Definitions

# arithmetic

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

First compute the arithmetic mean of x and y and call it a1. Next compute the geometric mean of x and y and call it g1; this is the square root of the product xy:

$a_1 = frac\left\{x+y\right\}\left\{2\right\}$

$g_1 = sqrt\left\{xy\right\}.$

Then iterate this operation with a1 taking the place of x and g1 taking the place of y. In this way, two sequences (an) and (gn) are defined:

$a_\left\{n+1\right\} = frac\left\{a_n + g_n\right\}\left\{2\right\}$

$g_\left\{n+1\right\} = sqrt\left\{a_n g_n\right\}.$

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).

## Example

To find the arithmetic-geometric mean of a0 = 24 and g0 = 6, first calculate their arithmetic mean and geometric mean, thus:

$a_1=frac\left\{24+6\right\}\left\{2\right\}=15,$

$g_1=sqrt\left\{24 times 6\right\}=12,$

and then iterate as follows:

$a_2=frac\left\{15+12\right\}\left\{2\right\}=13.5,$

$g_2=sqrt\left\{15 times 12\right\}=13.41640786500dots$ etc.

The first four iterations give the following values:

n an gn
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):

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

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.

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

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 sqrtdef 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```