Editing Approximation (section)

== 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}} |
}}