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Mathematical Markup Language (MathML) is a pair of mathematical markup languages, an application of
XML Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing data. It defines a set of rules for encoding electronic document, documents in a format that is both human-readable and Machine-r ...
for describing
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
notations and capturing both its structure and content. Its aim is to natively integrate mathematical formulae into
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pages and other documents. It is part of
HTML5 HTML5 (Hypertext Markup Language 5) is a markup language used for structuring and presenting hypertext documents on the World Wide Web. It was the fifth and final major HTML version that is now a retired World Wide Web Consortium (W3C) recommend ...
and standardised by
ISO The International Organization for Standardization (ISO ; ; ) is an independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. Me ...
/IEC since 2015.


History

Following some experiments in the Arena browser based on proposals for mathematical markup in HTML, MathML 1 was released as a W3C recommendation in April 1998 as the first
XML Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing data. It defines a set of rules for encoding electronic document, documents in a format that is both human-readable and Machine-r ...
language to be recommended by the
W3C The World Wide Web Consortium (W3C) is the main international standards organization for the World Wide Web. Founded in 1994 by Tim Berners-Lee, the consortium is made up of member organizations that maintain full-time staff working together in ...
. 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 1.1, Mozilla 1.0 and Opera 9.5. In October 2003, the second edition of MathML Version 2.0 was published as the final release by the W3C Math Working Group. MathML was originally designed before the finalization of XML 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 URL ''http://www.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 a
W3C recommendation The World Wide Web Consortium (W3C) is the main international standards organization for the World Wide Web. Founded in 1994 by Tim Berners-Lee, the consortium is made up of member organizations that maintain full-time staff working together in ...
on 20 October 2010. A recommendation of ''A MathML for CSS Profile'' was 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 with OpenMath. Other content elements are defined in terms of a transformation to the strict subset. New content elements include which associates bound variables () to expressions, for example a summation index. The new 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 in
WebKit WebKit is a browser engine primarily used in Apple's Safari web browser, as well as all web browsers on iOS and iPadOS. WebKit is also used by the PlayStation consoles starting with the PS3, the Tizen mobile operating systems, the Amazon K ...
around this same time, with a
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implementation 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 an ISO/IEC international 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 the
MathJax MathJax is a cross-browser JavaScript library that displays mathematical notation in web browsers, using MathML, LaTeX, and ASCIIMathML markup. MathJax is released as open-source software under the Apache License. The MathJax project started in ...
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.


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 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.


Presentation and semantics

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 A screen reader is a form of assistive technology (AT) that renders text and image content as speech or braille output. Screen readers are essential to blindness, blind people, and are useful to visually impaired people, Illiteracy, illiterate, ...
(e.g. using the VoiceOver in
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). JAWS from 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. 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. A valid MathML document typically consists of the XML declaration, DOCTYPE declaration, and document element. The document body then contains MathML expressions which appear in elements as needed in the document. Often, MathML will be embedded in more general documents, such as
HTML Hypertext Markup Language (HTML) is the standard markup language for documents designed to be displayed in a web browser. It defines the content and structure of web content. It is often assisted by technologies such as Cascading Style Sheets ( ...
,
DocBook DocBook is a Semantics (computer science), semantic markup language for technical documentation. It was originally intended for writing technical documents related to computer hardware and software, but it can be used for any other sort of docume ...
, or other
XML Extensible Markup Language (XML) is a markup language and file format for storing, transmitting, and reconstructing data. It defines a set of rules for encoding electronic document, documents in a format that is both human-readable and Machine-r ...
-based formats.


Presentation MathML

Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with . 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: * – identifiers; * – operators; * – numbers; * – text. Note, however, that these token elements may be used as extension points, allowing markup in host languages. MathML in
HTML5 HTML5 (Hypertext Markup Language 5) is a markup language used for structuring and presenting hypertext documents on the World Wide Web. It was the fifth and final major HTML version that is now a retired World Wide Web Consortium (W3C) recommend ...
allows most inline HTML markup in mtext, and 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: * – a horizontal row of items; * , , and others – superscripts, limits over and under operators like sums, etc.; * – fractions; * and – roots; * – surrounding content with fences, such as parentheses. As usual in HTML and XML, many entities are available for specifying special symbols by name, such as and . 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: * (to distinguish \sin(x) from \sin\cdot x in \sin); * (to distinguish a_ from a_ in a_); * (vice versa); * (to distinguish 2 + \frac from 2 \cdot \frac in 2\tfrac). The full specification of MathML entities is closely coordinated with the corresponding specifications for use with HTML and XML in general. Thus, the expression a x^2+b x+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 (), operators (), or numbers (). Adding the token markup, the full form ends up as a x2 +bx +c A complete document that consists of just the MathML example above, is shown here: a x2 +bx +c


Content MathML

Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the element that represents function application. The function being applied is the first child element under , 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 and . Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML: , , etc. There are over a hundred different elements for different functions and operators. For example, represents \sin(x) and represents x+5. The elements representing operators and functions are empty elements, because their operands are the other elements under the containing . The expression a x^2+b x+c could be represented as a x 2 b x c Content MathML is nearly
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to expressions in a functional language such as Scheme and other dialects of
Lisp Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...
. 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 (times a (power x 2)) (times b x) c) This reflects the long-known close relationship between XML element structures, and
LISP Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized Polish notation#Explanation, prefix notation. Originally specified in the late 1950s, ...
or Scheme S-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 semanticsnames, descriptions and rules. A 2018 paper presented at the SIGIR conference 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 In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...
could be represented in Presentation MathML as an expression tree made up from layout elements like or : x = b ± b2 4ac 2 a x = \frac x = over This example uses the 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 as StarMath or the markup using
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...
syntax. The field is usually a MIME 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 as
web browser A web browser, often shortened to browser, is an application for accessing websites. When a user requests a web page from a particular website, the browser retrieves its files from a web server and then displays the page on the user's scr ...
s, 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 as
XHTML Extensible HyperText Markup Language (XHTML) is part of the family of XML markup languages which mirrors or extends versions of the widely used HyperText Markup Language (HTML), the language in which Web pages are formulated. While HTML, pr ...
files using XML namespaces. Example of MathML embedded in an XHTML file

Example of MathML embedded in an XHTML file

The area of a circle is π r 2 .

Inline MathML is also supported in
HTML5 HTML5 (Hypertext Markup Language 5) is a markup language used for structuring and presenting hypertext documents on the World Wide Web. It was the fifth and final major HTML version that is now a retired World Wide Web Consortium (W3C) recommend ...
files. There is no need to specify namespaces as there was in
XHTML Extensible HyperText Markup Language (XHTML) is part of the family of XML markup languages which mirrors or extends versions of the widely used HyperText Markup Language (HTML), the language in which Web pages are formulated. While HTML, pr ...
. Example of MathML embedded in an HTML5 file

Example of MathML embedded in an HTML5 file

The area of a circle is π r 2 .


Other standards

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 element. OpenMath ''content dictionaries'' can be used to define the meaning of elements. The following would define ''P''1(''x'') to be the first Legendre polynomial: P1 x 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. The
ISO The International Organization for Standardization (ISO ; ; ) is an independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. Me ...
/ IEC standard Office Open XML (OOXML) defines a different XML math syntax, derived from Microsoft Office products. However, it is partially compatible through XSL Transformations.


See also

* CSS *
List of document markup languages The following is a list of document markup languages. You may also find the List of markup languages of interest. Well-known document markup languages * HyperText Markup Language (HTML) – an ad hoc markup language that was originally created f ...
* Comparison of document markup languages * Formula editors * LaTeX2HTML * LaTeXML * KaTeXJavaScript library that converts LaTeX to MathML *
MathJax MathJax is a cross-browser JavaScript library that displays mathematical notation in web browsers, using MathML, LaTeX, and ASCIIMathML markup. MathJax is released as open-source software under the Apache License. The MathJax project started in ...
JavaScript library that converts LaTeX to MathML


References


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. * * – A collection of
XSLT XSLT (Extensible Stylesheet Language Transformations) is a language originally designed for transforming XML documents into other XML documents, or other formats such as HTML for web pages, plain text, or XSL Formatting Objects. These formats c ...
programs for handling MathML (e.g. converting Content MathML to Presentation MathML, converting Presentation MathML to
TeX Tex, TeX, TEX, may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Tex Earnhardt (1930–2020), U.S. businessman * Joe Tex (1933–1982), stage name of American soul singer ...
) {{DEFAULTSORT:Mathml Articles with example Scheme (programming language) code Mathematical markup languages World Wide Web Consortium standards XML markup languages XML-based standards