ISO/IEC 40314:2016

INTERNATIONAL ISO/IEC
STANDARD 40314
First edition
2016-03-01
Information technology —
Mathematical Markup Language
(MathML) Version 3.0 2nd Edition
Technologies de l’information — Langage de marquage
mathématique (MathML) Version 3.0 2e édition
Reference number
ISO/IEC 40314:2016(E)
©
ISO/IEC 2016

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ISO/IEC 40314:2016(E)

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ISO/IEC 40314:2016(E)
FOREWORD
ISO (the International Organization for Standardization) and IEC (the International Electrotechnical
Commission) form the specialized system for worldwide standardization. National bodies that are
members of ISO or IEC participate in the development of International Standards through technical
committees established by the respective organization to deal with particular fields of technical
activity. ISO and IEC technical committees collaborate in fields of mutual interest. Other
international organizations, governmental and non‐governmental, in liaison with ISO and IEC, also
take part in the work. In the field of information technology, ISO and IEC have established a joint
technical committee, ISO/IEC JTC 1.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for
the different types of document should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject
of patent rights. ISO and IEC shall not be held responsible for identifying any or all such patent
rights. Details of any patent rights identified during the development of the document will be in the
Introduction and/or on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does
not constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO's adherence to the WTO principles in the Technical
Barriers to Trade (TBT), see the following URL: Foreword — Supplementary information.
ISO/IEC 40314 was prepared by W3C and was adopted, under the PAS procedure, by Joint
Technical Committee ISO/IEC JTC 1, Information technology, in parallel with its approval by
national bodies of ISO and IEC.
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ISO/IEC 40314:2016(E)
Contents
1 Introduction 9
1.1 Mathematics and its Notation . 9
1.2 Origins and Goals . 10
1.2.1 Design Goals of MathML . 10
1.3 Overview . 11
1.4 A First Example. 11
2 MathML Fundamentals 14
2.1 MathML Syntax and Grammar . 14
2.1.1 General Considerations. 14
2.1.2 MathML and Namespaces . 14
2.1.3 Children versus Arguments. 15
2.1.4 MathML and Rendering . 15
2.1.5 MathML Attribute Values . 15
2.1.6 Attributes Shared by all MathML Elements . 20
2.1.7 Collapsing Whitespace in Input . 21
2.2 The Top-Level Element . 22
2.2.1 Attributes . 22
2.2.2 Deprecated Attributes . 24
2.3 Conformance . 24
2.3.1 MathML Conformance . 24
2.3.2 Handling of Errors . 27
2.3.3 Attributes for unspecified data . 27
3 Presentation Markup 28
3.1 Introduction . 28
3.1.1 What Presentation Elements Represent . 28
3.1.2 Terminology Used In This Chapter. 29
3.1.3 Required Arguments . 30
3.1.4 Elements with Special Behaviors. 31
3.1.5 Directionality . 32
3.1.6 Displaystyle and Scriptlevel . 33
3.1.7 Linebreaking of Expressions . 34
3.1.8 Warning about fine-tuning of presentation . 35
3.1.9 Summary of Presentation Elements . 37
3.1.10 Mathematics style attributes common to presentation elements . 38
3.2 Token Elements . 38
3.2.1 Token Element Content Characters,. 39
3.2.2 Mathematics style attributes common to token elements . 41
3.2.3 Identifier . 45
4
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ISO/IEC 40314:2016(E)
CONTENTS 5
3.2.4 Number . 46
3.2.5 Operator, Fence, Separator or Accent . 47
3.2.6 Text. 60
3.2.7 Space . 62
3.2.8 String Literal . 64
3.3 General Layout Schemata. 64
3.3.1 Horizontally Group Sub-Expressions . 64
3.3.2 Fractions . 67
3.3.3 Radicals, . 69
3.3.4 Style Change . 69
3.3.5 Error Message . 72
3.3.6 Adjust Space Around Content. 73
3.3.7 Making Sub-Expressions Invisible . 78
3.3.8 Expression Inside Pair of Fences . 80
3.3.9 Enclose Expression Inside Notation . 83
3.4 Script and Limit Schemata . 85
3.4.1 Subscript. 86
3.4.2 Superscript. 87
3.4.3 Subscript-superscript Pair . 87
3.4.4 Underscript . 88
3.4.5 Overscript . 89
3.4.6 Underscript-overscript Pair . 91
3.4.7 Prescripts and Tensor Indices,, 93
3.5 Tabular Math . 95
3.5.1 Table or Matrix . 95
3.5.2 Row in Table or Matrix . 99
3.5.3 Labeled Row in Table or Matrix . 99
3.5.4 Entry in Table or Matrix . 101
3.5.5 Alignment Markers, . 101
3.6 Elementary Math . 110
3.6.1 Stacks of Characters . 111
3.6.2 Long Division . 113
3.6.3 Group Rows with Similiar Positions . 114
3.6.4 Rows in Elementary Math . 115
3.6.5 Carries, Borrows, and Crossouts . 115
3.6.6 A Single Carry . 116
3.6.7 Horizontal Line . 117
3.6.8 Elementary Math Examples . 118
3.7 Enlivening Expressions . 124
3.7.1 Bind Action to Sub-Expression . 124
3.8 Semantics and Presentation . 126
4 Content Markup 127
4.1 Introduction . 127
4.1.1 The Intent of Content Markup . 127
4.1.2 The Structure and Scope of Content MathML Expressions . 128
4.1.3 Strict Content MathML. 128
4.1.4 Content Dictionaries . 129
4.1.5 Content MathML Concepts . 130
4.2 Content MathML Elements Encoding Expression Structure . 131
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ISO/IEC 40314:2016(E)
6 CONTENTS
4.2.1 Numbers . 132
4.2.2 Content Identifiers . 138
4.2.3 Content Symbols . 140
4.2.4 String Literals . 142
4.2.5 Function Application. 143
4.2.6 Bindings and Bound Variables and. 146
4.2.7 Structure Sharing. 148
4.2.8 Attribution viasemantics . 150
4.2.9 Error Markup . 151
4.2.10 Encoded Bytes . 152
4.3 Content MathML for Specific Structures . 152
4.3.1 Container Markup . 153
4.3.2 Bindings with . 154
4.3.3 Qualifiers . 156
4.3.4 Operator Classes . 162
4.3.5 Non-strict Attributes . 169
4.4 Content MathML for Specific Operators and Constants . 170
4.4.1 Functions and Inverses . 170
4.4.2 Arithmetic, Algebra and Logic . 180
4.4.3 Relations . 200
4.4.4 Calculus and Vector Calculus . 205
4.4.5 Theory of Sets . 224
4.4.6 Sequences and Series . 233
4.4.7 Elementary classical functions . 243
4.4.8 Statistics . 247
4.4.9 Linear Algebra . 253
4.4.10 Constant and Symbol Elements . 260
4.5 Deprecated Content Elements. 268
4.5.1 Declare . 268
4.5.2 Relation . 268
4.5.3 Relation. 268
4.6 The Strict Content MathML Transformation . 268
5 Mixing Markup Languages for Mathematical Expressions 272
5.1 Annotation Framework . 272
5.1.1 Annotation elements . 272
5.1.2 Annotation keys . 273
5.1.3 Alternate representations . 274
5.1.4 Content equivalents. 275
5.1.5 Annotation references . 276
5.2 Elements for Semantic Annotations . 276
5.2.1 The element. 276
5.2.2 The element . 277
5.2.3 The element . 278
5.3 Combining Presentation and Content Markup . 281
5.3.1 Presentation Markup in Content Markup. 281
5.3.2 Content Markup in Presentation Markup. 282
5.4 Parallel Markup . 282
5.4.1 Top-level Parallel Markup . 282
5.4.2 Parallel Markup via Cross-References . 283
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ISO/IEC 40314:2016(E)
CONTENTS 7
6 Interactions with the Host Environment 286
6.1 Introduction . 286
6.2 Invoking MathML Processors. 286
6.2.1 Recognizing MathML in XML. 286
6.2.2 Recognizing MathML in HTML . 287
6.2.3 Resource Types for MathML Documents . 287
6.2.4 Names of MathML Encodings . 287
6.3 Transferring MathML . 288
6.3.1 Basic Transfer Flavor Names and Contents . 288
6.3.2 Recommended Behaviors when Transferring . 289
6.3.3 Discussion . 289
6.3.4 Examples . 290
6.4 Combining MathML and Other Formats . 292
6.4.1 Mixing MathML and XHTML . 294
6.4.2 Mixing MathML and non-XML contexts . 294
6.4.3 Mixing MathML and HTML . 294
6.4.4 Linking . 295
6.4.5 MathML and Graphical Markup . 296
6.5 Using CSS with MathML . 297
6.5.1 Order of processing attributes versus style sheets . 298
7 Characters, Entities and Fonts 299
7.1 Introduction . 299
7.2 Unicode Character Data. 299
7.3 Entity Declarations . 300
7.4 Special Characters Not in Unicode . 300
7.5 Mathematical Alphanumeric Symbols . 300
7.6 Non-Marking Characters . 303
7.7 Anomalous Mathematical Characters. 303
7.7.1 Keyboard Characters . 303
7.7.2 Pseudo-scripts . 304
7.7.3 Combining Characters . 306
A Parsing MathML 308
A.1 Use of MathML as Well-Formed XML. 308
A.2 Using the RelaxNG Schema for MathML3. 308
A.2.1 Full MathML . 309
A.2.2 Elements Common to Presentation and Content MathML. 309
A.2.3 The Grammar for Presentation MathML . 311
A.2.4 The Grammar for Strict Content MathML3 . 323
A.2.5 The Grammar for Content MathML . 325
A.2.6 MathML as a module in a RelaxNG Schema . 332
A.3 Using the MathML DTD . 333
A.3.1 Document Validation Issues . 333
A.3.2 Attribute values in the MathML DTD . 333
A.3.3 DOCTYPE declaration for MathML . 334
A.4 Using the MathML XML Schema . 334
A.4.1 Associating the MathML schema with MathML fragments . 334
A.5 Parsing MathML in XHTML . 334
A.6 Parsing MathML in HTML . 334
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ISO/IEC 40314:2016(E)
8 CONTENTS
B Media Types Registrations 335
B.1 Selection of Media Types for MathML Instances . 335
B.2 Media type for Generic MathML . 336
B.3 Media type for Presentation MathML . 337
B.4 Media type for Content MathML . 338
C Operator Dictionary (Non-Normative) 340
C.1 Indexing of the operator dictionary . 340
C.2 Format of operator dictionary entries . 340
C.3 Notes onlspace andrspace attributes . 341
C.4 Operator dictionary entries . 341
D Glossary (Non-Normative) 379
E Working Group Membership and Acknowledgments (Non-Normative) 383
E.1 The Math Working Group Membership . 383
E.2 Acknowledgments . 386
F Changes (Non-Normative) 387
F.1 Changes between MathML 3.0 First Edition and Second Edition . 387
F.2 Changes between MathML 2.0 Second Edition and MathML 3.0 . 390
G Normative References 391
H References (Non-Normative) 393
I Index (Non-Normative) 395
I.1 MathML Elements . 395
I.2 MathML Attributes . 400
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ISO/IEC 40314:2016(E)
Chapter 1
Introduction
1.1 Mathematics and its Notation
A distinguishing feature of mathematics is the use of a complex and highly evolved system of two-
dimensional symbolic notation. As J. R. Pierce writes in his book on communication theory, math-
ematics and its notation should not be viewed as one and the same thing [Pierce1961]. Mathematical
ideas can exist independently of the notation that represents them. However, the relation between mean-
ing and notation is subtle, and part of the power of mathematics to describe and analyze derives from
its ability to represent and manipulate ideas in symbolic form. The challenge before a Mathematical
Markup Language (MathML) in enabling mathematics on the World Wide Web is to capture both no-
tation and content (that is, its meaning) in such a way that documents can utilize the highly evolved
notation of written and printed mathematics as well as the new potential for interconnectivity in elec-
tronic media.
Mathematical notation evolves constantly as people continue to innovate in ways of approaching and
expressing ideas. Even the common notation of arithmetic has gone through an amazing variety of
styles, including many defunct ones advocated by leading mathematical figures of their day [Cajori1928].
Modern mathematical notation is the product of centuries of refinement, and the notational conventions
for high-quality typesetting are quite complicated and subtle. For example, variables and letters which
stand for numbers are usually typeset today in a special mathematical italic font subtly distinct from
the usual text italic; this seems to have been introduced in Europe in the late sixteenth century. Spacing
around symbols for operations such as +, -,× and / is slightly different from that of text, to reflect con-
ventions about operator precedence that have evolved over centuries. Entire books have been devoted to
the conventions of mathematical typesetting, from the alignment of superscripts and subscripts, to rules
for choosing parenthesis sizes, and on to specialized notational practices for subfields of mathematics.
The manuals describing the nuances of present-day computer typesetting and composition systems can
run to hundreds of pages.
Notational conventions in mathematics, and in printed text in general, guide the eye and make printed
expressions much easier to read and understand. Though we usually take them for granted, we, as mod-
ern readers, rely on numerous conventions such as paragraphs, capital letters, font families and cases,
and even the device of decimal-like numbering of sections such as is used in this document. Such nota-
tional conventions are perhaps even more important for electronic media, where one must contend with
the difficulties of on-screen reading. Appropriate standards coupled with computers enable a broaden-
ing of access to mathematics beyond the world of print. The markup methods for mathematics in use
just before the Web rose to prominence importantly included T X (also written TeX)[Knuth1986] and
E
approaches based on SGML ([AAP-math], [Poppelier1992] and [ISO-12083]).
It is remarkable how widespread the current conventions of mathematical notation have become. The
general two-dimensional layout, and most of the same symbols, are used in all modern mathematical
communications, whether the participants are, say, European, writing left-to-right, or Middle-Eastern,
9
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ISO/IEC 40314:2016(E)
10 Chapter 1. Introduction
writing right-to-left. Of course, conventions for the symbols used, particularly those naming functions
and variables, may tend to favor a local language and script. The largest variation from the most com-
mon is a form used in some Arabic-speaking communities which lays out the entire mathematical
notation from right-to-left, roughly in mirror image of the European tradition.
However, there is more to putting mathematics on the Web than merely finding ways of displaying
traditional mathematical notation in a Web browser. The Web represents a fundamental change in the
underlying metaphor for knowledge storage, a change in which interconnection plays a central role.
It has become important to find ways of communicating mathematics which facilitate automatic pro-
cessing, searching and indexing, and reuse in other mathematical applications and contexts. With this
advance in communication technology, there is an opportunity to expand our ability to represent, en-
code, and ultimately to communicate our mathematical insights and understanding with each other. We
believe that MathML as specified below is an important step in developing mathematics on the Web.
1.2 Origins and Goals
1.2.1 Design Goals of MathML
MathML has been designed from the beginning with the following ultimate goals in mind.
MathML should ideally:
• Encode mathematical material suitable for all educational and scientific communication.
• Encode both mathematical notation and mathematical meaning.
• Facilitate conversion to and from other mathematical formats, both presentational and se-
mantic. Output formats should include:
– graphical displays
– speech synthesizers
– input for computer algebra systems
– other mathematics typesetting languages, such as T X
E
– plain text displays, e.g. VT100 emulators
– international print media, including braille
It is recognized that conversion to and from other notational systems or media may entail
loss of information in the process.
• Allow the passing of information intended for specific renderers and applications.
• Support efficient browsing of lengthy expressions.
• Provide for extensibility.
• Be well suited to templates and other common techniques for editing formulas.
• Be legible to humans, and simple for software to generate and process.
No matter how successfully MathML achieves its goals as a markup language, it is clear that MathML
is useful only if it is implemented well. The W3C Math Working Group has identified a short list of
additional implementation goals. These goals attempt to describe concisely the minimal functionality
MathML rendering and processing software should try to provide.
• MathML expressions in HTML (and XHTML) pages should render properly in popular Web
browsers, in accordance with reader and author viewing preferences, and at the highest qual-
ity possible given the capabilities of the platform.
• HTML (and XHTML) documents containing MathML expressions should print properly and
at high-quality printer resolutions.
• MathML expressions in Web pages should be able to react to user gestures, such those as
with a mouse, and to coordinate communication with other applications through the browser.
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ISO/IEC 40314:2016(E)
1.3. Overview 11
• Mathematical expression editors and converters should be developed to facilitate the creation
of Web pages containing MathML expressions.
The extent to which these goals are ultimately met depends on the cooperation and support of browser
vendors and other developers. The W3C Math Working Group has continued to work with other work-
ing groups of the W3C, and outside the W3C, to ensure that the needs of the scientific community will
be met. MathML 2 and its implementations showed considerable progress in this area over the situation
that obtained at the time of the MathML 1.0 Recommendation (April 1998) [MathML1]. MathML3
and the developing Web are expected to allow much more.
1.3 Overview
MathML is a markup language for describing mathematics. It is usually expressed in XML syntax,
although HTML and other syntaxes are possible. A special aspect of MathML is that there are two
main strains of markup: Presentation markup, discussed in Chapter 3, is used to display mathematical
expressions; and Content markup, discussed in Chapter 4, is used to convey mathematical meaning.
Content markup is specified in particular detail. This specification makes use of an XML format called
Content Dictionaries This format has been developed by the OpenMath Society, [OpenMath2004] with
the dictionaries being used by this specification involving joint development by the OpenMath Society
and the W3C Math Working Group.
Fundamentals common to both strains of markup are covered in Chapter 2, while the means for com-
bining these strains, as well as external markup, into single MathML objects are discussed in Chapter 5.
How MathML interacts with applications is covered in Chapter 6. Finally, a discussion of special sym-
bols, and issues regarding characters, entities and fonts, is given in Chapter 7.
1.4 A First Example
The quadratic formula provides a simple but instructive illustration of MathML markup.
√
2
−b± b − 4ac
x=
2a
MathML offers two flavors of markup of this formula. The first is the style which emphasizes the actual
presentation of a formula, the two-dimensional layout in which the symbols are arranged. An example
of this type is given just below. The second flavor emphasizes the mathematical content and an example
of it follows the first one.

x
=



-
b

±

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ISO/IEC 40314:2016(E)
12 Chapter 1. Introduction


b
2

-

4
⁢
a
⁢
c





2
⁢
a



Consider the superscript 2 in this formula. It represents the squaring operation here, but the meaning
of a superscript in other situations depends on the context. A letter with a superscript can be used to
signify a particular component of a vector, or maybe the superscript just labels a different type of some
structure. Similarly two letters written one just after the other could signify two variables multiplied
together, as they do in the quadratic formula, or they could be two letters making up the name of a single
variable. What is called Content Markup in MathML allows closer specification of the mathematical
meaning of many common formulas. The quadratic formula given in this style of markup is as follows.


x






b







b
2
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ISO/IEC 40314:2016(E)
1.4. A First Example 13



4
a
c






2
a



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ISO/IEC 40314:2016(E)
Chapter 2
MathML Fundamentals
2.1 MathML Syntax and Grammar
2.1.1 General Considerations
The basic ‘syntax’ of MathML is defined using XML syntax, but other syntaxes that can encode labeled
trees are possible. Notably the HTML parser may also be used with MathML. Upon this, we layer a
‘grammar’, being the rules for allowed elements, the order in which they can appear, and how they
may be contained within each other, as well as additional syntactic rules for the values of attributes.
These rules are defined by this specification, and formalized by a RelaxNG schema [RELAX-NG].
The RelaxNG Schema is normative, but a DTD (Document Type Definition) and an XML Schema
[XMLSchemas] are provided for continuity (they were normative for MathML2). See Appendix A.
MathML’s character set consists of legal characters as specified by Unicode [Unicode], further restricted
by the characters not allowed in XML. The use of Unicode characters for mathematics is discussed in
Chapter 7.
The following sections discuss the general aspects of the MathML grammar as well as describe the
syntaxes used for attribute values.
2.1.2 MathML and Namespaces
An XML namespace [Namespaces] is a collection of names identified by a URI. The URI for the
MathML namespace is:
http://www.w3.org/1998/Math/MathML
To declare a namespace when using the XML serialisation of MathML, one uses anxmlns attribute, or
an attribute with anxmlns prefix. When thexmlns attribute is used alone, it sets the default namespace
for the element on which it appears, and for any child elements. For example:

.

When the xmlns attribute is used as a prefix, it declares a prefix which can then be used to explicitly
associate other elements and attributes with a particular namespace. When embedding MathML within
XHTML, one might use:

...
.
...

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ISO/IEC 40314:2016(E)
2.1. MathML Syntax and Grammar 15
HTML
...