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An approximation is anything that is intentionally similar but not exactly equal to something else.


Etymology and usage

The word ''approximation'' is derived from
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
''approximatus'', from ''proximus'' meaning ''very near'' and the
prefix A prefix is an affix which is placed before the stem of a word. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the word to which it is affixed. Prefixes, like other affixes, can b ...
''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. 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. 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
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, it is also frequently applied to such things as mathematical functions,
shape A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
s, and
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
s. 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 Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
prevents use of exact representations. The type of approximation used depends on the available
information Information is an Abstraction, abstract concept that refers to something which has the power Communication, to inform. At the most fundamental level, it pertains to the Interpretation (philosophy), interpretation (perhaps Interpretation (log ...
, 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.


Mathematics

Approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
is a branch of mathematics, and a quantitative part of
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
.
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
deals with approximations of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s. 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 In computing, a roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Ro ...
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. 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum 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.


Typography

The approximately equals sign, ≈, was introduced by British mathematician
Alfred Greenhill Sir Alfred George Greenhill (29 November 1847 in London – 10 February 1927 in London), was a British mathematician. George Greenhill was educated at Christ's Hospital School and from there he went to St John's College, Cambridge in 1866. In ...
in 1892, in his book ''Applications of Elliptic Functions''.


LaTeX symbols

Typical meanings of
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...
symbols. * \approx (\approx) : approximate equality, like \pi \approx 3.14. * \not\approx (\not\approx) : inequality, despite any approximation (1 \not\approx 2). * \simeq (\simeq) : function asymptotic equivalence, like f(n) \simeq 3n^2 . ** Thus, \pi \simeq 3.14 is wrong under this definition, despite wide use. * \sim (\sim) : function proportionality; the f(n) used in \simeq is f(n) \sim n^2 . * \cong (\cong) : figure congruence, like \Delta ABC \cong \Delta A'B'C' . * \eqsim (\eqsim) : equal up to a constant. * \lessapprox (\lessapprox) and \gtrapprox (\gtrapprox) : either an inequality holds or approximate equality.


Unicode

Approximate equalities denoted by wavy or dotted symbols.


Science

Approximation arises naturally in
scientific experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when ...
s. 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 The history of science covers the development of science from ancient history, ancient times to the present. It encompasses all three major branches of science: natural science, natural, social science, social, and formal science, formal. Pr ...
shows that earlier theories and laws can be ''approximations'' to some deeper set of laws. Under the
correspondence principle In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
, a new scientific theory should reproduce the results of older, well-established, theories in those domains where the old theories work. 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 A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
often approximate the shape of the Earth as a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
even though more accurate representations are possible, because many physical characteristics (e.g.,
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
) 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. An approximate solution is effected by performing
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
s. 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 Philosophy of science is the branch of philosophy concerned with the foundations, methods, and implications of science. Amongst its central questions are the difference between science and non-science, the reliability of scientific theories, ...
accept that empirical
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...
s are always ''approximations'' — they do not perfectly represent what is being measured.


Law

Within the
European Union The European Union (EU) is a supranational union, supranational political union, political and economic union of Member state of the European Union, member states that are Geography of the European Union, located primarily in Europe. The u ...
(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,European Commission
Guide to the Approximation of European Union Environmental Legislation
last updated 2 August 2019, accessed 15 November 2022
and as a continuing process when required by an
EU Directive A directive is a legal act of the European Union that requires Member state of the European Union, member states to achieve particular goals without dictating how the member states achieve those goals. A directive's goals have to be made the go ...
. ''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".EUR-Lex
Directive (EU) 2015/2436 of the European Parliament and of the Council of 16 December 2015 to approximate the laws of the Member States relating to trade marks (recast) (Text with EEA relevance)
published 23 December 2015, accessed 15 November 2022
The
European Commission The European Commission (EC) is the primary Executive (government), executive arm of the European Union (EU). It operates as a cabinet government, with a number of European Commissioner, members of the Commission (directorial system, informall ...
describes approximation of law as "a unique obligation of membership in the European Union".


See also

* * * * * * Double tilde (disambiguation)Various meanings of ~~ or ≈ * * * * * * * * * * * * * * *


References


External links

* {{Authority control Numerical analysis Equivalence (mathematics) Comparison (mathematical)