- $Mu(x,y)\; =\; frac\{pi\}\{4\}\; cdot\; frac\{x\; +\; y\}\{K\; left(left(frac\{x\; -\; y\}\{x\; +\; y\}\; right)^2\; right)\; \}$

where K(x) is the complete elliptic integral of the first kind.

The reciprocal of the arithmetic-geometric mean of 1 and the square root of 2 is called Gauss's constant.

- $frac\{1\}\{Mu(1,\; sqrt\{2\})\}\; =\; G\; =\; 0.8346268dots$

named after Carl Friedrich Gauss.

The geometric-harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. The arithmetic-harmonic mean can be similarly defined, but takes the same value as the geometric mean.

## Implementation in Python

The following example code in Python computes the arithmetic-geometric mean of two positive real numbers:from math import sqrt

def avg(a, b, delta=None):

if None==delta:

delta=(a+b)/2*1E-10

if(abs(b-a)>delta):

return avg((a+b)/2.0, sqrt(a*b), delta)

else:

return (a+b)/2.0

## See also

## References

- Jonathan Borwein, Peter Borwein, Pi and the AGM. A study in analytic number theory and computational complexity. Reprint of the 1987 original. Canadian Mathematical Society Series of Monographs and Advanced Texts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp. ISBN 0-471-31515-X

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Last updated on Friday October 10, 2008 at 04:53:00 PDT (GMT -0700)

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