MathML is a language for expressing mathematical notations, both their presentation (how a notation looks on a page) and their semantics or content (what a notation means).

This page presents an overview of what I have used from presentation MathML version 3.0: of the 39 elements, and a few of their attributes. For details, additional elements, and additional attributes for the elements presented here, or for content MathML, see the W3C definition [MathML 3.0]. Each discussion in this document of an element m ends with a link to m's section in the definition (→W3C definition of m).

Organization and coverage

MathML, like other recursively-defined languages, contains compound elements which contain other elements, and tokens which do not.

Overview of MathML constructs
Group	Function	Element	Examples
Token elements	Identifier	mi	$x$	<mi>x</mi>
			$π$	<mi>π</mi>
			$\sin$	<mi>sin</mi>
	Number	mn	$123$	<mn>123</mn>
	Operator, fence, separator	mo	$+$	<mo>+</mo>
			$($	<mo>(</mo>
			$,$	<mo>,</mo>
	Text	mtext	$Thus$	<mtext>Thus</mtext>
	Space	mspace		<mspace width='1em' />

Compound elements	Horizontal expression	mrow	$1 + 2$	<mrow> <mn>1</mn><mo>+</mo> <mn>2</mn> </mrow>
	Fenced and separated list	mfenced	$(1, 2)$	<mfenced> <mn>1</mn><mn>2</mn> </mfenced>
	Fraction	mfrac	$\frac{1}{2}$	<mfrac><mn>1</mn><mn>2</mn></mfrac>
	Sub-, superscript	msub	$a_{1}$	<msub><mi>a</mi><mn>1</mn></msub>
		msup	$b^{2}$	<msup><mi>b</mi><mn>2</mn></msup>
		msubsup	$c_{3}^{4}$	<msubsup> <mi>c</mi><mn>3</mn><mn>4</mn> </msubsup>
	Radical	msqrt	$\sqrt{3}$	<msqrt><mn>3</mn></msqrt>
	Radical	mroot	$\sqrt[3]{4}$	<mroot><mn>4</mn><mn>3</mn></mroot>
	Tables and matrices	mtable	$(\begin{matrix} 2 & -1 \\ 1 & 0 \end{matrix})$ $\begin{matrix} a & = & u v \\ b & = & u^{2} - v^{2} \end{matrix}$ →examples
		mtr
		mtd

Top-level element		math

The following token elements are not discussed here: maction, maligngroup, malignmark, mglyph ¹, ms.

Token element mo is discussed here but not its use for mathematical accents.

The following compound elements are not discussed here: menclose, mlabeledtr, mlongdiv, mmultiscripts, mpadded, mphantom, mprescripts, mscarries, mscarry, msgroup, msline, mstack, mstyle, munder, mover, munderover, none.

Note that some of the elements not discussed here are not supported by most or in some cases any browsers at this writing (2019), or are poorly supported.

The namespace for MathML is http://www.w3.org/1998/Math/MathML.

The <math> element

All MathML must be within a <math> element. (To save space, many of this page's examples do not show it.)

A <math> element may not be contained within another <math> element.

The <math> element takes several attributes (see the W3C definition of math for specifics).

Attribute	Values	Effect	Default
display	'block' 'inline'	Displays its contents' presentation as a block or inline	'inline'

The <math> element does not take the ordinary HTML attributes such as class and style.

→W3C definition of math

Token elements

Identifiers: mi

<mi>S</mi> presents identifier $S$ , which can be a single letter or other character or a sequence of them.

Not obvious

Function names like $\sin$ should be presented with mi. (See the function application operator.)

Symbolic constants should be presented with mi; here are some examples:

Sym.	UTF-8 for it		In MathML
Sym.	Oct &#	Hex &#x	In MathML
$π$	π	π	<mi> π </mi>
$ⅇ$	ⅇ	ⅇ	<mi> ⅇ<!--DOUBLE-STRUCK ITALIC SMALL E--> </mi>
$ⅈ$	ⅈ	ⅈ	<mi> ⅈ<!--DOUBLE-STRUCK ITALIC SMALL I--> </mi>

$\dots$	…	…	<mi> … </mi>

A horizontal ellipsis as an element in a list should be presented with mi.

→W3C definition of mi

Numbers: mn

<mn>12</mn> presents the number $12$ .

→W3C definition of mn

Operators, fences, and separators: mo

According to the W3C specification, mo is used for so many kinds of things because operators required sophisticated rendering and it was too tempting to reuse mo for other things that needed similar sophistication, namely fences, separators, and (not discussed here) mathematical accents.

→W3C definition of mo

Operator	In MathML
$-$	<mo>-</mo>
$>$	<mo>></mo>
$\geq$	<mo>≥</mo>
$=>$	<mo>=></mo>
$\sum$	<mo>∑</mo>
$.NOT.$	<mo>.NOT.</mo>

Operators

<mo>o</mo> presents operator o, of one or more characters, with appropriate spacing, font size, etc.

Not obvious

mo is not used for functions, such as $\sin$ and $\cos$ (see the function application operator and mi).
The function application operator is an invisible character signifying the application of a named function (like sin or cos) to an operand. The function's name is presented with mi.
$\tan x$ :
```
  <math>
    <mi>tan</mi><mo>&ApplyFunction;</mo><mi>x</mi>
  </math>
```
The invisible times operator is an invisible character signifying the multiplication of two side-by-side factors.
$a b$ :
```
  <math>
    <mi>a</mi><mo>&InvisibleTimes;</mo><mi>a</mi>
  </math>
```
The invisible plus operator is an invisible character signifying the addition of two side-by-side addends, as in a mixed-fraction number.
$1 \frac{2}{3}$ :
```
  <math>
    <mn>1</mn> <mo>&#8292;</mo>
    <mfrac><mn>2</mn><mn>3</mn></mfrac>
  </math>
```

Sym.	utf-8 for it		In MathML
Sym.	Oct &#	Hex &#x	In MathML
function application	⁡	⁡	<mo>⁡</mo> <mo>⁡</mo>
invisible times	⁢	⁢	<mo>⁢</mo> <mo>⁢</mo>
invisible plus	⁤	⁤	<mo>⁤<!--INVISIBLE PLUS--></mo>

Fences

mo also presents fences such as ( and ).

(a + b)

using mrow for subexpression

a + b

  <math>
    <mo>(</mo>
    <mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow>
    <mo>)</mo>
  </math>

Note that $(a + b)$ may be equivalently presented using mfenced:

The fence='true' attribute should be given for fences that are not single ‘ ’ or double “ ” quotes, parentheses ( ), square brackets [ ], curly braces { }, absolute value bars | |, or any of about four dozen more listed in MathML's operator dictionary.

Separators

mo also presents separators such as commas.

Not obvious

The separator='true' attribute should be given for any separator other than comma, semicolon, or the invisible separator (⁣); see MathML's operator dictionary.

Sym.	utf-8 for it		In MathML
Sym.	Oct &#	Hex &#x	In MathML
invisible separator (invisible comma)	⁣	⁣	<mo>⁣</mo> <mo>⁣</mo>

a_{1, 2}

as for an array element in the first row and second column, presented using msub and mrow:

  <math>
    <msub>
      <mi>a</mi>
      <mrow>
        <mn>1</mn>
        <mo>&InvisibleComma:</mo>
        <mn>2</mn>
      </mrow>
    </msub>
  </math>

Text: mtext

<mtext>Commentary text</mtext> presents text $Commentary text$ .

→W3C definition of mtext

Space: mspace

<mspace width='1em' /> presents 1em of whitespace .

Widths may also be given in units of px, cm, mm, pt, and several others.

→W3C definition of mspace

Compound elements

Compound elements group mathematical subexpressions in useful ways.

Horizontally-grouped subexpressions: mrow

An mrow element groups its contents in a horizontal row, spaced appropriately as determined by what its contents are.

An mrow may contain other mrows (or other compound elements), and in fact often should. Each subexpression within an mrow should be contained in its own nested mrow. Each operator should be placed in its own mrow along with its operand(s), with the exception that a sequence of operators of the same precedence ² and their operands may be placed in a single mrow (for example, addition and subtraction may share an mrow, or multiplication and division, but not addition and multiplication).

a (b + c) = a b + a c

Each of these is a subexpression and should be in its own mrow:

$b + c$
$(b + c)$
$a (b + c)$
$a b$
$a c$
$a b + a c$

MathML

  <math>
    <mrow>
      <mi>a</mi>
      <mo>&InvisibleTimes;</mo>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mi>b</mi><mo>+</mo><mi>c</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
    </mrow>
    <mo>=</mo>
    <mrow>
      <mrow>
        <mi>a</mi>
        <mo>&InvisibleTimes;</mo>
        <mi>b</mi>
      </mrow>
      <mo>+</mo>
      <mrow>
        <mi>a</mi>
        <mo>&InvisibleTimes;</mo>
        <mi>c</mi>
      </mrow>
    </mrow>
  </math>

Several other MathML elements present their contents as if enclosed in an mrow:

math
mfenced
mfrac
mtd

and others not covered by this web page but described in the W3C definition §3.1.3.2.

→W3C definition of mrow

Horizontally-grouped subexpressions inside a pair of fences and separated by commas or other separators: mfenced

An mfenced element groups its contents in a horizontal row, separated by commas and enclosed in parentheses.

Different separators (or none) and different fences (or none) may be specified using attributes. With no separators and no fences, mfenced has the same effect as mrow.

Attribute	Value type	Effect of its value	Default
open	String	The opening fence	(
close	String	The closing fence	)
separators	String	The separator character(s); if more than one is given, they are used in sequence	,

$(a, b, c)$ <math><mfenced><mi>a</mi><mi>b</mi><mi>c</mi></mfenced></math>

[a: b: c]

<math>
  <mfenced open='[' close=']' separators=':,'>
    <mi>a</mi><mi>b</mi><mi>c</mi>
  </mfenced>
</math>

\max (d, e, f)

<math>
  <mi>max</mi>
  <mo>&#8289;<!--function application--></mo>
  <mfenced>
    <mi>d</mi><mi>e</mi><mi>f</mi>
  </mfenced>
</math>

→W3C definition of mfenced

Fractions: mfrac

<mfrac>N D</mfrac> presents the fraction with numerator N and denominator D.

The fraction is presented with a horizontal fraction bar, or with a slanted fraction bar if the attribute bevelled='true' is given.

$\frac{π}{2}$ <math><mfrac><mi>π</mi><mn>2</mn></mfrac></math>

\frac{1}{b + 2}

  <math>
    <mfrac>
      <mn>1</mn>
      <mrow><mi>b</mi><mo>+</mo><mn>2</mn></mrow>
    </mfrac>
  </math>

\frac{1}{b + 2}

  <math>
    <mfrac bevelled='true'>
      <mn>1</mn>
      <mrow><mi>b</mi><mo>+</mo><mn>2</mn></mrow>
    </mfrac>
  </math>

→W3C definition of mfrac

Sub- and superscripts: msub, msup, msubsup

Subscripts: msub

<msub>E B</msub> presents subexpression E with subscript B.

a_{1}

→W3C definition of msub

Superscripts: msup

<msub>E P</msub> presents subexpression E with superscript P.

b^{2 d}

  <math>
    <msup>
      <mi>b</mi>
      <mrow>
        <mn>2</mn>
        <mo>&InvisibleTimes#59</mo>
        <mi>d</mi>
      </mrow>
    </msup>
  </math>

→W3C definition of msup

Both sub- and superscripts: msubsup

<msubsup>E B P</msubsup> presents subexpression E with subscript B and superscript P.

c_{3}^{4}

  <math>
    <msubsup>
      <mi>c</mi>
      <mn>3</mn>
      <mn>4</mn>
    </msubsup>
  </math>

→W3C definition of msubsup

Radicals: msqrt, mroot

Square roots: msqrt

<msqrt>E</msqrt> presents the square root of subexpression E.

\sqrt{b^{2} - 4 a c}

MathML

  <math>
    <msqrt>
      <msup><mi>b</mi><mn>2</mn></msup>
      <mo>-</mo>
      <mrow>
        <mn>4</mn>
        <mo>&InvisibleTimes#59</mo>
        <mi>a</mi>
        <mo>&InvisibleTimes#59</mo>
        <mi>c</mi>
      </mrow>
    </msqrt>
  </math>

→W3C definition of msqrt

Roots in general: mroot

<mroot>E N</mroot> presents the Nth root of subexpression E.

\sqrt[4]{3}

→W3C definition of mroot

Tables and matrices: mtable, mtr, mtd

The mtable element presents a table, a matrix, or subexpressions whose layout can be specified in tabular form.

Analogous to an HTML table, an mtable element contains one or more mtr elements presenting the table's rows; each mtr element contains one or more mtd elements presenting the table's cells.

(\begin{matrix} 2 & -1 \\ 1 & 0 \end{matrix})

  <math>
    <mfenced>
      <mtable>
        <mtr><mtd>2</mtd><mtd>-1</mtd></mtr>
        <mtr><mtd>1</mtd><mtd>0</mtd></mtr>
      </mtable>
    </mfenced>
  </math>

\begin{matrix} a & = & u v \\ b & = & u^{2} - v^{2} \end{matrix}

  <math>
    <mtable side='right'>
      <mtr>
        <mtd><mtext>(1)</mtext></mtd>
        <mtd><mi>a</mi></mtd>
        <mtd><mo>=</mo></mtd>
        <mtd>
          <mrow>
            <mi>u</mi>
            <mo>&InvisibleTimes#59
            </mo><mi>v</mi>
          </mrow>
        </mtd>
      </mtr>
      <mtr>
        <mtd><mi>b</mi></mtd>
        <mtd><mo>=</mo></mtd>
        <mtd>
          <mrow>
            <msup><mi>u</mi><mn>2</mn></msup>
            <mo>-</mo>
            <msup><mi>v</mi><mn>2</mn></msup>
          </mrow>
        </mtd>
      </mtr>
    </mtable>
  </math>

→W3C definition of mtable, mtr, mtd

Common attributes

Of any element

Attribute	Value type	Effect of its value	Default
class	String	Associates the style class(es) with the element	none
style	String	Associates the style information with the element	none
href	URI	Makes the element a hyperlink	none

See [MathML 3.0] Using CSS with MathML to use class or style intelligently.

Elements also accept the id and xref attributes (for cross-referencing presentation and content elements), not discussed here.

Of any presentation element

Attribute	Value type	Effect of its value	Default
mathcolor	color	Sets the element's color	inherited
mathbackgroundcolor	color	Sets the element's background color	transparent

Colors may be #RGB, #RRGGBB, or an HTML color name.

Of any token element

Attribute	Value type	Effect of its value	Default
mathvariant	normal, bold, italic, double-struck, script, sans-serif, twelve others	Sets the element's font type	normal (except for mi)
mathsize	small normal big length	Sets the element's size	inherited

Token elements also accept the dir attribute, not discussed here.

Notes

MathML allows mglyph, which presents a nonstandard character using an image, to appear in mi, mn, mo, and mtext elements while still classifying them as token elements.

The precedence of two operators expresses which of them binds more tightly and thus which one is performed first during evaluation of an expression containing them. For example, multiplication is said to bind more tightly than addition, so the expression

$2 \times 3 + 4 \times 5$

is evaluated in the sequence multiplications then addition

$2 \times 3 + 4 \times 5 = (2 \times 3) + (4 \times 5) = 6 + 20 = 26$

rather than addition then multiplications

$2 \times 3 + 4 \times 5 \neq 2 \times (3 + 4) \times 5 = 2 \times 7 \times 5 = 70$

In contrast, addition and subtraction are said to bind with equal tightness. In such cases the operations are typically performed left-to-right, so that

Precedence	Operators
20	( )
40	, invisible comma
190	∨
200	∧
230	∀, ∃
240	∈, ⊂, ⊃
241	≤
242	≥
243	>
245	<
255	≠
260	=
265	∙ /
275	− +
390	invisible times
880	invisible plus

$1 + 2 - 3 + 4$

is evaluated

$1 + 2 - 3 + 4 = 3 - 3 + 4 = 0 + 4 = 4$

Operations may be listed in a total order by precedence, that is to say, for any two operators a and b, either a binds more tightly than b, b binds more tightly than a, or a and b bind equally tightly. Consequently the precedence of each in a set of operators is expressed as an integer. By custom, smaller integers mean tighter binding. and larger, less-tight.

The operator precedences for MathML are listed in a very, very long table, from which a dozen or so common operators have been drawn and appear in the brief table at right. (I don't know why multiplication seems to have been given two different precedences, one for ∙ and the other for invisible times.)

References

Mathematical Markup Language (MathML) Version 3.0 2nd Edition. W3C Recommendation 10 April 2014.

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thomas · alspaugh @ acm · org
2019May11Sa10:38
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