# MathML

Developed by World Wide Web Consortium Markup language XML 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 documents. It is a recommendation of the W3C math working group.

## 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 has 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.

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

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, 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 character 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

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

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

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 (both are valid):

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


although the less elegant:

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


is required to get consistent use of italics and even then OpenOffice.org Math produces inconsistent spacing when compared to the LaTeX output.

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 Template:Tag 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. MathML is not intended to be written or edited directly by humans.[1]

## 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.

eg. 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">
<meta name="language" content="en-US"/>
<meta name="description" content="Example of MathML embedded in an XHTML file"/>
<meta name="keywords" content="Example of MathML embedded in an XHTML file"/>
<title>Example of MathML embedded in an XHTML file</title>
<body>
<h1>Example of MathML embedded in an XHTML file</h1>
<p>
<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>
File:Math test.png
Image of MathML example (math-test.xhtml) using Firefox 3.5 with STIX Beta fonts

## Software support

### Editors

Some editors with native MathML support (including copy and paste of MathML) are Publicon from Wolfram Research and SciWriter from soft4science.

MathML is also supported by major office products such as OpenOffice.org, KOffice, and MS Office 2007, as well as by mathematical software products such as Mathematica 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 mathematical expressions to MathML are available, including converters between TeX and MathML.[2] 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.[3]

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.

### Web browsers

Of the major web browsers, those that directly support the format are recent versions of Gecko browsers (e.g., Firefox and Camino),[4][5] and the Opera web browser since version 9.5.

For Gecko-based browsers, the user is currently required to download special fonts in order to display MathML correctly; this is likely to change in late 2009 with the release of the STIX fonts.

Opera, since version 9.5, supports MathML for CSS profile,[6][7] but it is unable to position diacritical marks properly.[8] Before version 9.5 it required User JavaScript to emulate MathML support.[9]

Internet Explorer does not support MathML out of the box. Support can be added by installing the MathPlayer plugin.

### Web conversion

ASCIIMathML[10] 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[11] 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.

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[12] (Subset of LaTeX math syntax) to MathML conversion. It can also create bitmap images (Png, Jpg, Gif, etc.) from TeX-Math or MathML input.

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 propose a toolkit for building web pages that include interactive math (WebEQ Developers Suite,[13]).

## 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 embeded 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[14] through relatively simple XSL Transformations.

be-x-old:MathML