Mathematical Markup Language (MathML) Version 3.0

W3C Working Draft 27 April 2007

This version:
Latest MathML 3 version:
Latest MathML Recommendation:
David Carlisle, NAG
Patrick Ion, Mathematical Reviews, American Mathematical Society
Robert Miner, Design Science, Inc.


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.

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 First 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 First Public Working Draft specifies a new version of the the Mathematical Markup Language, MathML 3.0 [mathml] which is at present under active development. The Math WG hopes this outline draft will permit informed feedback on this most important part of its work. 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. During the period since its recommendation, a W3C Math Interest Group collected the experience with the deployment of MathML and identified issues with MathML that might be ameliorated. The rechartering af 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. However, the specification document will be almost completely rewritten to provide a coherent whole containing corrections to all the known errata and clarifications of issues that proved problematic and additions made. Throughout pains are being taken to distinguish the normative and non-normative aspects.

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 structure of this document is the same as that of the earlier MathML 2.0 Recommendation [MathML2]. MathML 2.0 itself was a revision of the earlier W3C Recommendation MathML 1.01 [MathML1]; MathML 3.0 itself is a revision of the W3C Recommendation MathML 2.0. It differs from it in that all chapters will have been updated, some new elements and attributes are being added and some have been deprecated. A current list of open issues, pointing into the relevant places in the draft, follows the Table of Contents.

Table of Contents

1 Introduction
    1.1 Mathematics and its Notation
    1.2 Origins and Goals
        1.2.1 The History of MathML
        1.2.2 Limitations of HTML
        1.2.3 Requirements for Mathematics Markup
        1.2.4 Design Goals of MathML
    1.3 The Role of MathML on the Web
        1.3.1 Layered Design of Mathematical Web Services
        1.3.2 Relation to Other Web Technology
2 MathML Fundamentals
    2.1 MathML Overview
        2.1.1 Taxonomy of MathML Elements
        2.1.2 Presentation Markup
        2.1.3 Content Markup
        2.1.4 Mixing Presentation and Content
    2.2 MathML in a Document
    2.3 Some MathML Examples
        2.3.1 Presentation Examples
        2.3.2 Content Examples
        2.3.3 Mixed Markup Examples
    2.4 MathML Syntax and Grammar
        2.4.1 MathML Syntax and Grammar
        2.4.2 An XML Syntax Primer
        2.4.3 Children versus Arguments
        2.4.4 MathML Attribute Values
        2.4.5 Attributes Shared by all MathML Elements
        2.4.6 Collapsing Whitespace in Input
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 Accessing glyphs for characters from MathML (mglyph)
    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 Tables and Matrices
        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
    3.6 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
4 Content Markup
    4.1 Introduction
        4.1.1 The Intent of Content Markup
        4.1.2 The Scope of Content Markup
        4.1.3 Basic Concepts of Content Markup
        4.1.4 The structure of MathML3 Content Expressions
        4.1.5 Canonical and Legacy Content MathML
    4.2 Canonical Content Markup
        4.2.1 Numbers
        4.2.2 Identifiers
        4.2.3 Symbols
        4.2.4 The MathML3 Content Dictionaries and Operators
        4.2.5 Function Application
        4.2.6 Bindings and Bound Variables
        4.2.7 Qualifiers
        4.2.8 Structure Sharing
        4.2.9 Semantic Mapping
        4.2.10 In Situ Error Markup
    4.3 Rendering of Content Elements
        4.3.1 General Rules
        4.3.2 Attributes Modifying Content Markup Rendering
    4.4 Legacy Markup in Content MathML
        4.4.1 Numbers with constant type
        4.4.2 Token Elements
        4.4.3 Tokens with Attributes
        4.4.4 Container Markup
        4.4.5 Domain of Application (domainofapplication) in Applications
        4.4.6 Domain of Application (domainofapplication) in Bindings
        4.4.7 Integrals with Calling patterns
        4.4.8 degree
        4.4.9 Upper and Lower Limits
        4.4.10 Lifted Associative Commutative Operators
        4.4.11 Declare (declare)
5 Combining Presentation and Content Markup
    5.1 Why Two Different Kinds of Markup?
    5.2 Mixed Markup
        5.2.1 Reasons to Mix Markup
        5.2.2 Combinations that are prohibited
        5.2.3 Presentation Markup Contained in Content Markup
        5.2.4 Content Markup Contained in Presentation Markup
    5.3 Parallel Markup
        5.3.1 Top-level Parallel Markup
        5.3.2 Fine-grained Parallel Markup
        5.3.3 Parallel Markup via Cross-References: id and xref
        5.3.4 Annotation Cross-References using XLink: id and href
    5.4 Tools, Style Sheets and Macros for Combined Markup
        5.4.1 Notational Style Sheets
        5.4.2 Content-Faithful Transformations
        5.4.3 Style Sheets for Extensions
6 Characters, Entities and Fonts
    6.1 Introduction
    6.2 MathML Characters
        6.2.1 Unicode Character Data
        6.2.2 Special Characters Not in Unicode
        6.2.3 Mathematical Alphanumeric Symbols Characters
        6.2.4 Non-Marking Characters
    6.3 Character Symbol Listings
        6.3.1 Special Constants
        6.3.2 Character Tables (ASCII format)
        6.3.3 Tables arranged by Unicode block
        6.3.4 Negated Mathematical Characters
        6.3.5 Variant Mathematical Characters
        6.3.6 Mathematical Alphanumeric Symbols
        6.3.7 MathML Character Names
7 The MathML Interface
    7.1 Embedding MathML in other Documents
        7.1.1 MathML and Namespaces
        7.1.2 The Top-Level math Element
        7.1.3 Invoking MathML Processors
        7.1.4 Mixing and Linking MathML and HTML
        7.1.5 MathML and Graphical Markup
        7.1.6 Using CSS with MathML
    7.2 Conformance
        7.2.1 MathML Conformance
        7.2.2 Handling of Errors
        7.2.3 Attributes for unspecified data
    7.3 Future Extensions
        7.3.1 Macros and Style Sheets
        7.3.2 XML Extensions to MathML
8 Document Object Model for MathML
    8.1 Introduction
        8.1.1 hasFeature String
        8.1.2 MathML DOM Extensions


A Parsing MathML
    A.1 Use of MathML as Well-Formed XML
    A.2 Using the MathML DTD
        A.2.1 DOCTYPE declaration for MathML
        A.2.2 DTD Parameters
        A.2.3 MathML as a DTD Module
        A.2.4 SGML
        A.2.5 The MathML DTD
    A.3 Using the MathML XML Schema
        A.3.1 Associating the MathML schema with MathML fragments
        A.3.2 Character entity references
B Content Markup Validation Grammar
C MathML3 Content Dictionaries
    C.1 About Content Dictionaries
    C.2 The MathML3 Content Dictionary Format
        C.2.1 Content Dictionaries
        C.2.2 Symbol Definitions
        C.2.3 Symbol Roles
        C.2.4 Default Rendering Specifications
D Document Object Model for MathML
    D.1 IDL Interfaces
        D.1.1 Miscellaneous Object Definitions
        D.1.2 Generic MathML Elements
        D.1.3 Presentation Elements
        D.1.4 Content Elements
    D.2 MathML DOM Tables
        D.2.1 Chart of MathML DOM Inheritance
        D.2.2 Table of Elements and MathML DOM Representations
E MathML Document Object Model Bindings (Non-Normative)
    E.1 MathML Document Object Model IDL Binding
    E.2 MathML Document Object Model Java Binding
    E.3 MathML Document Object Model ECMAScript Binding
F Operator Dictionary (Non-Normative)
    F.1 Format of operator dictionary entries
    F.2 Indexing of operator dictionary
    F.3 Choice of entity names
    F.4 Notes on lspace and rspace attributes
    F.5 Operator dictionary entries
G Sample CSS Style Sheet for MathML (Non-Normative)
H Glossary (Non-Normative)
I Working Group Membership and Acknowledgments (Non-Normative)
    I.1 The Math Working Group Membership
    I.2 Acknowledgments
J Changes (Non-Normative)
    J.1 Changes between MathML 2.0 Second Edition and MathML 3.0
K References (Non-Normative)
L Index (Non-Normative)
    L.1 MathML Elements
    L.2 MathML Attributes

Open Issues

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.

Presentation Markup

mrow-dir-att   wiki (member only)
Direction attribute for mrow
mpadded-att-names   wiki (member only)
mpadded attribute names
mpadded-clipping   wiki (member only)
Bounding Box as Clipping Rectangle
mpadded-rspace   wiki (member only)
rspace vs. advancewidth
mpadded-sign-conventions   wiki (member only)
mpadded sign conventions
mpadded-examples   wiki (member only)
mpadded examples
deprecate_maction   wiki (member only)
Deprecate maction
elem-add-img   wiki (member only)
Image Needed
elem-ldiv-img   wiki (member only)
Image Needed
elem-repeatingdec-img   wiki (member only)
Image Needed

Content Markup

legacy2canonical.xsl   wiki (member only)
Legacy to Canonical Transformation
encoding_value   wiki (member only)
encoding value
CD_encoding_table   wiki (member only)
CD encoding table?
integrals_om_mathml   wiki (member only)   ISSUE-9 (member only)
Sort out integrals between OpenMath and MathML
bvar_children   wiki (member only)
How many bound variables per bvar element?
share_presentation   wiki (member only)
share in Presentation MathML as well?
other_nowadays   wiki (member only)   ISSUE-3 (member only)
other is deprecated, delete the following
new_tokens_vs_csymbol   wiki (member only)
tokens for the new MathML3 symbols?
token_attribs   wiki (member only)   ISSUE-11 (member only)
Tokens with Attributes
dom_for_containers   wiki (member only)
MathML DOM for Container Elements
lifted_operators   wiki (member only)   ISSUE-8 (member only)
New Symbols for Lifted Operators

Combining Presentation and Content Markup

update_mixing   wiki (member only)
Update Combining Presentation and Content Markup

Characters, Entities and Fonts

regenerate_character_and_entity_tables   wiki (member only)
Character and entity tables need to be updated for MathML3 and ISO 9573-13

The MathML Interface

update_interface   wiki (member only)
Update Interface

Document Object Model for MathML

update_DOM_intro   wiki (member only)
Update DOM Introduction

Parsing MathML

update_schema   wiki (member only)
DTD and W3C XML Schema
relax_schema   wiki (member only)
Relax Schema
regenerate_character_and_entity_tables2   wiki (member only)
Character and tables need to be updated for MathML3 and ISO 9573-13

Content Markup Validation Grammar

update_content_grammar   wiki (member only)
Update Content Grammar
content_grammar_relax   wiki (member only)
Content Grammar relax NG

MathML3 Content Dictionaries

cds_normative   wiki (member only)
Are CDs normative in MathML3
mcd_dates   wiki (member only)   ISSUE-4 (member only)
Revision date for MathML3 Spec content dictionaries
official_mcd   wiki (member only)   ISSUE-5 (member only)
Approving bodies for CDs?
mathml_roles   wiki (member only)
More Roles for MathML?
mcd_rendering   wiki (member only)   ISSUE-6 (member only)
A MathML format for rendering?
algebraic_diff   wiki (member only)   ISSUE-10 (member only)
Algebraic differentiation operator
algebraic_ndiff   wiki (member only)   ISSUE-10 (member only)
nfold algebraic differentiation
type_on_set   wiki (member only)
Keep type attribute on set?
type_on_tendsto   wiki (member only)
keep type attribute on tendsto?

MathML Document Object Model Bindings

update_DOM_bindings   wiki (member only)
Update DOM bindings

Operator Dictionary

update_opdict   wiki (member only)
Update Operator Dictionary


update_changes   wiki (member only)
Changes between MathML 2 and 3 need to be logged