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An approximation is anything that is intentionally similar but not exactly equal to something else.
| | This article needs additional citations for verification .(April 2013) |
An approximation is anything that is intentionally similar but not exactly equal to something else.
The word approximation is derived from Latin approximatus, from proximus meaning very near and the prefix ad- (ad- before p becomes ap- by assimilation) meaning to. [1] Words like approximate, approximately and approximation are used especially in technical or scientific contexts. In everyday English, words such as roughly or around are used with a similar meaning. [2] It is often found abbreviated as approx.
The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock).
Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.
In science, approximation can refer to using a simpler process or model when the correct model is difficult to use. An approximate model is used to make calculations easier. Approximations might also be used if incomplete information prevents use of exact representations.
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.
Approximation theory is a branch of mathematics, and a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers.
Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. For example, 1.5 × 106 means that the true value of something being measured is 1,500,000 to the nearest hundred thousand (so the actual value is somewhere between 1,450,000 and 1,550,000); this is in contrast to the notation 1.500 × 106, which means that the true value is 1,500,000 to the nearest thousand (implying that the true value is somewhere between 1,499,500 and 1,500,500).
Numerical approximations sometimes result from using a small number of significant digits. Calculations are likely to involve rounding errors and other approximation errors. Log tables, slide rules and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results. [3] Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits.
Related to approximation of functions is the asymptotic value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum (k/2)+(k/4)+(k/8)+...(k/2^n) is asymptotically equal to k. No consistent notation is used throughout mathematics and some texts use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around.
| ≅ ≈ | |
|---|---|
Approximately equal to Almost equal to | |
| In Unicode | U+2245≅APPROXIMATELY EQUAL TO (≅, ≅) U+2248≈ALMOST EQUAL TO (≈, ≈, ≈, ≈, ≈, ≈) |
| Different from | |
| Different from | U+2242≂MINUS TILDE |
| Related | |
| See also | U+2249≉ NOT ALMOST EQUAL TO U+003D= EQUALS SIGN U+2243≃ASYMPTOTICALLY EQUAL TO |
The approximately equals sign, ≈, was introduced by British mathematician Alfred Greenhill. [4]
Symbols used in LaTeX markup.
\approx), usually to indicate approximation between numbers, like .
\not\approx), usually to indicate that numbers are not approximately equal (1 2).
\simeq), usually to indicate asymptotic equivalence between functions, like . So writing would be wrong under this definition, despite wide use.
\sim), usually to indicate proportionality between functions, the same of the line above will be .
\cong), usually to indicate congruence between figures, like .
\eqsim), usually to indicate that two quantities are equal up to constants.
\lessapprox) and (
\gtrapprox), usually to indicate that either the inequality holds or the two values are approximately equal.Symbols used to denote items that are approximately equal are wavy or dotted equals signs. [5]
Approximation arises naturally in scientific experiments. The predictions of a scientific theory can differ from actual measurements. This can be because there are factors in the real situation that are not included in the theory. For example, simple calculations may not include the effect of air resistance. Under these circumstances, the theory is an approximation to reality. Differences may also arise because of limitations in the measuring technique. In this case, the measurement is an approximation to the actual value.
The history of science shows that earlier theories and laws can be approximations to some deeper set of laws. Under the correspondence principle, a new scientific theory should reproduce the results of older, well-established, theories in those domains where the old theories work. [7] The old theory becomes an approximation to the new theory.
Some problems in physics are too complex to solve by direct analysis, or progress could be limited by available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly. Physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical characteristics (e.g., gravity) are much easier to calculate for a sphere than for other shapes.
Approximation is also used to analyze the motion of several planets orbiting a star. This is extremely difficult due to the complex interactions of the planets' gravitational effects on each other. [8] An approximate solution is effected by performing iterations. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained.
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 most common versions of philosophy of science accept that empirical measurements are always approximations — they do not perfectly represent what is being measured.
Within the European Union (EU), "approximation" refers to a process through which EU legislation is implemented and incorporated within Member States' national laws, despite variations in the existing legal framework in each country. Approximation is required as part of the pre-accession process for new member states, [9] and as a continuing process when required by an EU Directive. Approximation is a key word generally employed within the title of a directive, for example the Trade Marks Directive of 16 December 2015 serves "to approximate the laws of the Member States relating to trade marks". [10] The European Commission describes approximation of law as "a unique obligation of membership in the European Union". [9]
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric). If and , then (transitive).
The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.
The method of least squares is a parameter estimation method in regression analysis based on minimizing the sum of the squares of the residuals made in the results of each individual equation.
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter . The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration.
The tilde˜ or ~, is a grapheme with a number of uses. The name of the character came into English from Spanish, which in turn came from the Latin titulus, meaning 'title' or 'superscription'. Its primary use is as a diacritic (accent) in combination with a base letter; but, for historical reasons, it is also used in standalone form within a variety of contexts.
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation.
In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though and the left-exponent xb are common.
The equals sign or equal sign, also known as the equality sign, is the mathematical symbol =, which is used to indicate equality in some well-defined sense. In an equation, it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value.
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations (iterations).
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
Methods of computing square roots are algorithms for approximating the non-negative square root of a positive real number . Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed to some finite precision: these methods typically construct a series of increasingly accurate approximations.
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly.
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.
In numerical analysis, Gauss–Legendre quadrature is a form of Gaussian quadrature for approximating the definite integral of a function. For integrating over the interval [−1, 1], the rule takes the form:
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, statistics, and calculus.
≐ approaches a limit
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