The presentation of this document has been augmented to use XHTML rather than HTML, and to include all MathML examples as inlined MathML markup that should be rendered by a MathML enabled browser. The MathML examples that have been added, and which should be displayed in your browser, are marked like this.
Please refer to the errata for this document, which may include some normative corrections.
In addition to the HTML version, this document is also available in these non-normative formats: XHTML+MathML version and PDF version.
See also translations.
Copyright © 1998-2008 W3C^{®} (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.
This specification defines the Mathematical Markup Language, or MathML. MathML is an XML application 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 HTML has enabled this functionality for text.
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 not a User's Guide but rather a reference document.
MathML can be used to encode both mathematical notation and mathematical content. About thirty-five 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.
While MathML is human-readable, in all but the simplest cases, authors use equation editors, conversion programs, and other specialized software tools to generate MathML. Several versions of such MathML tools exist, and more, both freely available software and commercial products, are under development.
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 W3C technical reports index at http://www.w3.org/TR/.
This document is a W3C Public Working Draft produced by the W3C Math Working Group as part of the W3C Math Activity. The goals of the W3C Math Working Group are discussed in the W3C Math WG Charter (revised July 2006). A list of participants in the W3C Math Working Group is available.
Publication as a Working Draft does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.
This third Public Working Draft specifies a new version of the the Mathematical Markup Language, MathML 3.0 which is at present under active development. The Math WG hopes this draft will permit informed feedback. There is a description of some considerations underlying this work in the W3C Math WG's public Roadmap [roadmap]. Feedback should be sent to the Public W3C Math mailing list .
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 allows the revision to MathML 3.0 in the light of that experience, of other comments on the markup language, and of recent changes in specifications of the W3C and in the technological context. MathML 3.0 does 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 generated from formal content dictionaries. The MathML 3 Specification has also been restructured.
Public discussion of MathML and issues of support through the W3C
for mathematics on the Web takes place on the public mailing list of the Math Working
Group (list archives).
To subscribe send an email to www-math-request@w3.org
with the word subscribe
in the subject line.
Please report errors in this document to www-math@w3.org.
This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures 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 Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.
The basic chapter structure of this document is based on the earlier MathML 2.0 Recommendation [MathML2]. That MathML 2.0 itself was a revision of the earlier W3C Recommendation MathML 1.01 [MathML1]; 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. This Public Working Draft differs in structure from the initial Public Working Draft as renewed efforts to separate the formal from the explanatory have resulted in eight chapters not seven. Much has been moved to separate documents containing Primer material, material on Characters and Entities and on the MathML DOM. First Working Drafts of these documents will be published soon. A current list of open issues, pointing into the relevant places in the draft, follows the Table of Contents.
The present draft is an incremental one making public some of the results of Math Working Group work in recent months. The biggest difference this time is in Chapter 4, although there have been smaller ameliorations throughout the specification. A more detailed description of changes from the previous Recommendation follows.
With the second Working Draft, 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 incorporated into a MathML Primer being 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 is being edited to reflect the changes elsewhere in the document, and in the rapidly evolving Web environment. Some of their text used to go back to early days of the Web and XML, and its explanations are now commonplace.
Chapter 3, on presentation-oriented markup, in this draft adds new
material on linebreaking and on markup for elementary math
notations. Material introduced in the last draft revising the
mpadded
and maction
elements has been further revised as a result of active
discussion. It is possible it may undergo further modification.
In addition, the layout of schemata such
as that for long division and its associated
mcolumn
element have been carefully revised.
Earlier work, as recorded in the W3C Note Arabic
mathematical notation, has allowed clarification of the
relationship with bidirectional text and examples
with RTL text have been added.
Chapter 4, on content-oriented markup, contains major changes and additions in this Working Draft. The meaning of the actual content remains as before in principle, but a lot of work has been done on expressing it better. The text of this chapter is generated by filtered extraction from XML Content Dictionaries written in accordance with OpenMath. The details of the Content Dictionary format have been further specified and the generation procedure improved. It is expected that the Content Dictionaries will become a separate joint publication of the W3C and OpenMath referenced by the MathML3 specification. The Content Dictionaries are now publicly available in draft and much work has already been done on refining them. Their format is given in Chapter 8.
Chapter 5 is being refined as its purpose has been further clarified. This chapter deals with interrelations of parts of the MathML specification, especially with presentation and content markup.
Chapter 6 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 of Entity Definitions for Characters in XML, which incorporates those used for mathematics is becoming a public working draft [Entities]. It is expected that some new ancillary tables will be provided that reflect requests the Math WG has received.
Chapter 7 has been restored with a new and clearer purpose. This chapter looks outward to the larger world in which MathML must function.
Chapter 8 now specifies the format of MathML3 Content Dictionaries, as previously handled more briefly in sections 4.5 and 4.6. The DOM for MathML, previously in a chapter at this point, is being prepared as a separate specification.
The Appendices, of which there are eight shown, have not been fully reworked. Eventually what amount to revisions of the present appendices A, F, G, H, I and J are all that are expected to remain. Appendix A now contains the new RelaxNG schema for MathML3 as well as discussion of MathML3 DTD issues.
1 Introduction
1.1 Mathematics and its Notation
1.2 Origins and Goals
1.2.1 Design Goals of MathML
2 MathML Fundamentals
2.1 MathML Syntax and Grammar
2.1.1 MathML Syntax and Grammar
2.1.2 Children versus Arguments
2.1.3 MathML Attribute Values
2.1.4 Attributes Shared by all MathML Elements
2.1.5 Collapsing Whitespace in Input
2.2 Interfacing
2.3 Conformance
2.3.1 MathML Conformance
2.3.2 Handling of Errors
2.3.3 Attributes for unspecified data
2.4 Future Extensions
2.4.1 Macros and Style Sheets
2.4.2 XML Extensions to MathML
2.5 Embedding MathML in other Documents
2.5.1 MathML and Namespaces
2.5.2 The Top-Level
math Element
3 Presentation Markup
3.1 Introduction
3.1.1 What Presentation Elements Represent
3.1.2 Terminology Used In This Chapter
3.1.3 Required Arguments
3.1.4 Elements with Special Behaviors
3.1.5 Directionality
3.1.6 Summary of Presentation Elements
3.2 Token Elements
3.2.1 MathML characters in
token elements
3.2.2 Mathematics style attributes common to token
elements
3.2.3 Identifier (mi)
3.2.4 Number (mn)
3.2.5 Operator, Fence, Separator or Accent
(mo)
3.2.6 Text (mtext)
3.2.7 Space (mspace)
3.2.8 String Literal (ms)
3.2.9 Using images to represent
symbols (mglyph)
3.2.10 Line mline
3.3 General Layout Schemata
3.3.1 Horizontally Group Sub-Expressions
(mrow)
3.3.2 Fractions (mfrac)
3.3.3 Radicals (msqrt, mroot)
3.3.4 Style Change (mstyle)
3.3.5 Error Message (merror)
3.3.6 Adjust Space Around Content
(mpadded)
3.3.7 Making Sub-Expressions Invisible (mphantom)
3.3.8 Expression Inside Pair of Fences
(mfenced)
3.3.9 Enclose Expression Inside Notation
(menclose)
3.4 Script and Limit Schemata
3.4.1 Subscript (msub)
3.4.2 Superscript (msup)
3.4.3 Subscript-superscript Pair (msubsup)
3.4.4 Underscript (munder)
3.4.5 Overscript (mover)
3.4.6 Underscript-overscript Pair
(munderover)
3.4.7 Prescripts and Tensor Indices
(mmultiscripts)
3.5 Tabular Math
3.5.1 Table or Matrix
(mtable)
3.5.2 Row in Table or Matrix (mtr)
3.5.3 Labeled Row in Table or Matrix
(mlabeledtr)
3.5.4 Entry in Table or Matrix (mtd)
3.5.5 Alignment Markers
3.5.6 mcolumn
3.6 Enlivening Expressions
3.6.1 Bind Action to Sub-Expression
(maction)
3.7 Elementary Math
3.7.1 Addition, Subtraction, and Multiplication
3.7.2 Long Division
3.7.3 Repeating decimal
3.8 Semantics and Presentation
4 Content Markup
4.1 Introduction
4.2 Strict Content MathML
4.2.1 The structure of MathML3 Content Expressions
4.2.2 Encoding OpenMath Objects
4.2.3 Numbers
4.2.4 Symbols and Identifiers
4.2.5 Function Application
4.2.6 Bindings and Bound Variables
4.2.7 Qualifiers
4.2.8 Structure Sharing
4.2.9 Attribution via semantics
4.2.10 In Situ Error Markup
4.3 Pragmatic Content MathML
4.3.1 Operator Elements
4.3.2 Pragmatic Elements with Attributes
4.3.3 Bindings with apply
4.3.4 Container Markup for Constructor Symbols
4.3.5 Container Markup for Binding Constructors
4.3.6 Container Markup for Applicative Constructors
4.3.7 Symbols and Identifiers With Presentation MathML
4.3.8 Elementary MathML Types on Operator and Container Elements
4.3.9 Domain of Application
4.3.10 Domain of Application in Bindings
4.3.11 degree
4.3.12 Upper and Lower Limits
4.3.13 Lifted Associative Commutative Operators
4.3.14 Declare (declare)
4.4 The MathML3 Content Dictionaries and Operators
4.4.1 arith1
4.4.2 fns1
4.4.3 linalg1
4.4.4 logic1
4.4.5 nums1
4.4.6 relation1
4.4.7 set1
4.4.8 veccalc1
4.4.9 calculus1
4.4.10 integer1
4.4.11 linalg2
4.4.12 mathmlkeys
4.4.13 rounding1
4.4.14 setname1
4.4.15 complex1
4.4.16 interval1
4.4.17 linalg3
4.4.18 minmax1
4.4.19 piece1
4.4.20 error1
4.4.21 limit1
4.4.22 list1
4.4.23 multiset1
4.4.24 quant1
4.4.25 s_dist1
4.4.26 s_data1
4.4.27 transc1
5 Combining Presentation and Content Markup
5.1 Motivation
5.2 Semantic Annotations for Alternate Representations
5.2.1 The annotation-xml element
5.2.2 Annotation references
5.3 Semantic Annotations beyond Alternate Representations
5.4 Mixed Markup
5.4.1 Reasons to Mix Markup
5.4.2 Presentation Markup in Content Markup
5.4.3 Content Markup in Presentation Markup
5.5 Parallel Markup
5.5.1 Top-level Parallel Markup
5.5.2 Parallel Markup via Cross-References: xml:id and xref
6 Characters, Entities and Fonts
6.1 Introduction
6.2 Unicode Character Data
6.3 Entity Declarations
6.4 Special Characters Not in Unicode
6.5 Mathematical Alphanumeric Symbols
6.6 Non-Marking Characters
7 MathML interactions with the Wide World
7.1 Invoking MathML Processors: namespace, extensions, and mime-types
7.1.1 Recognizing MathML in an XML Model
7.1.2 Resource Types for MathML Documents
7.1.3 Names of MathML Encodings
7.2 Transferring MathML in Desktop Environments
7.2.1 Basic Transfer Flavors' Names and Contents
7.2.2 Recommended Behaviours when Transferring
7.2.3 Discussion
7.2.4 Examples
7.3 Combining MathML and Other Formats
7.3.1 Mixing MathML and HTML
7.3.2 Linking
7.3.3 Images
7.3.4 MathML and Graphical Markup
7.4 Using CSS with MathML
8 MathML3 Content Dictionaries
8.1 Introduction
8.2 MathML3 Content Dictionaries Infrastructure and Metadata
8.3 Symbol Declarations
8.4 Type Declarations
8.5 Symbol Roles
8.6 Rendering of Content Elements
8.6.1 Notation Specifications
8.6.2 Precedence-based Elisions
8.6.3 Rendering Cross-References for Parallel Markup
8.6.4 General Rules
8.6.5 Limitations and Extensions of Notation Documents
A Parsing MathML
A.1 Use of MathML as Well-Formed
XML
A.2 Using the RelaxNG Schema for MathML3
A.2.1 The Grammar for Presentation MathML
A.2.2 The Grammar for Strict Content MathML3
A.2.3 The Grammar for Pragmatic MathML
A.2.4 Full MathML
A.2.5 Full MathML with Deprecated Elements
A.2.6 Generated Grammar for Arity & Type Checking
A.2.7 MathML as a module in a RelaxNG Schema
A.3 Using the MathML DTD
A.4 Using the MathML XML Schema
B Operator Dictionary (Non-Normative)
B.1 Format of operator dictionary entries
B.2 Indexing of operator dictionary
B.3 Choice of entity names
B.4 Notes on lspace and
rspace attributes
B.5 Operator dictionary entries
C Sample CSS Style Sheet for MathML (Non-Normative)
D Glossary (Non-Normative)
E Working Group Membership and Acknowledgments (Non-Normative)
E.1 The Math Working Group Membership
E.2 Acknowledgments
F Changes (Non-Normative)
F.1 Changes between MathML 2.0 Second Edition and MathML 3.0
G References (Non-Normative)
H Index (Non-Normative)
H.1 MathML Elements
H.2 MathML Attributes
The following is a list of open issues which are highlighted in this draft. The issue name links to the text of the issue in this specification. There is also a W3C member-only link to the Math Working Group wiki. (Note that in many cases the wiki does not have a page discussing the issue, but will offer to create such pages on demand.) In some cases there is also a (member only) link to the Math Working Group's Issue tracking system.
mpadded
examples
maction
csymbol
cdbase
/cd
/name triplet
csymbol
with pMathML content