Copyright © 19982024 World Wide Web Consortium. W3C^{®} liability, trademark and permissive document license rules apply.
This specification defines the Mathematical Markup Language, or 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 [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 thirtyeight 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 humanreadable, 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.
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 [HTML] specifies the syntax for MathML in HTML. Unless explicitly noted, the examples in this specification are also valid HTML syntax.
This section describes the status of this document at the time of its publication. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at https://www.w3.org/TR/.
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 wwwmathrequest@w3.org
with the word subscribe
in the subject line.
Alternatively, report an issue at this specification's
GitHub repository.
A fuller discussion of the document's evolution can be found in I. Changes.
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This document was published by the Math Working Group as an Editor's Draft.
Publication as an Editor's Draft does not imply endorsement by W3C and its Members.
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 was produced by a group operating under the 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.
This document is governed by the 03 November 2023 W3C Process Document.
This section is nonnormative.
Mathematics and its notations have evolved over several centuries, or even millennia. To the experienced reader, mathematical notation conveys a large amount of information quickly and compactly. And yet, while the symbols and arrangements of the notations have a deep correspondence to the semantic structure and meaning of the mathematics being represented, the notation and semantics are not the same. The semantic symbols and structures are subtly distinct from those of the notation.
Thus, there is a need for a markup language which can represent both the traditional displayed notations of mathematics, as well as its semantic content. While the traditional rendering is useful to sighted readers, the markup language must also support accessibility. The semantic forms must support a variety of computational purposes. Both forms should be appropriate to all educational levels from elementary to research.
MathML is a markup language for describing mathematics. It uses XML syntax when used standalone or within other XML, or HTML syntax when used within HTML documents. Conceptually, MathML consists of two main strains of markup: Presentation markup is used to display mathematical expressions; and Content markup is used to convey mathematical meaning. These two strains, along with other external representations, can be combined using parallel markup.
This specification is organized as follows: 2. MathML Fundamentals discusses Fundamentals common to Presentation and Content markup; 3. Presentation Markup and 4. Content Markup cover Presentation and Content markup, respectively; 5. Annotating MathML: intent discusses how markup may be annotated, particularly for accessibility; 6. Annotating MathML: semantics discusses how markup may be annotated so that Presentation, Content and other formats may be combined; 7. Interactions with the Host Environment addresses how MathML interacts with applications; Finally, a discussion of special symbols, and issues regarding characters, entities and fonts, is given in 8. Characters, Entities and Fonts.
The specification of MathML is developed in two layers. MathML Core ([MathMLCore]) covers (most of) Presentation Markup, with the focus being the precise details of displaying mathematics in web browsers. MathML Full, this specification, extends MathML Core primarily by defining Content MathML, in 4. Content Markup. It also defines extensions to Presentation MathML consisting of additional attributes, elements or enhanced syntax of attributes. These are defined for compatibility with legacy MathML, as well as to cover 3.1.7 Linebreaking of Expressions, 3.6 Elementary Math and other aspects not included in level 1 of MathML Core but which may be incorporated into future versions of MathML Core.
This specification covers both MathML Core and its extensions; features common to both are indicated with , whereas extensions are indicated with .
It is intended that MathML Full is a proper superset of MathML Core. Moreover, it is intended that any valid Core Markup be considered as valid Full Markup as well. It is also intended that an otherwise conforming implementation of MathML Core, which also implements parts or all of the extensions of MathML Full, should continue to be considered a conforming implementation of MathML Core.
In addition to these two specifications, the Math WG group has developed the nonnormative Notes on MathML that contains additional examples and information to help understand best practices when using MathML.
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 [RELAXNGSCHEMA] in A. Parsing MathML. Derived schema in other formats, DTD (Document Type Definition) and XML Schema [XMLSchemas] are also provided.
MathML's character set consists of any Unicode characters [Unicode] allowed by the syntax being used. (See for example [XML] or [HTML].) The use of Unicode characters for mathematics is discussed in 8. Characters, Entities and Fonts.
The following sections discuss the general aspects of the MathML grammar as well as describe the syntaxes used for attribute values.
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 an xmlns
attribute, or an attribute with an xmlns
prefix.
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>...</mrow>
</math>
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 HTML using XML syntax, one might use:
<body xmlns:m="http://www.w3.org/1998/Math/MathML">
...
<m:math><m:mrow>...</m:mrow></m:math>
...
</body>
HTML does not support namespace extensibility in the same way. The HTML parser
has inbuilt knowledge of the HTML, SVG, and MathML namespaces. xmlns
attributes are
just treated as normal attributes. Thus, when using the HTML serialisation of MathML,
prefixed element names must not be used. xmlns
=http://www.w3.org/1998/Math/MathML
may be used on the math
element; it will be ignored by the HTML parser.
If a MathML expression is likely to be in contexts where it may be parsed by an XML
parser or an HTML parser, it SHOULD
use the following form to ensure maximum compatibility:
<math xmlns="http://www.w3.org/1998/Math/MathML">
...
</math>
There are presentation elements that conceptually accept only
a single argument, but which for convenience have been written to accept any number
of children;
then we infer an mrow
containing those children which acts as
the argument to the element in question; see 3.1.3.1 Inferred <mrow>
s.
In the detailed discussions of element syntax given with each element throughout the MathML specification, the number of arguments required and their order, as well as other constraints on the content, are specified. This information is also tabulated for the presentation elements in 3.1.3 Required Arguments.
Web Platform implementations of [MathMLCore] should follow the detailed layout rules specified in that document.
This document only recommends (i.e., does not require) specific ways of rendering Presentation MathML; this is in order to allow for mediumdependent rendering and for implementations not using the CSS based Web Platform.
MathML elements take attributes with values that further specialize
the meaning or effect of the element. Attribute names are shown in a
monospaced
font throughout this document. The meanings of attributes and their
allowed values are described within the specification of each element.
The syntax notation explained in this section is used in specifying allowed values.
To describe the MathMLspecific syntax of attribute values, the following conventions and notations are used for most attributes in the present document.
Notation  What it matches 

unsignedinteger  As defined in [MathMLCore], an integer , whose first character is neither
U+002D HYPHENMINUS character () nor
U+002B PLUS SIGN (+). 
positiveinteger  An unsignedinteger not consisting solely of "0"s (U+0030), representing a positive integer 
integer  an optional "" (U+002D), followed by an unsignedinteger, and representing an integer 
unsignednumber 
value as defined in
[CSSVALUES3] number , whose first character is neither
U+002D HYPHENMINUS character () nor
U+002B PLUS SIGN (+),
representing a nonnegative terminating decimal number
(a type of rational number) 
number  an optional prefix of "" (U+002D), followed by an unsigned number, representing a terminating decimal number (a type of rational number) 
character  a single nonwhitespace character 
string  an arbitrary, nonempty and finite, string of characters 
length  a length, as explained below, 2.1.5.2 Length Valued Attributes 
namedspace  a named length, namedspace, as explained in 2.1.5.2 Length Valued Attributes 
color  a color, using the syntax specified by [CSSColor3] 
id  an identifier, unique within the document; must satisfy the NAME syntax of the XML recommendation [XML] 
idref  an identifier referring to another element within the document; must satisfy the NAME syntax of the XML recommendation [XML] 
URI  a Uniform Resource Identifier [RFC3986]. Note that the attribute value is typed in the schema as anyURI which allows any sequence of XML characters. Systems needing to use this string as a URI must encode the bytes of the UTF8 encoding of any characters not allowed in URI using %HH encoding where HH are the byte value in hexadecimal. This ensures that such an attribute value may be interpreted as an IRI, or more generally a LEIRI; see [IRI]. 
italicized word  values as explained in the text for each attribute; see 2.1.5.3 Default values of attributes 
"literal"  quoted symbol, literally present in the attribute value (e.g. "+" or '+') 
The ‘types’ described above, except for string, may be combined into composite patterns using the following operators. The whole attribute value must be delimited by single (') or double (") quotation marks in the marked up document. Note that double quotation marks are often used in this specification to mark up literal expressions; an example is the "" in line 5 of the table above.
In the table below a form f means an instance of a type described in the table above. The combining operators are shown in order of precedence from highest to lowest:
Notation  What it matches 

$\left(f\right)$  same $f$ 
$f?$  an optional instance of $f$ 
$f*$  zero or more instances of $f$, with separating whitespace characters 
$f+$  one or more instances of $f$, with separating whitespace characters 
${f}_{1}{f}_{2}\cdots {f}_{n}$  one instance of each form ${f}_{i}$, in sequence, with no separating whitespace 
${f}_{1},{f}_{2},\dots ,{f}_{n}$  one instance of each form ${f}_{i}$, in sequence, with separating whitespace characters (but no commas) 
${f}_{1}\left{f}_{2}\right\cdots {f}_{n}$  any one of the specified forms ${f}_{i}$ 
The notation we have chosen here is in the style of the syntactical notation of the RelaxNG used for MathML's basic schema, A. Parsing MathML.
Since some applications are inconsistent about normalization of whitespace, for maximum interoperability it is advisable to use only a single whitespace character for separating parts of a value. Moreover, leading and trailing whitespace in attribute values should be avoided.
For most numerical attributes, only those in a subset of the expressible values are sensible; values outside this subset are not errors, unless otherwise specified, but rather are rounded up or down (at the discretion of the renderer) to the closest value within the allowed subset. The set of allowed values may depend on the renderer, and is not specified by MathML.
If a numerical value within an attribute value syntax description
is declared to allow a minus sign (''), e.g., number
or
integer
, it is not a syntax error when one is provided in
cases where a negative value is not sensible. Instead, the value
should be handled by the processing application as described in the
preceding paragraph. An explicit plus sign ('+') is not allowed as
part of a numerical value except when it is specifically listed in the
syntax (as a quoted '+' or "+"), and its presence can change the
meaning of the attribute value (as documented with each attribute
which permits it).
Most presentation elements have attributes that accept values
representing lengths to be used for size, spacing or similar properties.
[MathMLCore] accepts lengths only in the
<lengthpercentage>
syntax defined in [CSSVALUES3].
MathML Full extends length syntax by accepting also a namedspace
being one of:
Positive space  Negative space  Value 

veryverythinmathspace 
negativeveryverythinmathspace 
±1/18 em 
verythinmathspace 
negativeverythinmathspace 
±2/18 em 
thinmathspace 
negativethinmathspace 
±3/18 em 
mediummathspace 
negativemediummathspace 
±4/18 em 
thickmathspace 
negativethickmathspace 
±5/18 em 
verythickmathspace 
negativeverythickmathspace 
±6/18 em 
veryverythickmathspace 
negativeveryverythickmathspace 
±7/18 em 
In addition, the attributes on mpadded
allow three pseudounits, height
,
depth
, and width
(taking the place of one of the usual CSS units)
denoting the original dimensions of the content.
MathML 3 also allowed a deprecated usage with lengths specified as a number without a unit. This was interpreted as a multiple of the reference value. This form is considered invalid in MathML 4.
Two additional aspects of relative units must be clarified, however.
First, some elements such as 3.4 Script and Limit Schemata or mfrac
implicitly switch to smaller font sizes for some of their arguments.
Similarly, mstyle
can be used to explicitly change
the current font size. In such cases, the effective values of
an em
or ex
inside those contexts will be
different than outside. The second point is that the effective value
of an em
or ex
used for an attribute value
can be affected by changes to the current font size.
Thus, attributes that affect the current font size,
such as mathsize
and scriptlevel
, must be processed before
evaluating other length valued attributes.
Default values for MathML attributes are, in general, given along with the detailed descriptions of specific elements in the text. Default values shown in plain text in the tables of attributes for an element are literal, but when italicized are descriptions of how default values can be computed.
Default values described as inherited are taken from the
rendering environment, as described in 3.3.4 Style Change <mstyle>
,
or in some cases (which are described individually) taken from the values of other
attributes of surrounding elements, or from certain parts of those
values. The value used will always be one which could have been specified
explicitly, had it been known; it will never depend on the content or
attributes of the same element, only on its environment. (What it means
when used may, however, depend on those attributes or the content.)
Default values described as automatic should be computed by a MathML renderer in a way which will produce a highquality rendering; how to do this is not usually specified by the MathML specification. The value computed will always be one which could have been specified explicitly, had it been known, but it will usually depend on the element content and possibly on the context in which the element is rendered.
Other italicized descriptions of default values which appear in the tables of attributes are explained individually for each attribute.
The single or double quotes which are required around attribute values in an XML start tag are not shown in the tables of attribute value syntax for each element, but are around attribute values in examples in the text, so that the pieces of code shown are correct.
Note that, in general, there is no mechanism in MathML to simulate the
effect of not specifying attributes which are inherited or
automatic. Giving the words inherited
or
automatic
explicitly will not work, and is not generally
allowed. Furthermore, the mstyle
element (3.3.4 Style Change <mstyle>
)
can even be used to change the default values of presentation attributes
for its children.
Note also that these defaults describe the behavior of MathML applications when an attribute is not supplied; they do not indicate a value that will be filled in by an XML parser, as is sometimes mandated by DTDbased specifications.
In general, there are a number of
properties of MathML rendering that may be thought of as overall
properties of a document, or at least of sections of a large
document. Examples might be mathsize
(the math font
size: see 3.2.2 Mathematics style attributes common to token elements), or the
behavior in setting limits on operators such as integrals or sums
(e.g., movablelimits
or displaystyle
), or
upon breaking formulas over lines (e.g.
linebreakstyle
); for such attributes see several
elements in 3.2 Token Elements.
These may be thought to be inherited from some such
containing scope. Just above we have mentioned the setting of default
values of MathML attributes as inherited or
automatic; there is a third source of global default values
for behavior in rendering MathML, a MathML operator dictionary. A
default example is provided in B. Operator Dictionary.
This is also discussed in 3.2.5.6.1 The operator dictionary and examples are given in
3.2.5.2.1 Dictionarybased attributes.
In MathML, as in XML, whitespace
means simple spaces,
tabs, newlines, or carriage returns, i.e., characters with hexadecimal
Unicode codes U+0020, U+0009, U+000A, or
U+000D, respectively; see also the discussion of whitespace in Section 2.3 of
[XML].
MathML ignores whitespace occurring outside token elements.
Nonwhitespace characters are not allowed there. Whitespace occurring
within the content of token elements, except for <cs>
, is normalized as follows. All whitespace at the beginning and end of the content is
removed, and whitespace internal to content of the element is
collapsed canonically, i.e., each sequence of 1 or more
whitespace characters is replaced with one space character (U+0020, sometimes
called a blank character).
For example, <mo> ( </mo>
is equivalent to
<mo>(</mo>
, and
<mtext>
Theorem
1:
</mtext>
is equivalent to
<mtext>Theorem 1:</mtext>
or
<mtext>Theorem 1:</mtext>
.
Authors wishing to encode white space characters at the start or end of
the content of a token, or in sequences other than a single space, without
having them ignored, must use nonbreaking space U+00A0 (or nbsp
)
or other nonmarking characters that are not trimmed.
For example, compare the above use of an mtext
element
with
<mtext>
 <!nbsp>Theorem  <!nbsp>1:
</mtext>
When the first example is rendered, there is nothing before
Theorem
, one Unicode space character between Theorem
and
1:
, and nothing after 1:
. In the
second example, a single space character is to be rendered before
Theorem
; two spaces, one a Unicode space character and
one a Unicode nobreak space character, are to be rendered before
1:
; and there is nothing after the
1:
.
Note that the value of the xml:space
attribute is not relevant
in this situation since XML processors pass whitespace in tokens to a
MathML processor; it is the requirements of MathML processing which specify that
whitespace is trimmed and collapsed.
For whitespace occurring outside the content of the token elements
mi
, mn
, mo
, ms
, mtext
,
ci
, cn
, cs
, csymbol
and annotation
,
an mspace
element should be used, as opposed to an mtext
element containing
only whitespace entities.
MathML specifies a single toplevel or root math
element,
which encapsulates each instance of
MathML markup within a document. All other MathML content must be
contained in a math
element; in other words,
every valid MathML expression is wrapped in outer
<math>
tags. The math
element must always be the outermost element in a MathML expression;
it is an error for one math
element to contain
another. These considerations also apply when subexpressions are
passed between applications, such as for cutandpaste operations;
see 7.3 Transferring MathML.
The math
element can contain an arbitrary number
of child elements. They render by default as if they
were contained in an mrow
element.
The math
element accepts any of the attributes that can be set on
3.3.4 Style Change <mstyle>
, including the common attributes
specified in 2.1.6 Attributes Shared by all MathML Elements.
In particular, it accepts the dir
attribute for
setting the overall directionality; the math
element is usually
the most useful place to specify the directionality
(see 3.1.5 Directionality for further discussion).
Note that the dir
attribute defaults to ltr
on the math
element (but inherits on all other elements
which accept the dir
attribute); this provides for backward
compatibility with MathML 2.0 which had no notion of directionality.
Also, it accepts the mathbackground
attribute in the same sense
as mstyle
and other presentation elements to set the background
color of the bounding box, rather than specifying a default for the attribute
(see 3.1.9 Mathematics attributes common to presentation elements).
In addition to those attributes, the math
element accepts:
Name  values  default 

display  "block"  "inline"  inline 
specifies whether the enclosed MathML expression should be rendered
as a separate vertical block (in display style)
or inline, aligned with adjacent text.
When display =block , displaystyle is initialized
to true ,
whereas when display =inline , displaystyle
is initialized to false ;
in both cases scriptlevel is initialized to 0
(see 3.1.6 Displaystyle and Scriptlevel).
Moreover, when the math element is embedded in a larger document,
a block math element should be treated as a block element as appropriate
for the document type (typically as a new vertical block),
whereas an inline math element should be treated as inline
(typically exactly as if it were a sequence of words in normal text).
In particular, this applies to spacing and linebreaking: for instance,
there should not be spaces or line breaks inserted between inline math
and any immediately following punctuation.
When the display attribute is missing, a rendering agent is free to initialize
as appropriate to the context.


maxwidth  length  available width 
specifies the maximum width to be used for linebreaking. The default is the maximum width available in the surrounding environment. If that value cannot be determined, the renderer should assume an infinite rendering width.  
overflow  "linebreak"  "scroll"  "elide"  "truncate"  "scale"  linebreak 
specifies the preferred handing in cases where an expression is too long to fit in the allowed width. See the discussion below.  
altimg  URI  none 
provides a URI referring to an image to display as a fallback for user agents that do not support embedded MathML.  
altimgwidth  length  width of altimg 
specifies the width to display altimg , scaling the image if necessary;
see altimgheight .


altimgheight  length  height of altimg 
specifies the height to display altimg , scaling the image if necessary;
if only one of the attributes altimgwidth and altimgheight
are given, the scaling should preserve the image's aspect ratio;
if neither attribute is given, the image should be shown at its natural size.


altimgvalign  length  "top"  "middle"  "bottom"  0ex 
specifies the vertical alignment of the image with respect to adjacent inline material.
A positive value of altimgvalign shifts the bottom of the image above the
current baseline, while a negative value lowers it.
The keyword "top" aligns the top of the image with the top of adjacent inline material;
"center" aligns the middle of the image to the middle of adjacent material;
"bottom" aligns the bottom of the image to the bottom of adjacent material
(not necessarily the baseline). This attribute only has effect
when display =inline .
By default, the bottom of the image aligns to the baseline.


alttext  string  none 
provides a textual alternative as a fallback for user agents that do not support embedded MathML or images.  
cdgroup  URI  none 
specifies a CD group file that acts as a catalogue of CD bases for locating
OpenMath content dictionaries of csymbol , annotation , and
annotationxml elements in this math element; see 4.2.3 Content Symbols <csymbol> . When no cdgroup attribute is explicitly specified, the
document format embedding this math element may provide a method for determining
CD bases. Otherwise the system must determine a CD base; in the absence of specific
information http://www.openmath.org/cd is assumed as the CD base for all
csymbol , annotation , and annotationxml elements. This is the
CD base for the collection of standard CDs maintained by the OpenMath Society.

In cases where size negotiation is not possible or fails
(for example in the case of an expression that is too long to fit in the allowed width),
the overflow
attribute is provided to suggest a processing method to the renderer.
Allowed values are:
Value  Meaning 

"linebreak"  The expression will be broken across several lines. See 3.1.7 Linebreaking of Expressions for further discussion. 
"scroll"  The window provides a viewport into the larger complete display of the mathematical expression. Horizontal or vertical scroll bars are added to the window as necessary to allow the viewport to be moved to a different position. 
"elide"  The display is abbreviated by removing enough of it so that
the remainder fits into the window. For example, a large polynomial
might have the first and last terms displayed with + ... +between them. Advanced renderers may provide a facility to zoom in on elided areas. 
"truncate"  The display is abbreviated by simply truncating it at the right and bottom borders. It is recommended that some indication of truncation is made to the viewer. 
"scale"  The fonts used to display the mathematical expression are chosen so that the full expression fits in the window. Note that this only happens if the expression is too large. In the case of a window larger than necessary, the expression is shown at its normal size within the larger window. 
This chapter specifies the presentation
elements of
MathML, which can be used to describe the layout structure of mathematical
notation.
Most of Presentation Markup is included in [MathMLCore]. That specification should be consulted for the precise details of displaying the elements and attributes that are part of core when displayed in web browsers. Outside of web browsers, MathML presentation elements only suggest (i.e. do not require) specific ways of rendering in order to allow for mediumdependent rendering and for individual preferences of style. Non browserbased renderers are free to use their own layout rules as long as the renderings are intelligible.
The names used for presentation elements are suggestive of their visual layout.
However, mathematical notation has a long history of being reused as new concepts are developed.
Because of this, an element such as mfrac
may not actually be a fraction and the
intent
attribute should be used to provide information for auditory renderings.
This chapter describes all of the presentation elements and attributes of MathML along with examples that might clarify usage.
The presentation elements are meant to express the syntactic
structure of mathematical notation in much the same way as titles, sections,
and paragraphs capture the higherlevel syntactic structure of a
textual document. Because of this, a single row of identifiers and operators
will often be represented by multiple nested mrow
elements rather than
a single mrow
. For example,
$x+a/b$
typically is represented as:
<mrow>
<mi> x </mi>
<mo> + </mo>
<mrow>
<mi> a </mi>
<mo> / </mo>
<mi> b </mi>
</mrow>
</mrow>
Similarly, superscripts are attached to the full expression constituting their base rather than to the just preceding character. This structure permits betterquality rendering of mathematics, especially when details of the rendering environment, such as display widths, are not known ahead of time to the document author. It also greatly eases automatic interpretation of the represented mathematical structures.
Certain characters are used to name identifiers or operators that in traditional notation render the same as other symbols or are rendered invisibly. For example, the characters U+2146, U+2147 and U+2148 represent differential d, exponential e and imaginary i, respectively and are semantically distinct from the same letters used as simple variables. Likewise, the characters U+2061, U+2062, U+2063 and U+2064 represent function application, invisible times, invisible comma and invisible plus . These usually render invisibly but represent significant information that may influence visual spacing and linebreaking, and may have distinct spoken renderings. Accordingly, authors should use these characters (or corresponding entities) wherever applicable.
The complete list of MathML entities is described in [Entities].
The presentation elements are divided into two classes. Token elements represent individual symbols, names, numbers, labels, etc. Layout schemata build expressions out of parts and can have only elements as content. These are subdivided into General Layout, Script and Limit, Tabular Math and Elementary Math schemata. There are also a few empty elements used only in conjunction with certain layout schemata.
All individual symbols
in a mathematical expression should be
represented by MathML token elements (e.g., <mn>24</mn>
).
The primary MathML token element
types are identifiers (mi,
e.g. variables or function names), numbers (mn), and
operators (mo,
including fences, such as parentheses, and separators, such
as commas). There are also token elements used to represent text or
whitespace that has more aesthetic than mathematical significance
and other elements representing string literals
for compatibility with
computer algebra systems.
The layout schemata specify the way in which subexpressions are built into larger expressions such as fraction and scripted expressions. Layout schemata attach special meaning to the number and/or positions of their children. A child of a layout schema is also called an argument of that element. As a consequence of the above definitions, the content of a layout schema consists exactly of a sequence of zero or more elements that are its arguments.
Many of the elements described herein require a specific number of arguments (always 1, 2, or 3). In the detailed descriptions of element syntax given below, the number of required arguments is implicitly indicated by giving names for the arguments at various positions. A few elements have additional requirements on the number or type of arguments, which are described with the individual element. For example, some elements accept sequences of zero or more arguments — that is, they are allowed to occur with no arguments at all.
Note that MathML elements encoding rendered space do
count as arguments of the elements in which they appear.
See 3.2.7 Space <mspace/>
for a discussion of the proper use of such
spacelike elements.
The elements listed in the following table as requiring 1*
argument (msqrt
, mstyle
, merror
,
mpadded
, mphantom
, menclose
,
mtd
, mscarry
,
and math
)
conceptually accept a single argument,
but actually accept any number of children.
If the number of children is 0 or is more than 1, they treat their contents
as a single inferred mrow
formed from all their children,
and treat this mrow
as the argument.
For example,
<msqrt>
<mo>  </mo>
<mn> 1 </mn>
</msqrt>
is treated as if it were
<msqrt>
<mrow>
<mo>  </mo>
<mn> 1 </mn>
</mrow>
</msqrt>
This feature allows MathML data not to contain (and its authors to
leave out) many mrow
elements that would otherwise be
necessary.
For convenience, here is a table of each element's argument count
requirements and the roles of individual arguments when these are
distinguished. An argument count of 1* indicates an inferred mrow
as described above.
Although the math
element is
not a presentation element, it is listed below for completeness.
Element  Required argument count  Argument roles (when these differ by position) 
mrow 
0 or more  
mfrac 
2  numerator denominator 
msqrt 
1*  
mroot 
2  base index 
mstyle 
1*  
merror 
1*  
mpadded 
1*  
mphantom 
1*  
mfenced 
0 or more  
menclose 
1*  
msub 
2  base subscript 
msup 
2  base superscript 
msubsup 
3  base subscript superscript 
munder 
2  base underscript 
mover 
2  base overscript 
munderover 
3  base underscript overscript 
mmultiscripts 
1 or more  base
(subscript superscript)*
[<mprescripts/>
(presubscript presuperscript)*] 
mtable 
0 or more rows  0 or more mtr elements 
mtr 
0 or more  0 or more mtd elements 
mtd 
1*  
mstack 
0 or more  
mlongdiv 
3 or more  divisor result dividend (msrow  msgroup  mscarries  msline)* 
msgroup 
0 or more  
msrow 
0 or more  
mscarries 
0 or more  
mscarry 
1*  
maction 
1 or more  depend on actiontype attribute 
math 
1* 
Certain MathML presentation elements exhibit special behaviors in certain contexts. Such special behaviors are discussed in the detailed element descriptions below. However, for convenience, some of the most important classes of special behavior are listed here.
Certain elements are considered spacelike; these are defined in
3.2.7 Space <mspace/>
. This definition affects some of the suggested rendering
rules for mo
elements (3.2.5 Operator, Fence, Separator or Accent
<mo>
).
Certain elements, e.g. msup
, are able to
embellish operators that are their first argument. These elements are
listed in 3.2.5 Operator, Fence, Separator or Accent
<mo>
, which precisely defines an embellished
operator
and explains how this affects the suggested rendering rules
for stretchy operators.
In the notations familiar to most readers, both the overall layout and the textual symbols are arranged from left to right (LTR). Yet, as alluded to in the introduction, mathematics written in Hebrew or in locales such as Morocco or Persia, the overall layout is used unchanged, but the embedded symbols (often Hebrew or Arabic) are written right to left (RTL). Moreover, in most of the Arabic speaking world, the notation is arranged entirely RTL; thus a superscript is still raised, but it follows the base on the left rather than the right.
MathML 3.0 therefore recognizes two distinct directionalities: the directionality of the text and symbols within token elements and the overall directionality represented by Layout Schemata. These two facets are discussed below.
Probably need to add a little discussion of vertical languages here (and their current lack of support)
The overall directionality for a formula, basically
the direction of the Layout Schemata, is specified by
the dir
attribute on the containing math
element
(see 2.2 The TopLevel
<math>
Element).
The default is ltr
. When dir
=rtl
is used, the layout is simply the mirror image of the conventional
European layout. That is, shifts up or down are unchanged,
but the progression in laying out is from right to left.
For example, in a RTL layout, sub and superscripts appear to the left of the base; the surd for a root appears at the right, with the bar continuing over the base to the left. The layout details for elements whose behavior depends on directionality are given in the discussion of the element. In those discussions, the terms leading and trailing are used to specify a side of an object when which side to use depends on the directionality; i.e. leading means left in LTR but right in RTL. The terms left and right may otherwise be safely assumed to mean left and right.
The overall directionality is usually set on the math
, but
may also be switched for an individual subexpression by using the dir
attribute on mrow
or mstyle
elements.
When not specified, all elements inherit the directionality of their container.
The text directionality comes into play for the MathML token elements
that can contain text (mtext
, mo
, mi
, mn
and ms
) and is determined by the Unicode properties of that text.
A token element containing exclusively LTR or RTL characters
is displayed straightforwardly in the given direction.
When a mixture of directions is involved, such as RTL Arabic
and LTR numbers, the Unicode bidirectional algorithm [Bidi]
should be applied. This algorithm specifies how runs of characters
with the same direction are processed and how the runs are (re)ordered.
The base, or initial, direction is given by the overall directionality
described above (3.1.5.1 Overall Directionality of Mathematics Formulas) and affects
how weakly directional characters are treated and how runs are nested.
(The dir
attribute is thus allowed on token elements to specify
the initial directionality that may be needed in rare cases.)
Any mglyph
or malignmark
elements appearing within
a token element are effectively neutral and have no effect
on ordering.
The important thing to notice is that the bidirectional algorithm is applied independently to the contents of each token element; each token element is an independent run of characters.
Other features of Unicode and scripts that should be respected are ‘mirroring’ and ‘glyph shaping’. Some Unicode characters are marked as being mirrored when presented in a RTL context; that is, the character is drawn as if it were mirrored or replaced by a corresponding character. Thus an opening parenthesis, ‘(’, in RTL will display as ‘)’. Conversely, the solidus (/ U+002F) is not marked as mirrored. Thus, an Arabic author that desires the slash to be reversed in an inline division should explicitly use reverse solidus (\ U+005C) or an alternative such as the mirroring DIVISION SLASH (U+2215).
Additionally, calligraphic scripts such as Arabic blend, or connect sequences of characters together, changing their appearance. As this can have a significant impact on readability, as well as aesthetics, it is important to apply such shaping if possible. Glyph shaping, like directionality, applies to each token element's contents individually.
Note that for the transfinite cardinals represented by Hebrew characters, the code points U+2135U+2138 (ALEF SYMBOL, BET SYMBOL, GIMEL SYMBOL, DALET SYMBOL) should be used in MathML, not the alphabetic lookalike code points. These code points are strong lefttoright.
Socalled ‘displayed’ formulas, those appearing on a line by themselves,
typically make more generous use of vertical space than inline formulas,
which should blend into the adjacent text without intruding into
neighboring lines. For example, in a displayed summation, the limits
are placed above and below the summation symbol, while when it appears inline
the limits would appear in the sub and superscript position.
For similar reasons, sub and superscripts,
nested fractions and other constructs typically display in a
smaller size than the main part of the formula.
MathML implicitly associates with every presentation node
a displaystyle
and scriptlevel
reflecting whether
a more expansive vertical layout applies and the level of scripting
in the current context.
These values are
initialized by the math
element
according to the display
attribute.
They are automatically adjusted by the
various script and limit schemata elements,
and the elements
mfrac
and
mroot
,
which typically set displaystyle
false and increment scriptlevel
for some or all of their arguments.
(See the description for each element for the specific rules used.)
They also may be set explicitly via the displaystyle
and scriptlevel
attributes on the mstyle
element
or the displaystyle
attribute of mtable
.
In all other cases, they are inherited from the node's parent.
The displaystyle
affects the amount of vertical space used to lay out a formula:
when true, the more spacious layout of displayed equations is used,
whereas when false a more compact layout of inline formula is used.
This primarily affects the interpretation
of the largeop
and movablelimits
attributes of
the mo
element.
However, more sophisticated renderers are free to use
this attribute to render more or less compactly.
The main effect of scriptlevel
is to control the font size.
Typically, the higher the scriptlevel
, the smaller the font size.
(Nonvisual renderers can respond to the font size in an analogous way for their medium.)
Whenever the scriptlevel
is changed, whether automatically or explicitly,
the current font size is multiplied by the value of
scriptsizemultiplier
to the power of the change in scriptlevel
.
However, changes to the font size due to scriptlevel
changes should
never reduce the size below scriptminsize
to prevent scripts
becoming unreadably small.
The default scriptsizemultiplier
is approximately the square root of 1/2
whereas scriptminsize
defaults to 8 points;
these values may be changed on mstyle
; see 3.3.4 Style Change <mstyle>
.
Note that the scriptlevel
attribute of mstyle
allows arbitrary
values of scriptlevel
to be obtained, including negative values which
result in increased font sizes.
The changes to the font size due to scriptlevel
should be viewed
as being imposed from ‘outside’ the node.
This means that the effect of scriptlevel
is applied
before an explicit mathsize
(see 3.2.2 Mathematics style attributes common to token elements)
on a token child of mfrac
.
Thus, the mathsize
effectively overrides the effect of scriptlevel
.
However, that change to scriptlevel
changes the current font size,
which affects the meaning of an em
length
(see 2.1.5.2 Length Valued Attributes)
and so the scriptlevel
still may have an effect in such cases.
Note also that since mathsize
is not constrained by scriptminsize
,
such direct changes to font size can result in scripts smaller than scriptminsize
.
Note that direct changes to current font size, whether by
CSS or by the mathsize
attribute (see 3.2.2 Mathematics style attributes common to token elements),
have no effect on the value of scriptlevel
.
TeX's \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle
correspond to displaystyle
and scriptlevel
as
true
and 0
,
false
and 0
,
false
and 1
,
and false
and 2
, respectively.
Thus, math
's
display
=block
corresponds to \displaystyle,
while display
=inline
corresponds to \textstyle.
MathML provides support for both automatic and manual (forced)
linebreaking of expressions to break excessively long
expressions into several lines.
All such linebreaks take place within mrow
(including inferred mrow
; see 3.1.3.1 Inferred <mrow>
s)
or mfenced
.
The breaks typically take place at mo
elements
and also, for backwards compatibility, at mspace
.
Renderers may also choose to place automatic linebreaks at other points
such as between adjacent mi
elements or even within a token element
such as a very long mn
element. MathML does not provide a means to
specify such linebreaks, but if a renderer chooses to linebreak at such a point,
it should indent the following line according to the
indentation attributes
that are in effect at that point.
Automatic linebreaking occurs when the containing math
element
has overflow
=linebreak
and the display engine determines that there is not enough space available to
display the entire formula. The available width must therefore be known
to the renderer. Like font properties, one is assumed to be inherited from the environment
in which the MathML element lives. If no width can be determined, an
infinite width should be assumed. Inside of an mtable
,
each column has some width. This width may be specified as an attribute
or determined by the contents. This width should be used as the
line wrapping width for linebreaking, and each entry in an mtable
is linewrapped as needed.
Forced linebreaks are specified by using
linebreak
=newline
on an mo
or mspace
element.
Both automatic and manual linebreaking can occur within the same formula.
Automatic linebreaking of subexpressions of mfrac
, msqrt
, mroot
and menclose
and the various script elements is not required.
Renderers are free to ignore forced breaks within those elements if they choose.
Attributes on mo
and possibly on mspace
elements control
linebreaking and indentation of the following line. The aspects of linebreaking
that can be controlled are:
Where — attributes determine the desirability of
a linebreak at a specific operator or space, in particular whether a
break is required or inhibited. These can only be set on
mo
and mspace
elements.
(See 3.2.5.2.2 Linebreaking attributes.)
Operator Display/Position — when a linebreak occurs,
determines whether the operator will appear
at the end of the line, at the beginning of the next line, or in both positions;
and how much vertical space should be added after the linebreak.
These attributes can be set on mo
elements or inherited from
mstyle
or math
elements.
(See 3.2.5.2.2 Linebreaking attributes.)
Indentation — determines the indentation of the
line following a linebreak, including indenting so that the next line aligns
with some point in a previous line.
These attributes can be set on mo
elements or
inherited from mstyle
or math
elements.
(See 3.2.5.2.3 Indentation attributes.)
When a math element appears in an inline context, it may obey whatever paragraph flow
rules
are employed by the document's text rendering engine.
Such rules are necessarily outside of the scope of this specification.
Alternatively, it may use the value of the math
element's overflow attribute.
(See 2.2.1 Attributes.)
The following example demonstrates forced linebreaks and forced alignment:
<mrow>
<mrow> <mi>f</mi> <mo>⁡<!ApplyFunction></mo> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow>
<mo id='eq1equals'>=</mo>
<mrow>
<msup>
<mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow>
<mn>4</mn>
</msup>
<mo linebreak='newline' linebreakstyle='before'
indentalign='id' indenttarget='eq1equals'>=</mo>
<mrow>
<msup> <mi>x</mi> <mn>4</mn> </msup>
<mo id='eq1plus'>+</mo>
<mrow> <mn>4</mn> <mo>⁢<!InvisibleTimes></mo> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow>
<mo>+</mo>
<mrow> <mn>6</mn> <mo>⁢<!InvisibleTimes></mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow>
<mo linebreak='newline' linebreakstyle='before'
indentalignlast='id' indenttarget='eq1plus'>+</mo>
<mrow> <mn>4</mn> <mo>⁢<!InvisibleTimes></mo> <mi>x</mi> </mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mrow>
</mrow>
This displays as
Note that because indentalignlast
defaults to indentalign
,
in the above example indentalign
could have been used in place of
indentalignlast
. Also, the specifying linebreakstyle='before'
is not needed because that is the default value.
mi 
identifier 
mn 
number 
mo 
operator, fence, or separator 
mtext 
text 
mspace 
space 
ms 
string literal 
Additionally, the mglyph
element
may be used within Token elements to represent nonstandard symbols as images.
mrow 
group any number of subexpressions horizontally 
mfrac 
form a fraction from two subexpressions 
msqrt 
form a square root (radical without an index) 
mroot 
form a radical with specified index 
mstyle 
style change 
merror 
enclose a syntax error message from a preprocessor 
mpadded 
adjust space around content 
mphantom 
make content invisible but preserve its size 
mfenced 
surround content with a pair of fences 
menclose 
enclose content with a stretching symbol such as a long division sign 
msub 
attach a subscript to a base 
msup 
attach a superscript to a base 
msubsup 
attach a subscriptsuperscript pair to a base 
munder 
attach an underscript to a base 
mover 
attach an overscript to a base 
munderover 
attach an underscriptoverscript pair to a base 
mmultiscripts 
attach prescripts and tensor indices to a base 
mtable 
table or matrix 
mtr 
row in a table or matrix 
mtd 
one entry in a table or matrix 
maligngroup and
malignmark 
alignment markers 
mstack 
columns of aligned characters 
mlongdiv 
similar to msgroup, with the addition of a divisor and result 
msgroup 
a group of rows in an mstack that are shifted by similar amounts 
msrow 
a row in an mstack 
mscarries 
row in an mstack whose contents represent carries or borrows 
mscarry 
one entry in an mscarries 
msline 
horizontal line inside of mstack 
maction 
bind actions to a subexpression 
In addition to the attributes listed in 2.1.6 Attributes Shared by all MathML Elements, all MathML presentation elements accept the following classes of attribute.
Presentation elements also accept all the Global Attributes specified by [MathMLCore].
These attributes include the following two attributes that are primarily intended for visual media.
They are not expected to affect the intended semantics of displayed
expressions, but are for use in highlighting or drawing attention
to the affected subexpressions. For example, a red "x" is not assumed
to be semantically different than a black "x", in contrast to
variables with different mathvariant
values (see 3.2.2 Mathematics style attributes common to token elements).
Name  values  default 
mathcolor  color  inherited 
Specifies the foreground color to use when drawing the components of this element,
such as the content for token elements or any lines, surds, or other decorations.
It also establishes the default mathcolor used for child elements
when used on a layout element.


mathbackground  color  "transparent"  transparent 
Specifies the background color to be used to fill in the bounding box of the element and its children. The default, "transparent", lets the background color, if any, used in the current rendering context to show through. 
Since MathML expressions are often embedded in a textual data format such as HTML, the MathML renderer should inherit the foreground color used in the context in which the MathML appears. Note, however, that MathML (in contrast to [MathMLCore]) doesn't specify the mechanism by which style information is inherited from the rendering environment. See 3.2.2 Mathematics style attributes common to token elements for more details.
Note that the suggested MathML visual rendering rules do not define the
precise extent of the region whose background is affected by the
mathbackground
attribute,
except that, when the content does not have
negative dimensions and its drawing region should not overlap with other
drawing due to surrounding negative spacing, should lie
behind all the drawing done to render the content, and should not lie behind any of
the drawing done to render surrounding expressions. The effect of overlap
of drawing regions caused by negative spacing on the extent of the
region affected by the mathbackground
attribute is not
defined by these rules.
Token elements in presentation markup are broadly intended to represent the smallest units of mathematical notation which carry meaning. Tokens are roughly analogous to words in text. However, because of the precise, symbolic nature of mathematical notation, the various categories and properties of token elements figure prominently in MathML markup. By contrast, in textual data, individual words rarely need to be marked up or styled specially.
Token elements represent
identifiers (mi
),
numbers (mn
),
operators (mo
),
text (mtext
),
strings (ms
)
and spacing (mspace
).
The mglyph
element
may be used within token elements
to represent nonstandard symbols by images.
Preceding detailed discussion of the individual elements,
the next two subsections discuss the allowable content of
token elements and the attributes common to them.
Character data in MathML markup is only allowed to occur as part of
the content of token elements. Whitespace between elements is ignored.
With the exception of the empty mspace
element,
token elements can contain any sequence of zero or more Unicode characters,
or mglyph
or
malignmark
elements.
The mglyph
element is used
to represent nonstandard characters or symbols by images;
the malignmark
element establishes an alignment point for use within
table constructs, and is otherwise invisible (see 3.5.4 Alignment Markers
<maligngroup/>
, <malignmark/>
).
Characters can be either represented directly as Unicode character data, or indirectly via numeric or character entity references. Unicode contains a number of lookalike characters. See [MathMLNotes] for a discussion of which characters are appropriate to use in which circumstance.
Token elements (other than mspace
) should
be rendered as their content, if any (i.e. in the visual case, as a
closelyspaced horizontal row of standard glyphs for the characters
or images for the mglyph
s in their content).
An mspace
element is rendered as a blank space of a width determined by its attributes.
Rendering algorithms should also take into account the
mathematics style attributes as described below, and modify surrounding
spacing by rules or attributes specific to each type of token
element. The directional characteristics of the content must
also be respected (see 3.1.5.2 Bidirectional Layout in Token Elements).
mglyph
is not supported in [MathMLCore].
In a Web Platform Context it is recommended that the HTML img
element is used. This is allowed in token elements when MathML is embedded in (X)HTML.
For existing MathML using mglyph
a Javascript polyfill
is provided for Web documents that implements mglyph
using img
.
The mglyph
element provides a mechanism
for displaying images to represent nonstandard symbols.
It may be used within the content of the token elements
mi
, mn
, mo
, mtext
or ms
where existing Unicode characters are not adequate.
Unicode defines a large number of characters used in mathematics and, in most cases, glyphs representing these characters are widely available in a variety of fonts. Although these characters should meet almost all users needs, MathML recognizes that mathematics is not static and that new characters and symbols are added when convenient. Characters that become well accepted will likely be eventually incorporated by the Unicode Consortium or other standards bodies, but that is often a lengthy process.
Note that the glyph's src
attribute uniquely identifies the mglyph
;
two mglyph
s with the same values for src
should
be considered identical by applications that must determine whether
two characters/glyphs are identical.
The mglyph
element accepts the attributes listed in
3.1.9 Mathematics attributes common to presentation elements, but note that mathcolor
has no effect.
The background color, mathbackground
, should show through
if the specified image has transparency.
mglyph
also accepts the additional attributes listed here.
Name  values  default 
src  URI  required 
Specifies the location of the image resource; it may be a URI relative to the baseURI of the source of the MathML, if any.  
width  length  from image 
Specifies the desired width of the glyph; see height .


height  length  from image 
Specifies the desired height of the glyph.
If only one of width and height are given,
the image should be scaled to preserve the aspect ratio;
if neither are given, the image should be displayed at its natural size.


valign  length  0ex 
Specifies the baseline alignment point of the image with respect to the current baseline. A positive value shifts the bottom of the image above the current baseline while a negative value lowers it. A value of 0 (the default) means that the baseline of the image is at the bottom of the image.  
alt  string  required 
Provides an alternate name for the glyph. If the specified image can't be found or displayed, the renderer may use this name in a warning message or some unknown glyph notation. The name might also be used by an audio renderer or symbol processing system and should be chosen to be descriptive. 
The following example illustrates how a researcher might use
the mglyph
construct with a set of images to work
with braid group notation.
<mrow>
<mi><mglyph src="mybraid23" alt="2 3 braid"/></mi>
<mo>+</mo>
<mi><mglyph src="mybraid132" alt="1 3 2 braid"/></mi>
<mo>=</mo>
<mi><mglyph src="mybraid13" alt="1 3 braid"/></mi>
</mrow>
This might render as:
In addition to the attributes defined for all presentation elements
(3.1.9 Mathematics attributes common to presentation elements), MathML includes two mathematics style attributes
as well as a directionality attribute
valid on all presentation token elements,
as well as the math
and mstyle
elements;
dir
is also valid on mrow
elements.
The attributes are:
Name  values  default 
mathvariant  "normal"  "bold"  "italic"  "bolditalic"  "doublestruck"  "boldfraktur"  "script"  "boldscript"  "fraktur"  "sansserif"  "boldsansserif"  "sansserifitalic"  "sansserifbolditalic"  "monospace"  "initial"  "tailed"  "looped"  "stretched"  normal (except on <mi> ) 
Specifies the logical class of the token. Note that this class is more than styling, it typically conveys semantic intent; see the discussion below.  
mathsize  "small"  "normal"  "big"  length  inherited 
Specifies the size to display the token content.
The values small and big choose a size
smaller or larger than the current font size, but leave the exact proportions
unspecified; normal is allowed for completeness, but since
it is equivalent to 100% or 1em , it has no effect.


dir  "ltr"  "rtl"  inherited 
specifies the initial directionality for text within the token:
ltr (Left To Right) or rtl (Right To Left).
This attribute should only be needed in rare cases involving weak or neutral characters;
see 3.1.5.1 Overall Directionality of Mathematics Formulas for further discussion.
It has no effect on mspace .

The mathvariant
attribute defines logical classes of token
elements. Each class provides a collection of typographicallyrelated
symbolic tokens. Each token has a specific meaning within a given
mathematical expression and, therefore, needs to be visually
distinguished and protected from inadvertent documentwide style
changes which might change its meaning. Each token is identified
by the combination of the mathvariant
attribute value
and the character data in the token element.
When MathML rendering takes place in an environment where CSS is
available, the mathematics style attributes can be viewed as
predefined selectors for CSS style rules.
See 7.5 Using CSS with MathML for discussion of the
interaction of MathML and CSS.
Also, see [MathMLforCSS] for discussion of rendering MathML by CSS
and a sample CSS style sheet.
When CSS is not available, it is up to the internal style mechanism of the rendering
application
to visually distinguish the different logical classes.
Most MathML renderers will probably want to rely on some degree on additional,
internal style processing algorithms.
In particular, the mathvariant
attribute does not follow the CSS inheritance model;
the default value is normal
(nonslanted)
for all tokens except for mi
with singlecharacter content.
See 3.2.3 Identifier <mi>
for details.
Renderers have complete freedom in
mapping mathematics style attributes to specific rendering properties.
However, in practice, the mathematics style attribute names and values
suggest obvious typographical properties, and renderers should attempt
to respect these natural interpretations as far as possible. For
example, it is reasonable to render a token with the
mathvariant
attribute set to sansserif
in
Helvetica or Arial. However, rendering the token in a Times Roman
font could be seriously misleading and should be avoided.
In principle, any mathvariant
value may be used with any
character data to define a specific symbolic token. In practice,
only certain combinations of character data and mathvariant
values will be visually distinguished by a given renderer. For example,
there is no clearcut rendering for a "fraktur alpha" or a "bold italic
Kanji" character, and the mathvariant
values "initial",
"tailed", "looped", and "stretched" are appropriate only for Arabic
characters.
Certain combinations of character data and mathvariant
values are equivalent to assigned Unicode code points that encode
mathematical alphanumeric symbols. These Unicode code points are
the ones in the
Arabic Mathematical Alphabetic Symbols block U+1EE00 to U+1EEFF,
Mathematical Alphanumeric Symbols block U+1D400 to U+1D7FF,
listed in the Unicode standard, and the ones in the
Letterlike
Symbols range U+2100 to U+214F that represent "holes" in the
alphabets in the SMP, listed in 8.2 Mathematical Alphanumeric Symbols.
These characters are described in detail in section 2.2 of
UTR #25.
The description of each such character in the Unicode standard
provides an unstyled character to which it would be equivalent
except for a font change that corresponds to a mathvariant
value. A token element that uses the unstyled character in combination
with the corresponding mathvariant
value is equivalent to a
token element that uses the mathematical alphanumeric symbol character
without the mathvariant
attribute. Note that the appearance
of a mathematical alphanumeric symbol character should not be altered
by surrounding mathvariant
or other style declarations.
Renderers should support those combinations of character data and
mathvariant
values that correspond to Unicode characters,
and that they can visually distinguish using available font characters.
Renderers may ignore or support those combinations of character data
and mathvariant
values that do not correspond to an assigned
Unicode code point, and authors should recognize that support for
mathematical symbols that do not correspond to assigned Unicode code
points may vary widely from one renderer to another.
Since MathML expressions are often embedded in a textual data
format such as HTML, the surrounding text and the MathML must share
rendering attributes such as font size, so that the renderings will be
compatible in style. For this reason, most attribute values affecting
text rendering are inherited from the rendering environment, as shown
in the default
column in the table above. (In
cases where the surrounding text and the MathML are being rendered by
separate software, e.g. a browser and a plugin, it is also important
for the rendering environment to provide the MathML renderer with
additional information, such as the baseline position of surrounding
text, which is not specified by any MathML attributes.)
Note, however, that MathML doesn't specify the mechanism by which
style information is inherited from the rendering environment.
If the requested mathsize
of the current font is not available, the
renderer should approximate it in the manner likely to lead to the
most intelligible, highest quality rendering.
Note that many MathML elements automatically change the font size
in some of their children; see the discussion in 3.1.6 Displaystyle and Scriptlevel.
MathML can be combined with other formats as described in
7.4 Combining MathML and Other Formats.
The recommendation is to embed other formats in MathML by extending the MathML
schema to allow additional elements to be children of the mtext
element or
other leaf elements as appropriate to the role they serve in the expression
(see 3.2.3 Identifier <mi>
, 3.2.4 Number <mn>
, and 3.2.5 Operator, Fence, Separator or Accent
<mo>
).
The directionality, font size, and other font attributes should inherit from
those that would be used for characters of the containing leaf element
(see 3.2.2 Mathematics style attributes common to token elements).
Here is an example of embedding SVG inside of mtext in an HTML context:
<mtable>
<mtr>
<mtd>
<mtext><input type="text" placeholder="what shape is this?"/></mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<mtext>
<svg xmlns="http://www.w3.org/2000/svg" width="4cm" height="4cm" viewBox="0 0 400 400">
<rect x="1" y="1" width="398" height="398" style="fill:none; stroke:blue"/>
<path d="M 100 100 L 300 100 L 200 300 z" style="fill:red; stroke:blue; strokewidth:3"/>
</svg>
</mtext>
</mtd>
</mtr>
</mtable>
<mi>
An mi
element represents a symbolic name or
arbitrary text that should be rendered as an identifier. Identifiers
can include variables, function names, and symbolic constants.
A typical graphical renderer would render an mi
element
as its content (see 3.2.1
Token Element Content Characters, <mglyph/>
),
with no extra spacing around it (except spacing associated with
neighboring elements).
Not all mathematical identifiers
are represented by
mi
elements — for example, subscripted or primed
variables should be represented using msub
or
msup
respectively. Conversely, arbitrary text
playing the role of a term
(such as an ellipsis in a summed series)
should be represented using an mi
element.
It should be stressed that mi
is a
presentation element, and as such, it only indicates that its content
should be rendered as an identifier. In the majority of cases, the
contents of an mi
will actually represent a
mathematical identifier such as a variable or function name. However,
as the preceding paragraph indicates, the correspondence between
notations that should render as identifiers and notations that are
actually intended to represent mathematical identifiers is not
perfect. For an element whose semantics is guaranteed to be that of an
identifier, see the description of ci
in
4. Content Markup.
mi
elements accept the attributes listed in
3.2.2 Mathematics style attributes common to token elements, but in one case with a different default value:
Name  values  default 
mathvariant  "normal"  "bold"  "italic"  "bolditalic"  "doublestruck"  "boldfraktur"  "script"  "boldscript"  "fraktur"  "sansserif"  "boldsansserif"  "sansserifitalic"  "sansserifbolditalic"  "monospace"  "initial"  "tailed"  "looped"  "stretched"  (depends on content; described below) 
Specifies the logical class of the token.
The default is normal (nonslanted) unless the content
is a single character, in which case it would be italic .

Note that for purposes of determining equivalences of Math
Alphanumeric Symbol
characters (see 8.2 Mathematical Alphanumeric Symbols)
the value of the mathvariant
attribute should be resolved first,
including the special defaulting behavior described above.
<mi>x</mi>
<mi>D</mi>
<mi>sin</mi>
<mi mathvariant='script'>L</mi>
<mi></mi>
An mi
element with no content is allowed;
<mi></mi>
might, for example, be used by an
expression editor
to represent a location in a MathML expression
which requires a term
(according to conventional syntax for
mathematics) but does not yet contain one.
Identifiers include function names such as
sin
. Expressions such as sin x
should be written using the character U+2061
(entity af
or ApplyFunction
) as shown below;
see also the discussion of invisible operators in 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
<mrow>
<mi> sin </mi>
<mo> ⁡<!ApplyFunction> </mo>
<mi> x </mi>
</mrow>
Miscellaneous text that should be treated as a term
can also be
represented by an mi
element, as in:
<mrow>
<mn> 1 </mn>
<mo> + </mo>
<mi> … </mi>
<mo> + </mo>
<mi> n </mi>
</mrow>
When an mi
is used in such exceptional
situations, explicitly setting the mathvariant
attribute
may give better results than the default behavior of some
renderers.
The names of symbolic constants should be represented as
mi
elements:
<mi> π </mi>
<mi> ⅈ </mi>
<mi> ⅇ </mi>
<mn>
An mn
element represents a numeric
literal
or other data that should be rendered as a numeric
literal. Generally speaking, a numeric literal is a sequence of digits,
perhaps including a decimal point, representing an unsigned integer or real
number.
A typical graphical renderer would render an mn
element as
its content (see 3.2.1
Token Element Content Characters, <mglyph/>
), with no extra spacing around them
(except spacing from neighboring elements such as mo
).
mn
elements are typically rendered in an unslanted font.
The mathematical concept of a number
can be quite
subtle and involved, depending on the context. As a consequence, not all
mathematical numbers should be represented using mn
; examples of mathematical numbers that should be
represented differently are shown below, and include
complex numbers, ratios of numbers shown as fractions, and names of numeric
constants.
Conversely, since mn
is a presentation
element, there are a few situations where it may be desirable to include
arbitrary text in the content of an mn
that
should merely render as a numeric literal, even though that content
may not be unambiguously interpretable as a number according to any
particular standard encoding of numbers as character sequences. As a
general rule, however, the mn
element should be
reserved for situations where its content is actually intended to
represent a numeric quantity in some fashion. For an element whose
semantics are guaranteed to be that of a particular kind of
mathematical number, see the description of cn
in
4. Content Markup.
mn
elements accept the attributes listed in 3.2.2 Mathematics style attributes common to token elements.
<mn> 2 </mn>
<mn> 0.123 </mn>
<mn> 1,000,000 </mn>
<mn> 2.1e10 </mn>
<mn> 0xFFEF </mn>
<mn> MCMLXIX </mn>
<mn> twentyone </mn>
Many mathematical numbers should be represented using presentation
elements other than mn
alone; this includes
complex numbers, negative numbers, ratios of numbers shown as fractions, and
names of numeric constants.
<mrow>
<mn> 2 </mn>
<mo> + </mo>
<mrow>
<mn> 3 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<mi> ⅈ </mi>
</mrow>
</mrow>
<mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac>
<mrow><mo></mo><mn>2</mn></mrow>
<mi> π </mi>
<mi> ⅇ </mi>
<mo>
An mo
element represents an operator or
anything that should be rendered as an operator. In general, the
notational conventions for mathematical operators are quite
complicated, and therefore MathML provides a relatively sophisticated
mechanism for specifying the rendering behavior of an
mo
element. As a consequence, in MathML the list
of things that should render as an operator
includes a number of
notations that are not mathematical operators in the ordinary
sense. Besides ordinary operators with infix, prefix, or postfix
forms, these include fence characters such as braces, parentheses, and
absolute value
bars; separators
such as comma and semicolon; and
mathematical accents such as a bar or tilde over a symbol.
We will use the term "operator" in this chapter to refer to operators in this broad
sense.
Typical graphical renderers show all mo
elements as the content (see 3.2.1
Token Element Content Characters, <mglyph/>
),
with additional spacing around the element determined by its attributes and
further described below.
Renderers without access to complete fonts for the MathML character
set may choose to render an mo
element as
not precisely the characters in its content in some cases. For example,
<mo> ≤ </mo>
might be rendered as
<=
to a terminal. However, as a general rule,
renderers should attempt to render the content of an
mo
element as literally as possible.
That is,
<mo> ≤ </mo>
and
<mo> <= </mo>
should render differently.
The first one should render as a single character
representing a lessthanorequalto sign, and the second one as the
twocharacter sequence <=
.
A key feature of the mo
element is that its
default attribute values are set on a casebycase basis from an
operator dictionary
as explained below. In particular, default
value for stretch
, symmetric
and
accent
can usually be found in the operator dictionary
and therefore need not be specified on each mo
element.
Note that some mathematical operators are represented not by mo
elements alone, but by mo
elements embellished
with (for example) surrounding
superscripts; this is further described below. Conversely, as presentation
elements, mo
elements can contain arbitrary text,
even when that text has no standard interpretation as an operator; for an
example, see the discussion Mixing text and mathematics
in
3.2.6 Text <mtext>
. See also 4. Content Markup for
definitions of MathML content elements that are guaranteed to have the
semantics of specific mathematical operators.
Note also that linebreaking, as discussed in
3.1.7 Linebreaking of Expressions, usually takes place at operators
(either before or after, depending on local conventions).
Thus, mo
accepts attributes to encode the desirability
of breaking at a particular operator, as well as attributes
describing the treatment of the operator and indentation in case
a linebreak is made at that operator.
mo
elements accept
the attributes listed in 3.2.2 Mathematics style attributes common to token elements
and the additional attributes listed here.
Since the display of operators is so critical in mathematics,
the mo
element accepts a large number of attributes;
these are described in the next three subsections.
Most attributes get their default values from an enclosing
mstyle
element, math
element,
from the containing document,
or from
3.2.5.6.1 The operator dictionary.
When a value that is listed as inherited
is not explicitly given on an
mo
, mstyle
element, math
element, or found in the operator
dictionary for a given mo
element, the default value shown in
parentheses is used.
Name  values  default 
form  "prefix"  "infix"  "postfix"  set by position of operator in an mrow 
Specifies the role of the operator in the enclosing expression.
This role and the operator content affect the lookup of the operator in the operator
dictionary
which affects the spacing and other default properties;
see 3.2.5.6.2 Default value of the form attribute.


lspace  length  set by dictionary (thickmathspace) 
Specifies the leading space appearing before the operator; see 3.2.5.6.4 Spacing around an operator. (Note that before is on the right in a RTL context; see 3.1.5 Directionality.)  
rspace  length  set by dictionary (thickmathspace) 
Specifies the trailing space appearing after the operator; see 3.2.5.6.4 Spacing around an operator. (Note that after is on the left in a RTL context; see 3.1.5 Directionality.)  
stretchy  "true"  "false"  set by dictionary (false) 
Specifies whether the operator should stretch to the size of adjacent material; see 3.2.5.7 Stretching of operators, fences and accents.  
symmetric  "true"  "false"  set by dictionary (false) 
Specifies whether the operator should be kept symmetric around the math axis when stretchy. Note this property only applies to vertically stretched symbols. See 3.2.5.7 Stretching of operators, fences and accents.  
maxsize  length  set by dictionary (unbounded) 
Specifies the maximum size of the operator when stretchy; see 3.2.5.7 Stretching of operators, fences and accents. If not given, the maximum size is unbounded. Unitless or percentage values indicate a multiple of the reference size, being the size of the unstretched glyph. MathML 4 deprecates "infinity" as possible value as it is the same as not providing a value.  
minsize  length  set by dictionary (100%) 
Specifies the minimum size of the operator when stretchy; see 3.2.5.7 Stretching of operators, fences and accents. Unitless or percentage values indicate a multiple of the reference size, being the size of the unstretched glyph.  
largeop  "true"  "false"  set by dictionary (false) 
Specifies whether the operator is considered a ‘large’ operator,
that is, whether it should be drawn larger than normal when
displaystyle =true
(similar to using TeX's \displaystyle ).
Examples of large operators include U+222B and U+220F
(entities int and prod ).
See 3.1.6 Displaystyle and Scriptlevel for more discussion.


movablelimits  "true"  "false"  set by dictionary (false) 
Specifies whether under and overscripts attached to
this operator ‘move’ to the more compact sub and superscript positions
when displaystyle is false.
Examples of operators that typically have movablelimits =true
are U+2211 and U+220F
(entitites sum , prod ),
as well as lim .
See 3.1.6 Displaystyle and Scriptlevel for more discussion.


accent  "true"  "false"  set by dictionary (false) 
Specifies whether this operator should be treated as an accent (diacritical mark)
when used as an underscript or overscript;
see munder ,
mover
and munderover .
Note: for compatibility with MathML Core, use accent =true on
the enclosing mover and munderover in place of this attribute.

The following attributes affect when a linebreak does or does not occur, and the appearance of the linebreak when it does occur.
Name  values  default 
linebreak  "auto"  "newline"  "nobreak"  "goodbreak"  "badbreak"  auto 
Specifies the desirability of a linebreak occurring at this operator:
the default auto indicates the renderer should use its default
linebreaking algorithm to determine whether to break;
newline is used to force a linebreak;
for automatic linebreaking, nobreak forbids a break;
goodbreak suggests a good position;
badbreak suggests a poor position.


lineleading  length  inherited (100%) 
Specifies the amount of vertical space to use after a linebreak. For tall lines, it is often clearer to use more leading at linebreaks. Rendering agents are free to choose an appropriate default.  
linebreakstyle  "before"  "after"  "duplicate"  "infixlinebreakstyle"  set by dictionary (before) 
Specifies whether a linebreak occurs ‘before’ or ‘after’ the operator
when a linebreak occurs on this operator; or whether the operator is duplicated.
before causes the operator to appear at the beginning of the new line
(but possibly indented);
after causes it to appear at the end of the line before the break.
duplicate places the operator at both positions.
infixlinebreakstyle uses the value that has been specified for
infix operators; this value (one of before ,
after or duplicate ) can be specified by
the application or bound by mstyle
(before corresponds to the most common style of linebreaking).


linebreakmultchar  string  inherited (⁢) 
Specifies the character used to make an ⁢ operator visible at a linebreak.
For example, linebreakmultchar="·" would make the
multiplication visible as a center dot.

linebreak
values on adjacent mo
and mspace
elements do
not interact; linebreak
=nobreak
on an mo
does
not, in itself, inhibit a break on a preceding or following (possibly nested)
mo
or mspace
element and does not interact with the linebreakstyle
attribute value of the preceding or following mo
element.
It does prevent breaks from occurring on either side of the mo
element in all other situations.
The following attributes affect indentation of the lines making up a formula.
Primarily these attributes control the positioning of new lines following a linebreak,
whether automatic or manual. However, indentalignfirst
and indentshiftfirst
also control the positioning of a single line formula without any linebreaks.
When these attributes appear on mo
or mspace
they apply if a linebreak occurs
at that element.
When they appear on mstyle
or math
elements, they determine
defaults for the style to be used for any linebreaks occurring within.
Note that except for cases where heavily markedup manual linebreaking is desired,
many of these attributes are most useful when bound on an
mstyle
or math
element.
Note that since the rendering context, such as the available width and current font, is not always available to the author of the MathML, a renderer may ignore the values of these attributes if they result in a line in which the remaining width is too small to usefully display the expression or if they result in a line in which the remaining width exceeds the available linewrapping width.
Name  values  default 
indentalign  "left"  "center"  "right"  "auto"  "id"  inherited (auto) 
Specifies the positioning of lines when linebreaking takes place within an mrow ;
see below for discussion of the attribute values.


indentshift  length  inherited (0) 
Specifies an additional indentation offset relative to the position determined
by indentalign .
When the value is a percentage value,
the value is relative to the
horizontal space that a MathML renderer has available, this is the current target
width as used for
linebreaking as specified in 3.1.7 Linebreaking of Expressions.
Note: numbers without units were allowed in MathML 3 and treated similarly to percentage values,
but unitless numbers are deprecated in MathML 4.


indenttarget  idref  inherited (none) 
Specifies the id of another element
whose horizontal position determines the position of indented lines
when indentalign =id .
Note that the identified element may be outside of the current
math element, allowing for interexpression alignment,
or may be within invisible content such as mphantom ;
it must appear before being referenced, however.
This may lead to an id being unavailable to a given renderer
or in a position that does not allow for alignment.
In such cases, the indentalign should revert to auto .


indentalignfirst  "left"  "center"  "right"  "auto"  "id"  "indentalign"  inherited (indentalign) 
Specifies the indentation style to use for the first line of a formula;
the value indentalign (the default) means
to indent the same way as used for the general line.


indentshiftfirst  length  "indentshift"  inherited (indentshift) 
Specifies the offset to use for the first line of a formula;
the value indentshift (the default) means
to use the same offset as used for the general line.
Percentage values and numbers without unit are interpreted as described for indentshift .


indentalignlast  "left"  "center"  "right"  "auto"  "id"  "indentalign"  inherited (indentalign) 
Specifies the indentation style to use for the last line when a linebreak
occurs within a given mrow ;
the value indentalign (the default) means
to indent the same way as used for the general line.
When there are exactly two lines, the value of this attribute should
be used for the second line in preference to indentalign .


indentshiftlast  length  "indentshift"  inherited (indentshift) 
Specifies the offset to use for the last line when a linebreak
occurs within a given mrow ;
the value indentshift (the default) means
to indent the same way as used for the general line.
When there are exactly two lines, the value of this attribute should
be used for the second line in preference to indentshift .
Percentage values and numbers without unit are interpreted as described for indentshift .

The legal values of indentalign are:
Value  Meaning 
left  Align the left side of the next line to the left side of the line wrapping width 
center  Align the center of the next line to the center of the line wrapping width 
right  Align the right side of the next line to the right side of the line wrapping width 
auto  (default) indent using the renderer's default indenting style; this may be a fixed amount or one that varies with the depth of the element in the mrow nesting or some other similar method. 
id  Align the left side of the next line to the left side of the element
referenced by the idref
(given by indenttarget );
if no such element exists, use auto as the indentalign value 
<mo> + </mo>
<mo> < </mo>
<mo> ≤ </mo>
<mo> <= </mo>
<mo> ++ </mo>
<mo> ∑ </mo>
<mo> .NOT. </mo>
<mo> and </mo>
<mo> ⁢<!InvisibleTimes> </mo>
<mo mathvariant='bold'> + </mo>
Note that the mo
elements in these examples
don't need explicit stretchy
or symmetric
attributes,
since these can be found using the
operator dictionary as described below. Some of these examples could also
be encoded using the mfenced
element described in
3.3.8 Expression Inside Pair of Fences
<mfenced>
.
(a+b)
<mrow>
<mo> ( </mo>
<mrow>
<mi> a </mi>
<mo> + </mo>
<mi> b </mi>
</mrow>
<mo> ) </mo>
</mrow>
[0,1)
<mrow>
<mo> [ </mo>
<mrow>
<mn> 0 </mn>
<mo> , </mo>
<mn> 1 </mn>
</mrow>
<mo> ) </mo>
</mrow>
f(x,y)
<mrow>
<mi> f </mi>
<mo> ⁡<!ApplyFunction> </mo>
<mrow>
<mo> ( </mo>
<mrow>
<mi> x </mi>
<mo> , </mo>
<mi> y </mi>
</mrow>
<mo> ) </mo>
</mrow>
</mrow>
Certain operators that are invisible
in traditional
mathematical notation should be represented using specific characters (or entity
references) within mo
elements, rather than simply
by nothing. The characters used for these invisible
operators
are:
Character  Entity name  Short name 
U+2061  ApplyFunction 
af 
U+2062  InvisibleTimes 
it 
U+2063  InvisibleComma 
ic 
U+2064 
The MathML representations of the examples in the above table are:
<mrow>
<mi> f </mi>
<mo> ⁡<!ApplyFunction> </mo>
<mrow>
<mo> ( </mo>
<mi> x </mi>
<mo> ) </mo>
</mrow>
</mrow>
<mrow>
<mi> sin </mi>
<mo> ⁡<!ApplyFunction> </mo>
<mi> x </mi>
</mrow>
<mrow>
<mi> x </mi>
<mo> ⁢<!InvisibleTimes> </mo>
<mi> y </mi>
</mrow>
<msub>
<mi> m </mi>
<mrow>
<mn> 1 </mn>
<mo> ⁣<!InvisibleComma> </mo>
<mn> 2 </mn>
</mrow>
</msub>
<mrow>
<mn> 2 </mn>
<mo> ⁤ </mo>
<mfrac>
<mn> 3 </mn>
<mn> 4 </mn>
</mfrac>
</mrow>
Typical visual rendering behaviors for mo
elements are more complex than for the other MathML token elements, so
the rules for rendering them are described in this separate
subsection.
Note that, like all rendering rules in MathML, these rules are suggestions rather than requirements. The description below is given to make the intended effect of the various rendering attributes as clear as possible. Detailed layout rules for browser implementations for operators are given in MathML Core.
Many mathematical symbols, such as an integral sign, a plus sign,
or a parenthesis, have a wellestablished, predictable, traditional
notational usage. Typically, this usage amounts to certain default
attribute values for mo
elements with specific
contents and a specific form
attribute. Since these
defaults vary from symbol to symbol, MathML anticipates that renderers
will have an operator dictionary
of default attributes for
mo
elements (see B. Operator Dictionary) indexed by each
mo
element's content and form
attribute. If an mo
element is not listed in the
dictionary, the default values shown in parentheses in the table of
attributes for mo
should be used, since these
values are typically acceptable for a generic operator.
Some operators are overloaded
, in the sense that they can occur
in more than one form (prefix, infix, or postfix), with possibly
different rendering properties for each form. For example, +
can be
either a prefix or an infix operator. Typically, a visual renderer
would add space around both sides of an infix operator, while only in
front of a prefix operator. The form
attribute allows
specification of which form to use, in case more than one form is
possible according to the operator dictionary and the default value
described below is not suitable.
The form
attribute does not usually have to be
specified explicitly, since there are effective heuristic rules for
inferring the value of the form
attribute from the
context. If it is not specified, and there is more than one possible
form in the dictionary for an mo
element with
given content, the renderer should choose which form to use as follows
(but see the exception for embellished operators, described later):
If the operator is the first argument in an mrow
with more than one argument
(ignoring all spacelike arguments (see 3.2.7 Space <mspace/>
) in the
determination of both the length and the first argument), the prefix form
is used;
if it is the last argument in an mrow
with more than one argument
(ignoring all spacelike arguments), the postfix
form is used;
if it is the only element in an implicit or explicit mrow
and if it is in a script position of one of the elements listed in 3.4 Script and Limit Schemata,
the postfix form is used;
in all other cases, including when the operator is not part of an
mrow
, the infix form is used.
Note that the mrow
discussed above may be inferred;
see 3.1.3.1 Inferred <mrow>
s.
Opening fences should have form
="prefix"
,
and closing fences should have form
="postfix"
;
separators are usually infix
, but not always,
depending on their surroundings. As with ordinary operators,
these values do not usually need to be specified explicitly.
If the operator does not occur in the dictionary with the specified
form, the renderer should use one of the forms that is available
there, in the order of preference: infix, postfix, prefix; if no forms
are available for the given mo
element content, the
renderer should use the defaults given in parentheses in the table of
attributes for mo
.
There is one exception to the above rules for choosing an mo
element's default form
attribute. An mo
element that is
embellished
by one or more nested subscripts, superscripts,
surrounding text or whitespace, or style changes behaves differently. It is
the embellished operator as a whole (this is defined precisely, below)
whose position in an mrow
is examined by the above
rules and whose surrounding spacing is affected by its form, not the mo
element at its core; however, the attributes
influencing this surrounding spacing are taken from the mo
element at the core (or from that element's
dictionary entry).
For example, the ${+}_{4}$
in
$a{+}_{4}b$
should be considered an infix operator as a whole, due to its position
in the middle of an mrow
, but its rendering
attributes should be taken from the mo
element
representing the
$+$
,
or when those are not specified explicitly,
from the operator dictionary entry for <mo form="infix"> +
</mo>
.
The precise definition of an embellished operator
is:
an mo
element;
or one of the elements
msub
,
msup
,
msubsup
,
munder
,
mover
,
munderover
,
mmultiscripts
,
mfrac
, or
semantics
(6.5 The <semantics>
element), whose first argument exists and is an embellished
operator;
or one of the elements
mstyle
,
mphantom
, or
mpadded
,
such that an mrow
containing the same
arguments would be an embellished operator;
or an maction
element whose selected
subexpression exists and is an embellished operator;
or an mrow
whose arguments consist (in any order)
of one embellished operator and zero or more spacelike elements.
Note that this definition permits nested embellishment only when there are no intervening enclosing elements not in the above list.
The above rules for choosing operator forms and defining
embellished operators are chosen so that in all ordinary cases it will
not be necessary for the author to specify a form
attribute.
The amount of horizontal space added around an operator (or embellished operator),
when it occurs in an mrow
, can be directly
specified by the lspace
and rspace
attributes. Note that lspace
and rspace
should
be interpreted as leading and trailing space, in the case of RTL direction.
By convention, operators that tend to bind tightly to their
arguments have smaller values for spacing than operators that tend to bind
less tightly. This convention should be followed in the operator dictionary
included with a MathML renderer.
Some renderers may choose to use no space around most operators appearing within subscripts or superscripts, as is done in TeX.
Nongraphical renderers should treat spacing attributes, and other rendering attributes described here, in analogous ways for their rendering medium. For example, more space might translate into a longer pause in an audio rendering.
Four attributes govern whether and how an operator (perhaps embellished)
stretches so that it matches the size of other elements: stretchy
, symmetric
, maxsize
, and minsize
. If an
operator has the attribute stretchy
=true
, then it (that is, each character in its content)
obeys the stretching rules listed below, given the constraints imposed by
the fonts and font rendering system. In practice, typical renderers will
only be able to stretch a small set of characters, and quite possibly will
only be able to generate a discrete set of character sizes.
There is no provision in MathML for specifying in which direction
(horizontal or vertical) to stretch a specific character or operator;
rather, when stretchy
=true
it
should be stretched in each direction for which stretching is possible
and reasonable for that character.
It is up to the renderer to know in which directions it is reasonable to
stretch a character, if it can stretch the character.
Most characters can be stretched in at most one direction
by typical renderers, but some renderers may be able to stretch certain
characters, such as diagonal arrows, in both directions independently.
The minsize
and maxsize
attributes limit the amount of stretching (in either direction). These two
attributes are given as multipliers of the operator's normal size in the
direction or directions of stretching, or as absolute sizes using units.
For example, if a character has maxsize
=300%
, then it
can grow to be no more than three times its normal (unstretched) size.
The symmetric
attribute governs whether the
height and
depth above and below the axis of the
character are forced to be equal
(by forcing both height and depth to become the maximum of the two).
An example of a situation where one might set
symmetric
=false
arises with parentheses around a matrix not aligned on the axis, which
frequently occurs when multiplying nonsquare matrices. In this case, one
wants the parentheses to stretch to cover the matrix, whereas stretching
the parentheses symmetrically would cause them to protrude beyond one edge
of the matrix. The symmetric
attribute only applies
to characters that stretch vertically (otherwise it is ignored).
If a stretchy mo
element is embellished (as defined
earlier in this section), the mo
element at its core is
stretched to a size based on the context of the embellished operator
as a whole, i.e. to the same size as if the embellishments were not
present. For example, the parentheses in the following example (which
would typically be set to be stretchy by the operator dictionary) will be
stretched to the same size as each other, and the same size they would
have if they were not underlined and overlined, and furthermore will
cover the same vertical interval:
<mrow>
<munder>
<mo> ( </mo>
<mo> _ </mo>
</munder>
<mfrac>
<mi> a </mi>
<mi> b </mi>
</mfrac>
<mover>
<mo> ) </mo>
<mo> ‾ </mo>
</mover>
</mrow>
Note that this means that the stretching rules given below must
refer to the context of the embellished operator as a whole, not just
to the mo
element itself.
This shows one way to set the maximum size of a parenthesis so that
it does not grow, even though its default value is
stretchy
=true
.
<mrow>
<mo maxsize="100%">(</mo>
<mfrac>
<msup><mi>a</mi><mn>2</mn></msup>
<msup><mi>b</mi><mn>2</mn></msup>
</mfrac>
<mo maxsize="100%">)</mo>
</mrow>
The above should render as as opposed to the default rendering .
Note that each parenthesis is sized independently; if only one of
them had maxsize
=100%
, they would render with different
sizes.
The general rules governing stretchy operators are:
If a stretchy operator is a direct subexpression of an mrow
element, or is the sole direct subexpression of an
mtd
element in some row of a table, then it should
stretch to cover the height and depth (above and below the axis) of the nonstretchy direct subexpressions in the
mrow
element or table row, unless stretching is
constrained by minsize
or maxsize
attributes.
In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
The preceding rules also apply in situations where the mrow
element is inferred.
The rules for symmetric stretching only apply if
symmetric
=true
and if the stretching occurs in an mrow
or in an mtr
whose rowalign
value is either baseline
or axis
.
The following algorithm specifies the height and depth of vertically stretched characters:
Let maxheight
and maxdepth
be the maximum height and depth of the
nonstretchy siblings within the same mrow
or mtr
.
Let axis be the height of the math axis above the baseline.
Note that even if a minsize
or maxsize
value is set on a stretchy operator,
it is not used in the initial calculation of the maximum height and depth of an mrow
.
If symmetric
=true
, then the computed height
and depth of the stretchy operator are:
height=max(maxheightaxis, maxdepth+axis) + axis
depth =max(maxheightaxis, maxdepth+axis)  axis
Otherwise the height and depth are:
height= maxheight
depth = maxdepth
If the total size = height+depth is less than minsize or greater than maxsize, increase or decrease both height and depth proportionately so that the effective size meets the constraint.
By default, most vertical arrows, along with most opening and closing fences are defined in the operator dictionary to stretch by default.
In the case of a stretchy operator in a table cell (i.e. within an
mtd
element), the above rules assume each cell of
the table row containing the stretchy operator covers exactly one row.
(Equivalently, the value of the rowspan
attribute is
assumed to be 1 for all the table cells in the table row, including
the cell containing the operator.) When this is not the case, the
operator should only be stretched vertically to cover those table
cells that are entirely within the set of table rows that the
operator's cell covers. Table cells that extend into rows not covered
by the stretchy operator's table cell should be ignored. See
3.5.3.2 Attributes for details about the rowspan
attribute.
If a stretchy operator, or an embellished stretchy operator,
is a direct subexpression of an munder
,
mover
, or munderover
element,
or if it is the sole direct subexpression of an mtd
element in some
column of a table (see mtable
), then it, or the mo
element at its core, should stretch to cover
the width of the other direct subexpressions in the given element (or
in the same table column), given the constraints mentioned above.
In the case of an embellished stretchy operator, the preceding rule applies to the stretchy operator at its core.
By default, most horizontal arrows and some accents stretch horizontally.
In the case of a stretchy operator in a table cell (i.e. within an
mtd
element), the above rules assume each cell of
the table column containing the stretchy operator covers exactly one
column. (Equivalently, the value of the columnspan
attribute is assumed to be 1 for all the table cells in the table row,
including the cell containing the operator.) When this is not the
case, the operator should only be stretched horizontally to cover
those table cells that are entirely within the set of table columns
that the operator's cell covers. Table cells that extend into columns
not covered by the stretchy operator's table cell should be
ignored. See 3.5.3.2 Attributes for details about the rowspan
attribute.
The rules for horizontal stretching include mtd
elements to allow arrows to stretch for use in commutative diagrams
laid out using mtable
. The rules for the horizontal
stretchiness include scripts to make examples such as the following
work:
<mrow>
<mi> x </mi>
<munder>
<mo> → </mo>
<mtext> maps to </mtext>
</munder>
<mi> y </mi>
</mrow>
If a stretchy operator is not required to stretch (i.e. if it is
not in one of the locations mentioned above, or if there are no other
expressions whose size it should stretch to match), then it has the
standard (unstretched) size determined by the font and current
mathsize
.
If a stretchy operator is required to stretch, but all other expressions
in the containing element (as described above) are also stretchy,
all elements that can stretch should grow to the maximum of the normal
unstretched sizes of all elements in the containing object, if they can
grow that large. If the value of minsize
or maxsize
prevents
that, then the specified (min or max) size is
used.
For example, in an mrow
containing nothing but
vertically stretchy operators, each of the operators should stretch to
the maximum of all of their normal unstretched sizes, provided no
other attributes are set that override this behavior. Of course,
limitations in fonts or font rendering may result in the final,
stretched sizes being only approximately the same.
<mtext>
An mtext
element is used to represent
arbitrary text that should be rendered as itself. In general, the
mtext
element is intended to denote commentary
text.
Note that text with a clearly defined notational role might be more appropriately marked up using mi or mo.
An mtext
element can also contain
renderable whitespace
, i.e. invisible characters that are
intended to alter the positioning of surrounding elements. In nongraphical
media, such characters are intended to have an analogous effect, such as
introducing positive or negative time delays or affecting rhythm in an
audio renderer. However, see 2.1.7 Collapsing Whitespace in Input.
mtext
elements accept the attributes listed in
3.2.2 Mathematics style attributes common to token elements.
See also the warnings about the legal grouping of spacelike elements
in 3.2.7 Space <mspace/>
, and about the use of
such elements for tweaking
in [MathMLNotes].
<mrow>
<mtext> Theorem 1: </mtext>
<mtext>  <!ThinSpace> </mtext>
<mtext>  <!ThickSpace> <!ThickSpace> </mtext>
<mtext> /* a comment */ </mtext>
</mrow>
<mspace/>
An mspace
empty element represents a blank
space of any desired size, as set by its attributes. It can also be
used to make linebreaking suggestions to a visual renderer.
Note that the default values for attributes have been chosen so that
they typically will have no effect on rendering. Thus, the mspace
element is generally used with one
or more attribute values explicitly specified.
Note the warning about the legal grouping of spacelike
elements
given below, and the warning about the use of such
elements for tweaking
in [MathMLNotes].
See also the other elements that can render as
whitespace, namely mtext
, mphantom
, and
maligngroup
.
In addition to the attributes listed below,
mspace
elements accept the attributes described in 3.2.2 Mathematics style attributes common to token elements,
but note that mathvariant
and mathcolor
have no effect and that
mathsize
only affects the interpretation of units in sizing
attributes (see 2.1.5.2 Length Valued Attributes).
mspace
also accepts the indentation attributes described in 3.2.5.2.3 Indentation attributes.
Name  values  default 
width  length  0em 
Specifies the desired width of the space.  
height  length  0ex 
Specifies the desired height (above the baseline) of the space.  
depth  length  0ex 
Specifies the desired depth (below the baseline) of the space. 
Linebreaking was originally specified on mspace
in MathML2,
but much greater control over linebreaking and indentation was add to mo
in MathML 3. Linebreaking on mspace
is deprecated in MathML 4.
<mspace height="3ex" depth="2ex"/>
A number of MathML presentation elements are spacelike
in the
sense that they typically render as whitespace, and do not affect the
mathematical meaning of the expressions in which they appear. As a
consequence, these elements often function in somewhat exceptional
ways in other MathML expressions. For example, spacelike elements are
handled specially in the suggested rendering rules for
mo
given in 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
The following MathML elements are defined to be spacelike
:
an mtext
, mspace
,
maligngroup
, or malignmark
element;
an mstyle
, mphantom
, or
mpadded
element, all of whose direct subexpressions
are spacelike;
a semantics
element whose first argument exists
and is spacelike;
an maction
element whose selected
subexpression exists and is spacelike;
an mrow
all of whose direct
subexpressions are spacelike.
Note that an mphantom
is not
automatically defined to be spacelike, unless its content is
spacelike. This is because operator spacing is affected by whether
adjacent elements are spacelike. Since the
mphantom
element is primarily intended as an aid
in aligning expressions, operators adjacent to an
mphantom
should behave as if they were adjacent
to the contents of the mphantom
,
rather than to an equivalently sized area of whitespace.
Authors who insert spacelike elements or
mphantom
elements into an existing MathML
expression should note that such elements are counted as
arguments, in elements that require a specific number of arguments,
or that interpret different argument positions differently.
Therefore, spacelike elements inserted into such a MathML element
should be grouped with a neighboring argument of that element by
introducing an mrow
for that purpose. For example,
to allow for vertical alignment on the right edge of the base of a
superscript, the expression
<msup>
<mi> x </mi>
<malignmark edge="right"/>
<mn> 2 </mn>
</msup>
is illegal, because msup
must have exactly 2 arguments;
the correct expression would be:
<msup>
<mrow>
<mi> x </mi>
<malignmark edge="right"/>
</mrow>
<mn> 2 </mn>
</msup>
See also the warning about tweaking
in
[MathMLNotes].
<ms>
The ms
element is used to represent
string literals
in expressions meant to be interpreted by
computer algebra systems or other systems containing programming
languages
. By default, string literals are displayed surrounded by
double quotes, with no extra spacing added around the string.
As explained in 3.2.6 Text <mtext>
, ordinary text
embedded in a mathematical expression should be marked up with mtext
,
or in some cases mo
or mi
, but never with ms
.
Note that the string literals encoded by ms
are made up of characters, mglyph
s and
malignmark
s rather than ASCII
strings
. For
example, <ms>&</ms>
represents a string
literal containing a single character, &
, and
<ms>&amp;</ms>
represents a string literal
containing 5 characters, the first one of which is
&
.
The content of ms
elements should be rendered with visible
escaping
of certain characters in the content,
including at least the left and right quoting
characters, and preferably whitespace other than individual
space characters. The intent is for the viewer to see that the
expression is a string literal, and to see exactly which characters
form its content. For example, <ms>double quote is
"</ms>
might be rendered as "double quote is \"".
Like all token elements, ms
does trim and
collapse whitespace in its content according to the rules of
2.1.7 Collapsing Whitespace in Input, so whitespace intended to remain in
the content should be encoded as described in that section.
ms
elements accept the attributes listed in
3.2.2 Mathematics style attributes common to token elements, and additionally:
Name  values  default 
lquote  string  U+0022 (entity quot ) 
Specifies the opening quote to enclose the content (not necessarily ‘left quote’ in RTL context).  
rquote  string  U+0022 (entity quot ) 
Specifies the closing quote to enclose the content (not necessarily ‘right quote’ in RTL context). 
Besides tokens there are several families of MathML presentation
elements. One family of elements deals with various
scripting
notations, such as subscript and
superscript. Another family is concerned with matrices and tables. The
remainder of the elements, discussed in this section, describe other basic
notations such as fractions and radicals, or deal with general functions
such as setting style properties and error handling.
<mrow>
An mrow
element is used to group together any
number of subexpressions, usually consisting of one or more mo
elements acting as operators
on one
or more other expressions that are their operands
.
Several elements automatically treat their arguments as if they were
contained in an mrow
element. See the discussion of
inferred mrow
s in 3.1.3 Required Arguments.
See also mfenced
(3.3.8 Expression Inside Pair of Fences
<mfenced>
),
which can effectively form an mrow
containing its arguments separated by commas.
mrow
elements are typically rendered visually
as a horizontal row of their arguments, left to right in the order in
which the arguments occur within a context with LTR directionality,
or right to left within a context with RTL directionality.
The dir
attribute can be used to specify
the directionality for a specific mrow
, otherwise it inherits the
directionality from the context. For aural agents, the arguments would be
rendered audibly as a sequence of renderings of
the arguments. The description in 3.2.5 Operator, Fence, Separator or Accent
<mo>
of suggested rendering
rules for mo
elements assumes that all horizontal
spacing between operators and their operands is added by the rendering
of mo
elements (or, more generally, embellished
operators), not by the rendering of the mrow
s
they are contained in.
MathML provides support for both automatic and manual
linebreaking of expressions (that is, to break excessively long
expressions into several lines). All such linebreaks take place
within mrow
s, whether they are explicitly marked up
in the document, or inferred (see 3.1.3.1 Inferred <mrow>
s),
although the control of linebreaking is effected through attributes
on other elements (see 3.1.7 Linebreaking of Expressions).
mrow
elements accept the attribute listed below in addition to
those listed in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
dir  "ltr"  "rtl"  inherited 
specifies the overall directionality ltr (Left To Right) or
rtl (Right To Left) to use to layout the children of the row.
See 3.1.5.1 Overall Directionality of Mathematics Formulas for further discussion.

Subexpressions should be grouped by the document author in the same way
as they are grouped in the mathematical interpretation of the expression;
that is, according to the underlying syntax tree
of the
expression. Specifically, operators and their mathematical arguments should
occur in a single mrow
; more than one operator
should occur directly in one mrow
only when they
can be considered (in a syntactic sense) to act together on the interleaved
arguments, e.g. for a single parenthesized term and its parentheses, for
chains of relational operators, or for sequences of terms separated by
+
and 
. A precise rule is given below.
Proper grouping has several purposes: it improves display by possibly affecting spacing; it allows for more intelligent linebreaking and indentation; and it simplifies possible semantic interpretation of presentation elements by computer algebra systems, and audio renderers.
Although improper grouping will sometimes result in suboptimal
renderings, and will often make interpretation other than pure visual
rendering difficult or impossible, any grouping of expressions using
mrow
is allowed in MathML syntax; that is,
renderers should not assume the rules for proper grouping will be
followed.
MathML renderers are required to treat an mrow
element containing exactly one argument as equivalent in all ways to
the single argument occurring alone, provided there are no attributes
on the mrow
element. If there are
attributes on the mrow
element, no
requirement of equivalence is imposed. This equivalence condition is
intended to simplify the implementation of MathMLgenerating software
such as templatebased authoring tools. It directly affects the
definitions of embellished operator and spacelike element and the
rules for determining the default value of the form
attribute of an mo
element;
see 3.2.5 Operator, Fence, Separator or Accent
<mo>
and 3.2.7 Space <mspace/>
. See also the discussion of equivalence of MathML
expressions in D.1 MathML Conformance.
A precise rule for when and how to nest subexpressions using
mrow
is especially desirable when generating
MathML automatically by conversion from other formats for displayed
mathematics, such as TeX, which don't always specify how subexpressions
nest. When a precise rule for grouping is desired, the following rule
should be used:
Two adjacent operators, possibly embellished, possibly separated by operands (i.e. anything
other than operators), should occur in the same
mrow
only when the leading operator has an infix or
prefix form (perhaps inferred), the following operator has an infix or
postfix form, and the operators have the same priority in the
operator dictionary (B. Operator Dictionary).
In all other cases, nested mrow
s should be used.
When forming a nested mrow
(during generation
of MathML) that includes just one of two successive operators with
the forms mentioned above (which means that either operator could in
principle act on the intervening operand or operands), it is necessary
to decide which operator acts on those operands directly (or would do
so, if they were present). Ideally, this should be determined from the
original expression; for example, in conversion from an
operatorprecedencebased format, it would be the operator with the
higher precedence.
Note that the above rule has no effect on whether any MathML expression is valid, only on the recommended way of generating MathML from other formats for displayed mathematics or directly from written notation.
(Some of the terminology used in stating the above rule is defined
in 3.2.5 Operator, Fence, Separator or Accent
<mo>
.)
As an example, 2x+yz should be written as:
<mrow>
<mrow>
<mn> 2 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<mi> x </mi>
</mrow>
<mo> + </mo>
<mi> y </mi>
<mo>  </mo>
<mi> z </mi>
</mrow>
The proper encoding of (x, y) furnishes a less obvious
example of nesting mrow
s:
<mrow>
<mo> ( </mo>
<mrow>
<mi> x </mi>
<mo> , </mo>
<mi> y </mi>
</mrow>
<mo> ) </mo>
</mrow>
In this case, a nested mrow
is required inside
the parentheses, since parentheses and commas, thought of as fence and
separator operators
, do not act together on their arguments.
<mfrac>
The mfrac
element is used for fractions. It can
also be used to mark up fractionlike objects such as binomial coefficients
and Legendre symbols. The syntax for mfrac
is
<mfrac> numerator denominator </mfrac>
The mfrac
element sets displaystyle
to false
, or if it
was already false increments scriptlevel
by 1,
within numerator and denominator.
(See 3.1.6 Displaystyle and Scriptlevel.)
mfrac
elements accept the attributes listed below
in addition to those listed in 3.1.9 Mathematics attributes common to presentation elements.
The fraction line, if any, should be drawn using the color specified by mathcolor
.
Name  values  default 
linethickness  length  "thin"  "medium"  "thick"  medium 
Specifies the thickness of the horizontal fraction bar, or rule. The default value is medium ;
thin is thinner, but visible;
thick is thicker.
The exact thickness of these is left up to the rendering agent.
However, if OpenType Math fonts are available then the renderer should set medium to
the value MATH.MathConstants.fractionRuleThickness
(the default in MathMLCore).
Note: MathML Core does only allow <lengthpercentage> values.


numalign  "left"  "center"  "right"  center 
Specifies the alignment of the numerator over the fraction.  
denomalign  "left"  "center"  "right"  center 
Specifies the alignment of the denominator under the fraction.  
bevelled  "true"  "false"  false 
Specifies whether the fraction should be displayed in a bevelled style (the numerator slightly raised, the denominator slightly lowered and both separated by a slash), rather than "build up" vertically. See below for an example. 
Thicker lines (e.g. linethickness
="thick") might be used with nested fractions;
a value of "0" renders without the bar such as for binomial coefficients.
In a RTL directionality context, the numerator leads (on the right),
the denominator follows (on the left) and the diagonal line slants upwards going from
right to left (see 3.1.5.1 Overall Directionality of Mathematics Formulas for clarification).
Although this format is an established convention, it is not universally
followed; for situations where a forward slash is desired in a RTL context,
alternative markup, such as an mo
within an mrow
should be used.
Here is an example which makes use of different values of linethickness
:
<mfrac linethickness="3px">
<mrow>
<mo> ( </mo>
<mfrac linethickness="0">
<mi> a </mi>
<mi> b </mi>
</mfrac>
<mo> ) </mo>
<mfrac>
<mi> a </mi>
<mi> b </mi>
</mfrac>
</mrow>
<mfrac>
<mi> c </mi>
<mi> d </mi>
</mfrac>
</mfrac>
This example illustrates bevelled fractions:
<mfrac>
<mn> 1 </mn>
<mrow>
<msup>
<mi> x </mi>
<mn> 3 </mn>
</msup>
<mo> + </mo>
<mfrac>
<mi> x </mi>
<mn> 3 </mn>
</mfrac>
</mrow>
</mfrac>
<mo> = </mo>
<mfrac bevelled="true">
<mn> 1 </mn>
<mrow>
<msup>
<mi> x </mi>
<mn> 3 </mn>
</msup>
<mo> + </mo>
<mfrac>
<mi> x </mi>
<mn> 3 </mn>
</mfrac>
</mrow>
</mfrac>
A more generic example is:
<mfrac>
<mrow>
<mn> 1 </mn>
<mo> + </mo>
<msqrt>
<mn> 5 </mn>
</msqrt>
</mrow>
<mn> 2 </mn>
</mfrac>
These elements construct radicals. The msqrt
element is
used for square roots, while the mroot
element is used
to draw radicals with indices, e.g. a cube root. The syntax for these
elements is:
<msqrt> base </msqrt>
<mroot> base index </mroot>
The mroot
element requires exactly 2 arguments.
However, msqrt
accepts a single argument, possibly
being an inferred mrow
of multiple children; see 3.1.3 Required Arguments.
The mroot
element increments scriptlevel
by 2,
and sets displaystyle
to false
, within
index, but leaves both attributes unchanged within base.
The msqrt
element leaves both
attributes unchanged within its argument.
(See 3.1.6 Displaystyle and Scriptlevel.)
Note that in a RTL directionality, the surd begins
on the right, rather than the left, along with the index in the case
of mroot
.
msqrt
and mroot
elements accept the attributes listed in
3.1.9 Mathematics attributes common to presentation elements. The surd and overbar should be drawn using the
color specified by mathcolor
.
Square roots and cube roots
<mrow>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
<mroot>
<mi>x</mi>
<mn>3</mn>
</mroot>
<mrow>
<mo>=</mo>
<msup>
<mi>x</mi>
<mrow>
<mrow>
<mn>1</mn>
<mo>/</mo>
<mn>2</mn>
</mrow>
<mo>+</mo>
<mrow>
<mn>1</mn>
<mo>/</mo>
<mn>3</mn>
</mrow>
</mrow>
</msup>
</mrow>
<mstyle>
The mstyle
element is used to make style
changes that affect the rendering of its
contents.
As a presentation element, it accepts
the attributes described in 3.1.9 Mathematics attributes common to presentation elements.
Additionally, it
can be given any attribute
accepted by any other presentation element, except for the
attributes described below.
Finally,
the mstyle
element can be given certain special
attributes listed in the next subsection.
The mstyle
element accepts a single argument,
possibly being an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
Loosely speaking, the effect of the mstyle
element
is to change the default value of an attribute for the elements it
contains. Style changes work in one of several ways, depending on
the way in which default values are specified for an attribute.
The cases are:
Some attributes, such as displaystyle
or
scriptlevel
(explained below), are inherited
from the surrounding context when they are not explicitly set. Specifying
such an attribute on an mstyle
element sets the
value that will be inherited by its child elements. Unless a child element
overrides this inherited value, it will pass it on to its children, and
they will pass it to their children, and so on. But if a child element does
override it, either by an explicit attribute setting or automatically (as
is common for scriptlevel
), the new (overriding)
value will be passed on to that element's children, and then to their
children, etc, unless it is again overridden.
Other attributes, such as linethickness
on
mfrac
, have default values that are not normally
inherited. That is, if the linethickness
attribute
is not set on the mfrac
element,
it will normally use the default value of medium
, even if it was
contained in a larger mfrac
element that set this
attribute to a different value. For attributes like this, specifying a
value with an mstyle
element has the effect of
changing the default value for all elements within its scope. The net
effect is that setting the attribute value with mstyle
propagates the change to all the elements it
contains directly or indirectly, except for the individual elements on
which the value is overridden. Unlike in the case of inherited attributes,
elements that explicitly override this attribute have no effect on this
attribute's value in their children.
Another group of attributes, such as stretchy
and form
, are
computed from operator dictionary information, position in the
enclosing mrow
, and other similar data. For
these attributes, a value specified by an enclosing mstyle
overrides the value that would normally be
computed.
Note that attribute values inherited from an
mstyle
in any manner affect a descendant element
in the mstyle
's content only if that attribute is
not given a value by the descendant element. On any element for
which the attribute is set explicitly, the value specified overrides the inherited
value. The only exception to this
rule is when the attribute value
is documented as
specifying an incremental change to the value inherited from that
element's context or rendering environment.
Note also that the difference between inherited and noninherited
attributes set by mstyle
, explained above, only
matters when the attribute is set on some element within the
mstyle
's contents that has descendants also
setting it. Thus it never matters for attributes, such as
mathsize
, which can only be set on token elements (or on
mstyle
itself).
MathML specifies that when
the attributes height
, depth
or width
are specified on an mstyle
element, they apply only to
mspace
elements, and not to the corresponding attributes of
mglyph
, mpadded
, or mtable
. Similarly, when
rowalign
or columnalign
are specified on an mstyle
element, they apply only to the
mtable
element, and not the mtr
,
mtd
, and maligngroup
elements.
When the lspace
attribute is set with mstyle
, it
applies only to the mo
element and not to mpadded
.
To be consistent, the voffset
attribute of the
mpadded
element can not be set on mstyle
.
When the align
attribute is set with mstyle
, it
applies only to the munder
, mover
, and munderover
elements, and not to the mtable
and mstack
elements.
The attributes src
and alt
on mglyph
,
and actiontype
on maction
, cannot be set on mstyle
.
As a presentation element, mstyle
directly accepts
the mathcolor
and mathbackground
attributes.
Thus, the mathbackground
specifies the color to fill the bounding
box of the mstyle
element itself; it does not
specify the default background color.
This is an incompatible change from MathML 2, but it is more useful
and intuitive. Since the default for mathcolor
is inherited,
this is no change in its behavior.
As stated above, mstyle
accepts all
attributes of all MathML presentation elements which do not have
required values. That is, all attributes which have an explicit
default value or a default value which is inherited or computed are
accepted by the mstyle
element.
This group of attributes is not accepted in MathML Core.
mstyle
elements accept the attributes listed in
3.1.9 Mathematics attributes common to presentation elements.
Additionally, mstyle
can be given the following special
attributes that are implicitly inherited by every MathML element as
part of its rendering environment:
Name  values  default 
scriptlevel  ( "+"  "" )? unsignedinteger  inherited 
Changes the scriptlevel in effect for the children.
When the value is given without a sign, it sets scriptlevel to the specified value;
when a sign is given, it increments ("+") or decrements ("") the current value.
(Note that large decrements can result in negative values of scriptlevel ,
but these values are considered legal.)
See 3.1.6 Displaystyle and Scriptlevel.


displaystyle  "true"  "false"  inherited 
Changes the displaystyle in effect for the children.
See 3.1.6 Displaystyle and Scriptlevel.


scriptsizemultiplier  number  0.71 
Specifies the multiplier to be used to adjust font size due
to changes in scriptlevel .
See 3.1.6 Displaystyle and Scriptlevel.


scriptminsize  length  8pt 
Specifies the minimum font size allowed due to changes in scriptlevel .
Note that this does not limit the font size due to changes to mathsize .
See 3.1.6 Displaystyle and Scriptlevel.


infixlinebreakstyle  "before"  "after"  "duplicate"  before 
Specifies the default linebreakstyle to use for infix operators; see 3.2.5.2.2 Linebreaking attributes  
decimalpoint  character  . 
Specifies the character used to determine the alignment point within
mstack
and
mtable columns
when the "decimalpoint" value is used to specify the alignment.
The default, ".", is the decimal separator used to separate the integral
and decimal fractional parts of floating point numbers in many countries.
(See 3.6 Elementary Math and 3.5.4 Alignment Markers
<maligngroup/> , <malignmark/> ).

If scriptlevel
is changed incrementally by an
mstyle
element that also sets certain other
attributes, the overall effect of the changes may depend on the order
in which they are processed. In such cases, the attributes in the
following list should be processed in the following order, regardless
of the order in which they occur in the XMLformat attribute list of
the mstyle
start tag:
scriptsizemultiplier
, scriptminsize
,
scriptlevel
, mathsize
.
In a continued fraction, the nested fractions should not shrink. Instead, they should remain the same size.
This can be accomplished by resetting displaystyle
and
scriptlevel
for the children of each mfrac
using mstyle
as shown below:
<mrow>
<mi>π</mi>
<mo>=</mo>
<mfrac>
<mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle>
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mstyle displaystyle="true" scriptlevel="0">
<msup> <mn>1</mn> <mn>2</mn> </msup>
</mstyle>
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>+</mo>
<mfrac>
<mstyle displaystyle="true" scriptlevel="0">
<msup> <mn>3</mn> <mn>2</mn> </msup>
</mstyle>
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>+</mo>
<mfrac>
<mstyle displaystyle="true" scriptlevel="0">
<msup> <mn>5</mn> <mn>2</mn> </msup>
</mstyle>
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>+</mo>
<mfrac>
<mstyle displaystyle="true" scriptlevel="0">
<msup> <mn>7</mn> <mn>2</mn> </msup>
</mstyle>
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo>+</mo>
<mo>⋱</mo>
</mstyle>
</mfrac>
</mstyle>
</mfrac>
</mstyle>
</mfrac>
</mstyle>
</mfrac>
</mstyle>
</mfrac>
</mrow>
<merror>
The merror
element displays its contents as an
error message
. This might be done, for example, by displaying the
contents in red, flashing the contents, or changing the background
color. The contents can be any expression or expression sequence.
merror
accepts
a single argument possibly being an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
The intent of this element is to provide a standard way for programs that generate MathML from other input to report syntax errors in their input. Since it is anticipated that preprocessors that parse input syntaxes designed for easy hand entry will be developed to generate MathML, it is important that they have the ability to indicate that a syntax error occurred at a certain point. See D.2 Handling of Errors.
The suggested use of merror
for reporting
syntax errors is for a preprocessor to replace the erroneous part of
its input with an merror
element containing a
description of the error, while processing the surrounding expressions
normally as far as possible. By this means, the error message will be
rendered where the erroneous input would have appeared, had it been
correct; this makes it easier for an author to determine from the
rendered output what portion of the input was in error.
No specific error message format is suggested here, but as with
error messages from any program, the format should be designed to make
as clear as possible (to a human viewer of the rendered error message)
what was wrong with the input and how it can be fixed. If the
erroneous input contains correctly formatted subsections, it may be
useful for these to be preprocessed normally and included in the error
message (within the contents of the merror
element), taking advantage of the ability of
merror
to contain arbitrary MathML expressions
rather than only text.
merror
elements accept the attributes listed in
3.1.9 Mathematics attributes common to presentation elements.
If a MathML syntaxchecking preprocessor received the input
<mfraction>
<mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow>
<mn> 2 </mn>
</mfraction>
which contains the nonMathML element mfraction
(presumably in place of the MathML element mfrac
),
it might generate the error message
<merror>
<mtext> Unrecognized element: mfraction; arguments were: </mtext>
<mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow>
<mtext> and </mtext>
<mn> 2 </mn>
</merror>
Note that the preprocessor's input is not, in this case, valid MathML, but the error message it outputs is valid MathML.
<mpadded>
An mpadded
element renders the same as its child content,
but with the size of the child's bounding box and the relative positioning
point of its content modified according to
mpadded
's attributes. It
does not rescale (stretch or shrink) its content. The name of the
element reflects the typical use of mpadded
to add padding,
or extra space, around its content. However, mpadded
can be
used to make more general adjustments of size and positioning, and some
combinations, e.g. negative padding, can cause the content of
mpadded
to overlap the rendering of neighboring content. See
[MathMLNotes] for warnings about several
potential pitfalls of this effect.
The mpadded
element accepts
a single argument which may be an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
It is suggested that audio renderers add (or shorten) time delays
based on the attributes representing horizontal space
(width
and lspace
).
mpadded
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
height  ( "+"  "" )? unsignednumber ( ("%" pseudounit?)  pseudounit  unit  namedspace )?  same as content 
Sets or increments the height of the mpadded element.
See below for discussion.


depth  ( "+"  "" )? unsignednumber (("%" pseudounit?)  pseudounit  unit  namedspace )?  same as content 
Sets or increments the depth of the mpadded element.
See below for discussion.


width  ( "+"  "" )? unsignednumber ( ("%" pseudounit?)  pseudounit  unit  namedspace )?  same as content 
Sets or increments the width of the mpadded element.
See below for discussion.


lspace  ( "+"  "" )? unsignednumber ( ("%" pseudounit?)  pseudounit  unit  namedspace )?  0em 
Sets the horizontal position of the child content. See below for discussion.  
voffset  ( "+"  "" )? unsignednumber ( ("%" pseudounit?)  pseudounit  unit  namedspace )?  0em 
Sets the vertical position of the child content. See below for discussion. 
Note: while [MathMLCore] supports the above attributes, it only allows the value to be a valid
<lengthpercentage>
.
Increments with the optional "+" or "" signs are not supported in MathML Core nor are pseudounits.
The pseudounit syntax symbol is described below.
Also, height
, depth
and
width
attributes are
referred to as size attributes, while lspace
and voffset
attributes
are position attributes.
These attributes specify the size of the bounding box of the mpadded
element relative to the size of the bounding box of its child content, and specify
the position of the child content of the mpadded
element relative to the
natural positioning of the mpadded
element. The typographical
layout parameters determined by these attributes are described in the next subsection.
Depending on the form of the attribute value, a dimension may be set to a new value,
or specified relative to the child content's corresponding dimension. Values may
be given as
multiples or percentages of any of the
dimensions of the normal rendering of the child content using socalled pseudounits,
or they can be set directly using standard units, see 2.1.5.2 Length Valued Attributes.
If the value of a size attribute begins with a +
or 
sign,
it specifies an increment or decrement to the corresponding
dimension by the following length value. Otherwise the corresponding
dimension is set directly to the following length value.
Note that since a leading minus sign indicates a decrement, the size
attributes (height
, depth
, width
)
cannot be set directly to negative values. In addition, specifying a
decrement that would produce a net negative value for these attributes
has the same effect as
setting the attribute to zero. In other words, the effective
bounding box of an mpadded
element always has nonnegative
dimensions. However, negative values are allowed for the relative positioning
attributes lspace
and voffset
.
Length values (excluding any sign) can be specified in several formats.
Each format begins with an unsignednumber,
which may be followed by
a %
sign (effectively scaling the number)
and an optional pseudounit,
by a pseudounit alone,
or by a unit (excepting %
).
The possible pseudounits are the keywords height
,
depth
, and width
. They represent the length of the samenamed dimension of the
mpadded
element's child content.
For any of these length formats, the resulting length
is the product of the number (possibly including the %
)
and the following pseudounit,
unit,
namedspace
or the default value for the attribute if no such unit or space is given.
Some examples of attribute formats using pseudounits (explicit or
default) are as follows: depth="100%height"
and
depth="1.0height"
both set the depth of the
mpadded
element to the height of its content.
depth="105%"
sets the depth to 1.05 times the content's
depth, and either depth="+100%"
or
depth="200%"
sets the depth to twice the content's
depth.
The rules given above imply that all of the following attribute settings have the same effect, which is to leave the content's dimensions unchanged:
<mpadded width="+0em"> ... </mpadded>
<mpadded width="+0%"> ... </mpadded>
<mpadded width="0em"> ... </mpadded>
<mpadded width="0height"> ... </mpadded>
<mpadded width="100%"> ... </mpadded>
<mpadded width="100%width"> ... </mpadded>
<mpadded width="1width"> ... </mpadded>
<mpadded width="1.0width"> ... </mpadded>
<mpadded> ... </mpadded>
Note that the examples in the Version 2 of the MathML specification showed spaces within the attribute values, suggesting that this was the intended format. Formally, spaces are not allowed within these values, but implementers may wish to ignore such spaces to maximize backward compatibility.
The content of an mpadded
element defines a fragment of mathematical
notation, such as a character, fraction, or expression, that can be regarded as
a single typographical element with a natural positioning point relative to its
natural bounding box.
The size of the bounding box of an mpadded
element is
defined as the size of the bounding box of its content, except as
modified by the mpadded
element's
height
, depth
, and
width
attributes. The natural positioning point of the
child content of the mpadded
element is located to coincide
with the natural positioning point of the mpadded
element,
except as modified by the lspace
and voffset
attributes. Thus, the size attributes of mpadded
can be used
to expand or shrink the apparent bounding box of its content, and the
position attributes of mpadded
can be used to move the
content relative to the bounding box (and hence also neighboring elements).
Note that MathML doesn't define the precise relationship between "ink",
bounding boxes and positioning points, which are implementation
specific. Thus, absolute values for mpadded attributes may not be
portable between implementations.
The height
attribute specifies the vertical extent of the
bounding box of the mpadded
element above its baseline.
Increasing the height
increases the space between the baseline
of the mpadded
element and the content above it, and introduces
padding above the rendering of the child content. Decreasing the
height
reduces the space between the baseline of the
mpadded
element and the content above it, and removes
space above the rendering of the child content. Decreasing the
height
may cause content above the mpadded
element to overlap the rendering of the child content, and should
generally be avoided.
The depth
attribute specifies the vertical extent of the
bounding box of the mpadded
element below its baseline.
Increasing the depth
increases the space between the baseline
of the mpadded
element and the content below it, and introduces
padding below the rendering of the child content. Decreasing the
depth
reduces the space between the baseline of the mpadded
element and the content below it, and removes space below the rendering
of the child content. Decreasing the depth
may cause content
below the mpadded
element to overlap the rendering of the child
content, and should generally be avoided.
The width
attribute specifies the horizontal distance
between the positioning point of the mpadded
element and the
positioning point of the following content.
Increasing the width
increases the space between the
positioning point of the mpadded
element and the content
that follows it, and introduces padding after the rendering of the
child content. Decreasing the width
reduces the space
between the positioning point of the mpadded
element and
the content that follows it, and removes space after the rendering
of the child content. Setting the width
to zero causes
following content to be positioned at the positioning point of the
mpadded
element. Decreasing the width
should
generally be avoided, as it may cause overprinting of the following
content.
The lspace
attribute ("leading" space;
see 3.1.5.1 Overall Directionality of Mathematics Formulas) specifies the horizontal
location of the positioning point of the child content with respect to
the positioning point of the mpadded
element. By default they
coincide, and therefore absolute values for lspace have the same effect
as relative values.
Positive values for the lspace
attribute increase the space
between the preceding content and the child content, and introduce padding
before the rendering of the child content. Negative values for the
lspace
attributes reduce the space between the preceding
content and the child content, and may cause overprinting of the
preceding content, and should generally be avoided. Note that the
lspace
attribute does not affect the width
of
the mpadded
element, and so the lspace
attribute
will also affect the space between the child content and following
content, and may cause overprinting of the following content, unless
the width
is adjusted accordingly.
The voffset
attribute specifies the vertical location
of the positioning point of the child content with respect to the
positioning point of the mpadded
element. Positive values
for the voffset
attribute raise the rendering of the child
content above the baseline. Negative values for the voffset
attribute lower the rendering of the child content below the baseline.
In either case, the voffset
attribute may cause overprinting
of neighboring content, which should generally be avoided. Note that
the voffset
attribute does not affect the height
or depth
of the mpadded
element, and so the voffset
attribute will also affect the space between the child content and neighboring
content, and may cause overprinting of the neighboring content, unless the
height
or depth
is adjusted accordingly.
MathML renderers should ensure that, except for the effects of the
attributes, the relative spacing between the contents of the
mpadded
element and surrounding MathML elements would
not be modified by replacing an mpadded
element with an
mrow
element with the same content, even if linebreaking
occurs within the mpadded
element. MathML does not define
how nondefault attribute values of an mpadded
element interact
with the linebreaking algorithm.
The effects of the size and position attributes are illustrated
below. The following diagram illustrates the use of lspace
and voffset
to shift the position of child content without
modifying the mpadded
bounding box.
The corresponding MathML is:
<mrow>
<mi>x</mi>
<mpadded lspace="0.2em" voffset="0.3ex">
<mi>y</mi>
</mpadded>
<mi>z</mi>
</mrow>
The next diagram illustrates the use of
width
, height
and depth
to modifying the mpadded
bounding box without changing the relative position
of the child content.
The corresponding MathML is:
<mrow>
<mi>x</mi>
<mpadded width="+90%width" height="+0.3ex" depth="+0.3ex">
<mi>y</mi>
</mpadded>
<mi>z</mi>
</mrow>
The final diagram illustrates the generic use of mpadded
to modify both
the bounding box and relative position of child content.
The corresponding MathML is:
<mrow>
<mi>x</mi>
<mpadded lspace="0.3em" width="+0.6em">
<mi>y</mi>
</mpadded>
<mi>z</mi>
</mrow>
<mphantom>
The mphantom
element renders invisibly, but
with the same size and other dimensions, including baseline position,
that its contents would have if they were rendered
normally. mphantom
can be used to align parts of
an expression by invisibly duplicating subexpressions.
The mphantom
element accepts
a single argument possibly being an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
Note that it is possible to wrap both an
mphantom
and an mpadded
element around one MathML expression, as in
<mphantom><mpadded attributesettings>
... </mpadded></mphantom>
, to change its size and make it
invisible at the same time.
MathML renderers should ensure that the relative spacing between
the contents of an mphantom
element and the
surrounding MathML elements is the same as it would be if the
mphantom
element were replaced by an
mrow
element with the same content. This holds
even if linebreaking occurs within the mphantom
element.
For the above reason, mphantom
is
not considered spacelike (3.2.7 Space <mspace/>
) unless its
content is spacelike, since the suggested rendering rules for
operators are affected by whether nearby elements are spacelike. Even
so, the warning about the legal grouping of spacelike elements may
apply to uses of mphantom
.
mphantom
elements accept the attributes listed in
3.1.9 Mathematics attributes common to presentation elements (the mathcolor
has no effect).
There is one situation where the preceding rules for rendering an
mphantom
may not give the desired effect. When an
mphantom
is wrapped around a subsequence of the
arguments of an mrow
, the default determination
of the form
attribute for an mo
element within the subsequence can change. (See the default value of
the form
attribute described in 3.2.5 Operator, Fence, Separator or Accent
<mo>
.) It may be
necessary to add an explicit form
attribute to such an
mo
in these cases. This is illustrated in the
following example.
In this example, mphantom
is used to ensure
alignment of corresponding parts of the numerator and denominator of a
fraction:
<mfrac>
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mrow>
<mi> x </mi>
<mphantom>
<mo form="infix"> + </mo>
<mi> y </mi>
</mphantom>
<mo> + </mo>
<mi> z </mi>
</mrow>
</mfrac>
This would render as something like
rather than as
The explicit attribute setting form
="infix"
on the
mo
element inside the mphantom
sets the
form
attribute to what it would have been in the absence of the
surrounding mphantom
. This is necessary since
otherwise, the +
sign would be interpreted as a prefix
operator, which might have slightly different spacing.
Alternatively, this problem could be avoided without any explicit
attribute settings, by wrapping each of the arguments
<mo>+</mo>
and <mi>y</mi>
in its
own mphantom
element, i.e.
<mfrac>
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mrow>
<mi> x </mi>
<mphantom>
<mo> + </mo>
</mphantom>
<mphantom>
<mi> y </mi>
</mphantom>
<mo> + </mo>
<mi> z </mi>
</mrow>
</mfrac>
The mfenced
element provides a convenient form
in which to express common constructs involving fences (i.e. braces,
brackets, and parentheses), possibly including separators (such as
comma) between the arguments.
For example, <mfenced> <mi>x</mi> </mfenced>
renders as (x)
and is equivalent to
<mrow> <mo> ( </mo> <mi>x</mi> <mo> ) </mo> </mrow>
and <mfenced> <mi>x</mi> <mi>y</mi> </mfenced>
renders as (x, y)
and is equivalent to
<mrow>
<mo> ( </mo>
<mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow>
<mo> ) </mo>
</mrow>
Individual fences or separators are represented using
mo
elements, as described in 3.2.5 Operator, Fence, Separator or Accent
<mo>
. Thus, any mfenced
element is completely equivalent to an expanded form described below.
While mfenced
might be more convenient for authors or authoring software,
only the expanded form is supported in [MathMLCore].
A renderer that supports this recommendation is required to
render either of these forms in exactly the same way.
In general, an mfenced
element can contain
zero or more arguments, and will enclose them between fences in an
mrow
; if there is more than one argument, it will
insert separators between adjacent arguments, using an additional
nested mrow
around the arguments and separators
for proper grouping (3.3.1 Horizontally Group SubExpressions
<mrow>
). The general expanded form is
shown below. The fences and separators will be parentheses and comma
by default, but can be changed using attributes, as shown in the
following table.
mfenced
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
The delimiters and separators should be drawn using the color specified by mathcolor
.
Name  values  default 
open  string  ( 
Specifies the opening delimiter.
Since it is used as the content of an mo element, any whitespace
will be trimmed and collapsed as described in 2.1.7 Collapsing Whitespace in Input.


close  string  ) 
Specifies the closing delimiter.
Since it is used as the content of an mo element, any whitespace
will be trimmed and collapsed as described in 2.1.7 Collapsing Whitespace in Input.


separators  string  , 
Specifies a sequence of zero or more separator characters, optionally separated by
whitespace.
Each pair of arguments is displayed separated by the corresponding separator
(none appears after the last argument).
If there are too many separators, the excess are ignored;
if there are too few, the last separator is repeated.
Any whitespace within separators is ignored.

A generic mfenced
element, with all attributes
explicit, looks as follows:
<mfenced open="openingfence"
close="closingfence"
separators="sep#1 sep#2 ... sep#(n1)" >
arg#1
...
arg#n
</mfenced>
In an RTL directionality context, since the initial text
direction is RTL, characters in the open
and close
attributes that have a mirroring counterpart will be rendered in that
mirrored form. In particular, the default values will render correctly
as a parenthesized sequence in both LTR and RTL contexts.
The general mfenced
element shown above is
equivalent to the following expanded form:
<mrow>
<mo fence="true"> openingfence </mo>
<mrow>
arg#1
<mo separator="true"> sep#1 </mo>
...
<mo separator="true"> sep#(n1) </mo>
arg#n
</mrow>
<mo fence="true"> closingfence </mo>
</mrow>
Each argument except the last is followed by a separator. The inner
mrow
is added for proper grouping, as described in
3.3.1 Horizontally Group SubExpressions
<mrow>
.
When there is only one argument, the above form has no separators;
since <mrow> arg#1 </mrow>
is equivalent to
arg#1
(as described in 3.3.1 Horizontally Group SubExpressions
<mrow>
), this case is also equivalent to:
<mrow>
<mo fence="true"> openingfence </mo>
arg#1
<mo fence="true"> closingfence </mo>
</mrow>
If there are too many separator characters, the extra ones are
ignored. If separator characters are given, but there are too few, the
last one is repeated as necessary. Thus, the default value of
separators
="," is equivalent to
separators
=",,", separators
=",,,", etc. If
there are no separator characters provided but some are needed, for
example if separators
=" " or "" and there is more than
one argument, then no separator elements are inserted at all — that
is, the elements <mo separator="true"> sep#i
</mo>
are left out entirely. Note that this is different
from inserting separators consisting of mo
elements with empty content.
Finally, for the case with no arguments, i.e.
<mfenced open="openingfence"
close="closingfence"
separators="anything" >
</mfenced>
the equivalent expanded form is defined to include just
the fences within an mrow
:
<mrow>
<mo fence="true"> openingfence </mo>
<mo fence="true"> closingfence </mo>
</mrow>
Note that not all fenced expressions
can be encoded by an
mfenced
element. Such exceptional expressions
include those with an embellished
separator or fence or one
enclosed in an mstyle
element, a missing or extra
separator or fence, or a separator with multiple content
characters. In these cases, it is necessary to encode the expression
using an appropriately modified version of an expanded form. As
discussed above, it is always permissible to use the expanded form
directly, even when it is not necessary. In particular, authors cannot
be guaranteed that MathML preprocessors won't replace occurrences of
mfenced
with equivalent expanded forms.
Note that the equivalent expanded forms shown above include
attributes on the mo
elements that identify them as fences or
separators. Since the most common choices of fences and separators
already occur in the operator dictionary with those attributes,
authors would not normally need to specify those attributes explicitly
when using the expanded form directly. Also, the rules for the default
form
attribute (3.2.5 Operator, Fence, Separator or Accent
<mo>
) cause the
opening and closing fences to be effectively given the values
form
="prefix"
and
form
="postfix"
respectively, and the
separators to be given the value
form
="infix"
.
Note that it would be incorrect to use mfenced
with a separator of, for instance, +
, as an abbreviation for an
expression using +
as an ordinary operator, e.g.
<mrow>
<mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi>
</mrow>
This is because the +
signs would be treated as separators,
not infix operators. That is, it would render as if they were marked up as
<mo separator="true">+</mo>
, which might therefore
render inappropriately.
<mfenced>
<mrow>
<mi> a </mi>
<mo> + </mo>
<mi> b </mi>
</mrow>
</mfenced>
Note that the above mrow
is necessary so that
the mfenced
has just one argument. Without it, this
would render incorrectly as (a, +,
b)
.
<mfenced open="[">
<mn> 0 </mn>
<mn> 1 </mn>
</mfenced>
<mrow>
<mi> f </mi>
<mo> ⁡<!ApplyFunction> </mo>
<mfenced>
<mi> x </mi>
<mi> y </mi>
</mfenced>
</mrow>
The menclose
element renders its content
inside the enclosing notation specified by its notation
attribute.
menclose
accepts
a single argument possibly being an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
menclose
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
The notations should be drawn using the color specified by mathcolor
.
The values allowed for notation
are openended.
Conforming renderers may ignore any value they do not handle, although
renderers are encouraged to render as many of the values listed below as
possible.
Name  values  default 
notation  (actuarial  phasorangle  box  roundedbox  circle 
left  right  top  bottom 
updiagonalstrike  downdiagonalstrike  verticalstrike  horizontalstrike  northeastarrow
 madruwb  text ) +

do nothing 
Specifies a space separated list of notations to be used to enclose the children.
See below for a description of each type of notation.
MathML 4 deprecates the use of longdiv and radical .
These notations duplicate functionality provided by mlongdiv and msqrt respectively;
those elements should be used instead.
The default has been changed so that if no notation is given,
or if it is an empty string,
then menclose should not draw.

Any number of values can be given for
notation
separated by whitespace; all of those given and
understood by a MathML renderer should be rendered.
Each should be rendered as if the others were not present; they should not nest one
inside of the other. For example,
notation
="circle box
" should
result in circle and a box around the contents of menclose
; the circle and box may overlap. This is shown in the first example below.
Of the predefined notations, only phasorangle
is
affected by the directionality (see 3.1.5.1 Overall Directionality of Mathematics Formulas):
When notation
is specified as
actuarial
, the contents are drawn enclosed by an
actuarial symbol. A similar result can be achieved
with the value top right
.
The values box
,
roundedbox
, and circle
should
enclose the contents as indicated by the values. The amount of
distance between the box, roundedbox, or circle, and the contents are
not specified by MathML, and left to the renderer. In practice,
paddings on each side of 0.4em in the horizontal direction and .5ex in
the vertical direction seem to work well.
The values left
,
right
, top
and
bottom
should result in lines drawn on those sides of
the contents. The values northeastarrow
,
updiagonalstrike
,
downdiagonalstrike
, verticalstrike
and horizontalstrike
should result in the indicated
strikeout lines being superimposed over the content of the
menclose
, e.g. a strikeout that extends from the lower left
corner to the upper right corner of the menclose
element for
updiagonalstrike
, etc.
The value northeastarrow
is a recommended value to implement because it can be
used to implement TeX's \cancelto command. If a renderer implements other arrows for
menclose
, it is recommended that the arrow names are chosen from the following full set of
names for consistency and standardization among renderers:
uparrow
rightarrow
downarrow
leftarrow
northwestarrow
southwestarrow
southeastarrow
northeastarrow
updownarrow
leftrightarrow
northwestsoutheastarrow
northeastsouthwestarrow
The value madruwb
should generate an enclosure
representing an Arabic factorial (‘madruwb’ is the transliteration
of the Arabic مضروب for factorial).
This is shown in the third example below.
The baseline of an menclose
element is the baseline of its child (which might be an implied mrow
).
An example of using multiple attributes is
<menclose notation='circle box'>
<mi> x </mi><mo> + </mo><mi> y </mi>
</menclose>
An example of using menclose
for actuarial
notation is
<msub>
<mi>a</mi>
<mrow>
<menclose notation='actuarial'>
<mi>n</mi>
</menclose>
<mo>⁣<!InvisibleComma></mo>
<mi>i</mi>
</mrow>
</msub>
An example of phasorangle
, which is used in circuit analysis, is:
<mi>C</mi>
<mrow>
<menclose notation='phasorangle'>
<mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
</menclose>
</mrow>
An example of madruwb
is:
<menclose notation="madruwb">
<mn>12</mn>
</menclose>
The elements described in this section position one or more scripts around a base. Attaching various kinds of scripts and embellishments to symbols is a very common notational device in mathematics. For purely visual layout, a single generalpurpose element could suffice for positioning scripts and embellishments in any of the traditional script locations around a given base. However, in order to capture the abstract structure of common notation better, MathML provides several more specialized scripting elements.
In addition to sub/superscript elements, MathML has overscript
and underscript elements that place scripts above and below the base. These
elements can be used to place limits on large operators, or for placing
accents and lines above or below the base. The rules for rendering accents
differ from those for overscripts and underscripts, and this difference can
be controlled with the accent
and accentunder
attributes, as described in the appropriate
sections below.
Rendering of scripts is affected by the scriptlevel
and displaystyle
attributes, which are part of the environment inherited by the rendering
process of every MathML expression, and are described in 3.1.6 Displaystyle and Scriptlevel.
These attributes cannot be given explicitly on a scripting element, but can be
specified on the start tag of a surrounding mstyle
element if desired.
MathML also provides an element for attachment of tensor indices. Tensor indices are distinct from ordinary subscripts and superscripts in that they must align in vertical columns. Also, all the upper scripts should be baselinealigned and all the lower scripts should be baselinealigned. Tensor indices can also occur in prescript positions. Note that ordinary scripts follow the base (on the right in LTR context, but on the left in RTL context); prescripts precede the base (on the left (right) in LTR (RTL) context).
Because presentation elements should be used to describe the abstract
notational structure of expressions, it is important that the base
expression in all scripting
elements (i.e. the first
argument expression) should be the entire expression that is being
scripted, not just the trailing character. For example,
${\left(x+y\right)}^{2}$
should be written as:
<msup>
<mrow>
<mo> ( </mo>
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
</mrow>
<mo> ) </mo>
</mrow>
<mn> 2 </mn>
</msup>
<msub>
The msub
element attaches a subscript to a base using the syntax
<msub> base subscript </msub>
It increments scriptlevel
by 1, and sets displaystyle
to
false
, within subscript, but leaves both attributes
unchanged within base. (See 3.1.6 Displaystyle and Scriptlevel.)
msub
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
subscriptshift  length  automatic 
Specifies the minimum amount to shift the baseline of subscript down; the default is for the rendering agent to use its own positioning rules. 
<msup>
The msup
element attaches a superscript to a base using the syntax
<msup> base superscript </msup>
It increments scriptlevel
by 1, and sets displaystyle
to false
, within
superscript, but leaves both attributes unchanged within
base. (See 3.1.6 Displaystyle and Scriptlevel.)
msup
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
superscriptshift  length  automatic 
Specifies the minimum amount to shift the baseline of superscript up; the default is for the rendering agent to use its own positioning rules. 
<msubsup>
The msubsup
element is used to attach both a subscript and
superscript to a base expression.
<msubsup> base subscript superscript </msubsup>
It increments scriptlevel
by 1, and sets displaystyle
to
false
, within subscript and superscript,
but leaves both attributes unchanged within base.
(See 3.1.6 Displaystyle and Scriptlevel.)
Note that both scripts are positioned tight against the base as shown here
${x}_{1}^{2}$
versus the staggered positioning of nested scripts as shown here
${{x}_{1}}^{2}$;
the latter can be achieved by nesting an msub
inside an msup
.
msubsup
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
subscriptshift  length  automatic 
Specifies the minimum amount to shift the baseline of subscript down; the default is for the rendering agent to use its own positioning rules.  
superscriptshift  length  automatic 
Specifies the minimum amount to shift the baseline of superscript up; the default is for the rendering agent to use its own positioning rules. 
The msubsup
is most commonly used for adding
sub/superscript pairs to identifiers as illustrated above. However,
another important use is placing limits on certain large operators
whose limits are traditionally displayed in the script positions even
when rendered in display style. The most common of these is the
integral. For example,
would be represented as
<mrow>
<msubsup>
<mo> ∫ </mo>
<mn> 0 </mn>
<mn> 1 </mn>
</msubsup>
<mrow>
<msup>
<mi> ⅇ </mi>
<mi> x </mi>
</msup>
<mo> ⁢<!InvisibleTimes> </mo>
<mrow>
<mo> ⅆ </mo>
<mi> x </mi>
</mrow>
</mrow>
</mrow>
<munder>
The munder
element attaches an accent or limit placed under a base using the syntax
<munder> base underscript </munder>
It always sets displaystyle
to false
within the underscript,
but increments scriptlevel
by 1 only when accentunder
is false
.
Within base, it always leaves both attributes unchanged.
(See 3.1.6 Displaystyle and Scriptlevel.)
If base is an operator with movablelimits
=true
(or an embellished operator whose mo
element core has movablelimits
=true
),
and displaystyle
=false
,
then underscript is drawn in a subscript position.
In this case, the accentunder
attribute is ignored.
This is often used for limits on symbols such as U+2211 (entity sum
).
munder
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
accentunder  "true"  "false"  automatic 
Specifies whether underscript is drawn as an accentor as a limit. An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.


align  "left"  "right"  "center"  center 
Specifies whether the script is aligned left, center, or right under/over the base. As specified in 3.2.5.7.3 Horizontal Stretching Rules, the core of underscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire underscript. 
The default value of accentunder
is false, unless
underscript is an mo
element or an
embellished operator (see 3.2.5 Operator, Fence, Separator or Accent
<mo>
). If
underscript is an mo
element, the
value of its accent
attribute is used as the default
value of accentunder
. If underscript is an
embellished operator, the accent
attribute of the
mo
element at its core is used as the default
value. As with all attributes, an explicitly given value overrides
the default.
[MathMLCore] does not support the accent
attribute on 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
For compatibility with MathML Core, the accentunder
should be set on munder
.
An example demonstrating how accentunder
affects rendering:
<mrow>
<munder accentunder="true">
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mo> ⏟ </mo>
</munder>
<mtext> <!nbsp>versus <!nbsp></mtext>
<munder accentunder="false">
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mo> ⏟ </mo>
</munder>
</mrow>
<mover>
The mover
element attaches an accent or limit placed over a base using the syntax
<mover> base overscript </mover>
It always sets displaystyle
to false
within overscript,
but increments scriptlevel
by 1 only when accent
is false
.
Within base, it always leaves both attributes unchanged.
(See 3.1.6 Displaystyle and Scriptlevel.)
If base is an operator with movablelimits
=true
(or an embellished operator whose mo
element core has movablelimits
=true
),
and displaystyle
=false
,
then overscript is drawn in a superscript position.
In this case, the accent
attribute is ignored.
This is often used for limits on symbols such as U+2211 (entity sum
).
mover
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
accent  "true"  "false"  automatic 
Specifies whether overscript is drawn as an accentor as a limit. An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.


align  "left"  "right"  "center"  center 
Specifies whether the script is aligned left, center, or right under/over the base. As specified in 3.2.5.7.3 Horizontal Stretching Rules, the core of overscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire overscript. 
The difference between an accent versus limit is shown in the examples.
The default value of accent is false, unless
overscript is an mo
element or an
embellished operator (see 3.2.5 Operator, Fence, Separator or Accent
<mo>
). If
overscript is an mo
element, the value
of its accent
attribute is used as the default value
of accent
for mover
. If
overscript is an embellished operator, the accent
attribute of the mo
element at its core is used as the default value.
[MathMLCore] does not support the accent
attribute on 3.2.5 Operator, Fence, Separator or Accent
<mo>
.
For compatibility with MathML Core, the accentunder
should be set on munder
.
Two examples demonstrating how accent
affects rendering:
<mrow>
<mover accent="true">
<mi> x </mi>
<mo> ^ </mo>
</mover>
<mtext> <!nbsp>versus <!nbsp></mtext>
<mover accent="false">
<mi> x </mi>
<mo> ^ </mo>
</mover>
</mrow>
<mrow>
<mover accent="true">
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mo> ⏞ </mo>
</mover>
<mtext> <!nbsp>versus <!nbsp></mtext>
<mover accent="false">
<mrow>
<mi> x </mi>
<mo> + </mo>
<mi> y </mi>
<mo> + </mo>
<mi> z </mi>
</mrow>
<mo> ⏞ </mo>
</mover>
</mrow>
<munderover>
The munderover
element attaches accents or limits placed both over and under a base using the syntax
<munderover> base underscript overscript </munderover>
It always sets displaystyle
to false
within underscript and overscript,
but increments scriptlevel
by 1 only when
accentunder
or accent
, respectively, are false
.
Within base, it always leaves both attributes unchanged.
(see 3.1.6 Displaystyle and Scriptlevel).
If base is an operator with movablelimits
=true
(or an embellished operator whose mo
element core has movablelimits
=true
),
and displaystyle
=false
,
then underscript and overscript are drawn in a subscript and superscript position,
respectively. In this case, the accentunder
and accent
attributes are ignored.
This is often used for limits on symbols such as U+2211 (entity sum
).
munderover
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
accent  "true"  "false"  automatic 
Specifies whether overscript is drawn as an accentor as a limit. An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.


accentunder  "true"  "false"  automatic 
Specifies whether underscript is drawn as an accentor as a limit. An accent is drawn the same size as the base (without incrementing scriptlevel )
and is drawn closer to the base.


align  "left"  "right"  "center"  center 
Specifies whether the scripts are aligned left, center, or right under/over the base. As specified in 3.2.5.7.3 Horizontal Stretching Rules, the core of underscripts and overscripts that are embellished operators should stretch to cover the base, but the alignment is based on the entire underscript or overscript. 
The munderover
element is used instead of separate
munder
and mover
elements so that the
underscript and overscript are vertically spaced equally in relation
to the base and so that they follow the slant of the base as shown in the example.
The defaults for accent
and accentunder
are computed in the same way as for
munder
and
mover
, respectively.
This example shows the difference between nesting munder
inside
mover
and using munderover
when
movablelimits
=true
and in displaystyle
(which renders the same as msubsup
).
<mstyle displaystyle="false">
<mover>
<munder>
<mo>∑</mo>
<mi>i</mi>
</munder>
<mi>n</mi>
</mover>
<mo>+</mo>
<munderover>
<mo>∑</mo>
<mi>i</mi>
<mi>n</mi>
</munderover>
</mstyle>
<mmultiscripts>
,
<mprescripts/>
Presubscripts and tensor notations are represented by a single
element, mmultiscripts
, using the syntax:
<mmultiscripts>
base
(subscript superscript)*
[ <mprescripts/> (presubscript presuperscript)* ]
</mmultiscripts>
This element allows the representation of any number of verticallyaligned pairs of
subscripts
and superscripts, attached to one base expression. It supports both
postscripts and prescripts.
Missing scripts must be represented by a valid empty element denoting the empty subterm,
such as <mrow/>
.
(The element <none/>
was used in earlier MathML releases, but was equivalent
to an empty <mrow/>
).
All of the upper scripts should be baselinealigned and all the lower scripts should be baselinealigned.
The prescripts are optional, and when present are given after the postscripts. This order was chosen because prescripts are relatively rare compared to tensor notation.
The argument sequence consists of the base followed by zero or more
pairs of verticallyaligned subscripts and superscripts (in that
order) that represent all of the postscripts. This list is optionally
followed by an empty element mprescripts
and a
list of zero or more pairs of verticallyaligned presubscripts and
presuperscripts that represent all of the prescripts. The pair lists
for postscripts and prescripts are displayed in the same order as the
directional context (i.e. lefttoright order in LTR context). If
no subscript or superscript should be rendered in a given position,
then an empty element <mrow/>
should be used in
that position.
For each sub and superscript pair,
horizontalalignment of the elements in the pair should be
towards the base of the mmultiscripts
.
That is, prescripts should be right aligned,
and postscripts should be left aligned.
The base, subscripts, superscripts, the optional separator element
mprescripts
, the presubscripts, and the
presuperscripts are all direct subexpressions of the
mmultiscripts
element, i.e. they are all at the
same level of the expression tree. Whether a script argument is a
subscript or a superscript, or whether it is a presubscript or a
presuperscript is determined by whether it occurs in an evennumbered
or oddnumbered argument position, respectively, ignoring the empty
element mprescripts
itself when determining the
position. The first argument, the base, is considered to be in
position 1. The total number of arguments must be odd, if
mprescripts
is not given, or even, if it is.
The empty element mprescripts
is only allowed as direct subexpression
of mmultiscripts
.
Same as the attributes of msubsup
. See
3.4.3.2 Attributes.
The mmultiscripts
element increments scriptlevel
by 1, and sets displaystyle
to false
, within
each of its arguments except base, but leaves both attributes
unchanged within base. (See 3.1.6 Displaystyle and Scriptlevel.)
This example of a hypergeometric function demonstrates the use of pre and post subscripts:
<mrow>
<mmultiscripts>
<mi> F </mi>
<mn> 1 </mn>
<mrow/>
<mprescripts/>
<mn> 0 </mn>
<mrow/>
</mmultiscripts>
<mo> ⁡<!ApplyFunction> </mo>
<mrow>
<mo> ( </mo>
<mrow>
<mo> ; </mo>
<mi> a </mi>
<mo> ; </mo>
<mi> z </mi>
</mrow>
<mo> ) </mo>
</mrow>
</mrow>
This example shows a tensor. In the example, k and l are different indices
<mmultiscripts>
<mi> R </mi>
<mi> i </mi>
<mrow/>
<mrow/>
<mi> j </mi>
<mi> k </mi>
<mrow/>
<mi> l </mi>
<mrow/>
</mmultiscripts>
This example demonstrates alignment towards the base of the scripts:
<mmultiscripts>
<mi> X </mi>
<mn> 123 </mn>
<mn> 1 </mn>
<mprescripts/>
<mn> 123 </mn>
<mn> 1 </mn>
</mmultiscripts>
This final example of mmultiscripts
shows how the binomial
coefficient can be displayed in Arabic style
<mstyle dir="rtl">
<mmultiscripts><mo>ل</mo>
<mn>12</mn><mrow/>
<mprescripts/>
<mrow/><mn>5</mn>
</mmultiscripts>
</mstyle>
Matrices, arrays and other tablelike mathematical notation are marked
up using mtable
,
mtr
and
mtd
elements. These elements are similar to the
table
, tr
and td
elements of HTML, except that they provide
specialized attributes for the fine layout control
necessary for commutative diagrams, block matrices and so on.
While the twodimensional layouts used for elementary math such as addition and multiplication
are somewhat similar to tables, they differ in important ways.
For layout and for accessibility reasons, the mstack
and mlongdiv
elements discussed
in 3.6 Elementary Math should be used for elementary math notations.
<mtable>
A matrix or table is specified using the mtable
element. Inside of the mtable
element, only mtr
elements may appear.
Table rows that have fewer columns than other rows of the same
table (whether the other rows precede or follow them) are effectively
padded on the right (or left in RTL context) with empty mtd
elements so
that the number of columns in each row equals the maximum number of
columns in any row of the table. Note that the use of
mtd
elements with nondefault values of the
rowspan
or columnspan
attributes may affect
the number of mtd
elements that should be given
in subsequent mtr
elements to cover a given
number of columns.
MathML does not specify a table layout algorithm. In
particular, it is the responsibility of a MathML renderer to resolve
conflicts between the width
attribute and other
constraints on the width of a table, such as explicit values for columnwidth
attributes,
and minimum sizes for table cell contents. For a discussion of table layout algorithms,
see
Cascading
Style Sheets, level 2.
mtable
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Any rules drawn as part of the mtable
should be drawn using the color
specified by mathcolor
.
Name  values  default 
align  ("top"  "bottom"  "center"  "baseline"  "axis"), rownumber?  axis 
specifies the vertical alignment of the table with respect to its environment.
axis means to align the vertical center of the table on
the environment's axis.
(The axis of an equation is an alignment line used by typesetters.
It is the line on which a minus sign typically lies.)
center and baseline both mean to align the center of the table
on the environment's baseline.
top or bottom aligns the top or bottom of the table on the environment's baseline.
If the align attribute value ends with a rownumber,
the specified row (counting from 1 for the top row), rather than the table as a whole, is aligned in the way described above with the
exceptions noted below.
If rownumber is negative, it counts rows from the bottom.
When the value of rownumber is out of range or not an integer, it is ignored.
If a row number is specified and the alignment value is baseline or axis ,
the row's baseline or axis is used for alignment. Note this is only well defined when
the rowalign
value is baseline or axis ; MathML does not specify how
baseline or axis alignment should occur for other values of rowalign .


rowalign  ("top"  "bottom"  "center"  "baseline"  "axis") +  baseline 
specifies the vertical alignment of the cells with respect to other cells within the
same row:
top aligns the tops of each entry across the row;
bottom aligns the bottoms of the cells,
center centers the cells;
baseline aligns the baselines of the cells;
axis aligns the axis of each cells.
(See the note below about multiple values.)


columnalign  ("left"  "center"  "right") +  center 
specifies the horizontal alignment of the cells with respect to other cells within
the same column:
left aligns the left side of the cells;
center centers each cells;
right aligns the right side of the cells.
(See the note below about multiple values.)


alignmentscope  ("true"  "false") +  true 
[this attribute is described with the alignment elements, maligngroup and malignmark ,
in 3.5.4 Alignment Markers
<maligngroup/> , <malignmark/> .]


columnwidth  ("auto"  length  "fit") +  auto 
specifies how wide a column should be:
auto means that the column should be as wide as needed;
an explicit length means that the column is exactly that wide and the contents of
that column are made to fit
by linewrapping or clipping at the discretion of the renderer;
fit means that the page width
remaining after subtracting the auto or fixed width columns
is divided equally among the fit columns.
If insufficient room remains to hold the
contents of the fit columns, renderers may
linewrap or clip the contents of the fit columns.
Note that when the columnwidth is specified as
a percentage, the value is relative to the width of the table, not
as a percentage of the default (which is auto ). That
is, a renderer should try to adjust the width of the column so that it
covers the specified percentage of the entire table width.
(See the note below about multiple values.)


width  "auto"  length  auto 
specifies the desired width of the entire table and is intended for visual user agents.
When the value is a percentage value,
the value is relative to the
horizontal space that a MathML renderer has available,
this is the current target width as used for
linebreaking as specified in 3.1.7 Linebreaking of Expressions;
this allows the author to specify, for example, a table being full width
of the display.
When the value is auto , the MathML
renderer should calculate the table width from its contents using
whatever layout algorithm it chooses.
Note: numbers without units were allowed in MathML 3 and treated similarly to percentage values,
but unitless numbers are deprecated in MathML 4.


rowspacing  (length) +  1.0ex 
specifies how much space to add between rows. (See the note below about multiple values.)  
columnspacing  (length) +  0.8em 
specifies how much space to add between columns. (See the note below about multiple values.)  
rowlines  ("none"  "solid"  "dashed") +  none 
specifies whether and what kind of lines should be added between each row:
none means no lines;
solid means solid lines;
dashed means dashed lines (how the dashes are spaced is implementation dependent).
(See the note below about multiple values.)


columnlines  ("none"  "solid"  "dashed") +  none 
specifies whether and what kind of lines should be added between each column:
none means no lines;
solid means solid lines;
dashed means dashed lines (how the dashes are spaced is implementation dependent).
(See the note below about multiple values.)


frame  "none"  "solid"  "dashed"  none 
specifies whether and what kind of lines should be drawn around the table.
none means no lines;
solid means solid lines;
dashed means dashed lines (how the dashes are spaced is implementation dependent).


framespacing  length, length  0.4em 0.5ex 
specifies the additional spacing added between the table and frame,
if frame is not none .
The first value specifies the spacing on the right and left;
the second value specifies the spacing above and below.


equalrows  "true"  "false"  false 
specifies whether to force all rows to have the same total height.  
equalcolumns  "true"  "false"  false 
specifies whether to force all columns to have the same total width.  
displaystyle  "true"  "false"  false 
specifies the value of displaystyle within each cell
(scriptlevel is not changed);
see 3.1.6 Displaystyle and Scriptlevel.

In the above specifications for attributes affecting rows
(respectively, columns, or the gaps between rows or columns),
the notation (...)+
means that multiple values can be given for the attribute
as a space separated list (see 2.1.5 MathML Attribute Values).
In this context, a single value specifies the value to be used for all rows (resp.,
columns or gaps).
A list of values are taken to apply to corresponding rows (resp., columns or gaps)
in order, that is starting from the top row for rows or first column (left or right,
depending on directionality) for columns.
If there are more rows (resp., columns or gaps) than supplied values, the last value
is repeated as needed.
If there are too many values supplied, the excess are ignored.
Note that none of the areas occupied by lines
frame
, rowlines
and columnlines
,
nor the spacing framespacing
, rowspacing
or columnspacing
are counted as rows or columns.
The displaystyle
attribute is allowed on the mtable
element to set the inherited value of the attribute. If the attribute is
not present, the mtable
element sets displaystyle
to
false
within the table elements.
(See 3.1.6 Displaystyle and Scriptlevel.)
A 3 by 3 identity matrix could be represented as follows:
<mrow>
<mo> ( </mo>
<mtable>
<mtr>
<mtd> <mn>1</mn> </mtd>
<mtd> <mn>0</mn> </mtd>
<mtd> <mn>0</mn> </mtd>
</mtr>
<mtr>
<mtd> <mn>0</mn> </mtd>
<mtd> <mn>1</mn> </mtd>
<mtd> <mn>0</mn> </mtd>
</mtr>
<mtr>
<mtd> <mn>0</mn> </mtd>
<mtd> <mn>0</mn> </mtd>
<mtd> <mn>1</mn> </mtd>
</mtr>
</mtable>
<mo> ) </mo>
</mrow>
Note that the parentheses must be represented explicitly; they are not
part of the mtable
element's rendering. This allows
use of other surrounding fences, such as brackets, or none at all.
<mtr>
An mtr
element represents one row in a table
or matrix. An mtr
element is only allowed as a
direct subexpression of an mtable
element, and
specifies that its contents should form one row of the table. Each
argument of mtr
is placed in a different column
of the table, starting at the leftmost column in a LTR context or rightmost
column in a RTL context.
As described in 3.5.1 Table or Matrix
<mtable>
,
mtr
elements are
effectively padded with mtd
elements when they are shorter than other rows in a table.
mtr
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
rowalign  "top"  "bottom"  "center"  "baseline"  "axis"  inherited 
overrides, for this row, the vertical alignment of cells specified
by the rowalign attribute on the mtable .


columnalign  ("left"  "center"  "right") +  inherited 
overrides, for this row, the horizontal alignment of cells specified
by the columnalign attribute on the mtable .

Earlier versions of MathML specified an mlabeledtr
element
for numbered equations. In an mlabeledtr
, the
first mtd
represents the equation number and the remaining elements in the row
the equation being numbered. The side
and minlabelspacing
attributes of mtable
determines the placement of the equation
number.
This element was not widely implemented and is not specified in the current version, it is still valid
in the Legacy Schema.
In larger documents with many numbered equations, automatic
numbering becomes important. While automatic equation numbering and
automatically resolving references to equation numbers is outside the
scope of MathML, these problems can be addressed by the use of style
sheets or other means. In a CSS context, one could use an empty mtd
as the first child of a mtr
and use CSS counters and generated content
to fill in the equation number using a CSS style such as
body {counterreset: eqnum;}
mtd.eqnum {counterincrement: eqnum;}
mtd.eqnum:before {content: "(" counter(eqnum) ")"}
<mtd>
An mtd
element represents one entry, or cell, in a
table or matrix. An mtd
element is only
allowed as a direct subexpression of an mtr
element.
The mtd
element accepts
a single argument possibly being an inferred mrow
of multiple children;
see 3.1.3 Required Arguments.
mtd
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
rowspan  positiveinteger  1 
causes the cell to be treated as if it occupied the number of rows specified.
The corresponding mtd in the following rowspan 1 rows must be omitted.
The interpretation corresponds with the similar attributes for HTML tables.


columnspan  positiveinteger  1 
causes the cell to be treated as if it occupied the number of columns specified.
The following rowspan 1 mtd s must be omitted.
The interpretation corresponds with the similar attributes for HTML tables.


rowalign  "top"  "bottom"  "center"  "baseline"  "axis"  inherited 
specifies the vertical alignment of this cell, overriding any value
specified on the containing mrow and mtable .
See the rowalign attribute of mtable .


columnalign  "left"  "center"  "right"  inherited 
specifies the horizontal alignment of this cell, overriding any value
specified on the containing mrow and mtable .
See the columnalign attribute of mtable .

malignmark
and maligngroup
are not supported in [MathMLCore].
For most purposes it is recommended that alignment is implemented directly using mtable
columns. As noted in the following section these elements may be futher simplified or removed in a future version of MathML.
For existing MathML using malignmark
a Javascript polyfill
is provided.
With one significant exception, <maligngroup/>
and <malignmark/>
have had minimal adoption and implementation.
The one exception only uses the basics of alignment. Because of this, alignment in MathML is significantly
simplified to align with the current usage and make future implementation simplier. In particular, the following simplifications are made:
<maligngroup/>
and <malignmark/>
have been removed.groupalign
attribute previously allowed on
mtable
, mtr
, and
mtr
is removed<malignmark/>
used to be allowed anywhere, including inside of token elements;
it is now allowed in only the locations that <maligngroup/>
is allowed (see below)Alignment markers are spacelike elements (see 3.2.7 Space <mspace/>
) that can be used
to vertically align specified points within a column of MathML
expressions by the automatic insertion of the necessary amount of
horizontal space between specified subexpressions.
The discussion that follows will use the example of a set of simultaneous equations that should be rendered with vertical alignment of the coefficients and variables of each term, by inserting spacing somewhat like that shown here:
8.44x  +  55  y=  0 
3.1  x−  0.7y  =  −1.1 
If the example expressions shown above were arranged in a column but not aligned, they would appear as:
8.44x + 55.7y = 0 
3.1x − 50.7y = −1.1 
The expressions whose parts are to be aligned (each equation, in the
example above) must be given as the table elements (i.e. as the mtd
elements) of one column of an
mtable
. To avoid confusion, the term table
cell
rather than table element
will be used in the
remainder of this section.
All interactions between alignment elements are limited to the
mtable
column they arise in. That is, every
column of a table specified by an mtable
element
acts as an alignment scope
that contains within it all alignment
effects arising from its contents. It also excludes any interaction
between its own alignment elements and the alignment elements inside
any nested alignment scopes it might contain.
If there is only one alignment point, an alternative is to use linebreaking and indentation attributes
on mo
elements as described in 3.1.7 Linebreaking of Expressions.
An mtable
element can be given the attribute
alignmentscope
=false
to cause
its columns not to act as alignment scopes. This is discussed further at
the end of this section. Otherwise, the discussion in this section assumes
that this attribute has its default value of true
.
Each part of expression to be aligned should be in an maligngroup
.
The point of alignment is the left edge (right edge if for RTL) of the element that follows an maligngroup
element
unless an malignmark
element
is between maligngroup
elements. In that case, the left edge (right edge if for RTL) of the element that follows the
malignmark
is the point of alignment for that group.
If maligngroup
or maligngroup
occurs outside of an
mtable
, they are rendered with zero width.
In the example above, each equation would have one
maligngroup
element before each coefficient,
variable, and operator on the lefthand side, one before the
=
sign, and one before the constant on the righthand
side because these are the parts that should be aligned.
In general, a table cell containing n
maligngroup
elements contains n
alignment groups, with the ith group consisting of the
elements entirely after the ith
maligngroup
element and before the
(i+1)th; no element within the table cell's content
should occur entirely before its first
maligngroup
element.
Note that the division into alignment groups does not
necessarily fit the nested expression structure of the MathML
expression containing the groups — that is, it is permissible for one
alignment group to consist of the end of one
mrow
, all of another one, and the beginning of a
third one, for example. This can be seen in the MathML markup for the
example given at the end of this section.
Although alignment groups need not
coincide with the nested expression structure of layout schemata,
there are nonetheless restrictions on where maligngroup
and malignmark
elements are allowed within a table cell. These
elements may only be contained within elements (directly or indirectly) of the following types
(which are themselves contained in the table cell):
an mrow
element, including an inferred
mrow
such as the one formed by a multichild
mtd
element, but excluding mrow
which
contains a change of direction using the dir
attribute;
an mstyle
element
, but excluding those which change direction
using the dir
attribute;
an mphantom
element;
an mfenced
element;
an maction
element, though only its
selected subexpression is checked;
a semantics
element.
These restrictions are intended to ensure that alignment can be unambiguously specified, while avoiding complexities involving things like overscripts, radical signs and fraction bars. They also ensure that a simple algorithm suffices to accomplish the desired alignment.
For the table cells that are divided into alignment groups, every
element in their content must be part of exactly one alignment group,
except for the elements from the above list that contain
maligngroup
elements inside them and the
maligngroup
elements themselves. This means
that, within any table cell containing alignment groups, the first
complete element must be an maligngroup
element,
though this may be preceded by the start tags of other elements.
This requirement removes a potential confusion about how to align
elements before the first maligngroup
element,
and makes it easy to identify table cells that are left out of their
column's alignment process entirely.
It is not required that the table cells in a column that are divided into alignment groups each contain the same number of groups. If they don't, zerowidth alignment groups are effectively added on the right side (or left side, in a RTL context) of each table cell that has fewer groups than other table cells in the same column.
Do we want to tighten this so that all rows have the same number of maligngroup
elements?
Do we still want to allow rows without maligngroup
as described in this section?
Expressions in a column that are to have no alignment groups
should contain no maligngroup
elements. Expressions with no alignment groups are aligned using only
the columnalign
attribute that applies to the table
column as a whole. If such an expression is wider than the
column width needed for the table cells containing alignment groups,
all the table cells containing alignment groups will be shifted as a
unit within the column as described by the columnalign
attribute for that column. For example, a column heading with no
internal alignment could be added to the column of two equations given
above by preceding them with another table row containing an
mtext
element for the heading, and using the
default columnalign
="center" for the table, to
produce:
equations with aligned variables  
8.44x  +  55  y=  0  
3.1  x−  0.7y  =  −1.1 
or, with a shorter heading,
some equations  
8.44x  +  55  y=  0 
3.1  x−  0.7y  =  −1.1 
An malignmark
element anywhere within the
alignment group (except within another alignment scope wholly
contained inside it) overrides alignment at the start of an maligngroup
element.
The malignmark
element indicates that the
alignment point should occur on the left edge (right edge in a RTL context) of the following element.
Can malignmark
elements occur inside of tokens?
When an malignmark
element is provided within an
alignment group, it should only occur within the elements allowed for maligngroup
(see 3.5.4.3 Specifying alignment groups).
If there is more than one malignmark
element
in an alignment group, all but the first one will be ignored. MathML
applications may wish to provide a mode in which they will warn about
this situation, but it is not an error, and should trigger no warnings
by default. The rationale for this is that it would
be inconvenient to have to remove all
unnecessary malignmark
elements from
automatically generated data.
The above rules are sufficient to explain the MathML representation of the example given near the start of this section.
issue 180
One way to represent that in MathML is:
<mtable groupalign="{decimalpoint left left decimalpoint left left decimalpoint}">
<mtr>
<mtd>
<mrow>
<mrow>
<mrow>
<maligngroup/>
<mn> 8.44 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<maligngroup/>
<mi> x </mi>
</mrow>
<maligngroup/>
<mo> + </mo>
<mrow>
<maligngroup/>
<mn> 55 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<maligngroup/>
<mi> y </mi>
</mrow>
</mrow>
<maligngroup/>
<mo> = </mo>
<maligngroup/>
<mn> 0 </mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mrow>
<maligngroup/>
<mn> 3.1 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<maligngroup/>
<mi> x </mi>
</mrow>
<maligngroup/>
<mo>  </mo>
<mrow>
<maligngroup/>
<mn> 0.7 </mn>
<mo> ⁢<!InvisibleTimes> </mo>
<maligngroup/>
<mi> y </mi>
</mrow>
</mrow>
<maligngroup/>
<mo> = </mo>
<maligngroup/>
<mrow>
<mo>  </mo>
<mn> 1.1 </mn>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
A simple algorithm by which a MathML renderer can perform the
alignment specified in this section is given here. Since the alignment
specification is deterministic (except for the definition of the left
and right edges of a character), any correct MathML alignment
algorithm will have the same behavior as this one. Each
mtable
column (alignment scope) can be treated
independently; the algorithm given here applies to one
mtable
column, and takes into account the
alignment elements and the columnalign
attribute described
under mtable
(3.5.1 Table or Matrix
<mtable>
).
In an RTL context, switch left and right edges in the algorithm.
This algorithm should be verified by an implementation.
maligngroup
and malignmark
elements. The final rendering will be identical except for horizontal
shifts applied to each alignment group and/or table cell.malignmark
, otherwise the left edge),
and right edge are noted, allowing the width of
the group on each side of the alignment point (left and right) to be
determined. The sum of these two sidewidths, i.e. the sum of the widths to the left and right of the alignment point, will equal the width of the alignment group.
The widths of the table cells that contain no alignment groups were computed as part of the initial rendering, and may be different for each cell, and different from the single width used for cells containing alignment groups. The maximum of all the cell widths (for both kinds of cells) gives the width of the table column as a whole.
The position of each cell in the column is determined by the
applicable part of the value of the columnalign
attribute
of the innermost surrounding mtable
,
mtr
, or mtd
element that
has an explicit value for it, as described in the sections on those
elements. This may mean that the cells containing alignment groups
will be shifted within their column, in addition to their alignment
groups having been shifted within the cells as described above, but
since each such cell has the same width, it will be shifted the same
amount within the column, thus maintaining the vertical alignment of
the alignment points of the corresponding alignment groups in each
cell.
Mathematics used in the lower grades such as twodimensional addition, multiplication, and long division tends to be tabular in nature. However, the specific notations used varies among countries much more than for higher level math. Furthermore, elementary math often presents examples in some intermediate state and MathML must be able to capture these intermediate or intentionally missing partial forms. Indeed, these constructs represent memory aids or procedural guides, as much as they represent ‘mathematics’.
The elements used for basic alignments in elementary math are:
mstack
align rows of digits and operators
msgroup
groups rows with similar alignment
msrow
groups digits and operators into a row
msline
draws lines between rows of the stack
mscarries
annotates the following row with optional borrows/carries and/or crossouts
mscarry
a borrow/carry and/or crossout for a single digit
mlongdiv
specifies a divisor and a quotient for long division, along with a stack of the intermediate computations
mstack
and mlongdiv
are the parent elements for all elementary
math layout.
Any children of mstack
, mlongdiv
, and msgroup
,
besides msrow
, msgroup
, mscarries
and msline
,
are treated as if implicitly surrounded by an msrow
(see 3.6.4 Rows in Elementary Math <msrow>
for more details about rows).
Since the primary use of these stacking constructs is to
stack rows of numbers aligned on their digits,
and since numbers are always formatted lefttoright,
the columns of an mstack are always processed lefttoright;
the overall directionality in effect (i.e. the dir
attribute)
does not affect to the ordering of display of columns or carries in rows
and, in particular, does not affect the ordering of any operators within a row
(see 3.1.5 Directionality).
These elements are described in this section followed by examples of their use. In addition to twodimensional addition, subtraction, multiplication, and long division, these elements can be used to represent several notations used for repeating decimals.
A very simple example of twodimensional addition is shown below:
<mstack>
<mn>424</mn>
<msrow> <mo>+</mo> <mn>33</mn> </msrow>
<msline/>
</mstack>
Many more examples are given in 3.6.8 Elementary Math Examples.
mstack
is used to lay out rows of numbers that are aligned on each digit.
This is common in many elementary math notations such as 2D addition, subtraction,
and multiplication.
The children of an mstack
represent rows, or groups of them,
to be stacked each below the previous row; there can be any number of rows.
An msrow
represents a row;
an msgroup
groups a set of rows together
so that their horizontal alignment can be adjusted together;
an mscarries
represents a set of carries to be
applied to the following row;
an msline
represents a line separating rows.
Any other element is treated as if implicitly surrounded by msrow
.
Each row contains ‘digits’ that are placed into columns.
(see 3.6.4 Rows in Elementary Math <msrow>
for further details).
The stackalign
attribute together with
the position
and shift
attributes of msgroup
,
mscarries
, and msrow
determine
to which column a character belongs.
The width of a column is the maximum of the widths of each ‘digit’ in that
column — carries do not participate in the
width calculation; they are treated as having zero width.
If an element is too wide to fit into a column, it overflows into the adjacent
column(s) as determined by the charalign
attribute.
If there is no character in a column, its width is taken to be the width of a 0
in the current language (in many fonts, all digits have the same width).
The method for laying out an mstack is:
The ‘digits’ in a row are determined.
All of the digits in a row are initially aligned according to the stackalign
value.
Each row is positioned relative to that alignment based on the position
attribute (if any)
that controls that row.
The maximum width of the digits in a column are determined and
shorter and wider entries in that column are aligned according to
the charalign
attribute.
The width and height of the mstack element are computed based on the rows and columns. Any overflow from a column is not used as part of that computation.
The baseline of the mstack element is determined by the align
attribute.
mstack
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
align  ("top"  "bottom"  "center"  "baseline"  "axis"), rownumber?  baseline 
specifies the vertical alignment of the mstack with respect to its environment.
The legal values and their meanings are the same as that for mtable 's
align attribute.


stackalign  "left"  "center"  "right"  "decimalpoint"  decimalpoint 
specifies which column is used to horizontally align the rows.
For left , rows are aligned flush on the left;
similarly for right , rows are flush on the right;
for center , the middle column (or to the right of the middle, for an even number of columns)
is used for alignment.
Rows with nonzero position , or affected by a shift ,
are treated as if the
requisite number of empty columns were added on the appropriate side;
see 3.6.3 Group Rows with Similar Positions <msgroup> and 3.6.4 Rows in Elementary Math <msrow> .
For decimalpoint , the column used is the leftmost column in each
row that contains the decimalpoint character specified
using the decimalpoint attribute of mstyle (default ".").
If there is no decimalpoint character in the row, an implied decimal is assumed on
the right of the first number in the row;
see decimalpoint for a discussion
of decimalpoint .


charalign  "left"  "center"  "right"  right 
specifies the horizontal alignment of digits within a column.
If the content is larger than the column width, then it overflows the opposite side
from the alignment.
For example, for right , the content will overflow on the left side; for center,
it overflows on both sides.
This excess does not participate in the column width calculation, nor does it participate
in the overall width of the mstack .
In these cases, authors should take care to avoid collisions between column overflows.


charspacing  length  "loose"  "medium"  "tight"  medium 
specifies the amount of space to put between each column.
Larger spacing might be useful if carries are not placed above or are particularly
wide.
The keywords
loose , medium , and tight
automatically adjust spacing to when carries or other entries in a column are wide.
The three values allow authors to some flexibility in choosing what the layout looks
like
without having to figure out what values work well.
In all cases, the spacing between columns is a fixed amount and does not vary between
different columns.

Long division notation varies quite a bit around the world,
although the heart of the notation is often similar.
mlongdiv
is similar to mstack
and used to layout long division.
The first two children of mlongdiv
are the divisor and the result of the division, in that order.
The remaining children are treated as if they were children of mstack
.
The placement of these and the lines and separators used to display long division
are controlled
by the longdivstyle
attribute.
The result or divisor may be an elementary math element or may be an empty
<mrow/>
(the specific empty element
<none/>
used in MathML 3 is not used in this specification).
In particular, if msgroup
is used,
the elements in that group may or may not form their own mstack or be part of the
dividend's mstack
,
depending upon the value of the longdivstyle
attribute.
For example, in the US style for division,
the result is treated as part of the dividend's mstack
, but divisor is not.
MathML does not specify when the result and divisor form their own mstack
,
nor does it specify what should happen if msline
or other elementary math elements
are used for the result or divisor and they do not participate in the dividend's mstack
layout.
In the remainder of this section on elementary math, anything that is said about mstack
applies
to mlongdiv
unless stated otherwise.
mlongdiv
elements accept all of the attributes that mstack
elements
accept (including those specified in 3.1.9 Mathematics attributes common to presentation elements), along with the attribute listed below.
The values allowed for longdivstyle
are openended.
Conforming renderers may ignore any value they do not handle,
although renderers are encouraged to render as many of the values listed below as
possible.
Any rules drawn as part of division layout should be drawn using the color specified
by
mathcolor
.
Name  values  default 
longdivstyle  "lefttop"  "stackedrightright"  "mediumstackedrightright"  "shortstackedrightright"  "righttop"  "left/\right"  "left)(right"  ":right=right"  "stackedleftleft"  "stackedleftlinetop"  lefttop 
Controls the style of the long division layout. The names are meant as a rough mnemonic that describes the position of the divisor and result in relation to the dividend. 
See 3.6.8.3 Long Division for examples of how these notations are drawn. The values listed above are used for long division notations in different countries around the world:
lefttop
a notation that is commonly used in the United States, Great Britain, and elsewhere
stackedrightright
a notation that is commonly used in France and elsewhere
mediumrightright
a notation that is commonly used in Russia and elsewhere
shortstackedrightright
a notation that is commonly used in Brazil and elsewhere
righttop
a notation that is commonly used in China, Sweden, and elsewhere
left/\right
a notation that is commonly used in Netherlands
left)(right
a notation that is commonly used in India
:right=right
a notation that is commonly used in Germany
stackedleftleft
a notation that is commonly used in Arabic countries
stackedleftlinetop
a notation that is commonly used in Arabic countries
msgroup
is used to group rows inside of the mstack
and mlongdiv
elements
that have a similar position relative to the alignment of stack.
If not explicitly given, the children representing the stack in mstack
and mlongdiv
are treated as if they are implicitly surrounded by an msgroup
element.
msgroup
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
position  integer  0 
specifies the horizontal position of the rows within this group relative to
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values move each row towards the tens digit,
like multiplying by a power of 10,
effectively padding with empty columns on the right;
negative values move towards the ones digit,
effectively padding on the left.
The decimal point is counted as a column and should be taken into account for negative
values.


shift  integer  0 
specifies an incremental shift of position for successive children (rows or groups) within this group. The value is interpreted as with position, but specifies the position of each child (except the first) with respect to the previous child in the group. 
An msrow
represents a row in an mstack
.
In most cases it is implied by the context, but is useful
explicitly for putting multiple elements in a single row,
such as when placing an operator "+" or "" alongside a number
within an addition or subtraction.
If an mn
element is a child of msrow
(whether implicit or not), then the number is split into its digits
and the digits are placed into successive columns.
Any other element, with the exception of mstyle
is treated effectively
as a single digit occupying the next column.
An mstyle
is treated as if its children were
directly the children of the msrow
, but with their style affected
by the attributes of the mstyle
.
The empty element <mrow/>
may be used to create an empty column.
Note that a row is considered primarily as if it were a number,
which is always displayed lefttoright,
and so the directionality used to display the columns is always lefttoright;
textual bidirectionality within token elements (other than mn
) still applies,
as does the overall directionality within any children of the msrow
(which end up treated as single digits);
see 3.1.5 Directionality.
msrow
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
position  integer  0 
specifies the horizontal position of the rows within this group relative to
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values move each row towards the tens digit,
like multiplying by a power of 10,
effectively padding with empty columns on the right;
negative values move towards the ones digit,
effectively padding on the left.
The decimal point is counted as a column and should be taken into account for negative
values.

The mscarries
element is used for various annotations such as carries, borrows, and crossouts that
occur in elementary math.
The children are associated with elements in the following row of the mstack
.
It is an error for mscarries
to be the last element of an mstack
or mlongdiv
element. Each child of the mscarries
applies to the same column in the following row.
As these annotations are used to adorn what are treated as
numbers, the attachment of carries to columns proceeds from left to right;
the overall directionality does not apply to the ordering of the carries,
although it may apply to the contents of each carry;
see 3.1.5 Directionality.
Each child of mscarries
other than mscarry
or <mrow/>
is
treated as if implicitly surrounded by mscarry
;
the element <mrow/>
is used when no carry for a particular column is needed.
The element <none/>
was used in earlier MathML releases, but was equivalent to an empty <mrow/>
.
The mscarries
element sets displaystyle
to false
, and increments scriptlevel
by 1, so the children are
typically displayed in a smaller font. (See 3.1.6 Displaystyle and Scriptlevel.)
It also changes the default value of scriptsizemultiplier
.
The effect is that the inherited value of
scriptsizemultiplier
should still override the default value,
but the default value, inside mscarries
, should be 0.6
.
scriptsizemultiplier
can be set on the mscarries
element,
and the value should override the inherited value as usual.
If two rows of carries are adjacent to each other,
the first row of carries annotates the second (following) row as if the second row
had
location
=n
.
This means that the second row, even if it does not draw,
visually uses some (undefined by this specification) amount of space when displayed.
mscarries
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
position  integer  0 
specifies the horizontal position of the rows within this group relative to
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
The interpretation of the value is the same as position for msgroup or msrow ,
but it alters the association of each carry with the column below.
For example, position =1 would cause the rightmost carry to be associated with
the second digit column from the right.


location  "w"  "nw"  "n"  "ne"  "e"  "se"  "s"  "sw"  n 
specifies the location of the carry or borrow relative to the character below it in the associated column. Compass directions are used for the values; the default is to place the carry above the character.  
crossout  ("none"  "updiagonalstrike"  "downdiagonalstrike"  "verticalstrike"  "horizontalstrike")*  none 
specifies how the column content below each carry is "crossed out";
one or more values may be given and all values are drawn.
If none is given with other values, it is ignored.
See 3.6.8 Elementary Math Examples for examples of the different values.
The crossout is only applied for columns which have a corresponding
mscarry .
The crossouts should be drawn using the color specified by mathcolor .


scriptsizemultiplier  number  inherited (0.6) 
specifies the factor to change the font size by.
See 3.1.6 Displaystyle and Scriptlevel for a description of how this works with the scriptsize attribute.

mscarry
is used inside of mscarries
to
represent the carry for an individual column.
A carry is treated as if its width were zero; it does not participate in
the calculation of the width of its corresponding column;
as such, it may extend beyond the column boundaries.
Although it is usually implied, the element may be used explicitly to override the
location
and/or crossout
attributes of
the containing mscarries
.
It may also be useful with <mrow/>
as its content in order
to display no actual carry, but still enable a crossout
due to the enclosing mscarries
to be drawn for the given column.
The mscarry
element accepts the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
Name  values  default 
location  "w"  "nw"  "n"  "ne"  "e"  "se"  "s"  "sw"  inherited 
specifies the location of the carry or borrow relative to the character in the corresponding column in the row below it. Compass directions are used for the values.  
crossout  ("none"  "updiagonalstrike"  "downdiagonalstrike"  "verticalstrike"  "horizontalstrike")*  inherited 
specifies how the column content associated with the carry is "crossed out";
one or more values may be given and all values are drawn.
If none is given with other values, it is essentially ignored.
The crossout should be drawn using the color specified by mathcolor .

msline
draws a horizontal line inside of an mstack
element.
The position, length, and thickness of the line are specified as attributes.
If the length is specified, the line is positioned and drawn as if it were a number
with the given number of digits.
msline
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
The line should be drawn using the color specified by mathcolor
.
Name  values  default 
position  integer  0 
specifies the horizontal position of the rows within this group relative to
the position determined by the containing msgroup (according to its
position and shift attributes).
The resulting position value is relative to the column specified by stackalign of the containing mstack or mlongdiv .
Positive values move towards the tens digit (like multiplying by a power of 10);
negative values move towards the ones digit.
The decimal point is counted as a column and should be taken into account for negative
values.
Note that since the default line length spans the entire mstack ,
the position has no effect unless the length is specified as nonzero.


length  unsignedinteger  0 
Specifies the number of columns that should be spanned by the line.
A value of '0' (the default) means that all columns in
the row are spanned (in which case position and stackalign have no effect).


leftoverhang  length  0 
Specifies an extra amount that the line should overhang on the left of the leftmost column spanned by the line.  
rightoverhang  length  0 
Specifies an extra amount that the line should overhang on the right of the rightmost column spanned by the line.  
mslinethickness  length  "thin"  "medium"  "thick"  medium 
Specifies how thick the line should be drawn.
The line should have height=0, and depth=mslinethickness so that the top
of the msline is on the baseline of the surrounding context (if any).
(See 3.3.2 Fractions <mfrac> for discussion of the thickness keywords
medium , thin and thick .)

Twodimensional addition, subtraction, and multiplication typically involve numbers, carries/borrows, lines, and the sign of the operation.
Below is the example shown at the start of the section:
the digits inside the mn
elements each occupy a column as does the "+".
<mrow/>
is used to fill in the column under the "4" and make the "+" appear to the left of all of the operands.
Notice that no attributes are given on msline
causing it to span all of the columns.
<mstack>
<mn>424</mn>
<msrow> <mo>+</mo> <mrow/> <mn>33</mn> </msrow>
<msline/>
</mstack>
The next example illustrates how to put an operator on the right. Placing the operator on the right is standard in the Netherlands and some other countries. Notice that although there are a total of four columns in the example, because the default alignment is on the implied decimal point to the right of the numbers, it is not necessary to pad or shift any row.
<mstack>
<mn>123</mn>
<msrow> <mn>456</mn> <mo>+</mo> </msrow>
<msline/>
<mn>579</mn>
</mstack>
The following two examples illustrate the use of mscarries
,
mscarry
and using <mrow/>
to fill in a column.
The examples also illustrate two different ways of displaying a borrow.
<mstack>
<mscarries crossout='updiagonalstrike'>
<mn>2</mn> <mn>12</mn> <mscarry crossout='none'> <mrow/> </mscarry>
</mscarries>
<mn>2,327</mn>
<msrow> <mo></mo> <mn> 1,156</mn> </msrow>
<msline/>
<mn>1,171</mn>
</mstack>
<mstack>
<mscarries location='nw'>
<mrow/>
<mscarry crossout='updiagonalstrike' location='n'> <mn>2</mn> </mscarry>
<mn>1</mn>
<mrow/>
</mscarries>
<mn>2,327</mn>
<msrow> <mo></mo> <mn> 1,156</mn> </msrow>
<msline/>
<mn>1,171</mn>
</mstack>
The MathML for the second example uses mscarry
because a crossout should only happen on a single column:
The next example of subtraction shows a borrowed
amount that is underlined (the example is from a Swedish
source).
There are two things to notice:
an menclose
is used in the carry, and <mrow/>
is used for
the empty element so that mscarry
can be used to create a crossout.
<mstack>
<mscarries>
<mscarry crossout='updiagonalstrike'><mrow/></mscarry>
<menclose notation='bottom'> <mn>10</mn> </menclose>
</mscarries>
<mn>52</mn>
<msrow> <mo></mo> <mn> 7</mn> </msrow>
<msline/>
<mn>45</mn>
</mstack>
Below is a simple multiplication example that illustrates the use of msgroup
and
the shift
attribute. The first msgroup
is implied and doesn't
change the layout.
The second msgroup
could also be removed, but msrow
would be needed for last two children.
They msrow
would need to set the
position
or shift
attributes,
or would add <mrow/>
elements to pad the digits on the right.
<mstack>
<msgroup>
<mn>123</mn>
<msrow><mo>×</mo><mn>321</mn></msrow>
</msgroup>
<msline/>
<msgroup shift="1">
<mn>123</mn>
<mn>246</mn>
<mn>369</mn>
</msgroup>
<msline/>
</mstack>
The following is a more complicated example of multiplication that has multiple rows of carries. It also (somewhat artificially) includes commas (",") as digit separators. The encoding includes these separators in the spacing attribute value, along nonASCII values.
<mstack>
<mscarries><mn>1</mn><mn>1</mn><mrow/></mscarries>
<mscarries><mn>1</mn><mn>1</mn><mrow/></mscarries>
<mn>1,234</mn>
<msrow><mo>×</mo><mn>4,321</mn></msrow>
<msline/>
<mscarries position='2'>
<mn>1</mn>
<mrow/>
<mn>1</mn>
<mn>1</mn>
<mn>1</mn>
<mrow/>
<mn>1</mn>
</mscarries>
<msgroup shift="1">
<mn>1,234</mn>
<mn>24,68</mn>
<mn>370,2</mn>
<msrow position="1"> <mn>4,936</mn> </msrow>
</msgroup>
<msline/>
<mn>5,332,114</mn>
</mstack>
The notation used for long division varies considerably among countries. Most notations share the common characteristics of aligning intermediate results and drawing lines for the operands to be subtracted. Minus signs are sometimes shown for the intermediate calculations, and sometimes they are not. The line that is drawn varies in length depending upon the notation. The most apparent difference among the notations is that the position of the divisor varies, as does the location of the quotient, remainder, and intermediate terms.
The layout used is controlled by the longdivstyle
attribute. Below are examples for the values listed in 3.6.2.2 Attributes.
lefttop 
stackedrightright 
mediumstackedrightright 
shortstackedrightright 
righttop 





left/\right 
left)(right 
:right=right 
stackedleftleft 
stackedleftlinetop 





The MathML for the first example is shown below. It illustrates the use of nested
msgroup
s and how the position
is calculated in those usages.
<mlongdiv longdivstyle="lefttop">
<mn> 3 </mn>
<mn> 435.3</mn>
<mn> 1306</mn>
<msgroup position="2" shift="1">
<msgroup>
<mn> 12</mn>
<msline length="2"/>
</msgroup>
<msgroup>
<mn> 10</mn>
<mn> 9</mn>
<msline length="2"/>
</msgroup>
<msgroup>
<mn> 16</mn>
<mn> 15</mn>
<msline length="2"/>
<mn> 1.0</mn> <! aligns on '.', not the right edge ('0') >
</msgroup>
<msgroup position='1'> <! extra shift to move to the right of the "." >
<mn> 9</mn>
<msline length="3"/>
<mn> 1</mn>
</msgroup>
</msgroup>
</mlongdiv>
With the exception of the last example,
the encodings for the other examples are the same except that the values for
longdivstyle
differ and that a "," is used instead of a "." for the decimal point.
For the last example, the only difference from the other examples besides a different
value for
longdivstyle
is that Arabic numerals have been used in place of Latin numerals,
as shown below.
<mstyle decimalpoint="٫">
<mlongdiv longdivstyle="stackedleftlinetop">
<mn> ٣ </mn>
<mn> ٤٣٥٫٣</mn>
<mn> ١٣٠٦</mn>
<msgroup position="2" shift="1">
<msgroup>
<mn> ١٢</mn>
<msline length="2"/>
</msgroup>
<msgroup>
<mn> ١٠</mn>
<mn> ٩</mn>
<msline length="2"/>
</msgroup>
<msgroup>
<mn> ١٦</mn>
<mn> ١٥</mn>
<msline length="2"/>
<mn> ١٫٠</mn>
</msgroup>
<msgroup position='1'>
<mn> ٩</mn>
<msline length="3"/>
<mn> ١</mn>
</msgroup>
</msgroup>
</mlongdiv>
</mstyle>
Decimal numbers that have digits that repeat infinitely such as 1/3
(.3333...) are represented using several notations. One common notation
is to put a horizontal line over the digits that repeat (in Portugal an underline
is used).
Another notation involves putting dots over the digits that repeat.
The MathML for these involves using mstack
, msrow
, and msline
in a straightforward manner.
These notations are shown below:
<mstack stackalign="right">
<msline length="1"/>
<mn> 0.3333 </mn>
</mstack>
<mstack stackalign="right">
<msline length="6"/>
<mn> 0.142857 </mn>
</mstack>
<mstack stackalign="right">
<mn> 0.142857 </mn>
<msline length="6"/>
</mstack>
<mstack stackalign="right">
<msrow> <mo>.</mo> <mrow/><mrow/><mrow/><mrow/> <mo>.</mo> </msrow>
<mn> 0.142857 </mn>
</mstack>
The maction
element provides a mechanism for binding actions to expressions.
This element accepts any
number of subexpressions as arguments and the type of action that should happen
is controlled by the actiontype
attribute.
MathML 3 predefined the four actions:
toggle
,
statusline
,
statusline
, and
input
.
However, because the ability to implement any action depends very strongly on the platform,
MathML 4 no longer predefines what these actions do.
Furthermore, in the web environment events connected to javascript to perform actions are a
more powerful solution, although maction
provides a
convenient wrapper element on which to attach such an event.
Linking to other elements, either locally within the math
element or to some URL,
is not handled by maction
.
Instead, it is handled by adding a link directly on a MathML element as specified
in 7.4.4 Linking.
maction
elements accept the attributes listed
below in addition to those specified in 3.1.9 Mathematics attributes common to presentation elements.
By default, MathML applications that do not recognize the specified
actiontype
, or if the actiontype
attribute
is not present, should render the selected subexpression as
defined below. If no selected subexpression exists, it is a MathML
error; the appropriate rendering in that case is as described in
D.2 Handling of Errors.
Name  values  default 
actiontype  string  
Specifies what should happen for this element. The values allowed are openended. Conforming renderers may ignore any value they do not handle.  
selection  positiveinteger  1 
Specifies which child should be used for viewing. Its value should be between 1 and
the number of
children of the element. The specified child is referred to as the selected subexpressionof the maction element. If the value specified is out of range, it is an error. When the
selection attribute is not specified (including for
action types for which it makes no sense), its default value is 1, so
the selected subexpression will be the first subexpression.

If a MathML application responds to a user command to copy a MathML subexpression
to
the environment's clipboard
(see 7.3 Transferring MathML), any maction
elements present in what is copied should
be given selection
values that correspond to their selection
state in the MathML rendering at the time of the copy command.
When a MathML application receives a mouse event that may be processed by two or more nested maction elements, the innermost maction element of each action type should respond to the event.
The actiontype
values are openended. If another value is given and it requires additional attributes, which should begin with "data" or in XML they may be in a different namespace.
In the example,
nonstandard data attributes are being used to pass
additional information to renderers that support them.
The datacolor
attributes
might change the color of the characters in the presentation, while the
databackground
attribute might change the color of the background
behind the characters.
MathML uses the semantics
element to allow specifying semantic annotations to
presentation MathML elements; these can be content MathML or other notations. As
such,
semantics
should be considered part of both presentation MathML and content
MathML. All MathML processors should process the semantics
element, even if they
only process one of those subsets.
In semantic annotations a presentation MathML expression is typically the first child
of the semantics
element. However, it can also be given inside of an
annotationxml
element inside the semantics
element. If it is part of an
annotationxml
element, then
encoding
=application/mathmlpresentation+xml
or
encoding
=MathMLPresentation
may be used and presentation
MathML processors should use this value for the presentation.
See 6. Annotating MathML: semantics for more details about the
semantics
and annotationxml
elements.
There are currently "sample" renderings. Let's make this use intent
.
The purpose of Content Markup is to provide an explicit encoding of the underlying mathematical meaning of an expression, rather than any particular notation for the expression. Mathematical notation is at times ambiguous, contextdependent, and varies from community to community. In many cases, it is preferable to work directly with the underlying, formal, mathematical objects. Content Markup provides a rigorous, extensible semantic framework and a markup language for this purpose.
By encoding the underlying mathematical structure explicitly, without regard to how it is presented, it is possible to interchange information more precisely between systems that semantically process mathematical objects. Important application areas include computer algebra systems, automatic reasoning systems, industrial and scientific applications, multilingual translation systems, mathematical search, automated scoring of online assessments, and interactive textbooks.
This chapter presents an overview of basic concepts used to define Content Markup, describes a core collection of elements that comprise Strict Content Markup, and defines a full collection of elements to support common mathematical idioms. Strict Content Markup encodes general expression trees in a semantically rigorous way, while the full set of Content MathML elements provides backwardcompatibility with previous versions of Content Markup. The correspondence between full Content Markup and Strict Content Markup is defined in F. The Strict Content MathML Transformation, which details an algorithm to translate arbitrary Content Markup into Strict Content Markup.
Content MathML represents mathematical objects as expression
trees. In general, an expression tree is constructed by applying
an operator to a sequence of subexpressions. For example, the sum
x+y
can be constructed
as the application of the addition operator to two arguments
x and y, and the expression
cos(π)
as the application of the cosine function to the
number π.
The terminal nodes in an expression tree represent basic mathematical objects such as numbers, variables, arithmetic operations, and so on. The internal nodes in the tree represent function application or other mathematical constructions that build up compound objects.
MathML defines a relatively small number of commonplace mathematical constructs, chosen to be sufficient in a wide range of applications. In addition, it provides a mechanism to refer to concepts outside of the collection it defines, allowing them to be represented as well.
The defined set of content elements is designed to be adequate for simple coding of formulas typically used from kindergarten through the first two years of college in the United States, that is, up to ALevel or Baccalaureate level in Europe.
The primary role of the MathML content element set is to encode the mathematical structure of an expression independent of the notation used to present it. However, rendering issues cannot be ignored. There are many different approaches to render Content MathML formulae, ranging from native implementations of the MathML elements, to declarative notation definitions, to XSLT style sheets. Because rendering requirements for Content MathML vary widely, MathML does not provide a normative rendering specification. Instead, typical renderings are suggested by way of examples given using presentation markup.
The basic building blocks of Content MathML expressions are numbers, identifiers, and symbols. These building blocks are combined using function application and binding operators.
In the expression
$x+2$
,
the numeral $2$
represents a number with a fixed value. Content MathML uses the
cn
element to represent numerical quantities. The identifier
$x$
is a mathematical variable, that is, an identifier
that represents a quantity with no predetermined value. Content MathML
uses the
ci
element to represent variable identifiers.
The plus sign is an identifier that represents a fixed, externally
defined object, namely, the addition function. Such an identifier is
called a symbol, to distinguish it from a variable. Common
elementary functions and operators are all symbols in this sense.
Content MathML uses the
csymbol
element to represent symbols.
The fundamental way to combine numbers, variables, and symbols
is function application. Content MathML distinguishes between the
function itself (which may be a symbol such as the sine function,
a variable such as f, or some other expression)
and the result of applying the function to its arguments. The
apply
element groups the function with its arguments syntactically, and
represents the expression that results from applying the function
to its arguments.
In an expression, variables may be described as bound or
free variables. Bound variables have a special role within
the scope of a binding expression, and may be renamed consistently
within that scope without changing the meaning of the expression.
Free variables are those that are not bound within an expression.
Content MathML differentiates between the application of a function
to a free variable (e.g. f(x))
and an operation that binds a variable within a binding scope. The
bind
element is used to delineate the binding scope of a bound variable
and to group the binding operator with its bound variables, which
are supplied using the
bvar
element.
In Strict Content markup, the only way to perform variable binding
is to use the bind
element. In nonStrict
Content markup, other markup elements are provided that more closely
resemble wellknown idiomatic notations, such as limit
style
notations for sums and integrals. These constructs may implicitly
bind variables, such as the variable of integration, or the index variable
in a sum. MathML uses the term qualifier element to refer to
those elements used to represent the auxiliary data required by these
constructs.
Expressions involving qualifiers follow one of a small number of idiomatic patterns, each of which applies to a class of similar binding operators. For example, sums and products are in the same class because they use index variables following the same pattern. The Content MathML operator classes are described in detail in 4.3.4 Operator Classes.
Beginning in MathML 3, Strict Content MathML is defined as a minimal subset of Content MathML that is sufficient to represent the meaning of mathematical expressions using a uniform structure. The full Content MathML element set retains backward compatibility with MathML 2, and strikes a pragmatic balance between verbosity and formality.
Content MathML provides a considerable number of predefined functions
encoded as empty elements (e.g. sin
,
log
, etc.) and a variety of constructs for
forming compound objects (e.g. set
,
interval
, etc.). In contrast, Strict
Content MathML represents all known functions using a single element
(csymbol
) with an attribute that points
to its definition in an extensible content dictionary, and uses only
apply
and bind
elements to build up compound expressions. Token elements such as
cn
and ci
are
considered part of Strict Content MathML, but with a more restricted set
of attributes and with content restricted to text.
The formal semantics of Content MathML expressions are given by specifying equivalent Strict Content MathML expressions, which all have formal semantics defined in terms of content dictionaries. The exact correspondence between each nonStrict Content MathML structure and its Strict Content MathML equivalent is described in terms of rewrite rules that are used as part of the transformation algorithm given in F. The Strict Content MathML Transformation.
The algorithm described in F. The Strict Content MathML Transformation is complete in the sense that it gives every Content MathML expression a specific meaning in terms of a Strict Content MathML expression. In some cases, it gives a specific strict interpretation to an expression whose meaning was not sufficiently specified in MathML 2. The goal of this algorithm is to be faithful to natural mathematical intuitions, however, some edge cases may remain where the specific interpretation given by the algorithm may be inconsistent with earlier expectations.
A conformant MathML processor need not implement this algorithm. The existence of these transformation rules does not imply that a system must treat equivalent expressions identically. In particular, systems may give different presentation renderings for expressions that the transformation rules imply are mathematically equivalent. In general, Content MathML does not define any expectations for the computational behavior of the expressions it encodes, including, but not limited to, the equivalence of any specific expressions.
Strict Content MathML is designed to be compatible with OpenMath, a standard for representing formal mathematical objects and semantics. Strict Content MathML is an XML encoding of OpenMath Objects in the sense of [OpenMath]. The following table gives the correspondence between Strict Content MathML elements and their OpenMath equivalents.
Strict Content MathML  OpenMath 
cn 
OMI , OMF 
csymbol 
OMS 
ci 
OMV 
cs 
OMSTR 
apply 
OMA 
bind 
OMBIND 
bvar 
OMBVAR 
share 
OMR 
semantics 
OMATTR 
annotation ,
annotationxml 
OMATP , OMFOREIGN 
cerror 
OME 
cbytes 
OMB 
Any method to formalize the meaning of mathematical expressions
must be extensible, that is, it must provide the ability to define
new functions and symbols to expand the domain of discourse. Content
MathML uses the
csymbol
element to represent new symbols, and uses Content Dictionaries
to describe their mathematical semantics. The association between a symbol
and its semantic description is accomplished using the attributes of the
csymbol
element to point to the definition
of the symbol in a Content Dictionary.
The correspondence between operator elements in Content MathML and symbol definitions in Content Dictionaries is given in E.3 The Content MathML Operators. These definitions for predefined MathML operator symbols refer to Content Dictionaries developed by the OpenMath Society [OpenMath] in conjunction with the W3C Math Working Group. It is important to note that this information is informative, not normative. In general, the precise mathematical semantics of predefined symbols are not fully specified by the MathML Recommendation, and the only normative statements about symbol semantics are those present in the text of this chapter. The semantic definitions provided by the OpenMath Content Dictionaries are intended to be sufficient for most applications, and are generally compatible with the semantics specified for analogous constructs in this Recommendation. However, in contexts where highly precise semantics are required (e.g. communication between computer algebra systems, within formal systems such as theorem provers, etc.) it is the responsibility of the relevant community of practice to verify, extend or replace definitions provided by OpenMath Content Dictionaries as appropriate.
In this section we will present the elements for encoding the structure of content MathML expressions. These elements are the only ones used for the Strict Content MathML encoding. Concretely, we have
basic expressions, i.e. Numbers, string literals, encoded bytes, Symbols, and Identifiers.
derived expressions, i.e. function applications and binding expressions, and
Full Content MathML allows further elements presented in 4.3 Content MathML for Specific Structures and 4.3 Content MathML for Specific Structures, and allows a richer content model presented in this section. Differences in Strict and nonStrict usage of are highlighted in the sections discussing each of the Strict element below.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Cn  Cn  
Attributes  CommonAtt , type 
CommonAtt , DefEncAtt , type ?, base ? 

type Attribute Values 
integer 
real 
double 
hexdouble

integer 
real 
double 
hexdouble 
enotation 
rational 
complexcartesian 
complexpolar 
constant  text

default is real  
base Attribute Values 
integer

default is 10  
Content  text  (text  mglyph  sep  PresentationExpression )* 
The cn
element is the Content MathML element used to
represent numbers. Strict Content MathML supports integers, real numbers,
and double precision floating point numbers. In these types of numbers,
the content of cn
is text. Additionally, cn
supports rational numbers and complex numbers in which the different
parts are separated by use of the sep
element. Constructs
using sep
may be rewritten in Strict Content MathML as
constructs using apply
as described below.
The type
attribute specifies which kind of number is
represented in the cn
element. The default value is
real
. Each type implies that the content be of
a certain form, as detailed below.
The default rendering of the text content of cn
is the same as that of the Presentation element mn
, with suggested variants in the
case of attributes or sep
being used, as listed below.
In Strict Content MathML, the type
attribute is mandatory, and may only take the values
integer
, real
, hexdouble
or
double
:
An integer is represented by an optional sign followed by a string of
one or more decimal digits
.
A real number is presented in radix notation. Radix notation consists of an
optional sign (+
or 
) followed by a string of
digits possibly separated into an integer and a fractional part by a
decimal point. Some examples are 0.3, 1, and 31.56.
This type is used to mark up those doubleprecision
floating point numbers that can be represented in the IEEE 754
standard format [IEEE754]. This includes a subset of the (mathematical) real
numbers, negative zero, positive and negative real infinity
and a set of not a number
values. The lexical rules for
interpreting the text content of a cn
as an IEEE
double are specified by Section
3.1.2.5 of XML Schema Part 2: Datatypes Second Edition
[XMLSchemaDatatypes]. For example, 1E4, 1267.43233E12, 12.78e2,
12, 0, 0 and INF are all valid doubles in this format.
This type is used to directly represent the 64 bits of an
IEEE 754 doubleprecision floating point number as a 16 digit
hexadecimal number. Thus the number represents mantissa, exponent, and sign
from lowest to highest bits using a least significant byte ordering.
This consists of a string of 16 digits 09, AF.
The following example
represents a NaN value. Note that certain IEEE doubles, such as the
NaN in the example, cannot be represented in the lexical format for
the double
type.
<cn type="hexdouble">7F800000</cn>
Sample Presentation
<mn>0x7F800000</mn>
The base
attribute is used to specify how the content is
to be parsed. The attribute value is a base 10 positive integer
giving the value of base in which the text content of the cn
is to be interpreted. The base
attribute should only be
used on elements with type integer
or
real
. Its use on cn
elements of other type
is deprecated. The default value for base
is
10
.
Additional values for the type
attribute element for supporting
enotations for real numbers, rational numbers, complex numbers and selected important
constants. As with the integer
, real
,
double
and hexdouble
types, each of these types
implies that the content be of a certain form. If the type
attribute is
omitted, it defaults to real
.
Integers can be represented with respect to a base different from
10: If base
is present, it specifies (in base 10) the base for the digit encoding.
Thus base
='16' specifies a hexadecimal
encoding. When base
> 10, Latin letters (AZ, az) are used in
alphabetical order as digits. The case of letters used as digits is not
significant. The following example encodes the base 10 number 32736.
<cn base="16">7FE0</cn>
Sample Presentation
<msub><mn>7FE0</mn><mn>16</mn></msub>
When base
> 36, some integers cannot be represented using
numbers and letters alone. For example, while
<cn base="1000">10F</cn>
arguably represents the number written in base 10 as 1,000,015, the number
written in base 10 as 1,000,037 cannot be represented using letters and
numbers alone when base
is 1000. Consequently, support
for additional characters (if any) that may be used for digits when base
> 36 is application specific.
Real numbers can be represented with respect to a base
different than 10. If a base
attribute is present, then the digits are
interpreted as being digits computed relative to that base (in the same way as
described for type integer
).
A real number may be presented in scientific notation using this type. Such
numbers have two parts (a significand and an exponent)
separated by a <sep/>
element. The
first part is a real number, while the
second part is an integer exponent indicating a power of the base.
For example, <cn type="enotation">12.3<sep/>5</cn>
represents $12.3\times {10}^{5}$. The default presentation of this example is
12.3e5. Note that this type is primarily useful for backwards compatibility with
MathML 2, and in most cases, it is preferable to use the double
type, if the number to be represented is in the range of IEEE doubles:
A rational number is given as two integers to be used as the numerator and
denominator of a quotient. The numerator and denominator are
separated by <sep/>
.
<cn type="rational">22<sep/>7</cn>
Sample Presentation
<mrow><mn>22</mn><mo>/</mo><mn>7</mn></mrow>
A complex cartesian number is given as two numbers specifying the real and
imaginary parts. The real and imaginary parts are separated
by the <sep/>
element, and each part has
the format of a real number as described above.
<cn type="complexcartesian"> 12.3 <sep/> 5 </cn>
Sample Presentation
<mrow>
<mn>12.3</mn><mo>+</mo><mn>5</mn><mo>⁢<!InvisibleTimes></mo><mi>i</mi>
</mrow>
A complex polar number is given as two numbers specifying
the magnitude and angle. The magnitude and angle are separated
by the <sep/>
element, and each part has
the format of a real number as described above.
<cn type="complexpolar"> 2 <sep/> 3.1415 </cn>
Sample Presentation
<mrow>
<mn>2</mn>
<mo>⁢<!InvisibleTimes></mo>
<msup>
<mi>e</mi>
<mrow><mi>i</mi><mo>⁢<!InvisibleTimes></mo><mn>3.1415</mn></mrow>
</msup>
</mrow>
<mrow>
<mi>Polar</mi>
<mo>⁡<!ApplyFunction></mo>
<mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>3.1415</mn><mo>)</mo></mrow>
</mrow>
If the value type
is constant
,
then the content should be a Unicode representation of a
wellknown constant. Some important constants and their
common Unicode representations are listed below.
This cn
type is primarily for backward
compatibility with MathML 1.0. MathML 2.0 introduced many
empty elements, such as <pi/>
to
represent constants, and using these representations or
a Strict csymbol representation is preferred.
In addition to the additional values of the type attribute, the
content of cn
element can contain (in addition to the
sep
element allowed in Strict Content MathML) mglyph
elements to refer to characters not currently available in Unicode, or
a general presentation construct (see 3.1.8 Summary of Presentation Elements),
which is used for rendering (see 4.1.2 Content Expressions).
If a base
attribute is present, it specifies the base used for the digit
encoding of both integers. The use of base
with
rational
numbers is deprecated.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Ci  Ci 
Attributes  CommonAtt , type ? 
CommonAtt , DefEncAtt , type ? 
type Attribute Values 
integer 
rational 
real 
complex 
complexpolar 
complexcartesian 
constant 
function 
vector 
list 
set 
matrix

string 
Qualifiers 
BvarQ ,
DomainQ ,
degree ,
momentabout ,
logbase


Content  text  text  mglyph  PresentationExpression 
Content MathML uses the ci
element (mnemonic for content
identifier
) to construct a variable. Content identifiers
represent mathematical variables
which have
properties, but no fixed value. For example, x and y are variables
in the expression x+y
, and the variable
x would be represented as
<ci>x</ci>
In MathML, variables are distinguished from symbols, which have fixed, external definitions, and are represented by the csymbol element.
After white space normalization the content of a ci
element is interpreted as a
name that identifies it. Two variables are considered equal, if and only if their
names
are identical and in the same scope (see 4.2.6 Bindings and Bound Variables <bind>
and <bvar>
for a
discussion).
The ci
element uses the type
attribute to specify the basic type of
object that it represents. In Strict Content MathML, the set of permissible values
is
integer
, rational
, real
,
complex
, complexpolar
,
complexcartesian
, constant
, function
,
vector
, list
, set
, and matrix
. These values correspond
to the symbols
integer_type,
rational_type,
real_type,
complex_polar_type,
complex_cartesian_type,
constant_type,
fn_type,
vector_type,
list_type,
set_type, and
matrix_type in the
mathmltypes Content Dictionary: In this sense the following two expressions are considered equivalent:
<ci type="integer">n</ci>
<semantics>
<ci>n</ci>
<annotationxml cd="mathmltypes" name="type" encoding="MathMLContent">
<csymbol cd="mathmltypes">integer_type</csymbol>
</annotationxml>
</semantics>
Note that complex
should be considered
an alias for complexcartesian
and rewritten to the
same complex_cartesian_type
symbol. It is perhaps a more natural type name for use with
ci
as the distinction between cartesian and polar form really
only affects the interpretation of literals encoded with cn
.
The ci
element allows any string value for the type
attribute, in particular any of the names of the MathML container elements or their
type
values.
For a more advanced treatment of types, the type
attribute is
inappropriate. Advanced types require significant structure of their own (for example,
vector(complex)) and are probably best constructed as mathematical objects and
then associated with a MathML expression through use of the semantics
element. See [MathMLTypes] for more examples.
If the content of a ci
element consists of Presentation MathML, that
presentation is used. If no such tagging is supplied then the text
content is rendered as if it were the content of an mi
element. If an
application supports bidirectional text rendering, then the rendering follows the
Unicode bidirectional rendering.
The type
attribute can be interpreted to
provide rendering information. For example in
<ci type="vector">V</ci>
a renderer could display a bold V for the vector.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Csymbol  Csymbol 
Attributes  CommonAtt , cd 
CommonAtt , DefEncAtt , type ?, cd ? 
Content  SymbolName 
text  mglyph  PresentationExpression 
Qualifiers  BvarQ , DomainQ , degree , momentabout , logbase 
A csymbol
is used to refer to a specific,
mathematicallydefined concept with an external definition. In the
expression x+y
, the plus sign is
a symbol since it has a specific, external definition, namely the addition function.
MathML 3 calls such an identifier a
symbol. Elementary functions and common mathematical
operators are all examples of symbols. Note that the term
symbol
is used here in an abstract sense and has no
connection with any particular presentation of the construct on screen
or paper.
The csymbol
identifies the specific mathematical concept
it represents by referencing its definition via attributes.
Conceptually, a reference to an external definition is merely a URI,
i.e. a label uniquely identifying the definition. However, to be
useful for communication between user agents, external definitions
must be shared.
For this reason, several longstanding efforts have been organized to develop systematic, public repositories of mathematical definitions. Most notable of these, the OpenMath Society repository of Content Dictionaries (CDs) is extensive, open and active. In MathML 3, OpenMath CDs are the preferred source of external definitions. In particular, the definitions of predefined MathML 3 operators and functions are given in terms of OpenMath CDs.
MathML 3 provides two mechanisms for referencing external definitions or content
dictionaries. The first, using the cd
attribute, follows conventions
established by OpenMath specifically for referencing CDs. This is the
form required in Strict Content MathML. The second, using the
definitionURL
attribute, is backward compatible with MathML 2, and can be used
to reference CDs or any other source of definitions that can be
identified by a URI. It is described in the following section.
When referencing OpenMath CDs, the preferred method is to use the cd
attribute as follows. Abstractly, OpenMath symbol definitions are identified by a
triple
of values: a symbol name, a CD name, and a CD base,
which is a URI that disambiguates CDs of the same name. To associate such a triple
with a
csymbol
, the content of the csymbol
specifies the symbol name, and the
name of the Content Dictionary is given using the cd
attribute. The CD base is
determined either from the document embedding the math
element which contains the
csymbol
by a mechanism given by the embedding document format, or by system
defaults, or by the cdgroup
attribute, which is optionally specified on the
enclosing math
element; see 2.2.1 Attributes. In the absence
of specific information http://www.openmath.org/cd
is assumed as the CD base
for all csymbol
elements annotation
, and annotationxml
. This
is the CD base for the collection of standard CDs maintained by the OpenMath Society.
The cdgroup
specifies a URL to an OpenMath CD Group file. For a detailed
description of the format of a CD Group file, see Section 4.4.2 (CDGroups)
in [OpenMath]. Conceptually, a CD group file is a list of
pairs consisting of a CD name, and a corresponding CD base. When a csymbol
references a CD name using the cd
attribute, the name is looked up in the CD
Group file, and the associated CD base value is used for that csymbol
. When a CD
Group file is specified, but a referenced CD name does not appear in the group file,
or
there is an error in retrieving the group file, the referencing csymbol
is not
defined. However, the handling of the resulting error is not defined, and is the
responsibility of the user agent.
While references to external definitions are URIs, it is strongly recommended that CD files be retrievable at the location obtained by interpreting the URI as a URL. In particular, other properties of the symbol being defined may be available by inspecting the Content Dictionary specified. These include not only the symbol definition, but also examples and other formal properties. Note, however, that there are multiple encodings for OpenMath Content Dictionaries, and it is up to the user agent to correctly determine the encoding when retrieving a CD.
In addition to the forms described above, the csymbol
and element can contain
mglyph
elements to refer to characters not currently available in Unicode, or a
general presentation construct (see 3.1.8 Summary of Presentation Elements), which is used for
rendering (see 4.1.2 Content Expressions). In this case, when
writing to Strict Content MathML, the csymbol should be treated as a
ci
element, and rewritten using Rewrite: ci presentation mathml.
External definitions (in OpenMath CDs or elsewhere) may also be specified directly
for
a csymbol
using the definitionURL
attribute. When used to reference
OpenMath symbol definitions, the abstract triple of (symbol name, CD name, CD base)
is
mapped to a fullyqualified URI as follows:
URI = cdbase + '/' + cdname + '#' + symbolname
For example,
(plus, arith1, http://www.openmath.org/cd)
is mapped to
http://www.openmath.org/cd/arith1#plus
The resulting URI is specified as the value of the definitionURL
attribute.
This form of reference is useful for backwards compatibility with MathML2 and to
facilitate the use of Content MathML within URIbased frameworks (such as RDF [RDF] in the Semantic Web or OMDoc [OMDoc1.2]). Another benefit is
that the symbol name in the CD does not need to correspond to the content of the
csymbol
element. However, in general, this method results in much longer MathML
instances. Also, in situations where CDs are under development, the use of a CD Group
file allows the locations of CDs to change without a change to the markup. A third
drawback to definitionURL
is that unlike the cd
attribute, it is not
limited to referencing symbol definitions in OpenMath content dictionaries. Hence,
it is
not in general possible for a user agent to automatically determine the proper
interpretation for definitionURL
values without further information about the
context and community of practice in which the MathML instance occurs.
Both the cd
and definitionURL
mechanisms of external reference
may be used within a single MathML instance. However, when both a cd
and a
definitionURL
attribute are specified on a single csymbol
, the
cd
attribute takes precedence.
If the content of a csymbol
element is tagged using presentation tags,
that presentation is used. If no such tagging is supplied then the text
content is rendered as if it were the content of an mi
element. In
particular if an application supports bidirectional text rendering, then the
rendering follows the Unicode bidirectional rendering.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Cs  Cs 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content  text  text 
The cs
element encodes string literals
which may be used in Content MathML expressions.
The content of cs is text; no
Presentation MathML constructs are allowed even when used in
nonstrict markup. Specifically, cs
may not contain
mglyph
elements, and the content does not undergo white space
normalization.
Content MathML
<set>
<cs>A</cs><cs>B</cs><cs> </cs>
</set>
Sample Presentation
<mrow>
<mo>{</mo>
<ms>A</ms>
<mo>,</mo>
<ms>B</ms>
<mo>,</mo>
<ms>  </ms>
<mo>}</mo>
</mrow>
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Apply  Apply 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content  ContExp + 
ContExp +

(ContExp ,
BvarQ ,
Qualifier ?,
ContExp *) 
The most fundamental way of building a compound object in mathematics is by applying a function or an operator to some arguments.
In MathML, the apply
element is used to build an expression tree that
represents the application of a function or operator to its arguments. The
resulting tree corresponds to a complete mathematical expression. Roughly
speaking, this means a piece of mathematics that could be surrounded by
parentheses or logical brackets
without changing its meaning.
For example, (x + y) might be encoded as
<apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply>
The opening and closing tags of apply
specify exactly the scope of any
operator or function. The most typical way of using apply
is simple and
recursive. Symbolically, the content model can be described as:
<apply> op [ a b ...] </apply>
where the operands a, b, ... are MathML
expression trees themselves, and op is a MathML expression tree that
represents an operator or function. Note that apply
constructs can be
nested to arbitrary depth.
An apply
may in principle have any number of operands. For example,
(x + y + z) can be encoded as
<apply><csymbol cd="arith1">plus</csymbol>
<ci>x</ci>
<ci>y</ci>
<ci>z</ci>
</apply>
Note that MathML also allows applications without operands, e.g. to represent functions
like random()
, or currentdate()
.
Mathematical expressions involving a mixture of operations result in nested
occurrences of apply
. For example, a x + b
would be encoded as
<apply><csymbol cd="arith1">plus</csymbol>
<apply><csymbol cd="arith1">times</csymbol>
<ci>a</ci>
<ci>x</ci>
</apply>
<ci>b</ci>
</apply>
There is no need to introduce parentheses or to resort to
operator precedence in order to parse expressions correctly. The
apply
tags provide the proper grouping for the reuse
of the expressions within other constructs. Any expression
enclosed by an apply
element is welldefined, coherent
object whose interpretation does not depend on the surrounding
context. This is in sharp contrast to presentation markup,
where the same expression may have very different meanings in
different contexts. For example, an expression with a visual
rendering such as (F+G)(x)
might be a product, as in
<apply><csymbol cd="arith1">times</csymbol>
<apply><csymbol cd="arith1">plus</csymbol>
<ci>F</ci>
<ci>G</ci>
</apply>
<ci>x</ci>
</apply>
or it might indicate the application of the function F + G to the argument x. This is indicated by constructing the sum
<apply><csymbol cd="arith1">plus</csymbol><ci>F</ci><ci>G</ci></apply>
and applying it to the argument x as in
<apply>
<apply><csymbol cd="arith1">plus</csymbol>
<ci>F</ci>
<ci>G</ci>
</apply>
<ci>x</ci>
</apply>
In both cases, the interpretation of the outer apply
is
explicit and unambiguous, and does not change regardless of
where the expression is used.
The preceding example also illustrates that in an
apply
construct, both the function and the arguments
may be simple identifiers or more complicated expressions.
The apply
element is conceptually necessary in order to distinguish
between a function or operator, and an instance of its use. The expression
constructed by applying a function to 0 or more arguments is always an element from
the codomain of the function. Proper usage depends on the operator that is being
applied. For example, the plus
operator may have zero or more arguments,
while the minus
operator requires one or two arguments in order to be properly
formed.
Strict Content MathML applications are rendered as mathematical
function applications. If
<mi>F</mi>
denotes the rendering of
<ci>f</ci>
and
<mi>Ai</mi>
the rendering of
<ci>ai</ci>
, the sample
rendering of a simple application is as follows:
Content MathML
<apply><ci>f</ci>
<ci>a1</ci>
<ci>a2</ci>
<ci>...</ci>
<ci>an</ci>
</apply>
Sample Presentation
<mrow>
<mi>F</mi>
<mo>⁡<!ApplyFunction></mo>
<mrow>
<mo fence="true">(</mo>
<mi>A1</mi>
<mo separator="true">,</mo>
<mi>...</mi>
<mo separator="true">,</mo>
<mi>A2</mi>
<mo separator="true">,</mo>
<mi>An</mi>
<mo fence="true">)</mo>
</mrow>
</mrow>
NonStrict MathML applications may also be used with qualifiers. In the absence of
any more specific rendering rules for wellknown operators, rendering
should follow the sample presentation below, motivated by the typical
presentation for sum
. Let
<mi>Op</mi>
denote the rendering of
<ci>op</ci>
,
<mi>X</mi>
the rendering of
<ci>x</ci>
, and so on. Then:
Content MathML
<apply><ci>op</ci>
<bvar><ci>x</ci></bvar>
<domainofapplication><ci>d</ci></domainofapplication>
<ci>expressioninx</ci>
</apply>
Sample Presentation
<mrow>
<munder>
<mi>Op</mi>
<mrow><mi>X</mi><mo>∈</mo><mi>D</mi></mrow>
</munder>
<mo>⁡<!ApplyFunction></mo>
<mrow>
<mo fence="true">(</mo>
<mi>ExpressioninX</mi>
<mo fence="true">)</mo>
</mrow>
</mrow>
Many complex mathematical expressions are constructed with the use of bound
variables, and bound variables are an important concept of logic and formal
languages. Variables become bound in the scope of an expression through
the use of a quantifier. Informally, they can be thought of as the dummy variables
in expressions such as integrals, sums, products, and the logical quantifiers for
all
and there exists
. A bound variable is characterized by the property that
systematically renaming the variable (to a name not already appearing in the
expression) does not change the meaning of the expression.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Bind  Bind 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content 
ContExp ,
BvarQ *,
ContExp

ContExp ,
BvarQ *,
Qualifier *,
ContExp +

Binding expressions are represented as MathML expression trees using the bind
element. Its first child is a MathML expression that represents a binding operator,
for
example integral operator. This is followed by a nonempty list of bvar
elements denoting the bound variables, and then the final child which is a general
Content MathML expression, known as the body of the binding.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  BVar  BVar 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content  ci  semanticsci 
(ci  semanticsci ), degree ? 
degree ?, (ci  semanticsci )

The bvar
element is used to denote the bound variable of a binding
expression, e.g. in sums, products, and quantifiers or user defined functions.
The content of a bvar
element is an annotated variable,
i.e. either a content identifier represented by a ci
element or a
semantics
element whose first child is an annotated variable. The
name of an annotated variable of the second kind is the name of its first
child. The name of a bound variable is that of the annotated variable
in the bvar
element.
Bound variables are identified by comparing their names. Such
identification can be made explicit by placing an id
on the ci
element in the bvar
element and referring to it using the xref
attribute on all other instances. An example of this approach is
<bind><csymbol cd="quant1">forall</csymbol>
<bvar><ci id="varx">x</ci></bvar>
<apply><csymbol cd="relation1">lt</csymbol>
<ci xref="varx">x</ci>
<cn>1</cn>
</apply>
</bind>
This id
based approach is especially helpful when constructions
involving bound variables are nested.
It is sometimes necessary to associate additional
information with a bound variable. The information might be
something like a detailed mathematical type, an alternative
presentation or encoding or a domain of application. Such
associations are accomplished in the standard way by replacing
a ci
element (even inside the bvar
element)
by a semantics
element containing both the ci
and the additional information. Recognition of an instance of
the bound variable is still based on the actual ci
elements and not the semantics
elements or anything
else they may contain. The id
based approach
outlined above may still be used.
The following example encodes $\forall x.x+y=y+x$.
<bind><csymbol cd="quant1">forall</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><ci>y</ci></apply>
<apply><csymbol cd="arith1">plus</csymbol><ci>y</ci><ci>x</ci></apply>
</apply>
</bind>
In nonStrict Content markup, the bvar
element is used in
a number of idiomatic constructs. These are described in 4.3.3 Qualifiers and 4.3 Content MathML for Specific Structures.
It is a defining property of bound variables that they can be renamed
consistently in the scope of their parent bind
element.
This operation, sometimes known as αconversion,
preserves the semantics of the expression.
A bound variable $x$ may be renamed to say $y$ so long as $y$ does not occur free in the body of the binding, or in any annotations of the bound variable, $x$ to be renamed, or later bound variables.
If a bound variable
$x$
is renamed, all free occurrences of
$x$
in annotations in its bvar
element,
any following bvar
children of the bind
and in the expression in the body of the bind
should be renamed.
In the example in the previous section, note how renaming $x$ to $z$ produces the equivalent expression $\forall z.z+y=y+z$, whereas $x$ may not be renamed to $y$, as $y$ is free in the body of the binding and would be captured, producing the expression $\forall y.y+y=y+y$ which is not equivalent to the original expression.
If
<ci>b</ci>
and
<ci>s</ci>
are Content MathML expressions
that render as the Presentation MathML expressions
<mi>B</mi>
and
<mi>S</mi>
then the sample rendering of a binding element is as follows:
Content MathML
<bind><ci>b</ci>
<bvar><ci>x1</ci></bvar>
<bvar><ci>...</ci></bvar>
<bvar><ci>xn</ci></bvar>
<ci>s</ci>
</bind>
Sample Presentation
<mrow>
<mi>B</mi>
<mrow>
<mi>x1</mi>
<mo separator="true">,</mo>
<mi>...</mi>
<mo separator="true">,</mo>
<mi>xn</mi>
</mrow>
<mo separator="true">.</mo>
<mi>S</mi>
</mrow>
Content elements can be annotated with additional information via the
semantics
element. MathML uses the
semantics
element to wrap the annotated element and the
annotationxml
and annotation
elements used for representing the
annotations themselves. The use of the semantics
, annotation
and
annotationxml
is described in detail in 6. Annotating MathML: semantics.
The semantics
element is considered part of both
presentation MathML and Content MathML. MathML considers a semantics
element
(strict) Content MathML, if and only if its first child is (strict) Content MathML.
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Error  Error 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content 
csymbol , ContExp *

csymbol , ContExp *

A content error expression is made up of a csymbol
followed by a sequence of zero or more MathML expressions. The
initial expression must be a csymbol
indicating the kind of
error. Subsequent children, if present, indicate the context in
which the error occurred.
The cerror
element has no direct mathematical meaning.
Errors occur as the result of some action performed on an expression
tree and are thus of real interest only when some sort of
communication is taking place. Errors may occur inside other objects
and also inside other errors.
As an example, to encode a division by zero error, one might
employ a hypothetical aritherror
Content Dictionary
containing a DivisionByZero
symbol, as in the following
expression:
<cerror>
<csymbol cd="aritherror">DivisionByZero</csymbol>
<apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
</cerror>
Note that error markup generally should enclose only the smallest
erroneous subexpression. Thus a cerror
will often be a subexpression of
a bigger one, e.g.
<apply><csymbol cd="relation1">eq</csymbol>
<cerror>
<csymbol cd="aritherror">DivisionByZero</csymbol>
<apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
</cerror>
<cn>0</cn>
</apply>
The default presentation of a cerror
element is an
merror
expression whose first child is a presentation of the
error symbol, and whose subsequent children are the default
presentations of the remaining children of the cerror
. In
particular, if one of the remaining children of the cerror
is
a presentation MathML expression, it is used literally in the
corresponding merror
.
<cerror>
<csymbol cd="aritherror">DivisionByZero</csymbol>
<apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn>0</cn></apply>
</cerror>
Sample Presentation
<merror>
<mtext>DivisionByZero: </mtext>
<mfrac><mi>x</mi><mn>0</mn></mfrac>
</merror>
Note that when the context where an error occurs is so nonsensical that its default presentation would not be useful, an application may provide an alternative representation of the error context. For example:
<cerror>
<csymbol cd="error">Illegal bound variable</csymbol>
<cs> <bvar><plus/></bvar> </cs>
</cerror>
Schema Fragment (Strict)  Schema Fragment (Full)  

Class  Cbytes  Cbytes 
Attributes  CommonAtt 
CommonAtt , DefEncAtt 
Content  base64 
base64 
The content of cbytes
represents a stream of bytes as a
sequence of characters in Base64 encoding, that is it matches the
base64Binary data type defined in [XMLSchemaDatatypes]. All white space is ignored.
The cbytes
element is mainly used for OpenMath
compatibility, but may be used, as in OpenMath, to encapsulate output
from a system that may be hard to encode in MathML, such as binary
data relating to the internal state of a system, or image data.
The rendering of cbytes
is not expected to represent the
content and the proposed rendering is that of an empty
mrow
. Typically cbytes
is used in an
annotationxml
or is itself annotated with Presentation
MathML, so this default rendering should rarely be used.
The elements of Strict Content MathML described in the previous section are sufficient to encode logical assertions and expression structure, and they do so in a way that closely models the standard constructions of mathematical logic that underlie the foundations of mathematics. As a consequence, Strict markup can be used to represent all of mathematics, and is ideal for providing consistent mathematical semantics for all Content MathML expressions.
At the same time, many notational idioms of mathematics are not straightforward to represent directly with Strict Content markup. For example, standard notations for sums, integrals, sets, piecewise functions and many other common constructions require nonobvious technical devices, such as the introduction of lambda functions, to rigorously encode them using Strict markup. Consequently, in order to make Content MathML easier to use, a range of additional elements have been provided for encoding such idiomatic constructs more directly. This section discusses the general approach for encoding such idiomatic constructs, and their Strict Content equivalents. Specific constructions are discussed in detail in 4.3 Content MathML for Specific Structures.
Most idiomatic constructions which Content markup addresses fall
into about a dozen classes. Some of these classes, such as container elements, have
their own syntax. Similarly, a small number of nonStrict
constructions involve a single element with an exceptional syntax,
for example partialdiff
. These exceptional elements are
discussed on a casebycase basis in 4.3 Content MathML for Specific Structures. However, the majority of constructs consist of
classes of operator elements which all share a particular usage of
qualifiers.
These classes of operators are described in 4.3.4 Operator Classes.
In all cases, nonStrict expressions may be rewritten using only Strict markup. In most cases, the transformation is completely algorithmic, and may be automated. Rewrite rules for classes of nonStrict constructions are introduced and discussed later in this section, and rewrite rules for exceptional constructs involving a single operator are given in 4.3 Content MathML for Specific Structures. The complete algorithm for rewriting arbitrary Content MathML as Strict Content markup is summarized at the end of the Chapter in F. The Strict Content MathML Transformation.
Many mathematical structures are constructed from subparts or
parameters. For example, a set is a mathematical object that
contains a collection of elements, so it is natural for the
markup for a set to contain the markup for its constituent
elements. The markup for a set may define the set of elements
explicitly by enumerating them, or implicitly by rule that uses
qualifier elements. In either case, the markup for the elements is
contained in the markup for the set, and this style of
representation is called container markup in MathML. By
contrast, Strict markup represents an instance of a set as the
result of applying a function or constructor symbol to
arguments. In this style of markup, the markup for the set
construction is a sibling of the markup for the set elements in an
enclosing apply
element.
MathML provides container markup for the following mathematical constructs: sets, lists, intervals, vectors, matrices (two elements), piecewise functions (three elements) and lambda functions. There are corresponding constructor symbols in Strict markup for each of these, with the exception of lambda functions, which correspond to binding symbols in Strict markup.
The rewrite rules for obtaining equivalent Strict Content markup from container markup depend on the operator class of the particular operator involved. For details about a specific container element, obtain its operator class (and any applicable special case information) by consulting the syntax table and discussion for that element in E. The Content MathML Operators. Then apply the rewrite rules for that specific operator class as described in F. The Strict Content MathML Transformation.
The arguments to container elements that correspond to
constructors may be explicitly given as a sequence of child
elements, or implicitly given by a rule using qualifiers. The
exceptions are the interval
, piecewise
, piece
, and
otherwise
elements. The
arguments of these elements must be specified explicitly.
Here is an example of container markup with explicitly specified arguments:
<set><ci>a</ci><ci>b</ci><ci>c</ci></set>
This is equivalent to the following Strict Content MathML expression:
<apply><csymbol cd="set1">set</csymbol><ci>a</ci><ci>b</ci><ci>c</ci></apply>
Another example of container markup, where the list of arguments is given indirectly as an expression with a bound variable. The container markup for the set of even integers is:
<set>
<bvar><ci>x</ci></bvar>
<domainofapplication><integers/></domainofapplication>
<apply><times/><cn>2</cn><ci>x</ci></apply>
</set>
This may be written as follows in Strict Content MathML:
<apply><csymbol cd="set1">map</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="arith1">times</csymbol>
<cn>2</cn>
<ci>x</ci>
</apply>
</bind>
<csymbol cd="setname1">Z</csymbol>
</apply>
The lambda
element is a container element
corresponding to the lambda symbol
in the fns1 Content Dictionary. However, unlike the
container elements of the preceding section, which purely
construct mathematical objects from arguments, the lambda
element performs variable binding as well. Therefore, the child
elements of lambda
have distinguished roles. In
particular, a lambda
element must have at least one
bvar
child, optionally followed by qualifier elements, followed by a
Content MathML element. This basic difference between the
lambda
container and the other constructor container
elements is also reflected in the OpenMath symbols to which they
correspond. The constructor symbols have an OpenMath role of
application
, while the lambda symbol has a role of bind
.
This example shows the use of lambda
container element and the equivalent use of bind
in Strict Content MathML
<lambda><bvar><ci>x</ci></bvar><ci>x</ci></lambda>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>x</ci></bvar><ci>x</ci>
</bind>
MathML allows the use of the apply
element to perform
variable binding in nonStrict constructions instead of
the bind
element. This usage conserves backwards
compatibility with MathML 2. It also simplifies the encoding of
several constructs involving bound variables with qualifiers as
described below.
Use of the apply
element to bind variables is allowed
in two situations. First, when the operator to be applied is
itself a binding operator, the apply
element merely
substitutes for the bind
element. The logical quantifiers
<forall/>
, <exists/>
and the
container element lambda
are the primary examples of this
type.
The second situation arises when the operator being applied allows the use of bound variables with qualifiers. The most common examples are sums and integrals. In most of these cases, the variable binding is to some extent implicit in the notation, and the equivalent Strict representation requires the introduction of auxiliary constructs such as lambda expressions for formal correctness.
Because expressions using bound variables with qualifiers are
idiomatic in nature, and do not always involve true variable
binding, one cannot expect systematic renaming (alphaconversion)
of variables bound
with apply
to preserve meaning in
all cases. An example for this is the diff
element where
the bvar
term is technically not bound at all.
The following example illustrates the use of apply
with a binding operator. In these cases, the corresponding Strict
equivalent merely replaces the apply
element with a
bind
element:
<apply><forall/>
<bvar><ci>x</ci></bvar>
<apply><geq/><ci>x</ci><ci>x</ci></apply>
</apply>
The equivalent Strict expression is:
<bind><csymbol cd="logic1">forall</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="relation1">geq</csymbol><ci>x</ci><ci>x</ci></apply>
</bind>
In this example, the sum operator is not itself a binding operator, but bound variables with qualifiers are implicit in the standard notation, which is reflected in the nonStrict markup. In the equivalent Strict representation, it is necessary to convert the summand into a lambda expression, and recast the qualifiers as an argument expression:
<apply><sum/>
<bvar><ci>i</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><cn>100</cn></uplimit>
<apply><power/><ci>x</ci><ci>i</ci></apply>
</apply>
The equivalent Strict expression is:
<apply><csymbol cd="arith1">sum</csymbol>
<apply><csymbol cd="interval1">integer_interval</csymbol>
<cn>0</cn>
<cn>100</cn>
</apply>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>i</ci></bvar>
<apply><csymbol cd="arith1">power</csymbol>
<ci>x</ci>
<ci>i</ci>
</apply>
</bind>
</apply>
Many common mathematical constructs involve an operator together with some additional data. The additional data is either implicit in conventional notation, such as a bound variable, or thought of as part of the operator, as is the case with the limits of a definite integral. MathML 3 uses qualifier elements to represent the additional data in such cases.
Qualifier elements are always used in conjunction with operator or container
elements. Their meaning is idiomatic, and depends on the context in which they are
used. When used with an operator, qualifiers always follow the operator and precede
any arguments that are present. In all cases, if more than one qualifier is present,
they appear in the order bvar
, lowlimit
, uplimit
,
interval
, condition
, domainofapplication
, degree
,
momentabout
, logbase
.
The precise function of qualifier elements depends on the operator or container that they modify. The majority of use cases fall into one of several categories, discussed below, and usage notes for specific operators and qualifiers are given in 4.3 Content MathML for Specific Structures.
Class  qualifier 

Attributes  CommonAtt 
Content  ContExp 
(For the syntax of interval
see 4.3.10.3 Interval <interval>
.)
The primary use of domainofapplication
, interval
,
uplimit
, lowlimit
and condition
is to
restrict the values of a bound variable. The most general qualifier
is domainofapplication
. It is used to specify a set (perhaps
with additional structure, such as an ordering or metric) over which
an operation is to take place. The interval
qualifier, and
the pair lowlimit
and uplimit
also restrict a bound
variable to a set in the special case where the set is an
interval.
Note that interval
is only interpreted as a qualifier if it immediately
follows bvar
.
The condition
qualifier, like
domainofapplication
, is general, and can be used to restrict
bound variables to arbitrary sets. However, unlike the other
qualifiers, it restricts the bound variable by specifying a
Booleanvalued function of the bound variable. Thus,
condition
qualifiers always contain instances of the bound
variable, and thus require a preceding bvar
, while the other
qualifiers do not. The other qualifiers may even be used when no
variables are being bound, e.g. to indicate the restriction of a
function to a subdomain.
In most cases, any of the qualifiers capable of representing the
domain of interest can be used interchangeably. The most general
qualifier is domainofapplication
, and therefore has a
privileged role. It is the preferred form, unless there are
particular idiomatic reasons to use one of the other qualifiers,
e.g. limits for an integral. In MathML 3, the other forms are treated
as shorthand notations for domainofapplication
because they
may all be rewritten as equivalent domainofapplication
constructions. The rewrite rules to do this are given below. The other
qualifier elements are provided because they correspond to common
notations and map more easily to familiar presentations. Therefore,
in the situations where they naturally arise, they may be more
convenient and direct than domainofapplication
.
To illustrate these ideas, consider the following examples showing alternative
representations of a definite integral. Let $C$
denote the interval from 0 to 1,
and $f\left(x\right)={x}^{2}$. Then
domainofapplication
could be used to express the integral of a
function $f$ over
$C$ in this way:
<apply><int/>
<domainofapplication>
<ci type="set">C</ci>
</domainofapplication>
<ci type="function">f</ci>
</apply>
Note that no explicit bound variable is identified in this
encoding, and the integrand is a function. Alternatively, the
interval
qualifier could be used with an explicit bound variable:
<apply><int/>
<bvar><ci>x</ci></bvar>
<interval><cn>0</cn><cn>1</cn></interval>
<apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>
The pair lowlimit
and uplimit
can also be used.
This is perhaps the most standard
representation of this integral:
<apply><int/>
<bvar><ci>x</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><cn>1</cn></uplimit>
<apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>
Finally, here is the same integral, represented using
a condition
on the bound variable:
<apply><int/>
<bvar><ci>x</ci></bvar>
<condition>
<apply><and/>
<apply><leq/><cn>0</cn><ci>x</ci></apply>
<apply><leq/><ci>x</ci><cn>1</cn></apply>
</apply>
</condition>
<apply><power/><ci>x</ci><cn>2</cn></apply>
</apply>
Note the use of the explicit bound variable within the
condition
term. Note also that when a bound
variable is used, the integrand is an expression in the bound
variable, not a function.
The general technique of using a condition
element
together with domainofapplication
is quite powerful. For
example, to extend the previous example to a multivariate domain, one
may use an extra bound variable and a domain of application
corresponding to a cartesian product:
<apply><int/>
<bvar><ci>x</ci></bvar>
<bvar><ci>y</ci></bvar>
<domainofapplication>
<set>
<bvar><ci>t</ci></bvar>
<bvar><ci>u</ci></bvar>
<condition>
<apply><and/>
<apply><leq/><cn>0</cn><ci>t</ci></apply>
<apply><leq/><ci>t</ci><cn>1</cn></apply>
<apply><leq/><cn>0</cn><ci>u</ci></apply>
<apply><leq/><ci>u</ci><cn>1</cn></apply>
</apply>
</condition>
<list><ci>t</ci><ci>u</ci></list>
</set>
</domainofapplication>
<apply><times/>
<apply><power/><ci>x</ci><cn>2</cn></apply>
<apply><power/><ci>y</ci><cn>3</cn></apply>
</apply>
</apply>
Note that the order of the inner and outer bound variables is significant.
Class  qualifier 

Attributes  CommonAtt 
Content  ContExp 
The degree
element is a qualifier used to specify the
degree
or order
of an operation. MathML uses the
degree
element in this way in three contexts: to specify the degree of a
root, a moment, and in various derivatives. Rather than introduce special elements
for
each of these families, MathML provides a single general construct, the
degree
element in all three cases.
Note that the degree
qualifier is not used to restrict a bound variable in
the same sense of the qualifiers discussed above. Indeed, with roots and moments,
no
bound variable is involved at all, either explicitly or implicitly. In the case of
differentiation, the degree
element is used in conjunction with a
bvar
, but even in these cases, the variable may not be genuinely bound.
For the usage of degree
with the root
and moment
operators, see the discussion of those
operators below. The usage of degree
in differentiation is more complex. In
general, the degree
element indicates the order of the derivative with
respect to that variable. The degree element is allowed as the second child of a
bvar
element identifying a variable with respect to which the derivative is
being taken. Here is an example of a second derivative using the degree
qualifier:
<apply><diff/>
<bvar>
<ci>x</ci>
<degree><cn>2</cn></degree>
</bvar>
<apply><power/><ci>x</ci><cn>4</cn></apply>
</apply>
For details see 4.3.8.2 Differentiation <diff/>
and 4.3.8.3 Partial Differentiation <partialdiff/>
.
The qualifiers momentabout
and logbase
are
specialized elements specifically for use with the moment
and log
operators
respectively. See the descriptions of those operators below for their usage.
The Content MathML elements described in detail in the following sections may be broadly separated into classes. The class of each element is listed in the operator syntax table given in E.3 The Content MathML Operators. The class gives an indication of the general intended mathematical usage of the element, and also determines its usage as determined by the schema. Links to the operator syntax and schema class for each element are provided in the sections that introduce the elements.
The operator class also determines the applicable rewrite rules for mapping to Strict Content MathML. These rewrite rules are presented in detail in F. The Strict Content MathML Transformation. They include use cases applicable to specific operator classes, specialcase rewrite rules for individual elements, and a generic rewrite rule F.8 Rewrite operators used by operators from almost all operator classes.
The following sections present elements representing a core set of mathematical operators, functions and constants. Most are empty elements, covering the subject matter of standard mathematics curricula up to the level of calculus. The remaining elements are container elements for sets, intervals, vectors and so on. For brevity, all elements defined in this section are sometimes called operator elements.
Many MathML operators may be used with an arbitrary number of
arguments. The corresponding OpenMath symbols for elements in these classes
also take an arbitrary number of arguments.
In all such cases, either the arguments may be given
explicitly as children of the apply
or bind
element, or
the list may be specified implicitly via the use of qualifier
elements.
The plus
and times
elements represent the addition and multiplication operators. The
arguments are normally specified explicitly in the enclosing
apply
element. As an nary commutative operator, they can
be used with qualifiers to specify arguments, however,
this is discouraged, and the sum
or product
operators should be
used to represent such expressions instead.
Content MathML
<apply><plus/><ci>x</ci><ci>y</ci><ci>z</ci></apply>
Sample Presentation
<mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi></mrow>
The gcd
and lcm
elements represent the nary operators which return the greatest common divisor, or least common multiple of their arguments. The arguments may be explicitly specified in the enclosing apply element, or specified by quantifiers.
This default renderings are Englishlanguage locale specific: other locales may have different default renderings.
Content MathML
<apply><gcd/><ci>a</ci><ci>b</ci><ci>c</ci></apply>
Sample Presentation
<mrow>
<mi>gcd</mi>
<mo>⁡<!ApplyFunction></mo>
<mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow>
</mrow>
The sum
element represents the nary addition operator.
The terms of the sum are normally specified by rule through the use of
qualifiers. While it can be used with an explicit list of
arguments, this is strongly discouraged, and the plus
operator should be used instead in such situations.
The sum
operator may be used either with or without
explicit bound variables. When a bound variable is used, the
sum
element is followed by one or more bvar
elements giving the index variables, followed by qualifiers giving
the domain for the index variables. The final child in the enclosing
apply
is then an expression in the bound variables, and the
terms of the sum are obtained by evaluating this expression at each
point of the domain of the index variables. Depending on the
structure of the domain, the domain of summation is often given
by using uplimit
and lowlimit
to specify upper and
lower limits for the sum.
When no bound variables are explicitly given, the final child of
the enclosing apply
element must be a function, and the
terms of the sum are obtained by evaluating the function at
each point of the domain specified by qualifiers.
Content MathML
<apply><sum/>
<bvar><ci>x</ci></bvar>
<lowlimit><ci>a</ci></lowlimit>
<uplimit><ci>b</ci></uplimit>
<apply><ci>f</ci><ci>x</ci></apply>
</apply>
<apply><sum/>
<bvar><ci>x</ci></bvar>
<condition>
<apply><in/><ci>x</ci><ci type="set">B</ci></apply>
</condition>
<apply><ci type="function">f</ci><ci>x</ci></apply>
</apply>
<apply><sum/>
<domainofapplication>
<ci type="set">B</ci>
</domainofapplication>
<ci type="function">f</ci>
</apply>
Sample Presentation
<mrow>
<munderover>
<mo>∑</mo>
<mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow>
<mi>b</mi>
</munderover>
<mrow><mi>f</mi><mo>⁡<!ApplyFunction></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
</mrow>
<mrow>
<munder>
<mo>∑</mo>
<mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow>
</munder>
<mrow><mi>f</mi><mo>⁡<!ApplyFunction></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
</mrow>
<mrow><munder><mo>∑</mo><mi>B</mi></munder><mi>f</mi></mrow>
The product
element represents the nary multiplication operator.
The terms of the product are normally specified by rule through the use of
qualifiers. While it can be used with an explicit list of
arguments, this is strongly discouraged, and the times
operator should be used instead in such situations.
The product
operator may be used either with or without
explicit bound variables. When a bound variable is used, the
product
element is followed by one or more bvar
elements giving the index variables, followed by qualifiers giving
the domain for the index variables. The final child in the enclosing
apply
is then an expression in the bound variables, and the
terms of the product are obtained by evaluating this expression at
each point of the domain. Depending on the structure of the domain,
it is commonly given using uplimit
and lowlimit
qualifiers.
When no bound variables are explicitly given, the final child of
the enclosing apply
element must be a function, and the
terms of the product are obtained by evaluating the function
at each point of the domain specified by qualifiers.
Content MathML
<apply><product/>
<bvar><ci>x</ci></bvar>
<lowlimit><ci>a</ci></lowlimit>
<uplimit><ci>b</ci></uplimit>
<apply><ci type="function">f</ci>
<ci>x</ci>
</apply>
</apply>
<apply><product/>
<bvar><ci>x</ci></bvar>
<condition>
<apply><in/>
<ci>x</ci>
<ci type="set">B</ci>
</apply>
</condition>
<apply><ci>f</ci><ci>x</ci></apply>
</apply>
Sample Presentation
<mrow>
<munderover>
<mo>∏</mo>
<mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow>
<mi>b</mi>
</munderover>
<mrow><mi>f</mi><mo>⁡<!ApplyFunction></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
</mrow>
<mrow>
<munder>
<mo>∏</mo>
<mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow>
</munder>
<mrow><mi>f</mi><mo>⁡<!ApplyFunction></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow>
</mrow>
The compose
element represents the function
composition operator. Note that MathML makes no assumption about the domain
and codomain of the constituent functions in a composition; the domain of the
resulting composition may be empty.
The compose
element is a commutative nary operator. Consequently, it may be
lifted to the induced operator defined on a collection of arguments indexed by a (possibly
infinite) set by using qualifier elements as described in 4.3.5.4 Nary Functional Operators:
<compose/>
.
Content MathML
<apply><compose/><ci>f</ci><ci>g</ci><ci>h</ci></apply>
<apply><eq/>
<apply>
<apply><compose/><ci>f</ci><ci>g</ci></apply>
<ci>x</ci>
</apply>
<apply><ci>f</ci><apply><ci>g</ci><ci>x</ci></apply></apply>
</apply>
Sample Presentation
<mrow>
<mi>f</mi><mo>∘</mo><mi>g</mi><mo>∘</mo><mi>h</mi>
</mrow>
<mrow>
<mrow>
<mrow><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo></mrow>
<mo>⁡<!ApplyFunction></mo>
<mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow>
</mrow>
<mo>=</mo>
<mrow>
<mi>f</mi>
<mo>⁡<!ApplyFunction></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>⁡<!ApplyFunction></mo>
<mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
These elements represent
nary functions taking Boolean arguments and returning a Boolean value.
The arguments may be explicitly specified
in the enclosing apply
element, or specified via qualifier elements.
and
is true if all arguments are true, and false otherwise.
or
is true if any of the arguments are true, and false otherwise.
xor
is the logical exclusive or
function. It is true if there are an odd number of true arguments or false otherwise.
Content MathML
<apply><and/><ci>a</ci><ci>b</ci></apply>
<apply><and/>
<bvar><ci>i</ci></bvar>
<lowlimit><cn>0</cn></lowlimit>
<uplimit><ci>n</ci></uplimit>
<apply><gt/><apply><selector/><ci>a</ci><ci>i</ci></apply><cn>0</cn></apply>
</apply>
Strict Content MathML
<apply><csymbol cd="logic1">and</csymbol><ci>a</ci><ci>b</ci></apply>
<apply><csymbol cd="fns2">apply_to_list</csymbol>
<csymbol cd="logic1">and</csymbol>
<apply><csymbol cd="list1">map</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>i</ci></bvar>
<apply><csymbol cd="relation1">gt</csymbol>
<apply><csymbol cd="linalg1">vector_selector</csymbol>
<ci>i</ci>
<ci>a</ci>
</apply>
<cn>0</cn>
</apply>
</bind>
<apply><csymbol cd="interval1">integer_interval</csymbol>
<cn type="integer">0</cn>
<ci>n</ci>
</apply>
</apply>
</apply>
Sample Presentation
<mrow><mi>a</mi><mo>∧</mo><mi>b</mi></mrow>
<mrow>
<munderover>
<mo>⋀</mo>
<mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow>
<mi>n</mi>
</munderover>
<mrow>
<mo>(</mo>
<msub><mi>a</mi><mi>i</mi></msub>
<mo>></mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</mrow>
The selector
element is the operator for indexing into vectors, matrices
and lists. It accepts one or more arguments. The first argument identifies the vector,
matrix or list from which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.
When selector
is used with a single argument, it should be interpreted as
giving the sequence of all elements in the list, vector or matrix given. The ordering
of elements in the sequence for a matrix is understood to be first by column, then
by
row; so the resulting list is of matrix rows given entry by entry.
That is, for a matrix $\left({a}_{i,j}\right)$, where the indices denote row
and column, respectively, the ordering would be
${a}_{1,1}$,
${a}_{1,2}$, …,
${a}_{2,1}$,
${a}_{2,2}$, … etc.
When two arguments are given, and the first is a vector or list, the second argument specifies the index of an entry in the list or vector. If the first argument is a matrix then the second argument specifies the index of a matrix row.
When three arguments are given, the last one is ignored for a list or vector, and in the case of a matrix, the second and third arguments specify the row and column indices of the selected element.
Content MathML
<apply><selector/><ci type="vector">V</ci><cn>1</cn></apply>
<apply><eq/>
<apply><selector/>
<matrix>
<matrixrow><cn>1</cn><cn>2</cn></matrixrow>
<matrixrow><cn>3</cn><cn>4</cn></matrixrow>
</matrix>
<cn>1</cn>
</apply>
<matrix>
<matrixrow><cn>1</cn><cn>2</cn></matrixrow>
</matrix>
</apply>
Sample Presentation
<msub><mi>V</mi><mn>1</mn></msub>
<mrow>
<msub>
<mrow>
<mo>(</mo>
<mtable>
<mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr>
<mtr><mtd><mn>3</mn></mtd><mtd><mn>4</mn></mtd></mtr>
</mtable>
<mo>)</mo>
</mrow>
<mn>1</mn>
</msub>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable>
<mo>)</mo>
</mrow>
</mrow>
The union
element is used to denote the nary union of sets. It takes sets as arguments,
and denotes the set that contains all the elements that occur in any
of them.
The intersect
element is used to denote the nary union of sets. It takes sets as arguments,
and denotes the set that contains all the elements that occur in all
of them.
The cartesianproduct
element is used to represent the
Cartesian product operator.
Arguments may be explicitly specified in the enclosing apply
element, or
specified using qualifier elements as described in 4.3.5 Nary Operators.
Content MathML
<apply><union/><ci>A</ci><ci>B</ci></apply>
<apply><intersect/><ci>A</ci><ci>B</ci><ci>C</ci></apply>
<apply><cartesianproduct/><ci>A</ci><ci>B</ci></apply>
Sample Presentation
<mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow>
<mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>∩</mo><mi>C</mi></mrow>
<mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow>
Content MathML
<apply><union/>
<bvar><ci type="set">S</ci></bvar>
<domainofapplication>
<ci type="list">L</ci>
</domainofapplication>
<ci type="set"> S</ci>
</apply>
<apply><intersect/>
<bvar><ci type="set">S</ci></bvar>
<domainofapplication>
<ci type="list">L</ci>
</domainofapplication>
<ci type="set"> S</ci>
</apply>
Sample Presentation
<mrow><munder><mo>⋃</mo><mi>L</mi></munder><mi>S</mi></mrow>
<mrow><munder><mo>⋂</mo><mi>L</mi></munder><mi>S</mi></mrow>
A vector is an ordered ntuple of values representing an element of an ndimensional vector space.
For purposes of interaction with matrices and matrix multiplication, vectors are regarded as equivalent to a matrix consisting of a single column, and the transpose of a vector as a matrix consisting of a single row.
The components of a vector
may be given explicitly as
child elements, or specified by rule as described in 4.3.1.1 Container Markup for Constructor Symbols.
Content MathML
<vector>
<apply><plus/><ci>x</ci><ci>y</ci></apply>
<cn>3</cn>
<cn>7</cn>
</vector>
Sample Presentation
<mrow>
<mo>(</mo>
<mtable>
<mtr><mtd><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mtd></mtr>
<mtr><mtd><mn>3</mn></mtd></mtr>
<mtr><mtd><mn>7</mn></mtd></mtr>
</mtable>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
A matrix is regarded as made up of matrix rows, each of which can be thought of as a special type of vector.
Note that the behavior of the matrix
and matrixrow
elements is
substantially different from the mtable
and mtr
presentation
elements.
The matrix
element is a constructor
element, so the entries may be given explicitly as child elements,
or specified by rule as described in 4.3.1.1 Container Markup for Constructor Symbols. In the latter case, the
entries are specified by providing a function and a 2dimensional
domain of application. The entries of the matrix correspond to
the values obtained by evaluating the function at the points of
the domain.
Matrix rows are not directly rendered by themselves outside of the context of a matrix.
Content MathML
<matrix>
<bvar><ci type="integer">i</ci></bvar>
<bvar><ci type="integer">j</ci></bvar>
<condition>
<apply><and/>
<apply><in/>
<ci>i</ci>
<interval><ci>1</ci><ci>5</ci></interval>
</apply>
<apply><in/>
<ci>j</ci>
<interval><ci>5</ci><ci>9</ci></interval>
</apply>
</apply>
</condition>
<apply><power/><ci>i</ci><ci>j</ci></apply>
</matrix>
Sample Presentation
<mrow>
<mo>[</mo>
<msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub>
<mo></mo>
<mrow>
<msub><mi>m</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub>
<mo>=</mo>
<msup><mi>i</mi><mi>j</mi></msup>
</mrow>
<mo>;</mo>
<mrow>
<mrow>
<mi>i</mi>
<mo>∈</mo>
<mrow><mo>[</mo><mi>1</mi><mo>,</mo><mi>5</mi><mo>]</mo></mrow>
</mrow>
<mo>∧</mo>
<mrow>
<mi>j</mi>
<mo>∈</mo>
<mrow><mo>[</mo><mi>5</mi><mo>,</mo><mi>9</mi><mo>]</mo></mrow>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
The set
element represents the nary function which constructs a mathematical set from its arguments. The set
element takes the attribute type
which may have the values set
and multiset
. The members of the set to be constructed may be given explicitly as child elements of the constructor, or specified by rule as described in 4.3.1.1 Container Markup for Constructor Symbols. There is no implied ordering to the elements of a set.
The list
element represents the nary function which constructs a list from its arguments. Lists differ from sets in that there is an explicit order to the elements. The list
element takes the attribute order
which may have the values numeric
and lexicographic
. The list entries and order may be given explicitly or specified by rule as described in 4.3.1.1 Container Markup for Constructor Symbols.
Content MathML
<set>
<ci>a</ci><ci>b</ci><ci>c</ci>
</set>
<list>
<ci>a</ci><ci>b</ci><ci>c</ci>
</list>
Sample Presentation
<mrow>
<mo>{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>}</mo>
</mrow>
<mrow>
<mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo>
</mrow>
In general sets and lists can be constructed by providing a function and
a domain of application. The elements correspond to the
values obtained by evaluating the function at the points of the
domain. When this method is used for lists, the ordering of the list elements
may not be clear, so the kind of ordering may be specified by the
order
attribute. Two orders are supported: lexicographic
and numeric.
Content MathML
<set>
<bvar><ci>x</ci></bvar>
<condition>
<apply><lt/><ci>x</ci><cn>5</cn></apply>
</condition>
<ci>x</ci>
</set>
<list order="numeric">
<bvar><ci>x</ci></bvar>
<condition>
<apply><lt/><ci>x</ci><cn>5</cn></apply>
</condition>
</list>
<set>
<bvar><ci type="set">S</ci></bvar>
<condition>
<apply><in/><ci>S</ci><ci type="list">T</ci></apply>
</condition>
<ci>S</ci>
</set>
<set>
<bvar><ci> x </ci></bvar>
<condition>
<apply><and/>
<apply><lt/><