The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

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

Next compute the geometric mean of and and call it ; this is the square root of the product:

These two sequences converge to the same number, which is the arithmetic – geometric mean of and ; it is denoted by, or sometimes by.

To find the arithmetic – geometric mean of and, first calculate their arithmetic mean and geometric mean, thus:

An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

The covalent energy of a bond is approximately, by quantum mechanical calculations, the geometric mean of the two energies of covalent bonds of the same molecules ( which is approximately equal to the arithmetic mean-which is applied in the first formula above-as the energies are of the similar value, except for the highly electropositive elements i. e. when there is a larger difference of two dissociation energies, but the geometric mean is more accurate and almost always gives a positive excess energy, due to ionic bonding ), and there is an additional energy that comes from ionic factors, i. e. polar character of the bond.

It repeatedly replaces two numbers by their arithmetic and geometric mean, in order to approximate their arithmetic-geometric mean.

The values can be substituted into Legendre ’ s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with and.

A geometric mean is often used when comparing different items-finding a single " figure of merit " for these items-when each item has multiple properties that have different numeric ranges.

For example, the geometric mean can give a meaningful " average " to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability.

If an arithmetic mean was used instead of a geometric mean, the financial viability is given more weight because its numeric range is larger-so a small percentage change in the financial rating ( e. g. going from 80 to 90 ) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability ( e. g. going from 2 to 5 ).

In his youth he went to the continent and taught mathematics in Paris, where he published or edited, between the years 1612 and 1619, various geometric and algebraic tracts.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

In mathematics, a generalized mean, also known as power mean or Hölder mean ( named after Otto Hölder ), is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set.

Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems.

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.

* Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

Descartes ' influence in mathematics is equally apparent ; the Cartesian coordinate system — allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system ( and conversely, shapes to be described as equations ) — was named after him.

* Spring ( mathematics ), a geometric surface in the shape of a helically coiled tube

His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics.

In mathematics, physics, and engineering, a Euclidean vector ( sometimes called a geometric or spatial vector, or — as here — simply a vector ) is a geometric object that has a magnitude ( or length ) and direction and can be added to other vectors according to vector algebra.

When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors.

The case for a connection with Islamic mathematics is much stronger for the development of the geometric patterns with which arabesques are often combined in art.

Also his contribution to mathematics should be noted ; in 1658, he found the length of an arc of the cycloid using an exhaustion proof based on dissections to reduce the problem to summing segments of chords of a circle which are in geometric progression.

This philosophical view of mathematics ( see below ) has had a significant impact on Khayyám's celebrated approach and method in geometric algebra and in particular in solving cubic equations.

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis.

* Rotor ( mathematics ), an n-blade object in geometric algebra, which rotates another n-blade object about a fixed or translated point

It is presumed that this negative head was associated with some geometric factor of the assembly, since different readings were obtained with the same fluid and the only apparent difference was the assembly and disassembly of the apparatus.

In the field of computer graphics, an anisotropic surface will change in appearance as it is rotated about its geometric normal, as is the case with velvet.

In Aristotle this is categorized as the difference between ' arithmetic ' and ' geometric ' ( i. e. proportional ) equality.

For two geometric objects P and Q represented by the relations P ( x, y ) and Q ( x, y ) the intersection is the collection of all points ( x, y ) which are in both relations.

The true position ( or geometric position ) is the direction of the straight line between the observer and star at the instant of observation.

In calculus, this picture also gives a geometric proof of the derivative if one sets and interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, where the coefficient of the linear term ( in ) is the area of the n faces, each of dimension

Finite geometry is the study of geometric systems having only a finite number of points.

Two-dimensional ( unshaded ) cross-stitch in floral and geometric patterns, usually worked in black and red cotton floss on linen, is characteristic of folk embroidery in Eastern and Central Europe.

Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

The circumference of a circle is of special importance to geometric and trigonometric concepts.

Rendering is practically exclusively concerned with the particle aspect of light physics — known as geometric optics.

Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the x-axis.

The geometric definition of a constructible point is as follows.

The radius of curvature is introduced completely formally ( without need for geometric interpretation ) as:

However, in this approach the question of the change in radius of curvature with s is handled completely formally, consistent with a geometric interpretation, but not relying upon it, thereby avoiding any questions the image above might suggest about neglecting the variation in ρ.

This is a consequence of the fact that the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number.

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.