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(MathML) Mathematical Markup Language
Developed by World Wide Web Consortium
Type of format Markup language
Extended from XML
Standard W3C MathML

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.

History[edit]

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. As such, MathML markup is often not namespaced, and applications that deal with MathML, such as the Mozilla browsers, do not require a namespace. For applications that wish to namespace MathML, the recommended namespace URI is http://www.w3.org/1998/Math/MathML.

Presentation and semantics[edit]

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[edit]

Presentation MathML focuses on the display of an equation, and has about 30 elements, and 50 attributes. The elements all begin with m and include token element: <mi>x</mi> - identifiers; <mo>+</mo> - operators; <mn>2</mn> - number. Tokens are combined using layout elements that include: <mrow> - a row; <msup> - superscripts; <mfrac> - fractions. The attributes mainly control fine details of the presentation. A large number of entities are available that represent letters (&pi;), symbols (&RightArrow;) and some non-visible characters such as &InvisibleTimes; representing multiplication.

For valid XML documents as per the W3C specifications, a MathML file must have a doctype declaration:

  <?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">

Thus, the expression a x^2+b x+c could be represented in a valid MathML file as:

 <?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">
    <mrow>
      <mi>a</mi>
      <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>b</mi>
      <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
      <mi>x</mi>
      <mo>+</mo>
      <mi>c</mi>
    </mrow>
  </math>

Content MathML[edit]

Content MathML focuses on the semantic meaning of the expression. Central to Content MathML is the <apply> element that represents a function or operator, given in the first child, applied to the remaining child elements. 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 <ci> element represents an identifier, <cn> a number, and there are over a hundred different elements for different functions and operators. Content MathML uses only a few attributes.

The expression a x^2+b x+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>

Example and comparison to other formats[edit]

The well-known quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

would be marked up using LaTeX syntax like this:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

in troff/eqn like this:

x={-b +- sqrt{b sup 2 - 4ac}} over 2a

in OpenOffice.org 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 2a
x={-b +- sqrt {b^2 - 4ac}} over 2a

in 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">&#x2212;<!-- &minus; --></mo>
        <mi>b</mi>
        <mo>&#x00B1;<!-- &PlusMinus; --></mo>
        <msqrt>
          <msup>
            <mi>b</mi>
            <mn>2</mn>
          </msup>
          <mo>&#x2212;<!-- &minus; --></mo>
          <mn>4</mn>
          <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
          <mi>a</mi>
          <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
          <mi>c</mi>
        </msqrt>
      </mrow>
      <mrow>
        <mn>2</mn>
        <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
        <mi>a</mi>
      </mrow>
    </mfrac>
  </mrow>
</math>

In Content MathML, we cannot use &PlusMinus; (±) 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/multiops.html#plusminus">
          <mo>&PlusMinus;</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.[2]

Embedding MathML in XHTML files[edit]

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/xhtml-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>&#x2062;<!-- &InvisibleTimes; --></mo>
          <msup>
            <mi>x</mi>
            <mn>2</mn>
          </msup>
          <mo>+</mo>
          <mi>b</mi>
          <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
          <mi>x</mi>
          <mo>+</mo>
          <mi>c</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">&#x2212;<!-- &minus; --></mo>
              <mi>b</mi>
              <mo>&#x00B1;<!-- &PlusMinus; --></mo>
              <msqrt>
                <msup>
                  <mi>b</mi>
                  <mn>2</mn>
                </msup>
                <mo>&#x2212;<!-- &minus; --></mo>
                <mn>4</mn>
                <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
                <mi>a</mi>
                <mo>&#x2062;<!-- &InvisibleTimes; --></mo>
                <mi>c</mi>
              </msqrt>
            </mrow>
            <mrow>
              <mn>2</mn>
              <mo>&#x2062;<!-- &InvisibleTimes; --></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
Image of MathML example (math-test.xhtml) using Firefox 3.5 with STIX Beta fonts

Embedding MathML in HTML5 files[edit]

Inline MathML is also supported in HTML5 files in the upcoming versions of WebKit (Safari, Chrome), Gecko (Firefox) and Presto (Opera).

e.g. math-test.html:

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>Example of MathML embedded in an HTML5 file</title>
    <meta charset="utf-8" />
  </head>
  <body>
    <h1>Example of MathML embedded in an HTML5 file</h1>
    <p>
      Given the quadratic equation
      <math>
        <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>
      , the roots are given by
      <math>
        <mrow>
          <mi>x</mi>
          <mo>=</mo>
          <mfrac>
            <mrow>
              <mo form="prefix">&minus;</mo>
              <mi>b</mi>
              <mo>&PlusMinus;</mo>
              <msqrt>
                <msup>
                  <mi>b</mi>
                  <mn>2</mn>
                </msup>
                <mo>&minus;</mo>
                <mn>4</mn>
                <mo>&InvisibleTimes;</mo>
                <mi>a</mi>
                <mo>&InvisibleTimes;</mo>
                <mi>c</mi>
              </msqrt>
            </mrow>
            <mrow>
              <mn>2</mn>
              <mo>&InvisibleTimes;</mo>
              <mi>a</mi>
            </mrow>
          </mfrac>
        </mrow>
      </math>
      .
    </p>
  </body>
</html>

Software support[edit]

Editors[edit]

Some editors with native MathML support (including copy and paste of MathML) are MathFlow, MathMagic, and MathType from Design Science, Publicon from Wolfram Research, SciWriter from soft4science and WIRIS[3]. Full MathML editor list at W3C[4].

MathML is also supported by major office products such as OpenOffice.org (via OpenOffice.org 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[edit]

Several utilities for converting mathematical expressions to MathML are available, including converters between TeX and MathML.[5] ConTeXt does the reverse and uses TeX for typesetting MathML (usually resulting in PDF documents). MathType from Design Science allows users to create equations in a WYSIWYG window and export them as MathML. Also, Wolfram Research provides a web page to convert typed mathematical expressions to MathML,[6] and to render MathML as an image.[7]

GNU TeXmacs is a WYSIWYG editor with extensive support for mathematics. Converters exist for presentation MathML in both directions. TeXmacs can be used to write mathematical articles that are exported to XHTML with embedded MathML.

Another WYSIWYG equation editor MathMagic supports MathML saving and reading, including options for "TeX" encoding and other annotations. MathMagic also supports MathML batch conversion from/to TeX, LaTeX, EPS, PDF, PNG, GIF, JPEG, BMP, Microsoft Word equation, and other formats.

Web browsers[edit]

Of the major web browsers, recent versions of Gecko browsers (e.g., Firefox and Camino),[8] support MathML natively.

While the WebKit layout engine does have development version of MathML[9], this option is only available in version 5.1 and higher of Safari, which is based on WebKit.[10] Chrome, which is also based on WebKit, does not currently have this option enabled.[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 and IE8 can be added by installing the MathPlayer plugin. MathPlayer is not currently compatible with IE9.[11]

The KHTML-based Konqueror currently does not provide support for MathML.[16]

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 licence. The Cambria Math font supplied with Microsoft windows had a slightly more limited support.[11]

Web conversion[edit]

ASCIIMathML[17] 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[18] 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 formulea, can be used to translate LaTeX into MathML for direct interpretation by the browser.[19]

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[20] (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[21] 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[edit]

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 propose a toolkit for building web pages that include interactive math (WebEQ Developers Suite,[22]).

MathML Version 3[edit]

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[edit]

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/cd/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[23] through relatively simple XSL Transformations.

See also[edit]

References[edit]

  1. ^ a b Mathematical Markup Language Version 3.0 W3C Recommendation
  2. ^ Buswell, Steven; Devitt, Stan; Diaz, Angel; et al. (7 July 1999). "Mathematical Markup Language (MathML) 1.01 Specification (Abstract)". 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." 
  3. ^ WIRIS editor page describing the use of MathML
  4. ^ MathML Software - Editors at W3C
  5. ^ MathML and Mozilla
  6. ^ MathML Central: Convert Expression to MathML
  7. ^ http://www.mathmlcentral.com/Tools/FromMathML.jsp
  8. ^ Sidje, Roger B., Authoring MathML for Mozilla, Mozilla 
  9. ^ Bug 3251 - Implement MathML (master bug), WebKit 
  10. ^ "About Safari 5.1". Apple. July 20, 2011. Retrieved July 25, 2011. 
  11. ^ a b c Vismor, Timothy, Viewing Mathematics on the Internet, retrieved April 13 2011  Check date values in: |accessdate= (help)
  12. ^ McCathieNevile, Charles (2007-09-27), Can Kestrels do Math? MathML support in Opera Kestrel, Opera 
  13. ^ Olsen, Tommy A. (2007-11-16), Even more work, Opera 
  14. ^ http://my.opera.com/mathml/blog/show.dml/1460837#comments
  15. ^ UserJS for MathML 2.0
  16. ^ Bug 30526 - MathML support for Konqi 
  17. ^ ASCIIMathML: Math on the web for everyone
  18. ^ LaTeXMathML: a dynamic LaTeX mathematics to MathML converter
  19. ^ MathJax MathML Support
  20. ^ TeX-Math
  21. ^ jqMath - Put Math on the Web
  22. ^ WebEQ
  23. ^ David Carlisle (2007-05-09). "XHTML and MathML from Office 2007". David Carlisle. Retrieved 2007-09-20. 

External links[edit]

Specifications[edit]

Software[edit]

Miscellaneous[edit]