MathML fractions and roots

Relying on text containers, this article describes how to build more complex MathML expressions by nesting fractions and roots.

Prerequisites:	Basic software installed , basic knowledge of working with files , and HTML basics (study Introduction to HTML .)
Objective:	To get familiar with MathML elements used for writing fractions and square roots.

Subtrees of <mfrac >, <msqrt >and <mroot >

In the getting started with MathML article, we've already met the <mfrac > element to describe a fraction. Let's consider a basic example which adds new elements for roots (<msqrt > and <mroot > ):

html

Below is a screenshot of how it is rendered by a browser:

$Screenshot of mfrac, msqrt, mroot$

We already know that the <mfrac > element is rendered as a fraction: The first child (the numerator) is drawn above the second child (the denominator) separated by a horizontal bar.
The <msqrt > is rendered as a square root: its children are laid out like an <mrow > , prefixed by a root symbol √ and completely covered by an overbar.
Finally, the <mroot > element is rendered as an nth root: The first element is covered by the radical symbol while the second element is used as the degree of the root and rendered as a prefix superscript.

Active learning: nesting different elements

Here is a simple exercise to verify whether you understood the relation between a MathML subtree and its visual rendering. The document contains a MathML formula and you must check all subtrees corresponding to a subtree in that MathML formula. Once you are done, you can inspect the source of the MathML formula and verify if it matches your expectation.

                                        
                                            
                                                <!
                                                DOCTYPE
                                                html
                                                >
                                            
                                            
                                                
                                                    <
                                                    html
                                                
                                                lang
                                                
                                                    =
                                                    "
                                                    en-US"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    meta
                                                
                                                charset
                                                
                                                    =
                                                    "
                                                    utf-8"
                                                
                                                />
                                            
                                            
                                                
                                                    <
                                                    title
                                                
                                                >
                                            
                                            My page with math characters
                                            
                                                
                                                    </
                                                    title
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    mfrac1"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msqrt
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    msqrt1"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mroot
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    mroot1"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            4
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mroot
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    mroot2"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            5
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    mfrac2"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            6
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            7
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msqrt
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    msqrt2"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            8
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            9
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    ol
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    options"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    Verify the order of children in an mfrac!"
                                                
                                                >
                                            
                                            An mfrac with an mroot as its first child and an msqrt as its second child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    mroot2"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The '6 over 7'-th root of five."
                                                
                                                >
                                            
                                            An mroot with an mn as its first child and mfrac as its second child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    This formula does not contain any fraction inside a square root!"
                                                
                                                >
                                            
                                            An msqrt containing an mfrac element.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The square root of two."
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    msqrt1"
                                                
                                                >
                                            
                                            An msqrt with one mn child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    Verify the order of children in an mroot!"
                                                
                                                >
                                            
                                            An mroot with an mfrac as its first child and mn as its second child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The square root of 'eight minus nine'."
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    msqrt2"
                                                
                                                >
                                            
                                            An msqrt with the following list of children: mn, mo, mn.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The square root of two over the cubic root of four."
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    mfrac1"
                                                
                                                >
                                            
                                            An mfrac with an msqrt as its first child and an mroot as its second child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    mfrac must have exactly two children!"
                                                
                                                >
                                            
                                            An mfrac with the following list of children: msqrt, mn, msqrt.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    mroot must have exactly two children!"
                                                
                                                >
                                            
                                            An mroot with one mn child.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The fraction six over seven."
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    mfrac2"
                                                
                                                >
                                            
                                            An mfrac with two mn children.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    This formula does not contain any square root with more than two numbers!"
                                                
                                                >
                                            
                                            An msqrt with five mn children.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    checkbox"
                                                
                                                data-highlight
                                                
                                                    =
                                                    "
                                                    mroot1"
                                                
                                                data-comment
                                                
                                                    =
                                                    "
                                                    The cubic root of four."
                                                
                                                >
                                            
                                            An mroot with two mn children.

                                            
                                                
                                                    </
                                                    li
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    ol
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    strong
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    comment"
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    strong
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    strong
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    status"
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    strong
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    body
                                                
                                                >

                                        
                                            math
                                            {
                                            font-family
                                            :
                                            Latin Modern Math,
                                            STIX Two Math;
                                            font-size
                                            :
                                            200%;
                                            }
                                            math .highlight
                                            {
                                            background
                                            :
                                            pink;
                                            }
                                            math [id] .highlight
                                            {
                                            background
                                            :
                                            lightblue;
                                            }
                                            p
                                            {
                                            padding
                                            :
                                            0.5em;
                                            }

                                        
                                            const
                                            options =
                                            document.
                                            getElementById
                                            (
                                            "options"
                                            )
                                            ;
                                            const
                                            comment =
                                            document.
                                            getElementById
                                            (
                                            "comment"
                                            )
                                            ;
                                            const
                                            checkboxes =
                                            Array.
                                            from
                                            (
                                            options.
                                            getElementsByTagName
                                            (
                                            "input"
                                            )
                                            )
                                            ;
                                            const
                                            status =
                                            document.
                                            getElementById
                                            (
                                            "status"
                                            )
                                            ;
                                            function
                                            verifyOption
                                            (
                                            checkbox
                                            )
                                            {
                                            let
                                            mathml =
                                            checkbox.
                                            dataset.
                                            highlight;
                                            if
                                            (
                                            mathml)
                                            {
                                            mathml =
                                            document.
                                            getElementById
                                            (
                                            mathml)
                                            ;
                                            }
                                            if
                                            (
                                            checkbox.
                                            checked)
                                            {
                                            comment.
                                            textContent =
                                            checkbox.
                                            dataset.
                                            comment;
                                            if
                                            (
                                            mathml)
                                            {
                                            mathml.
                                            classList.
                                            add
                                            (
                                            "highlight"
                                            )
                                            ;
                                            }
                                            else
                                            {
                                            checkbox.
                                            checked =
                                            false
                                            ;
                                            }
                                            }
                                            else
                                            {
                                            comment.
                                            textContent =
                                            ""
                                            ;
                                            if
                                            (
                                            mathml)
                                            {
                                            mathml.
                                            classList.
                                            remove
                                            (
                                            "highlight"
                                            )
                                            ;
                                            }
                                            }
                                            const
                                            finished =
                                            checkboxes.
                                            every
                                            (
                                            (
                                            checkbox
                                            )
                                            =>
                                            !
                                            !
                                            checkbox.
                                            checked ===
                                            !
                                            !
                                            checkbox.
                                            dataset.
                                            highlight,
                                            )
                                            ;
                                            status.
                                            textContent =
                                            finished
?
                                            "Congratulations, you checked all the correct answers!"
                                            :
                                            ""
                                            ;
                                            }
                                            checkboxes.
                                            forEach
                                            (
                                            (
                                            checkbox
                                            )
                                            =>
                                            {
                                            checkbox.
                                            addEventListener
                                            (
                                            "change"
                                            ,
                                            (
                                            )
                                            =>
                                            {
                                            verifyOption
                                            (
                                            checkbox)
                                            ;
                                            }
                                            )
                                            ;
                                            }
                                            )
                                            ;

Stretchy radical symbols

As previously seen, the overbar of the <msqrt > and <mroot > elements stretches horizontally to cover their content. But actually the root symbol √ also stretches to be as tall as their content.

                                        
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                />

html

Warning: Special math fonts are generally required to make that stretching possible, the previous example relies on web fonts .

Fractions without bar

Some mathematical concepts are sometimes written using fraction-like notations such binomial coefficients or Legendre symbols . It is appropriate to use an <mfrac > element to markup such notations. For fraction-like notations that don't draw a horizontal bar, attach a linethickness="0" attribute to the <mfrac > element:

                                        
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                />

html

                                        
                                            
                                                
                                                    <
                                                    math
                                                
                                                display
                                                
                                                    =
                                                    "
                                                    block"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            (
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                linethickness
                                                
                                                    =
                                                    "
                                                    0"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            )
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            =
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ≠
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >

Note: Although the linethickness attribute can be used to specify an arbitrary thickness, it is better to keep the default value, which is calculated from parameters specified in the math font.

Summary

In this lesson, we've seen how to build fractions and roots using the <mfrac > , <msqrt > and <mroot > elements. We noticed some special feature of these elements, namely the fraction and radical symbol. We've seen how to use the linethickness attribute to draw fractions without bars. In the next article, we will continue with basic math notations and consider scripts .

Roots

MathML fractions and roots

Subtrees of <mfrac >, <msqrt >and <mroot >

Active learning: nesting different elements

Stretchy radical symbols

Fractions without bar

Summary

See also