Mathematical Markup Language (MathML) is a mathematical markup language, an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide Web pages and other documents. It is part of HTML5 and an ISO standard since 2015.
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.
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 URI 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.
Version 3 of the MathML specification was released as a W3C recommendation 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
<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.
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 MathPlayer plugin for Internet Explorer or Firefox, Opera 9.50 build 9656+ or the Fire Vox extension for Firefox). Newer versions of JAWS support MathML voicing as well as braille output.
Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with
m. A Presentation MathML expression is built up out of tokens that are combined using higher-level elements, which control their layout (there are also about 50 attributes, which mainly control fine details).
Token elements generally only contain characters (not other elements). They include:
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
<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 entities are available for specifying special symbols by name, such as
→. 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:
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:
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 <math> elements as needed in the document. Often, MathML will be embedded in more general documents, such as HTML, DocBook, or other XML schemas. A complete document that consists of just the MathML example above, is shown here:
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
cn. Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML:
power, etc. There are over a hundred different elements for different functions and operators.
Content MathML is nearly isomorphic to expressions in a functional language such as Scheme.
<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:
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. As proposed in, the semantic knowledge-base Wikidata can be used as OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.
The above equation could be represented in Presentation MathML as an expression tree made up from layout elements like mfrac or msqrt elements:<!-- 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">. -->
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 as StarMath or the markup using LaTeX syntax. The
encoding 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 case.
Although less compact than TeX, the XML structuring promises to make it widely usable and allows for instant display in applications such as Web browsers and facilitates an interpretation of its meaning in mathematical software products. MathML is not intended to be written or edited directly by humans.
MathML, being XML, can be embedded inside other XML files such as XHTML files using XML namespaces. Browsers such as Firefox 3+ and Opera 9.6+ (support incomplete) can display Presentation MathML embedded in XHTML.A rendering of the formula for a circle in MathML+XHTML using Firefox 22 on Mac OS X
While the WebKit layout engine has a development version of MathML, this feature is only available in version 5.1 and higher of Safari, Chrome 24 but not in later versions of Chrome. Google removed support of MathML claiming architectural security issues and low usage do not justify their engineering time. As of October 2013, the WebKit/Safari implementation has numerous bugs.
JavaFX/WebView. Also based on WebKit, the JavaFX embedded web browser supports MathML starting with JavaFX 8 Update 192 and JavaFX 11 versions. Support is broken, in JavaFX 8 previous versions, JavaFX 9 and JavaFX 10.
Internet Explorer does not support MathML natively. Support for IE6 through IE9 can be added by installing the MathPlayer plugin. IE10 has some crashing bugs with MathPlayer and Microsoft decided to completely disable in IE11 the binary plug-in interface that MathPlayer needs. MathPlayer has a license that may limit its use or distribution in commercial webpages and software. Using or distributing the MathPlayer plugin to display HTML content via the WebBrowser control in commercial software may also be forbidden by this license.
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 a slightly more limited support.
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 exists is overwhelmingly the result of unpaid volunteer time/work.
In 2015 the MathML Association was founded to support the adoption of the MathML standard.
Another standard called OpenMath that has been designed (largely by the same people who devised Content MathML) more specifically for storing formulae semantically can also be used to complement MathML. OpenMath data can be embedded in MathML using the
<annotation-xml encoding="OpenMath"> element. OpenMath content dictionaries can be used to define the meaning of
<csymbol> elements. The following would define P1(x) to be the first Legendre polynomial
The OMDoc format has been created for markup of larger mathematical structures than formulae, from statements like definitions, theorems, proofs, or example, to theories and 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/IEC standard Office Open XML (OOXML) defines a different XML math syntax, derived from Microsoft Office products. However, it is partially compatible through relatively simple XSL Transformations.