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 branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''approximatus'', from ''proximus'' meaning ''very near'' and the
prefix
A prefix is an affix which is placed before the Word stem, stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy'' ...
''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 original 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 c ...
s, it is also frequently applied to such things as
mathematical functions
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
,
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
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) a ...
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 abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
prevents use of exact representations.
The type of approximation used depends on the available
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
,
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, 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
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 by r ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 (e.g. ). The set of all ration ...
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 × 10
6 means that the approximation 1,500,000 has been measured to the nearest hundred thousand (the actual value is somewhere between 1,450,000 and 1,550,000), this is in contrast to the notation 1.500 × 10
6 which measures 1,500,000 to the nearest thousand (therefore giving a true value somewhere between 1,499,500 and 1,500,500).
It also is used when a number is
not rational, such as the number
π, which often is shortened to 3.14159, or 1.414 as the shortened form of .
Numerical approximation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
s sometimes result from using a small number of
significant digits
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expre ...
. Calculations are likely to involve
rounding errors and other
approximation error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
s.
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 tends to infinity. In projective geometry and related contexts, ...
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.
Typography
The approximately equals sign, ≈, was introduced by British mathematician
Alfred Greenhill
Sir Alfred George Greenhill, FRS FRAeS (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 ...
.
LaTeX symbols
Symbols used in
LaTeX
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
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.
Unicode
Symbols used to denote items that are approximately equal are wavy or dotted equals signs.
* : which is also sometimes used to indicate
proportionality
* : which is also sometimes used to indicate proportionality
* : another combination of "≈" and "=", which is used to indicate
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
or
congruence
*
*
*
*
* : yet another combination of "≈" and "=", used to indicate equivalence or approximate equivalence
* : which can be used to represent the approach of a variable, , to a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
; like the common syntax,
≐ 0
* : which is used like "
≈" or "
≃" in
Japan
Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the north ...
,
Taiwan
Taiwan, officially the Republic of China (ROC), is a country in East Asia, at the junction of the East and South China Seas in the northwestern Pacific Ocean, with the People's Republic of China (PRC) to the northwest, Japan to the nort ...
, and
Korea
Korea ( ko, 한국, or , ) is a peninsular region in East Asia. Since 1945, it has been divided at or near the 38th parallel, with North Korea (Democratic People's Republic of Korea) comprising its northern half and South Korea (Republic o ...
* : a reversed variation of
*
*
*
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 a ...
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 times to the present. It encompasses all three major branches of science: natural, social, and formal.
Science's earliest roots can be traced to Ancient Egypt and Meso ...
shows that earlier theories and laws can be ''approximations'' to some deeper set of laws. Under the
correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...
, 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 caus ...
often approximate the
shape of the Earth as a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. 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 () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
) 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 a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
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 ...
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 political and economic union of member states that are located primarily in Europe. The union has a total area of and an estimated total population of about 447million. The EU has often been des ...
(EU), "approximation" refers to a process through which EU legislation is implemented and incorporated within
Member States
A member state is a state that is a member of an international organization or of a federation or confederation.
Since the World Trade Organization (WTO) and the International Monetary Fund (IMF) include some members that are not sovereign 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
The European Union (EU) is a supranational political and economic union of member states that are located primarily in Europe. The union has a total area of and an estimated total population of about 447million. The EU has often been des ...
. ''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 executive of the European Union (EU). It operates as a cabinet government, with 27 members of the Commission (informally known as "Commissioners") headed by a President. It includes an administrative body o ...
describes approximation of law as "a unique obligation of membership in the European Union".
See also
*
*
*
*
*
*
Double tilde (disambiguation) Double tilde (~~ or ≈) may refer to:
*Approximation *Double negation ~(~
* Smart match operator in Perl, ~~
*Double binary NOT operator ( as used in languages like JavaScript and PHP as a quick way to cast variable as integer, where it is ...
Various meanings of ~~ or ≈
*
*
*
*
*
*
*
*
*
*
*
*
*
*
References
External links
*
{{Authority control
Numerical analysis
Equivalence (mathematics)
Comparison (mathematical)