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<!DOCTYPE HTML>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Mathematical Markup Language (MathML) Version 4.0</title>
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<style type="text/css">
/** Table of Contents taken from html 5.2 spec
stop flicker in toc **/
.toc a {
/* More spacing; use padding to make it part of the click target. */
padding-top: 0.1rem;
/* Larger, more consistently-sized click target */
display: block;
/* Reverse color scheme */
color: black;
border-color: #3980B5;
border-bottom-width: 3px !important;
margin-bottom: 0px !important;
}
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
div.graphic{margin-left:2em}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
table.syntax th {
text-align: left;
}
table.attributes td { padding-left:0.5em; padding-right:0.5em; border: solid thin; }
table.attributes td.attname { white-space:nowrap; vertical-align:top;}
table.attributes td.attdesc { background-color:#F0F0FF; padding-left:2em; padding-right:2em}
th.uname {font-size: 50%; text-align:left;}
code { font-family: monospace; }
div.constraint,
div.issue,
div.note,
div.notice { margin-left: 2em; }
li p { margin-top: 0.3em;
margin-bottom: 0.3em; }
div.exampleInner pre { margin-left: 1em;
margin-top: 0em; margin-bottom: 0em}
div.exampleOuter {border: 4px double gray;
margin: 0em; padding: 0em}
div.exampleInner { background-color: #d5dee3;
border-top-width: 4px;
border-top-style: double;
border-top-color: #d3d3d3;
border-bottom-width: 4px;
border-bottom-style: double;
border-bottom-color: #d3d3d3;
padding: 4px; margin: 0em }
div.exampleWrapper { margin: 4px }
div.exampleHeader { font-weight: bold;
margin: 4px}
a.mainindex {font-weight: bold;}
li.sitem {list-style-type: none;}
.error { color: red }
div.mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example h5 {
margin-top: -0.3em;
margin-bottom: -0.5em;}
var.meta {background-color:green}
var.transmeta {background-color:red}
pre.mathml {padding: 0.5em;
background-color: #FFFFDD;}
pre.mathml-fragment {padding: 0.5em;
background-color: #FFFFDD;}
pre.strict-mathml {padding: 0.5em;
background-color: #FFFFDD;}
span.uname {color:#999900;font-size:75%;font-family:sans-serif;}
.minitoc { border-style: solid;
border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
border-width: 1px ;
color: #5D0091;
background: #F9F5DE;
border-color: red;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
.IDL-definition { border-style: solid;
border-width: 1px ;
color: #001000;
background: #E0FFE0;
border-color: #206020;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.baseline {vertical-align: baseline}
#eqnoc1 {width: 10%}
#eqnoc2 {width: 80%; text-align: center; }
#eqnoc3 {width: 10%; text-align: right; }
div.div1 {margin-bottom: 1em;}
.h3style {
text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2;
font-size: 125%;
}
h4 { text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2; }
h5 { text-align: left;
font-family: sans-serif;
font-weight: bold;
color: #0050B2; }
th {background: #E0FFE0;}
p, blockquote, h4 { font-family: sans-serif; }
dt, dd, dl, ul, li { font-family: sans-serif; }
pre, code { font-family: monospace }
a.termref {
text-decoration: none;
color: black;
}
.mathml-render {
font-family: serif;
font-size: 130%;
border: solid 4px green;
padding-left: 1em;
padding-right: 1em;
}
</style>
<link rel="stylesheet" type="text/css" href="https://www.w3.org/StyleSheets/TR/2016/W3C-ED.css"><script src="//www.w3.org/scripts/TR/2016/fixup.js"> </script></head>
<body>
<div class="head">
<p><a href="https://www.w3.org/"><img src="https://www.w3.org/StyleSheets/TR/2016/logos/W3C" alt="W3C" height="48" width="72"></a></p>
<h1><a id="title"></a>Mathematical Markup Language (MathML) Version 4.0</h1>
<h2><a id="w3c-doctype"></a>W3C Working Draft 26 January 2019</h2>
<dl>
<dt>This version:</dt>
<dd><a href="https://w3c.github.io/mathml/">https://w3c.github.io/mathml/</a></dd>
<dt>Latest MathML 3 version:</dt>
<dd>
<a href="https://www.w3.org/TR/MathML4/">https://www.w3.org/TR/MathML4/</a>
</dd>
<dt>Latest MathML Recommendation:</dt>
<dd>
<a href="https://www.w3.org/TR/MathML/">https://www.w3.org/TR/MathML/</a>
</dd>
<dt>Previous version:</dt>
<dd><a href="https://www.w3.org/TR/2014/REC-MathML3-20140410/">https://www.w3.org/TR/2014/REC-MathML3-20140410/</a></dd>
<dt>
Editors' version:</dt>
<dd><a href="https://w3c.github.io/mathml/">https://w3c.github.io/mathml/</a></dd>
<dt>Editors:</dt>
<dd>David Carlisle, NAG</dd>
<dd>Patrick Ion, Mathematical Reviews, American Mathematical Society</dd>
<dd>Robert Miner (deceased), Design Science, Inc.</dd>
<dt id="authors">Principal Authors:</dt>
<dd>Ron Ausbrooks, Stephen Buswell, David Carlisle, Giorgi Chavchanidze, Stéphane Dalmas, Stan Devitt, Angel Diaz, Sam Dooley, Roger Hunter, Patrick Ion, Michael Kohlhase, Azzeddine Lazrek, Paul Libbrecht, Bruce Miller, Robert Miner (deceased), Chris Rowley, Murray Sargent, Bruce Smith, Neil Soiffer, Robert Sutor, Stephen Watt</dd>
</dl>
<p>Please refer to the <a href="https://www.w3.org/Math/Documents/mathml3-errata.html"><strong>errata</strong></a>
for this document, which may include some normative corrections.</p>
<p>In addition to the <a href="Overview.html">HTML</a> version, this document is also available in these non-normative formats: <a href="Overview-d.html">diff marked HTML version</a>, <a href="Overview.xml">XHTML+MathML version</a>, <a href="mathml.html">single page HTML5+MathML version</a>, and <a href="mathml.pdf">PDF version</a>.</p>
<p>See also <a href="https://www.w3.org/2005/11/Translations/Query?titleMatch=MathML3"><strong>translations</strong></a>.</p>
<p class="copyright"><a href="https://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a> © 1998-2017 <a href="https://www.w3.org/"><abbr title="World Wide Web Consortium">W3C</abbr></a><sup>®</sup> (<a href="https://www.csail.mit.edu/"><abbr title="Massachusetts Institute of Technology">MIT</abbr></a>, <a href="https://www.ercim.eu/"><abbr title="European Research Consortium for Informatics and Mathematics">ERCIM</abbr></a>, <a href="https://www.keio.ac.jp/">Keio</a>, <a href="http://ev.buaa.edu.cn/">Beihang</a>). W3C <a href="https://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>, <a href="https://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a> and <a href="https://www.w3.org/Consortium/Legal/copyright-documents">document use</a> rules apply.</p>
<hr title="Separator for header">
</div>
<div>
<h2><a id="abstract"></a>Abstract</h2>
<p>
This specification defines the Mathematical Markup Language, or
<a id="td-mathml"></a>MathML. MathML is a markup language
for describing mathematical notation and capturing
both its structure and content. The goal of MathML is to enable
mathematics to be served, received, and processed on the World Wide
Web, just as <a id="td-html"></a>HTML has
enabled this functionality for text.
</p>
<p>
This specification of the markup language MathML is intended
primarily for a readership consisting of those who will be
developing or implementing renderers or editors using it, or
software that will communicate using MathML as a protocol for input
or output. It is <em>not</em> a User's Guide but rather a
reference document.
</p>
<p>
MathML can be used to encode both mathematical notation and
mathematical content. About thirty-eight of the MathML tags describe
abstract notational structures, while another about one hundred and
seventy provide a way of unambiguously specifying the intended
meaning of an expression. Additional chapters discuss how the MathML
content and presentation elements interact, and how MathML renderers
might be implemented and should interact with browsers. Finally,
this document addresses the issue of special characters used for
mathematics, their handling in MathML, their presence in Unicode,
and their relation to fonts.
</p>
<p>
While MathML is human-readable, authors typically will
use equation editors, conversion
programs, and other specialized software tools to generate
MathML. Several versions of such MathML tools exist,
both freely available software and commercial
products, and more are under development.
</p>
<p>MathML was originally specified as an XML application and most of the
examples in this specification assume that syntax. Other syntaxes are possible most
notably
<a href="appendixg.html#HTML5">[HTML5]</a> specifies the syntax for MathML in HTML. Unless explictly noted,
the examples in this specification are also valid HTML syntax.</p>
</div>
<div>
<h2><a id="status"></a>Status of this Document</h2>
<p><strong>This document is an editors' copy that has no official standing. (Date: 2019-01-26)</strong></p>
<p><em> This section describes the status of this document at the time
of its publication. Other documents may supersede this document. A
list of current W3C publications and the latest revision of this
technical report can be found in the <a href="https://www.w3.org/TR/">W3C technical reports index</a> at
https://www.w3.org/TR/.</em>
</p>
<p>
This document was produced by the <a href="https://www.w3.org/Math/">W3C Math Working Group</a> as
a <a href="https://www.w3.org/Consortium/Process/tr#RecsW3C">Recommendation</a>
and is part of the
W3C <a href="https://www.w3.org/Math/Activity">Math Activity</a>.
The goals of the W3C Math Working Group are discussed in the <a href="https://www.w3.org/Math/Documents/Charter2006.html"> W3C Math WG
Charter</a> (revised July 2006). The authors of this document are
the W3C Math Working Group members. A list of <a href="appendixe.html">participants in the W3C Math Working Group</a>
is available.
</p>
<p>This document has been reviewed by W3C Members, by software
developers, and by other W3C groups and interested parties, and is
endorsed by the Director as a W3C Recommendation. It is a stable
document and may be used as reference material or cited from another
document. W3C's role in making the Recommendation is to draw attention
to the specification and to promote its widespread deployment. This
enhances the functionality and interoperability of the Web.</p>
<p>
All reported errata to
the first edition have been addressed in this addition, and a full change log appears
in
<a href="appendixf.html">Appendix F Changes</a>. The diff-marked version linked in the frontmatter highlights all changes between
the first and second editions. In addition to incorporating errata, the main change
in this addition is to recognise that MathML parsing is also specified in <a href="appendixg.html#HTML5">[HTML5]</a> and where necessary to note where HTML and XML usage differ.
</p>
<p>
The Working Group maintains a comprehensive
<a href="https://www.w3.org/Math/testsuite/">Test Suite</a>. This
is publicly available and developers are encouraged to submit their results for display.
The <a href="https://www.w3.org/Math/testsuite/results/tests.html">Test Results</a>
are public. They show at least two interoperable implementations for each
essential test. Further details may be found in the
<a href="https://www.w3.org/Math/Documents/mml3-implementation-report.html">Implementation Report</a>.</p>
<p>
The MathML 2.0 (Second Edition) specification has been a W3C
Recommendation since 2001. After its
recommendation, a W3C Math Interest Group collected
reports of experience
with the deployment of MathML and identified issues with MathML that
might be ameliorated. The rechartering of a Math Working Group
did not signal any change in the overall design of MathML.
The major additions in MathML 3 are support for
bidirectional layout, better linebreaking and explicit positioning,
elementary math notations, and a new strict content MathML vocabulary
with well-defined semantics. The MathML 3 Specification has also been
restructured.
</p>
<p> This document was produced by a group operating under
the <a href="https://www.w3.org/Consortium/Patent-Policy-20040205/">5 February 2004 W3C Patent Policy</a>. W3C maintains a
<a href="https://www.w3.org/2004/01/pp-impl/35549/status" rel="disclosure">public list of any patent disclosures</a> made in connection with
the deliverables of the group; that page also includes instructions
for disclosing a patent. An individual who has actual knowledge of
a patent which the individual believes contains
<a href="https://www.w3.org/Consortium/Patent-Policy-20040205/#def-essential">Essential Claim(s)</a>
must disclose the information in accordance with
<a href="https://www.w3.org/Consortium/Patent-Policy-20040205/#sec-Disclosure">section 6 of the W3C Patent Policy</a>.</p>
<p>Public discussion of MathML and issues of support through the W3C
for mathematics on the Web takes place on <a href="mailto:[email protected]"> the public mailing list of the Math Working
Group</a> (<a href="https://lists.w3.org/Archives/Public/www-math/">list archives</a>).
To subscribe send an email to <a href="mailto:[email protected]">[email protected]</a>
with the word <code>subscribe</code> in the subject line.
Alternatively, report an issue at this specification's
<a href="https://github.com/w3c/mathml">GitHub repository</a>.
</p>
<p>
The basic chapter structure of this document is based on the earlier
MathML 2.0 Recommendation <a href="appendixh.html#MathML2">[MathML2]</a>. That MathML 2.0
itself was a revision of the earlier W3C Recommendation MathML 1.01
<a href="appendixh.html#MathML1">[MathML1]</a>; MathML 3.0 is a revision of the W3C
Recommendation MathML 2.0. It differs from it in that all previous
chapters have been updated, some new
elements and attributes added and some
deprecated.
Much has been moved to separate documents containing explanatory
material, material on characters and entities and on the MathML DOM.
The discussion of character entities has led to the document
<em>XML Entity Definitions for Characters</em> <a href="appendixg.html#Entities">[Entities]</a>,
which is now a W3C Recommendation. The concern with
use of CSS with MathML has led to the document <em>A MathML for CSS Profile</em>
<a href="appendixh.html#MathMLforCSS">[MathMLforCSS]</a>, which was a W3C Recommendation
accompanying MathML 3.0.
</p>
<p>
The biggest differences from MathML 2.0 (Second Edition) are in Chapters 4 and 5,
although there have been
smaller improvements throughout the specification. A more detailed
description of changes from the previous Recommendation follows.
</p>
<ul>
<li>
<p>
Much of the non-normative explication
that formerly was found in Chapters 1 and 2, and many examples from
elsewhere in the previous MathML specifications, were removed
from the MathML3 specification and planned to be incorporated into a MathML Primer
to be prepared as a separate document. It is expected this will help
the use of this formal MathML3 specification as a reference document
in implementations, and offer the new user better help in
understanding MathML's deployment. The remaining content of Chapters 1
and 2 has been edited to reflect the changes elsewhere in the
document, and in the rapidly evolving Web environment. Some of the
text in them went back to early days of the Web and XML, and its
explanations are now commonplace.
</p>
</li>
<li>
<p>
Chapter 3, on presentation-oriented markup, adds new material on linebreaking,
and on markup for elementary math notations used in many
countries (<code>mstack</code>, <code>mlongdiv</code> and other
associated elements). Other changes include revisions to
the <code>mglyph</code>, <code>mpadded</code> and <code>maction</code>
elements and significant unification and cleanup of attribute
values.
Earlier work, as recorded in the W3C Note <a href="https://www.w3.org/TR/2006/NOTE-arabic-math-20060131/">Arabic
mathematical notation</a>, has allowed clarification of the
relationship with bidirectional text and examples
with RTL text have been added.
</p>
</li>
<li>
<p>
Chapter 4, on content-oriented markup, contains major changes and additions.
The meaning of the actual content remains as before in principle,
but a lot of work has been done on expressing it better. A few new elements have been
added.
</p>
</li>
<li>
<p>
Chapter 5 has been refined as its purpose has been further
clarified to deal with the mixing of markup languages. This chapter deals,
in particular, with interrelations of parts of the
MathML specification, especially with presentation and content
markup.
</p>
</li>
<li>
<p>
Chapter 6 is a new addition which deals with the issues of interaction
of MathML with a host environment. This chapter deals with interrelations
of the MathML specification with XML and HTML, but in the context of deployment
on the Web. In particular there is a discussion of the interaction of CSS
with MathML.
</p>
</li>
<li>
<p>
Chapter 7 replaces the previous Chapter 6, and has been rewritten
and reorganized to reflect the new
situation in regard to Unicode, and the changed
W3C context with regard to named character entities. The
new W3C specification <em>XML Entity Definitions for
Characters</em>, which incorporates those used for mathematics has
become a a W3C Recommendation, <a href="appendixg.html#Entities">[Entities]</a>.</p>
</li>
<li>
<p>
The Appendices, of which there are eight shown, have been
reworked.
Appendix A now contains the new RelaxNG schema
for MathML3 as well as discussion of MathML3 DTD issues.
Appendix B addresses media types associated with MathML and
implicitly constitutes a request for the registration of three new ones,
as is now standard for work from the W3C.
Appendix C contains a new simplified and reconsidered Operator Dictionary.
Appendices D, E, F, G and H contain similar non-normative material to
that in the previous specification, now appropriately updated.
</p>
</li>
<li>
<p>
A fuller discussion of the document's evolution can be found in
<a href="appendixf.html">Appendix F Changes</a>.
</p>
</li>
</ul>
</div>
<nav id="toc">
<h2 id="tochead">Table of Contents</h2>
<ol class="toc">
<li><a href="chapter1.html"><span class="secno">1 </span> Introduction</a>
<ol class="toc">
<li><a href="chapter1.html#intro.notation"><span class="secno">1.1 </span> Mathematics and its Notation</a>
</li>
<li><a href="chapter1.html#intro.origin"><span class="secno">1.2 </span> Origins and Goals</a>
<ol class="toc">
<li><a href="chapter1.html#intro.goals"><span class="secno">1.2.1 </span> Design Goals of MathML</a>
</li>
</ol>
</li>
<li><a href="chapter1.html#intro.overview"><span class="secno">1.3 </span> Overview</a>
</li>
<li><a href="chapter1.html#intro.example"><span class="secno">1.4 </span> A First Example</a>
</li>
</ol>
</li>
<li><a href="chapter2.html"><span class="secno">2 </span> MathML Fundamentals</a>
<ol class="toc">
<li><a href="chapter2.html#fund.syntax"><span class="secno">2.1 </span> MathML Syntax and Grammar</a>
<ol class="toc">
<li><a href="chapter2.html#fund.xmlgeneral"><span class="secno">2.1.1 </span> General Considerations</a>
</li>
<li><a href="chapter2.html#interf.namespace"><span class="secno">2.1.2 </span> MathML and Namespaces</a>
</li>
<li><a href="chapter2.html#fund.xmlsyntax"><span class="secno">2.1.3 </span> Children versus Arguments</a>
</li>
<li><a href="chapter2.html#fund.renderingmodel"><span class="secno">2.1.4 </span> MathML and Rendering</a>
</li>
<li><a href="chapter2.html#fund.attval"><span class="secno">2.1.5 </span> MathML Attribute Values</a>
<ol class="toc">
<li><a href="chapter2.html#id.2.1.5.1"><span class="secno">2.1.5.1 </span> Syntax notation used in the MathML specification</a>
</li>
<li><a href="chapter2.html#fund.units"><span class="secno">2.1.5.2 </span> Length Valued Attributes</a>
</li>
<li><a href="chapter2.html#fund.color"><span class="secno">2.1.5.3 </span> Color Valued Attributes</a>
</li>
<li><a href="chapter2.html#fund.defaults"><span class="secno">2.1.5.4 </span> Default values of attributes</a>
</li>
</ol>
</li>
<li><a href="chapter2.html#fund.globatt"><span class="secno">2.1.6 </span> Attributes Shared by all MathML Elements</a>
</li>
<li><a href="chapter2.html#fund.collapse"><span class="secno">2.1.7 </span> Collapsing Whitespace in Input</a>
</li>
</ol>
</li>
<li><a href="chapter2.html#interf.toplevel"><span class="secno">2.2 </span> The Top-Level
<code><math></code> Element</a>
<ol class="toc">
<li><a href="chapter2.html#interf.toplevel.atts"><span class="secno">2.2.1 </span> Attributes</a>
</li>
<li><a href="chapter2.html#id.2.2.2"><span class="secno">2.2.2 </span> Deprecated Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter2.html#interf.genproc"><span class="secno">2.3 </span> Conformance</a>
<ol class="toc">
<li><a href="chapter2.html#fund.mathmlconf"><span class="secno">2.3.1 </span> MathML Conformance</a>
<ol class="toc">
<li><a href="chapter2.html#interf.testsuite"><span class="secno">2.3.1.1 </span> MathML Test Suite and Validator</a>
</li>
<li><a href="chapter2.html#interf.deprec"><span class="secno">2.3.1.2 </span> Deprecated MathML 1.x and MathML 2.x Features</a>
</li>
<li><a href="chapter2.html#interf.extension"><span class="secno">2.3.1.3 </span> MathML
Extension Mechanisms and Conformance</a>
</li>
</ol>
</li>
<li><a href="chapter2.html#interf.error"><span class="secno">2.3.2 </span> Handling of Errors</a>
</li>
<li><a href="chapter2.html#interf.unspecified"><span class="secno">2.3.3 </span> Attributes for unspecified data</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html"><span class="secno">3 </span> Presentation Markup</a>
<ol class="toc">
<li><a href="chapter3.html#presm.intro"><span class="secno">3.1 </span> Introduction</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.1"><span class="secno">3.1.1 </span> What Presentation Elements Represent</a>
</li>
<li><a href="chapter3.html#id.3.1.2"><span class="secno">3.1.2 </span> Terminology Used In This Chapter</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.2.1"><span class="secno">3.1.2.1 </span> Types of presentation elements</a>
</li>
<li><a href="chapter3.html#id.3.1.2.2"><span class="secno">3.1.2.2 </span> Terminology for other classes of elements and their relationships</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.reqarg"><span class="secno">3.1.3 </span> Required Arguments</a>
<ol class="toc">
<li><a href="chapter3.html#presm.inferredmrow"><span class="secno">3.1.3.1 </span> Inferred <code><mrow></code>s</a>
</li>
<li><a href="chapter3.html#id.3.1.3.2"><span class="secno">3.1.3.2 </span> Table of argument requirements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#id.3.1.4"><span class="secno">3.1.4 </span> Elements with Special Behaviors</a>
</li>
<li><a href="chapter3.html#presm.bidi"><span class="secno">3.1.5 </span> Directionality</a>
<ol class="toc">
<li><a href="chapter3.html#presm.bidi.math"><span class="secno">3.1.5.1 </span> Overall Directionality of Mathematics Formulas</a>
</li>
<li><a href="chapter3.html#presm.bidi.token"><span class="secno">3.1.5.2 </span> Bidirectional Layout in Token Elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.scriptlevel"><span class="secno">3.1.6 </span> Displaystyle and Scriptlevel</a>
</li>
<li><a href="chapter3.html#presm.linebreaking"><span class="secno">3.1.7 </span> Linebreaking of Expressions</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.7.1"><span class="secno">3.1.7.1 </span> Control of Linebreaks</a>
</li>
<li><a href="chapter3.html#presm.lbalgorithm"><span class="secno">3.1.7.2 </span> Automatic Linebreaking Algorithm (Informative)</a>
</li>
<li><a href="chapter3.html#presm.inline.lbalgorithm"><span class="secno">3.1.7.3 </span> Linebreaking Algorithm for Inline Expressions (Informative)</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.warnfinetuning"><span class="secno">3.1.8 </span> Warning about fine-tuning of presentation</a>
<ol class="toc">
<li><a href="chapter3.html#presm.warntweaking"><span class="secno">3.1.8.1 </span> Warning: non-portability of tweaking</a>
</li>
<li><a href="chapter3.html#presm.warnspacing"><span class="secno">3.1.8.2 </span> Warning: spacing should not be used to convey meaning</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.summary"><span class="secno">3.1.9 </span> Summary of Presentation Elements</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.9.1"><span class="secno">3.1.9.1 </span> Token Elements</a>
</li>
<li><a href="chapter3.html#id.3.1.9.2"><span class="secno">3.1.9.2 </span> General Layout Schemata</a>
</li>
<li><a href="chapter3.html#id.3.1.9.3"><span class="secno">3.1.9.3 </span> Script and Limit Schemata</a>
</li>
<li><a href="chapter3.html#id.3.1.9.4"><span class="secno">3.1.9.4 </span> Tables and Matrices</a>
</li>
<li><a href="chapter3.html#id.3.1.9.5"><span class="secno">3.1.9.5 </span> Elementary Math Layout</a>
</li>
<li><a href="chapter3.html#id.3.1.9.6"><span class="secno">3.1.9.6 </span> Enlivening Expressions</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.presatt"><span class="secno">3.1.10 </span> Mathematics style attributes common to presentation elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.tokel"><span class="secno">3.2 </span> Token Elements</a>
<ol class="toc">
<li><a href="chapter3.html#presm.tokenchars"><span class="secno">3.2.1 </span>
Token Element Content Characters, <code><mglyph/></code></a>
<ol class="toc">
<li><a href="chapter3.html#presm.symbolchars"><span class="secno">3.2.1.1 </span> Alphanumeric symbol characters</a>
</li>
<li><a href="chapter3.html#presm.mglyph"><span class="secno">3.2.1.2 </span> Using images to represent
symbols <code><mglyph/></code></a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.commatt"><span class="secno">3.2.2 </span> Mathematics style attributes common to token elements</a>
<ol class="toc">
<li><a href="chapter3.html#presm.deprecatt"><span class="secno">3.2.2.1 </span> Deprecated style attributes on token elements</a>
</li>
<li><a href="chapter3.html#presm.html.insertion"><span class="secno">3.2.2.2 </span> Embedding HTML in MathML</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mi"><span class="secno">3.2.3 </span> Identifier <code><mi></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.3.1"><span class="secno">3.2.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.3.2"><span class="secno">3.2.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#presm.mi.examples"><span class="secno">3.2.3.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mn"><span class="secno">3.2.4 </span> Number <code><mn></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.4.1"><span class="secno">3.2.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.4.2"><span class="secno">3.2.4.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#presm.mn.examples"><span class="secno">3.2.4.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#presm.not.single.mn"><span class="secno">3.2.4.4 </span> Numbers that should not be written
using <code><mn></code> alone</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mo"><span class="secno">3.2.5 </span> Operator, Fence, Separator or Accent
<code><mo></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.5.1"><span class="secno">3.2.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mo.attrs"><span class="secno">3.2.5.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.5.3"><span class="secno">3.2.5.3 </span> Examples with ordinary operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.4"><span class="secno">3.2.5.4 </span> Examples with fences and separators</a>
</li>
<li><a href="chapter3.html#presm.invisibleops"><span class="secno">3.2.5.5 </span> Invisible operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.6"><span class="secno">3.2.5.6 </span> Names for other special operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.7"><span class="secno">3.2.5.7 </span> Detailed rendering rules for <code><mo></code> elements</a>
</li>
<li><a href="chapter3.html#presm.op.stretch"><span class="secno">3.2.5.8 </span> Stretching of operators, fences and accents</a>
</li>
<li><a href="chapter3.html#presm.mo.linebreaks"><span class="secno">3.2.5.9 </span> Examples of Linebreaking</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtext"><span class="secno">3.2.6 </span> Text <code><mtext></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.6.1"><span class="secno">3.2.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.6.2"><span class="secno">3.2.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.6.3"><span class="secno">3.2.6.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#presm.mixtextmath"><span class="secno">3.2.6.4 </span> Mixing text and mathematics</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mspace"><span class="secno">3.2.7 </span> Space <code><mspace/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.7.1"><span class="secno">3.2.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.7.2"><span class="secno">3.2.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.7.3"><span class="secno">3.2.7.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#id.3.2.7.4"><span class="secno">3.2.7.4 </span> Definition of space-like elements</a>
</li>
<li><a href="chapter3.html#id.3.2.7.5"><span class="secno">3.2.7.5 </span> Legal grouping of space-like elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.ms"><span class="secno">3.2.8 </span> String Literal <code><ms></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.8.1"><span class="secno">3.2.8.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.8.2"><span class="secno">3.2.8.2 </span> Attributes</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.genlayout"><span class="secno">3.3 </span> General Layout Schemata</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mrow"><span class="secno">3.3.1 </span> Horizontally Group Sub-Expressions
<code><mrow></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.1.1"><span class="secno">3.3.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.1.2"><span class="secno">3.3.1.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.1.3"><span class="secno">3.3.1.3 </span> Proper grouping of sub-expressions using <code><mrow></code></a>
</li>
<li><a href="chapter3.html#id.3.3.1.4"><span class="secno">3.3.1.4 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mfrac"><span class="secno">3.3.2 </span> Fractions <code><mfrac></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.2.1"><span class="secno">3.3.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.2.2"><span class="secno">3.3.2.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.2.3"><span class="secno">3.3.2.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mroot"><span class="secno">3.3.3 </span> Radicals <code><msqrt></code>, <code><mroot></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.3.1"><span class="secno">3.3.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.3.2"><span class="secno">3.3.3.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mstyle"><span class="secno">3.3.4 </span> Style Change <code><mstyle></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.4.1"><span class="secno">3.3.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mstyle.attrs"><span class="secno">3.3.4.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.4.3"><span class="secno">3.3.4.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.merror"><span class="secno">3.3.5 </span> Error Message <code><merror></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.5.1"><span class="secno">3.3.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.5.2"><span class="secno">3.3.5.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.5.3"><span class="secno">3.3.5.3 </span> Example</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mpadded"><span class="secno">3.3.6 </span> Adjust Space Around Content
<code><mpadded></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.6.1"><span class="secno">3.3.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mpaddedatt"><span class="secno">3.3.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.6.3"><span class="secno">3.3.6.3 </span> Meanings of size and position attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mphantom"><span class="secno">3.3.7 </span> Making Sub-Expressions Invisible <code><mphantom></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.7.1"><span class="secno">3.3.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.7.2"><span class="secno">3.3.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.7.3"><span class="secno">3.3.7.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mfenced"><span class="secno">3.3.8 </span> Expression Inside Pair of Fences
<code><mfenced></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.8.1"><span class="secno">3.3.8.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.8.2"><span class="secno">3.3.8.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.8.3"><span class="secno">3.3.8.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.menclose"><span class="secno">3.3.9 </span> Enclose Expression Inside Notation
<code><menclose></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.9.1"><span class="secno">3.3.9.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.9.2"><span class="secno">3.3.9.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.9.3"><span class="secno">3.3.9.3 </span> Examples</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.scrlim"><span class="secno">3.4 </span> Script and Limit Schemata</a>
<ol class="toc">
<li><a href="chapter3.html#presm.msub"><span class="secno">3.4.1 </span> Subscript <code><msub></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.1.1"><span class="secno">3.4.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.1.2"><span class="secno">3.4.1.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msup"><span class="secno">3.4.2 </span> Superscript <code><msup></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.2.1"><span class="secno">3.4.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.2.2"><span class="secno">3.4.2.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msubsup"><span class="secno">3.4.3 </span> Subscript-superscript Pair <code><msubsup></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.3.1"><span class="secno">3.4.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.subsupatt"><span class="secno">3.4.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.3.3"><span class="secno">3.4.3.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.munder"><span class="secno">3.4.4 </span> Underscript <code><munder></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.4.1"><span class="secno">3.4.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.4.2"><span class="secno">3.4.4.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.4.3"><span class="secno">3.4.4.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mover"><span class="secno">3.4.5 </span> Overscript <code><mover></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.5.1"><span class="secno">3.4.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.5.2"><span class="secno">3.4.5.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.5.3"><span class="secno">3.4.5.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.munderover"><span class="secno">3.4.6 </span> Underscript-overscript Pair
<code><munderover></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.6.1"><span class="secno">3.4.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.6.2"><span class="secno">3.4.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.6.3"><span class="secno">3.4.6.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mmultiscripts"><span class="secno">3.4.7 </span> Prescripts and Tensor Indices
<code><mmultiscripts></code>,
<code><mprescripts/></code>,
<code><none/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.7.1"><span class="secno">3.4.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.7.2"><span class="secno">3.4.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.7.3"><span class="secno">3.4.7.3 </span> Examples</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.tabmat"><span class="secno">3.5 </span> Tabular Math</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mtable"><span class="secno">3.5.1 </span> Table or Matrix
<code><mtable></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.1.1"><span class="secno">3.5.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mtable.attrs"><span class="secno">3.5.1.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.1.3"><span class="secno">3.5.1.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtr"><span class="secno">3.5.2 </span> Row in Table or Matrix <code><mtr></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.2.1"><span class="secno">3.5.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.2.2"><span class="secno">3.5.2.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mlabeledtr"><span class="secno">3.5.3 </span> Labeled Row in Table or Matrix
<code><mlabeledtr></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.3.1"><span class="secno">3.5.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.3.2"><span class="secno">3.5.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.3.3"><span class="secno">3.5.3.3 </span> Equation Numbering</a>
</li>
<li><a href="chapter3.html#id.3.5.3.4"><span class="secno">3.5.3.4 </span> Example</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtd"><span class="secno">3.5.4 </span> Entry in Table or Matrix <code><mtd></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.4.1"><span class="secno">3.5.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mtdatts"><span class="secno">3.5.4.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.malign"><span class="secno">3.5.5 </span> Alignment Markers
<code><maligngroup/></code>, <code><malignmark/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.5.1"><span class="secno">3.5.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.5.2"><span class="secno">3.5.5.2 </span> Specifying alignment groups</a>
</li>
<li><a href="chapter3.html#id.3.5.5.3"><span class="secno">3.5.5.3 </span> Table cells that are not divided into alignment groups</a>
</li>
<li><a href="chapter3.html#id.3.5.5.4"><span class="secno">3.5.5.4 </span> Specifying alignment points using <code><malignmark/></code></a>
</li>
<li><a href="chapter3.html#id.3.5.5.5"><span class="secno">3.5.5.5 </span> <code><malignmark/></code> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.5.6"><span class="secno">3.5.5.6 </span> <code><maligngroup/></code> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.5.7"><span class="secno">3.5.5.7 </span> Inheritance of groupalign values</a>
</li>
<li><a href="chapter3.html#id.3.5.5.8"><span class="secno">3.5.5.8 </span> MathML representation of an alignment example</a>
</li>
<li><a href="chapter3.html#id.3.5.5.9"><span class="secno">3.5.5.9 </span> Further details of alignment elements</a>
</li>
<li><a href="chapter3.html#id.3.5.5.10"><span class="secno">3.5.5.10 </span> A simple alignment algorithm</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.elementary"><span class="secno">3.6 </span> Elementary Math</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mstack"><span class="secno">3.6.1 </span> Stacks of Characters <code><mstack></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.1.1"><span class="secno">3.6.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.1.2"><span class="secno">3.6.1.2 </span> Attributes</a>
</li>