Mathematical Markup Language (MathML) is 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 a recommendation of the W3C math working group.
MathML is not intended to be written or edited directly by humans, but exceptions are more likely to be made for Presentation MathML than for Content MathML.
History
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. 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.[1]
MathML 3.0 was officially released as a W3C Recommendation on 21 October 2010, as a revision of MathML 2.0. MathML 3 is backward compatible with MathML 2.
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. Note that 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.
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 (e.g. using the MathPlayer plugin for Internet Explorer, Opera 9.50 build 9656+ or the Fire Vox extension for Firefox).
Presentation MathML
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:
<mi>x</mi> – identifiers;<mo>+</mo> – operators;<mn>2</mn> – numbers.<mtext>non zero</mtext> – text.
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:
<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, many entities are available for specifying special symbols by name, such as π and →. An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as ⁢ for implicit multiplication. They are: U+2061 FUNCTION APPLICATION; U+2062 INVISIBLE TIMES; U+2063 INVISIBLE SEPARATOR; and U+2064 INVISIBLE PLUS. The full specification of MathML entities is [1], and is closely coordinated with the corresponding specifications for use with HTML and XML in general at [2].
Thus, the expression 
requires two layout elements: one to create the overall horizontal row and one for the superscripted exponent. Including only the layout elements and the (not yet marked up) bare tokens, the structure looks like this:
<mrow> a ⁢ <msup>x 2</msup> + b ⁢ x + c </mrow>
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>⁢</mo> <msup><mi>x</mi><mn>2</mn></msup> <mo>+</mo><mi>b</mi><mo>⁢</mo><mi>x</mi> <mo>+</mo><mi>c</mi> </mrow>
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:
<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/m athml2.dtd"> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>a</mi> <mo>⁢</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <mo>⁢ </mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> </math>
Content 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 (see [3]).
For example, <apply><sin/><ci>x& lt;/ci></apply> represents 
and <apply><plus/><ci>x </ci><cn>5</cn>< /apply> represents 
. The elements representing operators and functions are empty elements, because their operands are the other elements under the containing <apply>.
The expression 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 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:
(plus (times a (power x 2)) (times b x) c)
This reflects the long-known close relationship between XML element structures, and LISP or Scheme S-expressions[2][3]
Example and comparison to other formats
The well-known quadratic formula:
would be marked up using TeX syntax like this:
x={-b \pm \sqrt{b^2 - 4ac} \over 2a}in troff/eqn like this:
x={-b +- sqrt{b sup 2 – 4ac}} over 2ain OpenOffice Math like this (all three are valid):
x={-b plusminus sqrt {b^2 – 4 ac}} over {2 a}x={-b ± sqrt {b^2 – 4ac}} over 2ax={-b +- sqrt {b^2 – 4ac}} over 2ain ASCIIMathML like this:
x = (-b +- sqrt(b^2 – 4ac)) / (2a)
The above equation could be represented in Presentation MathML as an expression tree made up from layout elements like mfrac or msqrt elements:
<math mode="display" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mo form="prefix">−<!-- − --></mo> <mi>b</mi> <mo>±<!-- ± --></mo> <msqrt> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mo>⁢<!-- ⁢ --></mo> <mi>a</mi> <mo>⁢<!-- ⁢ --></mo> <mi>c</mi> </msqrt> </mrow> <mrow> <mn>2</mn> <mo>⁢<!-- ⁢ --></mo> <mi>a</mi> </mrow> </mfrac> </mrow></math>
In Content MathML, we cannot use ± (±) because it is a Presentation-only operator. To encode this statement in Content MathML, we must write both roots of the quadratic equation. This helps to illustrate the differences between Presentation and Content MathML.
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply><in/> <ci>x</ci> <set> <apply><divide/> <apply><plus/> <!-- upper root --> <apply><minus/> <ci>b</ci> </apply> <apply><root/> <apply><minus/> <apply><power/> <ci>b</ci> <cn>2</cn> </apply> <apply><times/> <cn>4</cn> <ci>a</ci> <ci>c</ci> </apply> </apply> </apply> </apply> <apply><times/> <cn>2</cn> <ci>a</ci> </apply> </apply> <apply><divide/> <apply><minus/> <!-- lower root --> <apply><minus/> <ci>b</ci> </apply> <apply><root/> <apply><minus/> <apply><power/> <ci>b</ci> <cn>2</cn> </apply> <apply><times/> <cn>4</cn> <ci>a</ci> <ci>c</ci> </apply> </apply> </apply> </apply> <apply><times/> <cn>2</cn> <ci>a</ci> </apply> </apply> </set> </apply></math>
The <annotation> element 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. Alternatively, the equation could be represented in Content MathML as an expression tree for the functional structure elements like apply (for function application) or eq (for the equality relation) elements:
<math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq/> <ci>x</ci> <apply> <frac/> <apply> <csymbol definitionURL="http://www.example.com/mathops/multi ops.html#plusminus"> <mo>±</mo> </csymbol> <apply> <minus/> <ci>b</ci> </apply> <apply> <power/> <apply> <minus/> <apply> <power/> <ci>b</ci> <cn>2</cn> </apply> <apply> <times/> <cn>4</cn> <ci>a</ci> <ci>c</ci> </apply> </apply> <cn>0.5</cn> </apply> </apply> <apply> <times/> <cn>2</cn> <ci>a</ci> </apply> </apply> </apply> <annotation encoding="TeX"> x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} </annotation> <annotation encoding="StarMath 5.0"> x={-b plusminus sqrt {b^2 - 4 ac}} over {2 a} </annotation></math>In the expression tree above, elements like times are defined by the MathML specification and stand for mathematical functions that are applied to sibling expressions that are interpreted as arguments. The csymbol element is a generic extension element that means whatever is specified in the document referred to in the definitionURL attribute.
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 a straightforward interpretation of its meaning in mathematical software products.[citation needed] MathML is not intended to be written or edited directly by humans.[4]
Embedding MathML in XHTML files
MathML, being XML, can be embedded inside other XML files such as XHTML files using XML namespaces. Recent browsers such as Firefox 3+ and Opera 9.6+ (support incomplete) can display Presentation MathML embedded in XHTML.
e.g. math-test.xhtml:
<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/x html-math11-f.dtd"><html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>Example of MathML embedded in an XHTML file</title> <meta name="description" content="Example of MathML embedded in an XHTML file"/> <meta name="keywords" content="Example of MathML embedded in an XHTML file"/> </head> <body> <h1>Example of MathML embedded in an XHTML file</h1> <p> Given the quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>a</mi> <mo>⁢<!-- ⁢ --></mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <mo>⁢<!-- ⁢ --></mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>0</mi> </mrow> </math> , the roots are given by <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mo form="prefix">−<!-- − --></mo> <mi>b</mi> <mo>±<!-- ± --></mo> <msqrt> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mo>⁢<!-- ⁢ --></mo> <mi>a</mi> <mo>⁢<!-- ⁢ --></mo> <mi>c</mi> </msqrt> </mrow> <mrow> <mn>2</mn> <mo>⁢<!-- ⁢ --></mo> <mi>a</mi> </mrow> </mfrac> </mrow> </math> . </p> </body></html>
Image of MathML example (math-test.xhtml) using
Firefox 3.5 with
STIX Beta fonts
Embedding MathML in HTML5 files
Inline MathML is also supported in HTML5 files in the current versions of WebKit (Safari, Chrome), Gecko (Firefox).
e.g. math-test.html:
<!DOCTYPE html><html lang="en"> <head> <meta charset="utf-8"> <title>Example of MathML embedded in an HTML5 file</title> </head> <body> <h1>Example of <abbr title="Mathematical Markup Language">MathML</abbr> embedded in an <abbr title="HyperText Markup Language">HTML</abbr>5 file</h1> <p> Given the quadratic equation <math> <mrow> <mi>a</mi> <mo>⁢</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <mo>⁢</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>0</mi> </mrow> </math>, the roots are given by <math> <mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mo form="prefix">−</mo> <mi>b</mi> <mo>±</mo> <msqrt> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>−</mo> <mn>4</mn> <mo>⁢</mo> <mi>a</mi> <mo>⁢</mo> <mi>c</mi> </msqrt> </mrow> <mrow> <mn>2</mn> <mo>⁢</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </math>. </p> </body></html>
Software support
Editors
Some editors with native MathML support (including copy and paste of MathML) are MathFlow and MathType from Design Science, MathMagic, Publicon from Wolfram Research, SciWriter from soft4science and WIRIS.[5] Full MathML editor list at W3C.[6]
MathML is also supported by major office products such as OpenOffice (via OpenOffice Math), KOffice, and MS Office 2007, as well as mathematical software products such as Mathematica, Maple and the Windows version of the Casio ClassPad 300. The W3C Browser/Editor Amaya can also be mentioned as a WYSIWYG MathML-as-is editor.
Firemath, an addon for Firefox, provides a WYSIWYG MathML editor.
Most editors will only produce presentation MathML. The MathDox formula editor is an OpenMath editor also providing presentation and content MathML. Formulator MathML Weaver uses WYSIWYG style to edit Presentation, Content and mixed markups of MathML.
Conversion
Several utilities for converting to and from MathML are available. W3.org maintains a list of MathML related software for download.[7]
Web browsers
Of the major web browsers, Gecko based browsers (e.g., Firefox and Camino) fully support MathML natively.[8]
While the WebKit layout engine has a development version of MathML,[9] this feature is only available in version 5.1 and higher of Safari,[citation needed] Chrome 24 on desktop platforms as well as Chrome on iOS.[10][11]
Opera, since version 9.5, supports MathML for CSS profile,[12][13] but is unable to position diacritical marks properly.[14] Prior to version 9.5 it required User JavaScript or custom stylesheets to emulate MathML support.[15]
Internet Explorer does not support MathML natively. Support in IE7, IE8 and IE9 can be added by installing the MathPlayer plugin.[16]
The KHTML-based Konqueror currently does not provide support for MathML.[17]
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.[18]
Web conversion
ASCIIMathML[19] provides a JavaScript library to rewrite a convenient Wiki-like text syntax used inline in web pages into MathML on the fly; it works in Gecko-based browsers, and Internet Explorer with MathPlayer. LaTeXMathML[20] does the same for (a subset of) the standard LaTeX mathematical syntax. ASCIIMathML syntax would also be quite familiar to anyone used to electronic scientific calculators.
MathJax, a JavaScript library for inline rendering of mathematical formulae, can be used to translate LaTeX into MathML for direct interpretation by the browser.[21]
Blahtex is a TeX-to-MathML converter intended for use with MediaWiki.
Equation Server for .NET from soft4science can be used on the server side (ASP.NET) for TeX-Math[22] (Subset of LaTeX math syntax) to MathML conversion. It can also create bitmap images (Png, Jpg, Gif, etc.) from TeX-Math or MathML input.
jqMath[23] is a JavaScript module that dynamically converts a simple TeX-like syntax to MathML if the browser supports it, else simple HTML and CSS.
LaTeXML is a perl utility to convert LaTeX documents to HTML, optionally either using MathML or converting mathematical expressions to bitmap images.
Support of software developers
Support of MathML format accelerates software application development in such various topics, as computer-aided education (distance learning, electronic textbooks and other classroom materials); automated creation of attractive reports; computer algebra systems; authoring, training, publishing tools (both for web and desktop-oriented), and many other applications for mathematics, science, business, economics, etc. Several software vendors propose a component edition of their MathML editors, thus providing the easy way for software developers to insert mathematics rendering/editing/processing functionality in their applications. For example, Formulator ActiveX Control from Hermitech Laboratory can be incorporated into an application as a MathML-as-is editor, Design Science offer a toolkit for building web pages that include interactive math (MathFlow Developers Suite,[24]).
MathML version 3
Version 3 of the MathML specification was released as a W3C Recommendation on 20 October 2010. A proposed recommendation of A MathML for CSS Profile has also been published; 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.[1]
Other standards
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
<apply> <csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/contrib/c d/orthpoly1.xhtml#legendreP"> <msub><mi>P</mi><mn>1</mn></msub> </csymbol> <ci>x</ci></apply>
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[25] through relatively simple XSL Transformations.
See also
References
- ^ a b Mathematical Markup Language Version 3.0 W3C Recommendation. W3.org. Retrieved on 9 May 2012.
- ^ Steven DeRose. The SGML FAQ Book: Understanding the Relationship of SGML and XML, Kluwer Academic Publishers, 1997. ISBN 978-0-7923-9943-8.
- ^ Canonical S-expressions#cite note-0
- ^ Buswell, Steven; Devitt, Stan; Diaz, Angel; et al. (7 July 1999). "Mathematical Markup Language (MathML) 1.01 Specification (Abstract)". http://www.w3.org/TR/REC-MathML/#abst ract. Retrieved 26 September 2006. "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."
- ^ WIRIS editor page describing the use of MathML. Wiris.com. Retrieved on 9 May 2012.
- ^ MathML Software – Editors at W3C. W3.org (24 April 2012). Retrieved on 9 May 2012.
- ^ MathML Implementations
- ^ Sidje, Roger B., Authoring MathML for Mozilla, Mozilla, http://www.mozilla.org/projects/mathm l/authoring.html
- ^ Bug 3251 – Implement MathML (master bug), WebKit, https://bugs.webkit.org/show_bug.cgi? id=3251
- ^ "MathML Support Coming To Chrome". BrowserFame. http://browserfame.com/900/mathml-in- chrome. Retrieved 16 October 2012.
- ^ "Stable Chrome 24 supports MathML and closes security holes". The H Online. http://www.h-online.com/open/news/ite m/Stable-Chrome-24-supports-MathML-an d-closes-security-holes-1781648.html. Retrieved 11 January 2013.
- ^ McCathieNevile, Charles (27 September 2007), Can Kestrels do Math? MathML support in Opera Kestrel, Opera, http://dev.opera.com/articles/view/ca n-kestrels-do-math-mathml-support-in/
- ^ Olsen, Tommy A. (16 November 2007), Even more work, Opera, http://my.opera.com/desktopteam/blog/ 2007/11/16/even-more-work
- ^ MathML – The Opera MathML blog. My.opera.com (1 November 2007). Retrieved on 9 May 2012.
- ^ UserJS for MathML 2.0. My.opera.com. Retrieved on 9 May 2012.
- ^ MathPlayer – version history, http://www.dessci.com/en/products/mat hplayer/versionhistory.htm
- ^ Bug 30526 – MathML support for Konqi, http://bugs.kde.org/show_bug.cgi?id=3 0526
- ^ Vismor, Timothy, Viewing Mathematics on the Internet, https://vismor.com/documents/site_imp lementation/viewing_mathematics/viewi ng_mathematics.php, retrieved 13 April 2011
- ^ ASCIIMathML: Math on the web for everyone. .chapman.edu. Retrieved on 9 May 2012.
- ^ LaTeXMathML: a dynamic LaTeX mathematics to MathML converter. Maths.nottingham.ac.uk. Retrieved on 9 May 2012.
- ^ MathJax MathML Support. Mathjax.org. Retrieved on 9 May 2012.
- ^ TeX-Math. TeX-Math. Retrieved on 9 May 2012.
- ^ jqMath – Put Math on the Web. Mathscribe.com. Retrieved on 9 May 2012.
- ^ MathFlow. Dessci.com. Retrieved on 9 May 2012.
- ^ David Carlisle (9 May 2007). "XHTML and MathML from Office 2007". David Carlisle. http://dpcarlisle.blogspot.com/2007/0 4/xhtml-and-mathml-from-office-20007. html. Retrieved 20 September 2007.
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