Mathematical Markup Language (MathML) Version 3.0

W3C Working Draft 10 June 2010

This version:
Latest MathML 3 version:
Latest MathML Recommendation:
Previous version:
David Carlisle, NAG
Patrick Ion, Mathematical Reviews, American Mathematical Society
Robert Miner, Design Science, Inc.
Principal Authors:
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, Chris Rowley, Murray Sargent, Bruce Smith, Neil Soiffer, Robert Sutor, Stephen Watt

In addition to the HTML version, this document is also available in these non-normative formats: diff marked HTML version, XHTML+MathML version, and PDF version.


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-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.

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.

Status of this Document

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 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 document is a Second Last Call Working Draft of the MathML 3.0 W3C specification. The Last Call period starts on 10 June 2010 and ends on 1 July 2010. The Math WG hopes that following the clear disposition of any comments this draft may provoke that it will put forward the MathML 3.0 specification for Proposed Recommendation status.

This Second Last Call Public Working Draft specifies a final version of the Mathematical Markup Language (MathML) Version 3.0 which has been under active development. It follows the immediately previous Candidate Recommendation draft dated 15 December 2009, which succeeded an earlier Last Call draft. During the Candidate Recommendation review period of testing and implementation the Math Working Group, with feedback from outside, came up with enough corrections of typos and clarifications, and a reformulation of an algorithm that this has an impact, albeit small, on the normative aspects of the specification. So the Math Working Group puts forward for active review this Second Last Call draft. The Working Group believes it has completed its work on revising MathML with this document. The differences from the Candidate Recommendation draft are explicitly detailed in a separate Last Call Change List, and in the diff-marked HTML version of the specification.

Feedback should be sent to the Public W3C Math mailing list. When sending an e-mail comment that you expect to be addressed in the final MathML specification, please put the text “second-last-call” in the subject line, preferably like this: “[MathML3-second-last-call] …summary of comment .

Since the testing and implementation activities provide satisfaction of the Exit Criteria for the Candidate Recommendation phase, the Working Group expects to demonstrate through interoperable implementations that the MathML 3.0 specification is worthy of passage to Proposed Recommendation status. The Working Group has been developing a comprehensive Test Suite, based on that for MathML 2.0 (Second Edition), which has been revised and extended to cover all features new with MathML 3.0. This is a publicly available. The Test Results will be made public incrementally during this specification's Candidate Recommendation period.

The Working Group expects to offer at least two interoperable implementations for each test before requesting advancement to PR. It is expected that a request to proceed to Proposed Recommendation status will not be made before 10 July 2010.

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.

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.

Feedback on this draft should be sent, preferably before 1 March 2010, to the Public W3C Math mailing list . When sending an e-mail comment that you expect to be addressed in the final MathML specification, please use a subject line like this: “[MathML3-CR] … summary of comment”.

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. 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 XML Entity Definitions for Characters [Entities], which is now a W3C Recommendation. The concern with use of CSS with MathML has led to the document A MathML for CSS Profile [MathMLforCSS], which is a Candidate Recommendation accompanying this one.

The present Candidate Recommendation makes public results of Math Working Group work in recent months. 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.

Table of Contents

1 Introduction
    1.1 Mathematics and its Notation
    1.2 Origins and Goals
        1.2.1 Design Goals of MathML
    1.3 Overview
    1.4 A First Example
2 MathML Fundamentals
    2.1 MathML Syntax and Grammar
        2.1.1 General Considerations
        2.1.2 MathML and Namespaces
        2.1.3 Children versus Arguments
        2.1.4 MathML and Rendering
        2.1.5 MathML Attribute Values
        2.1.6 Attributes Shared by all MathML Elements
        2.1.7 Collapsing Whitespace in Input
    2.2 The Top-Level math Element
        2.2.1 Attributes
        2.2.2 Deprecated Attributes
    2.3 Conformance
        2.3.1 MathML Conformance
        2.3.2 Handling of Errors
        2.3.3 Attributes for unspecified data
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 Displaystyle and Scriptlevel
        3.1.7 Linebreaking of Expressions
        3.1.8 Warning about fine-tuning of presentation
        3.1.9 Summary of Presentation Elements
        3.1.10 Mathematics style attributes common to 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>
    3.3 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 <maligngroup/>, <malignmark/>
    3.6 Elementary Math
        3.6.1 Stacks of Characters <mstack>
        3.6.2 Long Division <mlongdiv>
        3.6.3 Group Rows with Similiar Positions <msgroup>
        3.6.4 Rows in Elementary Math <msrow>
        3.6.5 Carries, Borrows, and Crossouts <mscarries>
        3.6.6 A Single Carry <mscarry>
        3.6.7 Horizontal Line <msline/>
        3.6.8 Elementary Math Examples
    3.7 Enlivening Expressions
        3.7.1 Bind Action to Sub-Expression <maction>
    3.8 Semantics and Presentation
4 Content Markup
    4.1 Introduction
        4.1.1 The Intent of Content Markup
        4.1.2 The Structure and Scope of Content MathML Expressions
        4.1.3 Strict Content MathML
        4.1.4 Content Dictionaries
        4.1.5 Content MathML Concepts
    4.2 Content MathML Elements Encoding Expression Structure
        4.2.1 Numbers <cn>
        4.2.2 Content Identifiers <ci>
        4.2.3 Content Symbols <csymbol>
        4.2.4 String Literals <cs>
        4.2.5 Function Application <apply>
        4.2.6 Bindings and Bound Variables <bind> and <bvar>
        4.2.7 Structure Sharing <share>
        4.2.8 Attribution via semantics
        4.2.9 Error Markup <cerror>
        4.2.10 Encoded Bytes <cbytes>
    4.3 Content MathML for Specific Structures
        4.3.1 Container Markup
        4.3.2 Bindings with <apply>
        4.3.3 Qualifiers
        4.3.4 Operator Classes
        4.3.5 Non-strict Attributes
    4.4 Content MathML for Specific Operators and Constants
        4.4.1 Functions and Inverses
        4.4.2 Arithmetic, Algebra and Logic
        4.4.3 Relations
        4.4.4 Calculus and Vector Calculus
        4.4.5 Theory of Sets
        4.4.6 Sequences and Series
        4.4.7 Elementary classical functions
        4.4.8 Statistics
        4.4.9 Linear Algebra
        4.4.10 Constant and Symbol Elements
    4.5 Deprecated Content Elements
        4.5.1 Declare <declare>
    4.6 The Strict Content MathML Transformation
5 Mixing Markup Languages for Mathematical Expressions
    5.1 Annotation Framework
        5.1.1 Annotation elements
        5.1.2 Annotation keys
        5.1.3 Alternate representations
        5.1.4 Content equivalents
        5.1.5 Annotation references
    5.2 Elements for Semantic Annotations
        5.2.1 The semantics element
        5.2.2 The annotation element
        5.2.3 The annotation-xml element
    5.3 Combining Presentation and Content Markup
        5.3.1 Presentation Markup in Content Markup
        5.3.2 Content Markup in Presentation Markup
    5.4 Parallel Markup
        5.4.1 Top-level Parallel Markup
        5.4.2 Parallel Markup via Cross-References
6 Interactions with the Host Environment
    6.1 Introduction
    6.2 Invoking MathML Processors
        6.2.1 Recognizing MathML in XML
        6.2.2 Resource Types for MathML Documents
        6.2.3 Names of MathML Encodings
    6.3 Transferring MathML
        6.3.1 Basic Transfer Flavor Names and Contents
        6.3.2 Recommended Behaviors when Transferring
        6.3.3 Discussion
        6.3.4 Examples
    6.4 Combining MathML and Other Formats
        6.4.1 Mixing MathML and XHTML
        6.4.2 Mixing MathML and HTML, and other non-XML contexts
        6.4.3 Linking
        6.4.4 MathML and Graphical Markup
    6.5 Using CSS with MathML
        6.5.1 Order of processing attributes versus style sheets
        6.5.2 Layout engines that lack native MathML support
7 Characters, Entities and Fonts
    7.1 Introduction
    7.2 Unicode Character Data
    7.3 Entity Declarations
    7.4 Special Characters Not in Unicode
    7.5 Mathematical Alphanumeric Symbols
    7.6 Non-Marking Characters
    7.7 Anomalous Mathematical Characters
        7.7.1 Keyboard Characters
        7.7.2 Pseudo-scripts
        7.7.3 Combining Characters


A Parsing MathML
    A.1 Use of MathML as Well-Formed XML
    A.2 Using the RelaxNG Schema for MathML3
        A.2.1 Full MathML
        A.2.2 Elements Common to Presentation and Content MathML
        A.2.3 The Grammar for Presentation MathML
        A.2.4 The Grammar for Strict Content MathML3
        A.2.5 The Grammar for Content MathML
        A.2.6 MathML as a module in a RelaxNG Schema
    A.3 Using the MathML DTD
        A.3.1 Document Validation Issues
        A.3.2 Attribute values in the MathML DTD
        A.3.3 DOCTYPE declaration for MathML
    A.4 Using the MathML XML Schema
        A.4.1 Associating the MathML schema with MathML fragments
B Media Types Registrations
    B.1 Selection of Media Types for MathML Instances
    B.2 Media type for Generic MathML
    B.3 Media type for Presentation MathML
    B.4 Media type for Content MathML
C Operator Dictionary (Non-Normative)
    C.1 Indexing of the operator dictionary
    C.2 Format of operator dictionary entries
    C.3 Notes on lspace and rspace attributes
    C.4 Operator dictionary entries
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 Normative References
H References (Non-Normative)
I Index (Non-Normative)
    I.1 MathML Elements
    I.2 MathML Attributes