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MathML
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{{short description|Format for expressing mathematical formulae}}

factoids
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 ISO/IEC DIS 40314 since 2015.

## 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.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 URIweblink 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 on 20 October 2010. A recommendation of A MathML for CSS Profile was later released on 7 June 2011;WEB, A MathML for CSS Profile,weblink W3C, 25 July 2013, 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 {{tag|bind|o}} which associates bound variables ({{tag|bvar|o}}) to expressions, for example a summation index. The new {{tag|share|o}} 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.Mathematical Markup Language Version 3.0 W3C Recommendation. W3.org. Retrieved on 9 May 2012.The Second Edition of MathML 3.0 was published as a W3C Recommendation on April 10, 2014.MathML Version 3.0 2nd Edition. W3.org. Retrieved on 8 July 2014. The specification was approved as an ISO/IEC international standard 40314:2015 on June 23, 2015.W3C MathML 3.0 Approved as ISO/IEC International Standard. W3.org. Retrieved on July 12, 2015.

## 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 â€œContent MathMLâ€). 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:
• x â€“ identifiers;
• + â€“ operators;
• 2 â€“ numbers.
• non zero â€“ 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
non zerois 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:
• {{tag|mrow|o}} â€“ a horizontal row of items;
• {{tag|msup|o}}, {{tag|munderover|o}}, and others â€“ superscripts, limits over and under operators like sums, etc.;
• {{tag|mfrac|o}} â€“ fractions;
• {{tag|msqrt|o}} and {{tag|mroot|o}} â€“ roots;
• {{tag|mfenced|o}} - 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 &pi; and &RightArrow;. An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as &InvisibleTimes; 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 entitiesweblink is closely coordinated with the corresponding specifications for use with HTML and XMLweblink 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. Including only the layout elements and the (not yet marked up) bare tokens, the structure looks like this:

a ⁢ x 2
+ b ⁢ x
+ c

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 ⁢ x2
+b⁢x
+c

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 $elements as needed in the document. Often, MathML will be embedded in more general documents, such asHTML,DocBook, or otherXMLschemas. A complete document that consists of just the MathML example above, is shown here:$

a

x
2

+
b

x
+
c

### Content MathML

Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the {{tag|apply|o}} element that represents function application. The function being applied is the first child element under {{tag|apply|o}}, 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 operatorweblink example, {{code|2=xml|x}} represents sin(x) and {{code|2=xml|x5}} represents x+5. The elements representing operators and functions are empty elements, because their operands are the other elements under the containing {{tag|apply|o}}.The expression a x^2+b x+c could be represented as

a

x
2

b
x

c
Content MathML is nearly isomorphic to expressions in a functional language such as Scheme. {{tag|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.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

#### Wikidata annotation in Content MathML

According to the OM SocietyWEBSITE,weblink , OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics - names, descriptions and rules. As proposed inJOURNAL, Moritz Schubotz, Philipp Scharpf, Bela Gipp., Representing Mathematical Formulae in Content MathML using Wikidata.,weblink 2018, Birndl@sigir, , the semantic knowledge-base WikidataWEBSITE,weblink can be used as OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.

## Example and comparison to other formats

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 â€“ 4ac}} over 2a
in Apache OpenOffice Math and LibreOffice Math like this (all three are valid):
x={-b plusminus sqrt {b^2 â€“ 4 ac}} over {2 a}
x={-b +- sqrt {b^2 â€“ 4ac}} over 2ax={-b Â± sqrt {b^2 â€“ 4ac}} over 2ain AsciiMath 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:

x
=

b
±

b
2

4

a

c

2

a

x=frac{-bpmsqrt{b^2-4ac}}{2a}

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

This example uses the {{tag|annotation|open}} 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.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.WEB,weblink Mathematical Markup Language (MathML) 1.01 Specification (Abstract), 26 September 2006, Buswellauthor2=Devitt, Stan, Diaz, Angel, 7 July 1999, 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., etal,

## Embedding MathML in HTML/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.

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

.

(File:MathMLxhtml.png|nofloat|435px) A rendering of the formula for a circle in MathML+XHTML using Firefox 22 on Mac OS X
{{clear}}Inline MathML is also supported in HTML5 files in the current versions of WebKit (Safari and JavaFX/WebView ), Gecko (Firefox). There is no need to specify namespaces like in the XHTML.

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 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 element. OpenMath content dictionaries can be used to define the meaning of elements. The following would define P1(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, 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{{Citation| url=http://dpcarlisle.blogspot.com/2007/04/xhtml-and-mathml-from-office-20007.html| title=XHTML and MathML from Office 2007| author=David Carlisle| publisher=David Carlisle| date=9 May 2007 postscript=}} through relatively simple XSL Transformations.

## References

{{Reflist|30em}}

### Specifications

{{W3C Standards}}{{Web browsers}}{{Use dmy dates|date=May 2012}}{{Authority control}}

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