MathML

For a guide on using MathML and math tags on Wikipedia, seeHelp:Displaying a formula.

MathML

Mathematical Markup Language
Abbreviation	MathML
Native name	Mathematical Markup Language ISO/IEC 40314
Status	W3C Recommendation
First published	April 1998(1998-04)
Latest version	3.0 April 10, 2014(2014-04-10)
Organization	W3C,ISO,IEC
Committee	W3C Math Working Group ISO/IEC JTC 1
Editors	David Carlisle Patrick Ion Robert Miner Frédéric Wang
Authors	Principal authors 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
Base standards	XML
Related standards	OpenMath,Office Open XML,OMDoc
Website	W3C:www.w3.org/TR/MathML/ ISO:www.iso.org/standard/58439.html

Mathematical Markup Language(MathML) is amathematical markup language,an application ofXMLfor describingmathematicalnotations and capturing both its structure and content, and is one of a number ofmathematical markup languages.Its aim is to natively integrate mathematical formulae intoWorld Wide Webpages and other documents. It is part ofHTML5and standardised byISO/IEC since 2015.

History

Following some experiments in theArena browserbased on proposals for mathematical markup in HTML,MathML 1 was released as a W3C recommendation in April 1998 as the firstXMLlanguage to be recommended by theW3C.Version 1.01 of the format was released in July 1999 and version 2.0 appeared in February 2001. Implementations of the specification appeared inAmaya 1.1,Mozilla 1.0andOpera 9.5.In October 2003, the second edition of MathML Version 2.0 was published as the final release by theW3C Math Working Group.

MathML was originally designed before the finalization ofXML namespaces.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 URLhttp:// w3.org/1998/Math/MathML.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.

MathML version 3

Version 3 of the MathML specification was released as aW3C recommendationon 20 October 2010. A recommendation ofA MathML for CSS Profilewas later released on 7 June 2011;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 withOpenMath.Other content elements are defined in terms of a transformation to the strict subset. New content elements include<bind>which associates bound variables (<bvar>) to expressions, for example a summation index. The new<share>element allows structure sharing.

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.An implementation of MathML 2 landed inWebKitaround this same time,with aChromiumimplementation following a couple of years later,although that implementation was removed from Chromium after less than a year.

The Second Edition of MathML 3.0 was published as a W3C Recommendation on 10 April 2014.The specification was approved as anISO/IECinternational standard 40314:2015 on 23 June 2015.Also in 2015, the MathML Association was founded to support the adoption of the MathML standard.At that time, according to a member of theMathJaxteam, 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.

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.”MathML Core set itself apart from MathML 3.0 by including detailed rendering rules and integration withCSS,automated browser support testing resources, and focusing on a fundamental subset of MathML. An implementation was added to Chromium at the beginning of 2023.

Presentation and semantics

Generic MathML

Filename extension	.mml
Internet media type	application/mathml+xml
Type code	MML
Uniform Type Identifier (UTI)	public.mathml
UTI conformation	public.xml
Developed by	World Wide Web Consortium
Type of format	Mathematical markup language
Extended from	XML
Extended to	Presentation MathML Content MathML
Standard	W3C MathML ISO/IEC 40314
Open format?	Yes

MathML deals not only with thepresentationbut also themeaningof 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 ofscreen readers(e.g. using theVoiceOverinSafari).JAWSfrom version 16 onward supports MathML voicing as well as braille output.

The quality of rendering of MathML in a browser depends on the installed fonts. TheSTIX Fonts projecthave released a comprehensive set of mathematical fonts under an open license. TheCambria Mathfont supplied with Microsoft Windows had slightly more limited support.

A valid MathML document typically consists of the XML declaration,DOCTYPEdeclaration, and document element. The document body then contains MathML expressions which appear in<math>elements as needed in the document. Often, MathML will be embedded in more general documents, such asHTML,DocBook,or otherXML-based formats.

Presentation MathML

Presentation MathML

Internet media type	application/mathml-presentation+xml
Type code	MMLp
Uniform Type Identifier (UTI)	public.mathml.presentation
UTI conformation	public.mathml
Extended from	Generic MathML

Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin withm.A Presentation MathML expression is built up out oftokensthat 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:

• <mi>x</mi>– identifiers;
• <mo>+</mo>– operators;
• <mn>2</mn>– numbers;
• <mtext>suchthat</mtext>– text.

Note, however, that these token elements may be used as extension points, allowing markup in host languages. MathML inHTML5allows most inline HTML markup in mtext, and<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:

• <mrow>– a horizontal row of items;
• <msup>,<munderover>,and others – superscripts, limits over and under operators like sums, etc.;
• <mfrac>– fractions;
• <msqrt>and<mroot>– roots;
• <mfenced>– surrounding content with fences, such as parentheses.

As usual in HTML and XML, manyentitiesare available for specifying special symbols by name, such asπand→.An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as⁢(or the shorthand⁢) for implicit multiplication. They are:

• U+2061FUNCTION APPLICATION(to distinguish$\sin(x)$from$\sin \cdot x$in$\sin {x}$);
• U+2062INVISIBLE TIMES(to distinguish$a_{m\times n}$from$a_{m,n}$in$a_{mn}$);
• U+2063INVISIBLE SEPARATOR(vice versa);
• U+2064INVISIBLE PLUS(to distinguish$2+{\frac {1}{3}}$from$2\cdot {\frac {1}{3}}$in$2{\tfrac {1}{3}}$).

The full specification of MathML entitiesis closely coordinated with the corresponding specifications for use with HTML and XML in general.

Thus, the expression$ax^{2}+bx+c$requires two layout elements: one to create the overall horizontal row and one for the superscripted exponent. However, the individual tokens also have to be identified as identifiers (<mi>), operators (<mo>), or numbers (<mn>). Adding the token markup, the full form ends up as

<mrow>
<mi>a</mi><mo>&InvisibleTimes;</mo><msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo><mi>b</mi><mo>&InvisibleTimes;</mo><mi>x</mi>
<mo>+</mo><mi>c</mi>
</mrow>

A complete document that consists of just the MathML example above, is shown here:

<?xml version= "1.0" encoding= "UTF-8"?>
<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN" "http:// w3.org/Math/DTD/mathml2/mathml2.dtd" >
<mathxmlns="http:// w3.org/1998/Math/MathML">
<mrow>
<mi>a</mi><mo>&InvisibleTimes;</mo><msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo><mi>b</mi><mo>&InvisibleTimes;</mo><mi>x</mi>
<mo>+</mo><mi>c</mi>
</mrow>
</math>

Content MathML

Content MathML

Internet media type	application/mathml-content+xml
Type code	MMLc
Uniform Type Identifier (UTI)	public.mathml.content
UTI conformation	public.mathml
Extended from	Generic MathML

Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the<apply>element that represents function application. The function being applied is the first child element under<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<ci>and<cn>.Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML:<times>,<power>,etc. There are over a hundred different elements for different functions and operators.

For example,<apply><sin/><ci>x</ci></apply>represents$\sin(x)$and<apply><plus/><ci>x</ci><cn>5</cn></apply>represents$x+5$.The elements representing operators and functions are empty elements, because their operands are the other elements under the containing<apply>.

The expression$ax^{2}+bx+c$could be represented as

<math>
<apply>
<plus/>
<apply>
<times/>
<ci>a</ci>
<apply>
<power/>
<ci>x</ci>
<cn>2</cn>
</apply>
</apply>
<apply>
<times/>
<ci>b</ci>
<ci>x</ci>
</apply>
<ci>c</ci>
</apply>
</math>

Content MathML is nearlyisomorphictoexpressionsin afunctional languagesuch asSchemeand other dialects ofLisp.<apply>...</apply>amounts to Scheme's(...),and the many operator and function elements amount to Scheme functions. With this trivial literal transformation, plus un-tagging the individual tokens, the example above becomes:

(plus
(timesa(powerx2))
(timesbx)
c)

This reflects the long-known close relationship between XML element structures, andLISPor SchemeS-expressions.

Wikidata annotation in Content MathML

According to the OM Society,OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics – names, descriptions and rules. A 2018 paper presented at theSIGIRconferenceproposed that the semantic knowledge baseWikidatacould 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-knownquadratic formulacould be represented in Presentation MathML as an expression tree made up from layout elements like<mfrac>or<msqrt>:

<mathmode="display"xmlns="http:// w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<moform="prefix">&minus;</mo>
<mi>b</mi>
<mo>&pm;</mo>
<msqrt>
<msup><mi>b</mi><mn>2</mn></msup>
<mo>&minus;</mo>
<mn>4</mn><mo>&it;</mo><mi>a</mi><mo>&it;</mo><mi>c</mi>
</msqrt>
</mrow>
<mrow>
<mn>2</mn>
<mo>&it;</mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<annotationencoding="application/x-tex"><!-- TeX -->
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
</annotation>
<annotationencoding="StarMath 5.0">
x={-bplusminussqrt{b^2-4ac}}over{2a}
</annotation>
<!-- More annotations can be written: application/x-troff-eqn for eqn, application/x-asciimath for AsciiMath... -->
<!-- Semantic MathML go under <annotation-xml encoding= "MathML-Content" >. -->
</semantics>
</math>

This example uses the<annotation>element, which can be used to embed a semantic annotation in non-XML format, for example to store the formula in the format used by an equation editor such asStarMathor the markup usingLaTeXsyntax. Theencodingfield is usually aMIME type,although most of the equation encodings don't have such a registration; freeform text may be used in such cases.

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 asweb browsers,and facilitates an interpretation of its meaning in mathematical software products. MathML is not intended to be written or edited directly by humans.

Embedding MathML in HTML/XHTML files

MathML, being XML, can be embedded inside other XML files such asXHTMLfiles using XML namespaces.

<?xml version= "1.0" encoding= "UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
"http:// w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd" >
<htmlxmlns="http:// w3.org/1999/xhtml"xml:lang="en">
<head>
<title>Example of MathML embedded in an XHTML file</title>
<metaname="description"content="Example of MathML embedded in an XHTML file"/>
</head>
<body>
<h1>Example of MathML embedded in an XHTML file</h1>
<p>
The area of a circle is
<mathxmlns="http:// w3.org/1998/Math/MathML">
<mi>&#x03C0;<!-- π --></mi>
<mo>&#x2062;<!-- &InvisibleTimes; --></mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</math>.
</p>
</body>
</html>

A rendering of the formula for a circle in MathML+XHTML using Firefox 22 on Mac OS X

Inline MathML is also supported inHTML5files. There is no need to specify namespaces as there was inXHTML.

<!DOCTYPE html>
<htmllang="en">
<head>
<metacharset="utf-8">
<title>Example of MathML embedded in an HTML5 file</title>
</head>
<body>
<h1>Example of MathML embedded in an HTML5 file</h1>
<p>
The area of a circle is
<math>
<mi>&pi;</mi>
<mo>&InvisibleTimes;</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</math>.
</p>
</body>
</html>

Other standards

Another standard calledOpenMaththat 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<annotation-xmlencoding="OpenMath">element. OpenMathcontent dictionariescan be used to define the meaning of<csymbol>elements. The following would defineP₁(x) to be the firstLegendre polynomial:

<apply>
<csymbolencoding="OpenMath"definitionURL="http:// openmath.org/cd/contrib/cd/orthpoly1.xhtml#legendreP">
<msub><mi>P</mi><mn>1</mn></msub>
</csymbol>
<ci>x</ci>
</apply>

TheOMDocformat 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.

TheISO/IECstandardOffice Open XML(OOXML) defines a different XML math syntax, derived fromMicrosoft Officeproducts. However, it is partially compatiblethroughXSL Transformations.

See also

• CSS
• List of document markup languages
• Comparison of document markup languages
• Formula editors
• LaTeX2HTML
• LaTeXML
• KaTeX– JavaScript library that converts LaTeX to MathML
• MathJax– JavaScript library that converts LaTeX to MathML

Further reading

Specifications

• W3C Recommendation: Mathematical Markup Language (MathML) 1.01 Specification
• W3C Recommendation: Mathematical Markup Language (MathML) Version 2.0 (Second Edition)
• W3C Recommendation: Mathematical Markup Language (MathML) Version 3.0 (Third Edition)

External links

• W3C Math Home– Contains the specifications, a FAQ, and a list of supporting software.
• Pavi, Sandhu (12 December 2002)."The MathML Handbook".Charles River Media.Retrieved2 October2015.
• web-xslt– A collection ofXSLTprograms for handling MathML (e.g. converting Content MathML to Presentation MathML, converting Presentation MathML toTeX)