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{{short description|Format for expressing mathematical formulae}}
{{
{{Use dmy dates|date=February 2021}}
{{Infobox technology standard
| title
| long_name
| native_name
| url = https://www.iso.org/standard/58439.html | title = ISO - ISO/IEC 40314:2016 - Information technology — Mathematical Markup Language (MathML) Version 3.0 2nd Edition | language = en | publication-date = 2016 | access-date = 2021-04-06 | }}</ref> }}
| native_name_lang
| status = W3C Recommendation<ref name="w3c-standard-2nd-ed">{{Cite web
| url = https://www.w3.org/TR/2014/REC-MathML3-20140410/
|
| publisher = W3C
| publication-date = 2014-04-10
| access-date = 2021-04-06
| editor-first1 = David | editor-last1 = Carlisle
| editor-first2 = Patrick | editor-last2 = Ion
| editor-first3 = Robert | editor-last3 = Miner
}}</ref>
|
| version = 3.0<ref name="w3c-standard-2nd-ed"/>
|
| organization = [[World Wide Web Consortium|W3C]], [[International Organization for Standardization|ISO]], [[International Electrotechnical Commission|IEC]]<ref name="iso-iec-standard"/>
| committee = {{ubl
| W3C Math Working Group
|
}}
| editors
| url = https://www.w3.org/TR/2022/WD-mathml-core-20220504/ | title = MathML Core | publisher = W3C | publication-date = 2022-05-04 | access-date = 2023-03-03 | editor-first1 = David | editor-last1 = Carlisle
| editor-first2 = Frédéric | editor-last2 = Wang
}}</ref>
}}
| authors
{{blist|Ron Ausbrooks|Stephen Buswell|David Carlisle|Giorgi Chavchanidze|Stéphane Dalmas|Stan Devitt|Angel Diaz|Sam Dooley|Roger Hunter|Patrick Ion|Michael Kohlhase|Azzeddine Lazrek|Paul Libbrecht|Bruce Miller|Robert Miner|Chris Rowley|Murray Sargent|Bruce Smith|Neil Soiffer|Robert Sutor|Stephen Watt}}
}}<ref name="w3c-standard-2nd-ed"/>
| website = {{ubl
| W3C: {{URL|https://www.w3.org/TR/MathML/}}
| ISO: {{URL|https://www.iso.org/standard/58439.html}}
}}
| base_standards = [[XML]]
| related_standards = [[OpenMath]], [[OpenDocument]], [[Office Open XML]], {{abbr|[[OMDoc]]|Open Mathematical Documents}}
| abbreviation = MathML
}}
'''Mathematical Markup Language''' ('''MathML''') is a pair of [[mathematical markup language]]s, an application of [[XML]] for describing [[mathematics|mathematical]] notations and capturing both its structure and content
== History ==
Following some experiments in the [[Arena browser]] based on proposals for mathematical markup in HTML,<ref>{{cite web
| title = 12 - Mathematical Equations
| url = https://www.w3.org/MarkUp/HTMLPlus/htmlplus_45.html
| date = 1993-11-08
}}</ref> MathML 1 was released as a W3C recommendation in April 1998 as the first [[XML]] language to be recommended by the [[W3C]]. Version 1.01 of the format was released in July 1999 and version 2.0 appeared in February 2001. Implementations of the specification appeared in [[Amaya (web editor)|Amaya 1.1]], [[Firefox|Mozilla 1.0]] and [[Opera 9.5]].<ref>{{cite web
| title = Mozilla 1.0 Released!
| url = http://www.mozillazine.org/articles/article2278.html
| date = 2002-06-05
| access-date = 2023-03-03
}}</ref><ref name="opera-kestrel">{{citation
| url = http://dev.opera.com/articles/view/can-kestrels-do-math-mathml-support-in/
| title = Can Kestrels do Math? MathML support in Opera Kestrel
| first = Charles | last = McCathieNevile
| date = 2007-09-27
| publisher = Opera
}}</ref> In October 2003, the second edition of MathML Version 2.0 was published as the final release by the [[World Wide Web Consortium|W3C Math Working Group]].
MathML was originally designed before the finalization of [[XML namespace]]s. However, it was assigned a namespace immediately after the Namespace Recommendation was completed, and for XML use, the elements should be in the namespace with namespace URL ''<nowiki>http://www.w3.org/1998/Math/MathML</nowiki>''. When MathML is used in HTML (as opposed to XML) this namespace is automatically inferred by the HTML parser and need not be specified in the document.<ref>{{cite web
| title = HTML Living Standard
| url = https://html.spec.whatwg.org/#parsing-main-inbody:mathml-namespace
| access-date = 2023-03-03
}}</ref>
=== MathML version 3 ===
Version 3 of the MathML specification was released as a [[World Wide Web Consortium#W3C recommendation (REC)|W3C recommendation]] on 20 October 2010. A recommendation of ''A MathML for CSS Profile'' was later released on 7 June 2011;<ref name=CSSProfile>{{cite web
| title = A MathML for CSS Profile
| url = https://www.w3.org/TR/mathml-for-css/
| publisher = W3C
| access-date = 2013-07-25
| date = 2011-06-07
}}</ref> this is a subset of MathML suitable for CSS formatting. Another subset, ''Strict Content MathML'', provides a subset of content MathML with a uniform structure and is designed to be compatible with [[OpenMath]]. Other content elements are defined in terms of a transformation to the strict subset. New content elements include {{code|<bind>|lang=XML}} which associates bound variables ({{code|<bvar>|lang=XML}}) to expressions, for example a summation index. The new {{code|<share>|lang=XML}} element allows structure sharing.<ref name="V3"/>
The development of MathML 3.0 went through a number of stages. In June 2006, the W3C rechartered the MathML Working Group to produce a MathML 3 Recommendation until February 2008, and in November 2008 extended the charter to April 2010. A sixth Working Draft of the MathML 3 revision was published in June 2009. On 10 August 2010 version 3 graduated to become a "Proposed Recommendation" rather than a draft.<ref name="V3">{{cite web
| url = https://www.w3.org/TR/MathML3/
| title = Mathematical Markup Language Version 3.0 W3C Recommendation
| website = W3.org
| access-date = 2012-05-09
}}</ref> An implementation of MathML 2 landed in [[WebKit]] around this same time,<ref>{{cite web
| title = Announcing…MathML!
| url = https://webkit.org/blog/1366/announcing%e2%80%a6mathml/
| date = 2010-08-17
| first = Beth | last = Dakin
| access-date = 2023-03-03
}}</ref> with a [[Chromium (web browser)|Chromium]] implementation following a couple of years later,<ref>{{cite web
| title = A web developer's guide to the latest Chrome Beta
| url = https://blog.chromium.org/2012/11/a-web-developers-guide-to-latest-chrome.html
| date = 2012-11-08
| access-date = 2023-03-03
}}</ref> although that implementation was removed from Chromium after less than a year.<ref>{{cite web
| title = Comment 32 on Issue 152430: Enabling support for MathML
| url = https://bugs.chromium.org/p/chromium/issues/detail?id=152430#c32
| date = 2013-02-05
| access-date = 2023-03-03
}}</ref>
The Second Edition of MathML 3.0 was published as a W3C Recommendation on 10 April 2014.<ref name="w3c-standard-2nd-ed"/> The specification was approved as an [[International Electrotechnical Commission|ISO/IEC]] international standard 40314:2015 on 23 June 2015.<ref>{{cite web
| url = https://www.w3.org/2015/06/mathmlpas.html.en
| title = W3C MathML 3.0 Approved as ISO/IEC International Standard
| website = W3.org
| date = 2015-06-23
| access-date = 2015-06-12
}}</ref> Also in 2015, the MathML Association was founded to support the adoption of the MathML standard.<ref>{{citation
| url = http://mathml-association.org/about.html
| archive-url = https://web.archive.org/web/20151002150927/http://mathml-association.org/about.html
| url-status = usurped
| archive-date = 2 October 2015
| title = Mondial Association for Tools Handling MathML
| access-date = 2016-06-20
| author1 = Deyan Ginev
| author2 = Michael Kohlhase
| author3 = Moritz Schubotz
| author4 = Raniere Silva
| author5 = Frédéric Wang
}}</ref> At that time, according to a member of the [[MathJax]] team, none of the major browser makers paid any of their developers for any MathML-rendering work; whatever support existed was overwhelmingly the result of unpaid volunteer time/work.<ref>{{cite web
| url = http://radar.oreilly.com/2013/11/mathml-forges-on.html
| title = MathML forges on
| first = Peter | last = Krautzberger
| work = oreilly.com
| date = 2013-11-01
| access-date = 2014-11-22
}}</ref>
=== MathML Core ===
In August 2021, a new specification called ''MathML Core'' was published, described as the "core subset of Mathematical Markup Language, or MathML, that is suitable for browser implementation."<ref>{{cite web
| title = MathML Core
| url = https://www.w3.org/TR/2022/WD-mathml-core-20220504/#abstract
| date = 2022-05-04
| access-date = 2023-03-03
}}</ref> MathML Core set itself apart from MathML 3.0 by including detailed rendering rules and integration with [[CSS]], automated browser support testing resources, and focusing on a fundamental subset of MathML. An implementation was added to Chromium at the beginning of 2023.<ref name="igalia-chrome-109">{{cite web
| url = https://www.igalia.com/2023/01/10/Igalia-Brings-MathML-Back-to-Chromium.html
| title = Igalia Brings MathML Back to Chromium
| date = 2023-01-10
| access-date = 2023-01-10
| publisher = Igalia News
}}</ref>
== {{anchor|Generic MathML}}Presentation and semantics ==
{{Infobox file format
| name = Generic MathML
| extension = .mml<ref name="mathml-media-types"/><ref name="w3c-mathml2-ch7">{{Cite web
| url = https://www.w3.org/TR/MathML2/chapter7.html#id.7.1.3
| title = The MathML Interface
| quote = The W3C Math Working Group recommends the standard file extension <code>.mml</code> used for browser registry.
| publisher = W3C
| publication-date = 2003-10-21
| access-date = 2023-09-02
}}</ref>
| mime = application/mathml+xml<ref name="mathml-media-types">{{Cite web
| url = https://w3c.github.io/mathml-docs/mathml-media-types/#media-types-mathml
| title = MathML Media-type Declarations
| first = Paul | last = Libbrecht
| publisher = W3C
| publication-date = 2023-09-01
| access-date = 2023-09-02
}}</ref>
| type code = <code>MML </code>
| uniform type = {{mono|public.mathml}}
| conforms to = {{mono|public.xml}}
| owner = [[World Wide Web Consortium]]
| type = [[Mathematical markup language]]
| extended from = [[XML]]
| extended to = {{ubl
| [[#Presentation MathML | Presentation MathML]]
| [[#Content MathML | Content MathML]]
}}
| standard = {{ubl
| {{URL|https://www.w3.org/Math/|W3C MathML}}<ref name="V3"/>
| {{URL|https://www.iso.org/standard/58439.html|ISO/IEC 40314}}<ref name="iso-iec-standard"/>
}}
| open = Yes
}}
MathML deals not only with the ''presentation'' but also the ''meaning'' of formula components (the latter part of MathML is known as "Content MathML"). Because the meaning of the equation is preserved separate from the presentation, how the content is communicated can be left up to the user. For example, web pages with MathML embedded in them can be viewed as normal web pages with many browsers, but visually impaired users can also have the same MathML read to them through the use of [[screen readers]] (e.g. using the [[VoiceOver]] in [[Safari (web browser)|Safari]]). [[JAWS (screen reader)|JAWS]] from version 16 onward supports MathML voicing as well as braille output.<ref>{{cite web
| title = JAWS Version 16
| url = https://support.freedomscientific.com/downloads/jaws/JAWSWhatsNew?version=16
| access-date = 2023-09-07
}}</ref>
The quality of rendering of MathML in a browser depends on the installed fonts. The [[STIX Fonts project]] have released a comprehensive set of mathematical fonts under an open license. The [[Cambria Math]] font supplied with Microsoft Windows had slightly more limited support.<ref name="Vismor">{{citation
| url = https://vismor.com/documents/site_implementation/viewing_mathematics/viewing_mathematics.php
| title = Viewing Mathematics on the Internet
| access-date = 2011-04-13
| first = Timothy | last = Vismor
}}</ref>
A valid MathML document typically consists of the XML declaration, [[Document type declaration|DOCTYPE]] declaration, and document element. The document body then contains MathML expressions which appear in {{tag|math|o|link=yes}} elements as needed in the document. Often, MathML will be embedded in more general documents, such as [[HTML]], [[DocBook]], or other [[XML]]-based formats.
=== Presentation MathML ===
{{Infobox file format
| name = Presentation MathML
| mime = application/mathml-presentation+xml<ref name="mathml-media-types"/>
| type code = <code>MMLp</code>
| uniform type = {{mono|public.mathml.presentation}}
| conforms to = {{mono|public.mathml}}
| extended from = [[#Generic MathML|Generic MathML]]
}}
Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with {{code|m}}. A Presentation MathML expression is built up out of ''tokens'' that are combined using higher-level elements, which control their layout. Finer details of presentation are affected by close to 50 attributes.
Token elements generally only contain characters (not other elements). They include:
* {{code|lang=XML|<mi>x</mi>}} – identifiers;
* {{code|lang=XML|<mo>+</mo>}} – operators;
* {{code|lang=XML|<mn>2</mn>}} – numbers
* {{code|lang=XML|<mtext>such
Note, however, that these token elements may be used as extension points, allowing markup in host languages.
MathML in [[HTML5]] allows most inline HTML markup in mtext, and {{code|lang=XML|<mtext><b>non</b> zero</mtext>}} is conforming, with the HTML markup being used within the MathML to mark up the embedded text (making the first word bold in this example).
These are combined using layout elements, that generally contain only elements. They include:
* {{
* {{
* {{
* {{
* {{
As usual in HTML and XML, many [[character entity reference|entities]] are available for specifying special symbols by name, such as {{code|lang=HTML|π}} and {{code|lang=HTML|→}}. An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as {{code|lang=HTML|⁢}} (or the shorthand {{code|lang=HTML|⁢}}) for implicit multiplication. They are:
* {{unichar|2061|FUNCTION APPLICATION}} (to distinguish <math>\sin(x)</math> from <math>\sin\cdot x</math> in <math>\sin{x}</math>);
* {{unichar|2062|INVISIBLE TIMES}} (to distinguish <math>a_{m\times n}</math> from <math>a_{m,n}</math> in <math>a_{mn}</math>);
* {{unichar|2063|INVISIBLE SEPARATOR}} (vice versa);
* {{unichar|2064|INVISIBLE PLUS}} (to distinguish <math>2 + \frac{1}{3}</math> from <math>2 \cdot \frac{1}{3}</math> in <math>2\tfrac{1}{3}</math>).
The full specification of MathML entities<ref>{{
| url = http://www.w3.org/TR/MathML3/chapter7.html | title = Characters, Entities and Fonts | website = }}</ref> is closely coordinated with the corresponding specifications for use with HTML and XML in general.<ref>{{ | url = http://www.w3.org/TR/xml-entity-names/Overview.html | title = XML Entity Definitions for Characters (2nd Edition) | website = }}</ref> Thus, the expression <math
<syntaxhighlight lang="xml">
</syntaxhighlight>
Line 154 ⟶ 267:
<syntaxhighlight lang="xml">
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/mathml2.dtd">
<math xmlns="http://www.w3.org/1998/Math/MathML">
</math>
</syntaxhighlight>
=== Content MathML ===
{{Infobox file format
| name = Content MathML
| mime = application/mathml-content+xml
| type code = <code>MMLc</code>
| uniform type = {{mono|public.mathml.content}}
| conforms to = {{mono|public.mathml}}
| extended from = [[#Generic MathML|Generic MathML]]
}}
Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the {{code|lang=XML|<apply>}} element that represents function application. The function being applied is the first child element under {{code|lang=XML|<apply>}}, and its operands or parameters are the remaining child elements. Content MathML uses only a few attributes.
Tokens such as identifiers and numbers are individually marked up, much as for Presentation MathML, but with elements such as {{code|lang=XML|<ci>}} and {{code|lang=XML|<cn>}}. Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML: {{code|lang=XML|<times>}}, {{code|lang=XML|<power>}}, etc. There are over a hundred different elements for different functions and operators.<ref>{{
| url = http://www.w3.org/TR/MathML3/chapter4.html#contm.opel | title = Content Markup | website = }}</ref> For example, {{code|
The expression <math>a x^2+b x+c</math> could be represented as
Line 176 ⟶ 300:
<syntaxhighlight lang="xml">
<math>
</math>
</syntaxhighlight>
Content MathML is nearly [[isomorphic]] to [[Binary expression tree|expressions]] in a [[Functional programming|functional language]] such as [[Scheme (programming language)|Scheme]] and other dialects of [[Lisp (programming language)|Lisp]]. {{
<syntaxhighlight lang="scheme">
(plus
Line 206 ⟶ 330:
This reflects the long-known close relationship between XML element structures, and [[Lisp (programming language)|LISP]] or Scheme [[S-expressions]].<ref>Steven DeRose. The SGML FAQ Book: Understanding the Relationship of SGML and XML, Kluwer Academic Publishers, 1997. {{isbn|978-0-7923-9943-8}}.</ref><ref>[[Canonical S-expressions#cite note-0]]</ref>
==== Wikidata annotation in Content MathML ====
According to the OM Society,<ref name="OpenMath">{{cite web
| url = https://www.openmath.org/
| title = OpenMath Home · OpenMath
| website = www.openmath.org
}}</ref> OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics{{snd}}names, descriptions and rules. A 2018 paper presented at the [[Special Interest Group on Information Retrieval|SIGIR]] conference<ref name="SchubotzScharpfGipp2018">{{cite journal
| first1 = Moritz | last1 = Schubotz
| first2 = Philipp | last2 = Scharpf
| first3 = Bela | last3 = Gipp
| title = Representing Mathematical Formulae in Content MathML using Wikidata.
| url = http://ceur-ws.org/Vol-2132/paper5.pdf
| date = 2018
| journal = Birndl@sigir
}}</ref> proposed that the semantic knowledge base [[Wikidata]] could be used as an OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.
== Example ==
The well-known [[quadratic formula]] could be represented in Presentation MathML as an expression tree made up from layout elements like {{code|lang=XML|<mfrac>}} or {{code|lang=XML|<msqrt>}}:
<syntaxhighlight lang="xml">
<math mode="display" xmlns="http://www.w3.org/1998/Math/MathML">
</math>
</syntaxhighlight>
This example uses the {{
Although less compact than other formats, the XML structuring of MathML makes its content widely usable and accessible, allows near-instant display in applications such as [[
| url =
| title = Mathematical Markup Language (MathML) 1.01 Specification (Abstract)
| access-date =
|
| first2 = Stan | last2 = Devitt
| first3 = Angel | last3 = Diaz
| display-authors = etal
| date = 1999-07-07
| quote = While MathML is human-readable it is anticipated that, in all but the simplest cases, authors will use equation editors, conversion programs, and other specialized software tools to generate MathML.
}}</ref>
== Embedding MathML in HTML/XHTML files ==
MathML, being XML, can be embedded inside other XML files such as [[XHTML]] files using XML namespaces.
<syntaxhighlight lang="
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
</html>
</syntaxhighlight>
[[File:MathMLxhtml.png|thumb|435px|A rendering of the formula for a circle in MathML+XHTML using Firefox 22 on Mac OS X]]
Line 299 ⟶ 431:
{{clear}}
<syntaxhighlight lang="
<!DOCTYPE html>
<html lang="en">
</html>
</syntaxhighlight>
== Other standards ==
The [[International Organization for Standardization|ISO]]/[[International Electrotechnical Commission|IEC]] standard [[OpenDocument]] (ODF) natively supports MathML for mathematical content. OpenDocument is the default file format in office suites like [[LibreOffice]] and [[Collabora Online]], the MathML content is stored in the ODF files (e.g., .odt, .ods, .odp). Microsoft Office does not support MathML in its default office suite XML document format. Instead, it defines a different XML math syntax derived from older [[Microsoft Office]] products.
Another standard called [[OpenMath]] that has been more specifically designed (largely by the same people who devised Content MathML) for storing formulae semantically can be used to complement MathML. OpenMath data can be embedded in MathML using the {{code|lang=XML|1=<annotation-xml encoding="OpenMath">}} element. OpenMath ''content dictionaries'' can be used to define the meaning of {{code|lang=XML|<csymbol>}} elements. The following would define ''P''<sub>1</sub>(''x'') to be the first [[Legendre polynomial]]:
<syntaxhighlight lang="xml">
<apply>
</apply>
</syntaxhighlight>
The [[OMDoc]] format has been created for markup of larger mathematical structures than formulae, from statements like definitions, theorems, proofs, and examples, to complete theories and even entire text books. Formulae in OMDoc documents can either be written in Content MathML or in OpenMath; for presentation, they are converted to Presentation MathML.
== See also ==
{{Portal|Internet|Mathematics}}
* [[CSS]]
* [[List of document markup languages]]
Line 353 ⟶ 480:
* [[KaTeX]]{{snd}}JavaScript library that converts LaTeX to MathML
* [[MathJax]]{{snd}}JavaScript library that converts LaTeX to MathML
* [[OpenDocument]] The ISO/IEC standard used by applications like [[LibreOffice]] and [[Collabora Online]] natively support MathML for mathematical content
== References ==
Line 368 ⟶ 493:
== External links ==
* [
* {{cite web
| url = https://www.data2type.de/en/xml-xslt-xslfo/math-ml/
| title = The MathML Handbook
| last = Pavi | first = Sandhu
| date = 2002-12-12
| publisher = [[Charles River Media]]
| access-date = 2015-10-02
}}
* {{GitHub|davidcarlisle/web-xslt|link=hidden}} – A collection of [[XSLT]] programs for handling MathML (e.g. converting Content MathML to Presentation MathML, converting Presentation MathML to [[TeX]])
{{W3C Standards}}
{{Web browsers}}
{{ISO standards}}
{{Authority control}}
{{DEFAULTSORT:Mathml}}
[[Category:
[[Category:Mathematical markup languages]]
[[Category:MathML| ]]
[[Category:World Wide Web Consortium standards]]
[[Category:XML markup languages]]
[[Category:XML-based standards]]
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