Editing Equals sign

{{Short description|Mathematical symbol of equality}}
{{About|the symbol|the concept of equality in mathematics|Equality (mathematics)}}
{{Redirect-several|dab=no|Double hyphen|Glossary of mathematical symbols#Equality, equivalence and similarity{{!}}Mathematical symbols § Equality}}
{{Infobox symbol
|mark={{=}}
|unicode = {{unichar|3D|Equals sign|html=}}
|see also = {{unichar|2260|Not equal to|nlink=}}<br />{{unichar|2248|almost equal to|nlink=}} <br />{{unichar|2261|identical to|nlink=Triple bar}}
}}
[[Image:2+2.svg|thumb|right|200px|A well-known [[Equality (mathematics)|equality]] featuring the equal sign]] 
The '''equals sign''' ([[British English]]) or '''equal sign''' ([[American English]]), also known as the '''equality sign''', is the [[mathematical symbol]] '''{{char|1==}}''', which is used to indicate [[equality (mathematics)|equality]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Equal |url=https://mathworld.wolfram.com/Equal.html |access-date=2020-08-09 |website=mathworld.wolfram.com |language=en |archive-date=2020-09-14 |archive-url=https://web.archive.org/web/20200914232729/https://mathworld.wolfram.com/Equal.html |url-status=live}}</ref> In an [[equation]] it is placed between two [[expression (mathematics)|expressions]] that have the same value, or for which one studies the conditions under which they have the same value.

In [[Unicode]] and [[ASCII]] it has the [[code point]] U+003D.<ref>{{cite web |title=C0 Controls and Basic Latin Range: 0000–007F |url=https://www.unicode.org/charts/PDF/U0000.pdf |page=0025{{snd}}0041 |publisher=Unicode Consortium |access-date=2021-03-29 |archive-date=2016-05-26 |archive-url=https://web.archive.org/web/20160526182105/http://www.unicode.org/charts/PDF/U0000.pdf |url-status=live }}</ref> It was invented in 1557 by the Welsh mathematician [[Robert Recorde]].

==History==
[[Image:First Equation Ever.png|thumb|The first use of an equals sign, equivalent to 14''x''+15=71 in modern notation. From ''[[The Whetstone of Witte]]'' (1557) by [[Robert Recorde]].]]
[[Image:Recorde - The Whetstone of Witte - equals.jpg|thumb|Recorde's introduction of "="]]

Prior to 16th century there was no common symbol for equality, and equality was usually expressed with a word, such as ''aequales, aequantur, esgale, faciunt, ghelijck'' or ''gleich,'' and sometimes by the abbreviated form ''aeq'', or simply {{angbr|æ}} and {{angbr|œ}}.<ref name=":0">{{cite encyclopedia |year=2002 |title=Robert Recorde |encyclopedia=[[MacTutor History of Mathematics Archive]] |url=https://mathshistory.st-andrews.ac.uk/Biographies/Recorde/ |access-date=19 October 2013 |url-status=live |archive-url=https://web.archive.org/web/20131129160351/http://www-history.mcs.st-and.ac.uk/Mathematicians/Recorde.html |archive-date=29 November 2013 |last2=Robertson |first2=E. F. |first1=J. J. |last1=O'Connor}}</ref> [[Diophantus]]'s use of {{angbr|ἴσ}}, short for {{lang|grc|ἴσος}} ({{tlit|grc|ísos}} 'equals'), in ''[[Arithmetica]]'' ({{circa|AD&nbsp;250}}) is considered one of the first uses of an equals sign.<ref>{{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |url=https://archive.org/details/isbn_9780309096577/page/34/mode/2up?q=Equals |title=Unknown Quantity: A Real And Imaginary History of Algebra |publisher=Joseph Henry Press |year=2006 |isbn=0-309-09657-X |page=35}}</ref>

The {{char|1==}} symbol, now universally accepted in mathematics for equality, was first recorded by the Welsh mathematician [[Robert Recorde]] in ''[[The Whetstone of Witte]]'' (1557).<ref>{{Cite web |title=The History of Equality Symbols in Math |url=https://sciencing.com/history-equality-symbols-math-8143072.html |access-date=2020-08-09 |website=Sciencing |date=24 April 2017 |language=en |archive-date=2020-09-14 |archive-url=https://web.archive.org/web/20200914232611/https://sciencing.com/history-equality-symbols-math-8143072.html |url-status=live }}</ref> The original form of the symbol was much wider than the present form. In his book Recorde explains his design of the "Gemowe lines" (meaning ''twin'' lines, from the [[Latin]] {{wikt-lang|la|gemellus}})<ref name="gemellus">See also [[wiktionary:geminus|geminus]] and [[Gemini (constellation)|Gemini]].</ref>
{{blockquote|
{{lang|en-emodeng|italics=yes|And to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or [[wikt:Gemini|Gemowe]] lines of one lengthe, thus: {{=}}, bicauſe noe .2. thynges, can be moare equalle.}} <br> And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or [[twin|duplicate]] lines of one [the same] length, thus: {{=}}, because no 2 things can be more equal.
|multiline=yes
|source={{cite book |last=Recorde |first=Robert |title=The Whetstone of Witte |location=London |publisher=John Kyngstone |date=1557 }} [https://archive.org/stream/TheWhetstoneOfWitte#page/n237/mode/2up the third page of the chapter "The rule of equation, commonly called Algebers Rule."]
}}
The symbol {{char|1==}} was not immediately popular. After its introduction by Recorde, it was not used again in print until 1618 (61 years later), in an anonymous Appendix in [[Edward Wright (mathematician)|Edward Wright's]] English translation of ''[[Mirifici Logarithmorum Canonis Descriptio|Descriptio]]'', by [[John Napier]]. It was not until 1631 that it received more than general recognition in England, being adopted as the symbol for equality in three influential works, [[Thomas Harriot]]'s [[Artis analyticae praxis|''Artis analyticae praxis'']], [[William Oughtred]]'s ''[[Clavis mathematicae]]'' and [[Richard Norwood]]'s ''[[Trigonometria]]''.<ref>{{Cite book |last=Cajori |first=Florian |url=https://archive.org/details/historyofmathema031756mbp/page/n319/mode/2up |title=A History Of Mathematical Notations Vol I |date=1928 |publisher=The Open Court Company, Publishers |others=Osmania University, Digital Library Of India}}</ref> Later used by [[John Wallis]], [[Isaac Barrow]] and [[Isaac Newton]], which helped it to spread to continental Europe.

=== Competing symbols ===
There were several other competing symbols for equality, especially outside of England around the 16th and 17th centuries, and Recorde's version made no significant influence in the European continent until 1650 or 1660. In 1559 the French monk [[Johannes Buteo]] published his ''Logistica'' using the symbol<math>\ [  </math> for equality. In 1571 [[Wilhelm Xylander]] published an edition of [[Arithmetica|Diophantus' ''Arithmetica'']] in which two parallel vertical lines {{char|{{!}}{{!}}}} were used for equality.{{Efn|He gives no clue as to the origin of the symbol. Moritz Cantor suggests that perhaps the Greek word ίσοι ("equal") was abbreviated in the manuscript used by Xylander, by the writing of only the two letters ίι.}} This version was adopted by several prominent writers, including [[Giovanni Camillo Glorioso|Giovanni Glorioso]], Cardinal [[Michelangelo Ricci]], and many French and Dutch mathematicians in the hundred years after, including [[René Descartes]] in 1621. 

A major competitor to Recorde's sign was Descartes' own symbol, introduced in his [[La Géométrie|''La Géométrie'']] (1637). In fact, Descartes himself used the sign {{char|1==}} for equality in a letter in 1640. Descartes does not give any reason for introducing his new symbol; however, [[Florian Cajori]] suggests it is because {{char|1==}} was also being used for a difference operation at the time. Due to the prominence of ''La Géométrie'', by 1675, Descartes' symbol gained favour over Recorde's in Europe, and most 17th-century writers on the continent either used Descartes' notation for equality or none at all. Around the turn of the 18th century, Recorde's notation gained favour rapidly. The dominating trend in mathematics of the time was [[History of calculus#Newton and Leibniz|differential and integral calculus]]. The fact that both Newton and [[Gottfried Wilhelm Leibniz]] used Recorde's symbol led to its general adoption.

==Usage in mathematics and computer programming==
In mathematics, the equals sign can be used as a simple statement of fact in a specific case ("{{nowrap|''x'' {{=}} 2}}"), or to create definitions ("{{nowrap|let ''x'' {{=}} 2}}"), conditional statements ("{{nowrap|if ''x'' {{=}} 2, then ...}}"), or to express a universal equivalence ("{{nowrap|(''x'' + 1)<sup>2</sup> {{=}} ''x''<sup>2</sup> + 2''x'' + 1}}").

The first important [[computer programming language]] to use the equals sign was the original version of [[Fortran]], FORTRAN&nbsp;I, designed in 1954 and implemented in 1957. In Fortran, {{char|1==}} serves as an [[assignment (computer science)|assignment]] operator: {{nowrap|{{code|1=X = 2}}}} sets the value of {{code|X}} to 2. This somewhat resembles the use of {{char|1==}} in a mathematical definition, but with different semantics: the expression following {{char|1==}} is evaluated first, and may refer to a previous value of {{code|X}}. For example, the assignment {{nowrap|{{code|1=X = X + 2}}}} increases the value of {{code|X}} by 2.

A rival programming-language usage was pioneered by the original version of [[ALGOL]], which was designed in 1958 and implemented in 1960. ALGOL included a [[relational operator]] that tested for equality, allowing constructions like {{nowrap|{{code|1=if x = 2}}}} with essentially the same meaning of {{char|1==}} as the conditional usage in mathematics. The equals sign was reserved for this usage.

Both usages have remained common in different programming languages into the early-21st century. As well as Fortran, {{char|1==}} is used for assignment in such languages as [[C (programming language)|C]], [[Perl]], [[Python (programming language)|Python]], [[awk|AWK]] and their descendants. But {{char|1==}} is used for equality and not assignment in the [[Pascal (programming language)|Pascal]] family, [[Ada (programming language)|Ada]], [[Eiffel (programming language)|Eiffel]], [[APL (programming language)|APL]], and other languages.

A few languages, such as [[BASIC]] and [[PL/I]], have used the equals sign to mean both assignment and equality, distinguished by context. However, in most languages where {{char|1==}} has one of these meanings, a different character or, more often, a sequence of characters is used for the other meaning. Following ALGOL, most languages that use {{char|1==}} for equality use {{char|1=:=}} for assignment, although APL, with its special character set, uses a left-pointing arrow.

Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the [[arithmetic IF]] statement) until FORTRAN{{nbsp}}IV was released in 1962, since when it has used the four characters {{code|.EQ.}} to test for equality. The language [[B (programming language)|B]] introduced the use of {{char|1===}} with this meaning, which has been copied by its descendant C and most later languages where {{char|1==}} means assignment.

Some languages additionally feature the "[[spaceship operator]]", or three-way comparison operator, {{char|1=<=>}}, to determine whether one value is less than, equal to, or greater than another.

===Several equals signs===
In some programming languages, <code>==</code> and <code>===</code> are used to check equality, so <code>1844&nbsp;== 1844</code> will return true.

In [[PHP]], the [[===|triple equals sign]], {{code|1====}}, denotes value and [[Data type|type]] equality,<ref>{{cite web
| url        = https://www.php.net/manual/en/language.operators.comparison.php
| title        = Comparison Operators
| website        = [[Php.net]]
| access-date        = 19 October 2013
| archive-date        = 19 October 2013
| archive-url        = https://web.archive.org/web/20131019192727/http://php.net/manual/en/language.operators.comparison.php
| url-status        = live
}}</ref> meaning that not only do the two expressions evaluate to equal values, but they are also of the same data type. For instance, the expression {{nowrap|{{code|1=0 == false}}}} is true, but {{nowrap|{{code|1=0 === false}}}} is not, because the number 0 is an integer value whereas false is a Boolean value.

[[JavaScript]] has the same semantics for {{code|1====}}, referred to as "equality without type coercion". However, in JavaScript the behavior of {{code|1===}} cannot be described by any simple consistent rules. The expression {{nowrap|{{code|1=0 == false}}}} is true, but {{nowrap|{{code|1=0 == undefined}}}} is false, even though both sides of the {{code|1===}} act the same in Boolean context. For this reason it is sometimes recommended to avoid the {{code|1===}} operator in JavaScript in favor of {{code|1====}}.<ref>{{cite web
 |last         = Crockford
 |first        = Doug
 |title        = JavaScript: The Good Parts
 |website      = [[YouTube]]
 |date = 27 February 2009
 |url          = https://www.youtube.com/watch?v=hQVTIJBZook
 |access-date  = 19 October 2013
 |archive-date = 4 November 2013
 |archive-url  = https://web.archive.org/web/20131104115604/http://www.youtube.com/watch?v=hQVTIJBZook
 |url-status   = live
}}</ref>

In Ruby, equality under {{code|1===}} requires both operands to be of identical type, e.g. {{nowrap|{{code|1=0 ==  false}}}} is false. The {{code|1====}} operator is flexible and may be defined arbitrarily for any given type. For example, a value of type {{code|Range}} is a range of integers, such as {{code|1800..1899}}. {{nowrap|{{code|1=(1800..1899) ==  1844}}}} is false, since the types are different (Range vs. Integer); however {{nowrap|{{code|1=(1800..1899) === 1844}}}} is true, since {{code|1====}} on {{code|Range}} values means "inclusion in the range".<ref>{{cite book
 |title        = [[why's (poignant) Guide to Ruby]]
 |chapter      = 5.1 This One's For the Disenfranchised
 |author       = why the lucky stiff
 |author-link  = why the lucky stiff
 |chapter-url  = http://mislav.uniqpath.com/poignant-guide/book/chapter-5.html#section1
 |access-date  = 19 October 2013
 |archive-date = 24 September 2015
 |archive-url  = https://web.archive.org/web/20150924063735/http://mislav.uniqpath.com/poignant-guide/book/chapter-5.html#section1
 |url-status   = dead
}}</ref> Under these semantics, <code>===</code> is [[Symmetric relation|non-symmetric]]; e.g. <code>1844&nbsp;===&nbsp;(1800..1899)</code> is false, since it is interpreted to mean <code>Integer#===</code> rather than <code>Range#===</code>.<ref>{{cite web
 |last         = Rasmussen
 |first        = Brett
 |title        = Don't Call it Case Equality
 |date         = 30 July 2009
 |url          = http://www.pmamediagroup.com/2009/07/dont-call-it-case-equality/
 |website      = pmamediagroup.com
 |access-date   = 19 October 2013
 |archive-url  = https://web.archive.org/web/20131021062941/http://www.pmamediagroup.com/2009/07/dont-call-it-case-equality/
 |archive-date = 21 October 2013
}}</ref>

==Other uses==
===Spelling===
====Tone letter====
The equals sign is also used as a grammatical [[tone letter]] in the orthographies of [[Budu language|Budu]] in the [[Congo-Kinshasa]], in [[Krumen language|Krumen]], [[Mwan language|Mwan]] and [[Dan language|Dan]] in the [[Ivory Coast]].<ref>{{cite book | author = Peter G. Constable | author2 = Lorna A. Priest | url = https://www.unicode.org/L2/L2006/06259r-mod-letters.pdf | title = Proposal to Encode Additional Orthographic and Modifier Characters | date = 31 July 2006 | access-date = 19 October 2013 | archive-date = 21 October 2013 | archive-url = https://web.archive.org/web/20131021002528/http://www.unicode.org/L2/L2006/06259r-mod-letters.pdf | url-status = live }}</ref><ref>{{cite book | editor = Hartell, Rhonda L. | year = 1993 | title = The Alphabets of Africa | location = Dakar | publisher = [[UNESCO]] and SIL | url = https://archive.org/details/rosettaproject_pbi_ortho-1 | access-date = 19 October 2013}}</ref> The Unicode character used for the tone letter ({{Unichar|A78A}})<ref>{{cite web | title = Unicode Latin Extended-D code chart | website = Unicode.org | url = https://www.unicode.org/charts/PDF/UA720.pdf | access-date = 19 October 2013 | archive-date = 25 March 2019 | archive-url = https://web.archive.org/web/20190325152831/http://unicode.org/charts/PDF/UA720.pdf | url-status = live }}</ref> is different from the mathematical symbol (U+003D).

====Personal names====
{{contains special characters|section=section}}
[[File:Assinatura do Santos Dumont 2.png|thumb|The signature of Santos-Dumont, showing a [[double hyphen]] that looks like an equals sign.]]

A possibly unique case of the equals sign of European usage in a person's name, specifically in a [[double-barreled name]], was by the aviation pioneer [[Alberto Santos-Dumont]], as he is also known not only to have often used a [[double hyphen]] {{char|&#x2E40;}} resembling an equal sign {{char|1==}} between his [[double-barreled name|two surnames]] in place of a hyphen, but also seems to have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.<ref>{{citation |mode=cs1 |last=Gray |first=Carroll F.|title=The 1906 Santos=Dumont No. 14bis |work=W.W.1 Aero: The Journal of the Early Aeroplane |number=194 |date=November 2006 |page=4}}</ref>

Instead of a double hyphen, the equals sign is sometimes used in [[Japanese language|Japanese]] as a separator between names. In [[Ojibwe language|Ojibwe]], the readily available equals sign on most keyboards is commonly used as a substitute for a double hyphen.

=== Linguistics ===
In linguistic [[interlinear gloss]]es, an equals sign is conventionally used to mark clitic boundaries: the equals sign is placed between the [[clitic]] and the word that the clitic is attached to.<ref>{{Cite web|title = Conventions for interlinear morpheme-by-morpheme glosses|url = https://www.eva.mpg.de/lingua/resources/glossing-rules.php|access-date = 2017-11-20|archive-date = 2019-08-04|archive-url = https://web.archive.org/web/20190804103429/https://www.eva.mpg.de/lingua/resources/glossing-rules.php|url-status = live}}</ref>

===Chemistry===
In [[chemical formula]]s, the two parallel lines denoting a [[double bond]] are commonly rendered using an equals sign (hence, a [[triple bond]] is commonly rendered using a [[triple bar]]).

===LGBT activism===
{{Expand section|date=July 2018}}
In recent years, the equals sign has been used to [[LGBT symbols|symbolize]] [[LGBT rights]]. It has been used since 1995 by the [[Human Rights Campaign]], which lobbies for [[marriage equality]], and subsequently by the [[United Nations Free & Equal]], which promotes [[LGBT rights at the United Nations]].<ref>[http://www.hrc.org/hrc-story/about-our-logo "HRC Story: Our Logo."] {{Webarchive|url=https://web.archive.org/web/20180718231624/http://www.hrc.org/hrc-story/about-our-logo |date=2018-07-18 }} 
 The Human Rights Campaign.  ''HRC.org'', Retrieved 4 December 2018.</ref>

===Telegrams and Telex===
In [[Morse code]] the equals sign is encoded by the letters B (-...) and T (-) run together (-...-).{{citation needed|date=August 2020}} The letters BT stand for Break Text, and are put between paragraphs, or groups of paragraphs in messages sent via [[Telex]],{{citation needed|date=August 2020}} a standardised tele-typewriter. The sign, used to mean Break Text, is given at the end of a [[telegram]] to separate the text of the message from the signature.{{citation needed|date=June 2022}}

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

===Approximately equal===
{{Main|Approximation#Typography}}

Symbols used to denote items that are ''[[approximately equal]]'' include the following:<ref name = "The Unicode Consortium">{{cite web | title = Mathematical Operators | website = Unicode.org | url = https://www.unicode.org/charts/PDF/U2200.pdf | access-date = 19 October 2013 | archive-date = 12 June 2018 | archive-url = https://web.archive.org/web/20180612210306/http://www.unicode.org/charts/PDF/U2200.pdf | url-status = live }}</ref>
* <span style="font-size: 200%;line-height:50%;">≈</span> ({{unichar|2248|almost equal to}}, [[LaTeX]] ''\approx'')
* <span style="font-size: 150%;line-height:50%">≃</span> ({{unichar|2243|asymptotically equal to|nlink=Asymptotic analysis}}, LaTeX ''\simeq''), a combination of {{char|≈}} and {{char|{{=}}}}, also used to indicate [[Asymptotic analysis|asymptotic equality]]
* <span style="font-size: 150%;line-height:50%">≅</span> ({{unichar|2245|approximately equal to}}, LaTeX ''\cong''), another combination of ≈ and =, which is also sometimes used to indicate [[isomorphism]] or [[Congruence relation|congruence]]
* <span style="font-size: 150%;line-height:50%">{{not a typo|∼}}</span> ({{unichar|223C|tilde operator|nlink=Tilde#Mathematics}}, LaTeX ''\sim''), which is also sometimes used to indicate [[proportionality (mathematics)|proportionality]] or [[similarity (geometry)|similarity]], being related by an [[equivalence relation]], or to indicate that a [[random variable]] is distributed according to a specific [[probability distribution]] (see also [[tilde]]), or alternatively between two quantities to indicate they are of the same [[order of magnitude]].
* <span style="font-size: 150%;line-height:50%">∽</span> ({{unichar|223D|reversed tilde}}, LaTeX ''\backsim''), which is also used to indicate [[proportionality (mathematics)|proportionality]]
* <span style="font-size: 150%;line-height:50%">≐</span> ({{unichar|2250|approaches the limit}}, LaTeX ''\doteq''), which can also be used to represent the approach of a variable to a [[Limit (mathematics)|limit]]
* <span style="font-size: 150%;line-height:50%">≒</span> ({{unichar|2252|Approximately equal to or the image of}}, LaTeX ''\fallingdotseq''), commonly used in Japan, Taiwan, and Korea.
* <span style="font-size: 150%;line-height:50%">≓</span> ({{unichar|2253|image of or approximately equal to}}, LaTeX ''\risingdotseq'')

In some areas of East Asia, such as Japan, "≒" is used to mean "the two terms are almost equal", but in other areas and specialized literature such as mathematics, "≃" is often used. In addition to its mathematical meaning, it is sometimes used in Japanese sentences with the intention of "almost the same".

===Not equal===
The symbol used to denote [[inequation]] (when items are not equal) is a [[Slash (punctuation)|slashed]] equal sign {{char|≠}} (U+2260). In [[LaTeX]], this is done with the "\neq" command.

Most programming languages, limiting themselves to the [[ASCII|7-bit ASCII]] [[character set]] and [[QWERTY|typeable characters]], use {{code|1=~=}}, {{code|1=!=}}, {{code|1=/=}}, or {{code|<>}} to represent their [[Boolean logic|Boolean]] [[inequality operator]].

===Identity===
The [[triple bar]] symbol {{char|≡}} (U+2261, LaTeX ''\equiv'') is often used to indicate an [[identity (mathematics)|identity]], a [[definition]] (which can also be represented by {{unichar|225D|equal to by definition}}  or {{unichar|2254|Colon equals}}), or a [[congruence relation]] in [[modular arithmetic]]. Also, in [[chemistry]], the triple bar can be used to represent a [[triple bond]] between atoms.

===Isomorphism===
The symbol {{char|≅}} is often used to indicate [[isomorphic]] algebraic structures or [[Congruence (geometry)|congruent]] geometric figures.

===In logic===
Equality of [[truth value]]s (through [[bi-implication]] or [[logical equivalence]]), may be denoted by various symbols including {{char|1==}}, {{char|~}}, and {{char|⇔}}.
===In geometry===
The symbol <math>\bumpeq </math> (LaTeX \bumpeq) is used to show two directed line segments have the same length and direction, [[equipollence (geometry)|equipollence]].

===Other related symbols <span class="anchor" id="Delta equal to"></span>===
Additional [[precomposed character|precomposed symbols]] with [[code point]]s in Unicode for notations related to the equal sign include the following:<ref name="The Unicode Consortium"/>

* <span style="font-size: 150%;line-height:50%;">≌</span> ({{unichar|224C|ALL EQUAL TO}})
* <span style="font-size: 150%;line-height:50%;">≔</span> ({{unichar|2254|COLON EQUALS}}) ([[Glossary of mathematical symbols#Equality, equivalence and similarity|used to define a symbol]] or [[assignment (computer science)|assign a variable]])
* <span style="font-size: 150%;line-height:50%;">≕</span> ({{unichar|2255|EQUALS COLON}}) (defines the symbol on the right-hand side)
* <span style="font-size: 150%;line-height:50%;">≖</span> ({{unichar|2256|RING IN EQUAL TO}})
* <span style="font-size: 150%;line-height:50%;">≗</span> ({{unichar|2257|RING EQUAL TO}})
<!-- U+2258 is mentioned already in the section "Identity" above. -->
* <span style="font-size: 150%;line-height:50%;">≘</span> ({{unichar|2258|CORRESPONDS TO}})
* <span style="font-size: 150%;line-height:50%;">≙</span> ({{unichar|2259|ESTIMATES}}) (the left-hand side is an [[estimator]] for the right-hand side)
* <span style="font-size: 150%;line-height:50%;">≚</span> ({{unichar|225A|EQUIANGULAR TO}})
* <span style="font-size: 150%;line-height:50%;">≛</span> ({{unichar|225B|STAR EQUALS}})
* <span style="font-size: 150%;line-height:50%;">≜</span> ({{unichar|225C|DELTA EQUAL TO}}) ([[Glossary of mathematical symbols#Equality, equivalence and similarity|used to define a symbol]])
<!-- U+225D is mentioned already in the section "Identity" above. -->
* <span style="font-size: 150%;line-height:50%;">≞</span> ({{unichar|225E|MEASURED BY}})
* <span style="font-size: 150%;line-height:50%;">≟</span> ({{unichar|225F|QUESTIONED EQUAL TO}})
* <span style="font-size: 150%;line-height:50%;">⩴</span> ({{unichar|2A74|DOUBLE COLON EQUAL}}) (see also [[Backus–Naur form]] for {{code|1=::=}})
* <span style="font-size: 150%;line-height:50%;">⩵</span> ({{unichar|2A75|TWO CONSECUTIVE EQUALS SIGNS}})
* <span style="font-size: 150%;line-height:50%;">⩶</span> ({{unichar|2A76|THREE CONSECUTIVE EQUALS SIGNS}})

==Incorrect usage==
The equals sign is sometimes used incorrectly within a mathematical argument to connect math steps in a non-standard way, rather than to show equality (especially by early mathematics students).

For example, if one were finding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might incorrectly write
:1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15.
Structurally, this is shorthand for
:([(1 + 2 = 3) + 3 = 6] + 4 = 10) + 5 = 15,
but the notation is incorrect, because each part of the equality has a different value. If interpreted strictly as it says, it would imply that
:3 = 6 = 10 = 15 = 15.
A correct version of the argument would be
:1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15.

This difficulty results from subtly different uses of the sign in education. In early, arithmetic-focused grades, the equals sign may be ''operational''; like the equal button on an electronic calculator, it demands the result of a calculation. Starting in algebra courses, the sign takes on a ''relational'' meaning of equality between two calculations. Confusion between the two uses of the sign sometimes persists at the university level.<ref>{{cite journal | url = https://tamu.academia.edu/SencerCorlu/Papers/522225/Capraro_R._M._Capraro_M._M._Yetkiner_Z._E._Corlu_M._S._Ozel_S._Ye_S._and_Kim_H._G._2011_._An_international_perspective_between_problem_types_in_textbooks_and_students_understanding_of_relational_equality._Mediterranean_Journal_for_Research_in_Mathematics_Education_An_International_Journal_10_187-213 | title = An International Perspective between Problem Types in Textbooks and Students' understanding of relational equality | last1 = Capraro | first1 = Robert M. | last2 = Capraro | first2 = Mary Margaret | last3 = Yetkiner | first3 = Ebrar Z. | last4 = Corlu | first4 = Sencer M. | last5 = Ozel | first5 = Serkan | last6 = Ye | first6 = Sun | last7 = Kim | first7 = Hae Gyu | journal = Mediterranean Journal for Research in Mathematics Education | volume = 10 | number = 1–2 | pages = 187–213 | year = 2011 | access-date = 19 October 2013 | archive-date = 26 April 2012 | archive-url = https://web.archive.org/web/20120426041729/http://tamu.academia.edu/SencerCorlu/Papers/522225/Capraro_R._M._Capraro_M._M._Yetkiner_Z._E._Corlu_M._S._Ozel_S._Ye_S._and_Kim_H._G._2011_._An_international_perspective_between_problem_types_in_textbooks_and_students_understanding_of_relational_equality._Mediterranean_Journal_for_Research_in_Mathematics_Education_An_International_Journal_10_187-213 | url-status = live }}</ref>

==Encodings==
* {{unichar|003d|equals sign|html=}}
Related symbols
* {{unichar|2260|not equal to|html=}}
*{{unichar|fe66|small equals sign|html=}}
*{{unichar|ff1d|fullwidth equals sign|html=}}
*{{unichar|1f7f0|heavy equals sign|html=}}
*{{unichar|224D}}
*{{unichar|226D}}
*{{unichar|2261}}
*{{unichar|2262}}
*{{unichar|2263}}

==See also==
* [[2 + 2 = 5]]
* [[Double hyphen]]
* [[Equality (mathematics)]]
* [[Logical equality]]
* [[Plus and minus signs]]

==Notes==
{{Reflist}}{{Notelist}}

==References==
*{{Cite book | author=Cajori, Florian | author-link=Florian Cajori | title=A History of Mathematical Notations | location=New York | publisher=Dover (reprint) | year=1993 | isbn=0-486-67766-4 | url-access=registration | url=https://archive.org/details/historyofmathema00cajo_0 }}
* Boyer, C. B.: ''A History of Mathematics'', 2nd ed. rev. by [[Uta Merzbach|Uta C. Merzbach]]. New York: Wiley, 1989 {{ISBN|0-471-09763-2}} (1991 pbk ed. {{ISBN|0-471-54397-7}})

==External links==
{{wiktionary|Appendix:Variations of "{{=}}"}}
{{wiktionary|{{=}}}}
*[http://jeff560.tripod.com/relation.html Earliest Uses of Symbols of Relation]
*[http://jeff560.tripod.com/witte.jpg Image of the page of ''The Whetstone of Witte'' on which the equal sign is introduced]
*[http://www.numericana.com/answer/symbol.htm#equal Scientific Symbols, Icons, Mathematical Symbols]
*[http://blog.plover.com/math/recorde.html Robert Recorde invents the equal sign]

{{DEFAULTSORT:Equal Sign}}
[[Category:Mathematical symbols]]
[[Category:1550s introductions]]
[[Category:1557 beginnings]]
[[Category:Definition]]
[[Category:Assignment operations]]
[[Category:Equivalence (mathematics)]]