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.

Branch of mathematics concerned with properties of and relations among integers. It is a popular subject among amateur mathematicians and students because of the wealth of seemingly simple problems that can be posed. Answers are much harder to come up with. It has been said that any unsolved mathematical problem of any interest more than a century old belongs to number theory. One of the best examples, recently solved, is Fermat's last theorem.

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Branch of mathematics that deals with the properties of numbers and ways of combining them through addition, subtraction, multiplication, and division. Initially it dealt only with the counting numbers, but its definition has broadened to include all real numbers. The most important arithmetic properties (where math.a and math.b are real numbers) are the commutative laws of addition and multiplication, math.a + math.b = math.b + math.a and math.amath.b = math.bmath.a; the associative laws of addition and multiplication, math.a + (math.b + math.c) = (math.a + math.b) + math.c and math.a(math.bmath.c) = (math.amath.b)math.c; and the distributive law, which connects addition and multiplication, math.a(math.b + math.c) = math.amath.b + math.amath.c. These properties include subtraction (addition of a negative number) and division (multiplication by a fraction).

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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:

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

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

Then iterate this operation with a₁ taking the place of x and g₁ 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).

Example

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

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

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

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	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):

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

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

• Inequality of arithmetic and geometric means

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