Editing Approximation (section)

===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}} |
}}