Editing Approximation

{{short description|Something roughly the same as something else}}
{{for|the sound change|Lenition}}
{{Redirect-distinguish|Approximate|Approximant}}
{{Refimprove|date=April 2013}}

An '''approximation''' is anything that is intentionally similar but not exactly [[equality (mathematics)|equal]] to something else.

==Etymology and usage==
The word ''approximation'' is derived from [[Latin]] ''approximatus'', from ''proximus'' meaning ''very near'' and the [[prefix]] ''ad-'' (''ad-'' before ''p'' becomes ap- by [[assimilation (phonology)|assimilation]]) meaning ''to''.<ref>The Concise Oxford Dictionary, ''Eighth edition 1990, {{ISBN|0-19-861243-5}}''</ref> 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.<ref>Longman Dictionary of Contemporary English, ''Pearson Education Ltd 2009, {{ISBN|978 1 4082 1532 6}}''</ref> 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]]s, it is also frequently applied to such things as [[Function (mathematics)|mathematical functions]], [[shape]]s, and [[physical law]]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]] prevents use of exact representations.

The type of approximation used depends on the available [[information]], [[Order of approximation|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]] is a branch of mathematics, and  a quantitative part of [[functional analysis]]. [[Diophantine approximation]] deals with approximations of [[real number]]s by [[rational number]]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<sup>6</sup> 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 × 10<sup>6</sup>, 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 approximation]]s sometimes result from using a small number of [[Significant figures|significant digits]]. Calculations are likely to involve [[Round-off error|rounding errors]] and other [[approximation error]]s. [[Logarithm|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.<ref>{{Cite web |url=http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |title=Numerical Computation Guide |access-date=2013-06-16 |archive-url=https://web.archive.org/web/20160406101256/http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |archive-date=2016-04-06 |url-status=dead }}</ref> 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 analysis|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 {{tmath|k/2+k/4+k/8+ \cdots +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 ===
{{See also|Glossary of mathematical symbols#Equality, equivalence and similarity}}
[[File:Approximately_Equal_Sign_-_Alfred_Greenhill.png|thumb|One of the first uses of the symbol (≈) "Approximately equal to." - [[Alfred George Greenhill|Alfred Greenhill]] (1892)]]
The '''approximately equals sign''', '''≈''', was introduced by British mathematician [[Alfred Greenhill]] in 1892, in his book ''Applications of Elliptic Functions''.<ref>{{Cite book |last=Greenhill |first=Alfred G. Sir |author-link=Alfred George Greenhill |url=https://quod.lib.umich.edu/u/umhistmath/ACQ7072.0001.001/355?rgn=full+text;view=pdf |title=The Applications of Elliptic Functions |publisher=[[MacMillan and Co]] |year=1892 |isbn=978-1163949573 |location=London |pages=340}}</ref><ref>{{Cite book |last1=Schilling |first1=Anne |url=https://doi.org/10.1142/9808 |title=Linear Algebra as an Introduction to Abstract Mathematics |last2=Nachtergaele |first2=Bruno |last3=Lankham |first3=Isaiah |date=January 2016 |publisher=[[LibreTexts]] |isbn=978-981-4723-79-4 |location=University of California, Davis |chapter=13.3: Some Common Mathematical Symbols and Abbreviations |doi=10.1142/9808 |chapter-url=https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/13%3A_Appendices/13.03%3A_Some_Common_Mathematical_Symbols_and_Abbreviations}}</ref>

{{Infobox symbol
|name=Approximately equal to<br/>Almost equal to 
|mark= ≅&nbsp;≈  
|unicode = {{unichar|2245|Approximately equal to|html=}}<br />{{unichar|2248|Almost equal to |html=}}
|see also = {{unichar|2249|Not almost equal to|nlink=Approximation#Unicode}}<br />{{unichar|3D|nlink=Equals sign}}<br />{{unichar|2243|Asymptotically equal to}}
|different from={{unichar|2242|Minus tilde}}
}}
====LaTeX symbols====
Typical meanings of [[LaTeX]] symbols.
* <math> \approx </math> (<code>\approx</code>) : approximate equality, like <math> \pi \approx 3.14</math>.
* <math> \not\approx </math> (<code>\not\approx</code>) : inequality, despite any approximation (<math>1 \not\approx 2</math>).
* <math> \simeq </math> (<code>\simeq</code>) : function asymptotic equivalence, like <math> f(n) \simeq 3n^2 </math>. 
** Thus, <math> \pi \simeq 3.14 </math> is wrong under this definition, despite wide use.
* <math> \sim </math> (<code>\sim</code>) : function proportionality; the <math>f(n)</math> used in <code>\simeq</code> is <math> f(n) \sim n^2 </math>.
* <math> \cong </math> (<code>\cong</code>) : figure congruence, like <math> \Delta ABC \cong \Delta A'B'C' </math>.
* <math> \eqsim </math> (<code>\eqsim</code>) : equal up to a constant.
* <math>\lessapprox</math> (<code>\lessapprox</code>) and <math>\gtrapprox</math> (<code>\gtrapprox</code>) : either an inequality holds or approximate equality.

====Unicode====
{{See also|Unicode mathematical operators}}

Approximate equalities denoted by wavy or dotted symbols.<ref>{{cite web| title =Mathematical Operators – Unicode| url =https://www.unicode.org/charts/PDF/U2200.pdf| access-date =2013-04-20}}</ref>
{{aligned table |cols=2|class=wikitable
| {{Unichar|223C|TILDE OPERATOR}} | Sometimes indicates [[proportionality (mathematics)|proportionality.]]
| {{Unichar|223D|REVERSED TILDE}} | Sometimes indicates proportionality.
| {{Unichar|2243|ASYMPTOTICALLY EQUAL TO}} | Combined "≈" and "{{=}}" representing [[Asymptotic analysis|asymptotic equality]].
| {{Unichar|2245|APPROXIMATELY EQUAL TO}} | Combined "≈" and "{{=}}" representing [[isomorphism]] or [[congruence relation|congruence]].
| {{unichar|2246|approximately but not actually equal to}} |
| {{unichar|2247|neither approximately nor actually equal to}} |
| {{Unichar|2248|ALMOST EQUAL TO}} |
| {{Unichar|2249|NOT ALMOST EQUAL TO}} |
| {{Unichar|224A|ALMOST EQUAL OR EQUAL TO}} | Combined "≈" and "{{=}}" representing equivalence or approximate equivalence.
| {{Unichar|2250|APPROACHES THE LIMIT}} | Represents a variable, like {{mvar|y}}, approaching a [[limit (mathematics)|limit]], for example, <math>\lim_{x \to \infty} y(x) \doteq 0</math>.<ref>{{cite book |title=D & D Standard Oil & Gas Abbreviator |year=2006 |publisher=PennWell 
 |url=https://books.google.com/books?id=7FPtZp8abSAC&dq=%22%E2%89%90%22+approach+limit&pg=PA366 |access-date=May 21, 2020 |quote=≐ approaches a limit |page=366|isbn=9781593701086 }}</ref>
| {{Unichar|2252|APPROXIMATELY EQUAL TO OR THE IMAGE OF}} | "<big>≈</big>" or "<big>≃</big>" equivalent in [[Japanese language|Japan]], [[Taiwanese Mandarin|Taiwan]], and [[Korean language|Korea]].
| {{Unichar|2253|IMAGE OF OR APPROXIMATELY EQUAL TO}} | Reversed variant of "≒" (U+2252).
| {{Unichar|225F|QUESTIONED EQUAL TO|nlink=≟}} |
| {{unichar|2A85|LESS-THAN OR APPROXIMATE}} |
| {{unichar|2A86|GREATER-THAN OR APPROXIMATE}} |
}}

== Science ==
Approximation arises naturally in [[scientific experiment]]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]] 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.<ref>[https://www.britannica.com/EBchecked/topic/138678/correspondence-principle Correspondence principle] – ''[[Encyclopædia Britannica]]''</ref> 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.<ref>[http://plus.maths.org/content/mathematical-mysteries-three-body-problem The three body problem]</ref> An approximate solution is effected by performing [[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 [[Perturbation theory|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 [[measurement]]s are always ''approximations'' — they do not perfectly represent what is being measured.

==Law==
Within the [[European Union]] (EU), "approximation" refers to a process through which EU legislation is implemented and incorporated within [[Member state of the European Union|Member States]]' national laws, despite variations in the existing legal framework in each country. Approximation is required as part of the [[EU accession|pre-accession process]] for new member states,<ref name=env>European Commission, [https://ec.europa.eu/environment/archives/guide/part1.htm Guide to the Approximation of European Union Environmental Legislation], last updated 2 August 2019, accessed 15 November 2022</ref> and as a continuing process when required by an [[Directive (European Union)|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".<ref>EUR-Lex, [https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A32015L2436&qid=1668514262360 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</ref> The [[European Commission]] describes approximation of law as "a unique obligation of membership in the European Union".<ref name=env />

== See also ==
{{div col}}
* {{Annotated link |Approximation algorithm}}
* {{Annotated link |Approximate computing}}
* {{Annotated link |Approximations of π}}
* {{Annotated link |Binomial approximation}}
* {{Annotated link |Congruence relation}}
* [[Double tilde (disambiguation)]]{{snd}}Various meanings of ~~ or ≈
* {{Annotated link |Estimation}}
* {{Annotated link |Fermi problem}}
* {{Annotated link |Idealization (philosophy of science)}}
* {{Annotated link |Least squares}}
* {{Annotated link |Linear approximation}}
* {{Annotated link |Newton's method}}
* {{Annotated link |Order of approximation}}
* {{Annotated link |Rough set}}
* {{Annotated link |Runge–Kutta methods}}
* {{Annotated link |Significant figures}}
* {{Annotated link |Small-angle approximation}}
* {{Annotated link |Successive-approximation ADC}}
* {{Annotated link |Taylor series}}
* {{Annotated link |Tolerance relation}}
* {{Annotated link |Intuition}}
{{div col end}}

== References ==
{{Reflist}}

== External links ==
{{Wiktionary|approximation}}
* {{Commons category-inline}}

{{Authority control}}

[[Category:Approximations| ]]
[[Category:Numerical analysis]]
[[Category:Equivalence (mathematics)]]
[[Category:Comparison (mathematical)]]
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