Approximation

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Approximation

An approximation (represented by the symbol ≈) is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.

Approximations may be used because incomplete information prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.

For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behaviours&mdash;e.g. gravity&mdash;are much easier to calculate for a sphere than for less regular shapes.

The problem consisting of two or more planets orbiting around a sun has no exact solution. Often, ignoring the gravitational effects of the planets gravitational pull on each other and assuming that the sun does not move achieve a good approximation. The use of perturbations to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.

The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.

cience

The scientific method is carried out with a constant interaction between scientific laws (theory) and empirical measurements, which are constantly compared to one another.

The approximation also refers to using a simpler process. This model is used to make predictions easier. The most common versions of philosophy of science accept that empirical measurements are always "approximations"&mdash;they do not perfectly represent what is being measured. The history of science indicates that the scientific laws commonly felt to be "true" at any time in history are only "approximations" to some deeper set of laws. For example, attempting to resolve a model using outdated physical laws alone incorporates an inherent source of error, which should be corrected by approximating the quantum effects not present in these laws.

Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements. This is the correspondence principle.

Mathematics

(formal) ~ (informal)
symbols representing approximation.
Approximation usually occurs when an exact form or an exact numerical number is unknown. Howeversome known form may exist and may be able to represent the real form so that no significant deviation can be found. It also is used when a number is not rational, such as the number π, which often is shortened to 3.14, or &radic;7 as ≈ 2.65.Numerical approximations sometimes result from using a small number of significant digits. Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximation to real numbers by rational numbers. The symbol "≈" means "approximately equal to"; tilde (~) and the Libra sign () are common alternatives.

ee also

* Approximation error
* Congruence
* Estimation
* Fermi estimate
* Linear approximation
* Newton's method
* Numerical analysis
* Orders of approximation
* Runge-Kutta methods
* Taylor series
* Least squares

References

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• approximation — [ aprɔksimasjɔ̃ ] n. f. • 1314 chir.; lat. médiév. approximatio, du bas lat. approximare « approcher » 1 ♦ (1740) Math. Calcul par lequel on approche d une grandeur réelle; détermination approchée. Méthodes par approximations dans le calcul d une …   Encyclopédie Universelle

• Approximation — (lat.: proximus, „der Nächste“) ist zunächst ein Synonym für Näherung; der Begriff wird in der Mathematik allerdings noch präzisiert. Es gibt vor allem zwei Gründe in der Mathematik, Näherungen zu untersuchen: Einmal könnte das Objekt des… …   Deutsch Wikipedia

• approximation — ap‧prox‧i‧ma‧tion [əˌprɒksˈmeɪʆn ǁ əˌprɑːk ] noun [countable] 1. a number, amount etc that is not exact, but is intended to give an idea of the real number, amount etc: • The figures are meant to be an approximation of the costs. 2. something… …   Financial and business terms

• Approximation — Ap*prox i*ma tion n. [Cf. F. approximation, LL. approximatio.] 1. The act of approximating; a drawing, advancing or being near; approach; also, the result of approximating. [1913 Webster] The largest capacity and the most noble dispositions are… …   The Collaborative International Dictionary of English

• approximation — APPROXIMATION. s. f. Terme de Mathématique. Opération par laquelle on approche toujours de plus en plus de la valeur d une quantité cherchée, sans la trouver exactement. Résoudre un problème par approximation. On dit: Un calcul par approximation …   Dictionnaire de l'Académie Française 1798

• Approximation — (lat.), Annäherung (s. d.); approximativ, annähernd …   Meyers Großes Konversations-Lexikon

• Approximation — (lat.), Annäherung, bei mathem. Rechnungen die Auffindung von Werten, welche dem wahren Wert einer Größe sehr nahe kommen; approximándo, annähernd, annäherungsweise, nach und nach; approximatīv, annähernd, ungefähr; der Wahrheit nahe kommend;… …   Kleines Konversations-Lexikon

• approximation — I noun approach, calculation, computation, contiguity, correspondence, estimate, estimation, inexactitude, inexactness, interpolation, likeness, measure, nearness, neighborhood, parity, propinquity, proximity, reckoning, resemblance, rough… …   Law dictionary

• approximation — (n.) early 15c., act of coming near or close, noun of action from APPROXIMATE (Cf. approximate) (v.). Meaning result of approximating is from 1650s …   Etymology dictionary

• approximation — [ə präk΄sə mā′shən] n. 1. the act or state of approximating, or coming close 2. an estimate, guess, or mathematical result that is approximately correct or close enough to exactness for a particular purpose …   English World dictionary

• Approximation — Une approximation est une représentation imprécise ayant toutefois un lien étroit avec la quantité ou l’objet qu’elle reflète : approximation d’un nombre (de Pi par 3.14, de la vitesse instantanée d’un véhicule par sa vitesse moyenne entre… …   Wikipédia en Français