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'''Mathematical Markup Language''' ('''MathML''') is a [[mathematical markup language]], an application of [[XML]] for describing [[mathematics|mathematical]] notations and capturing both its structure and content. It aims at integrating mathematical formulae into [[World Wide Web]] pages and other documents. It is part of [[HTML5]] and is a ISO/IEC standard [https://www.iso.org/standard/58439.html ISO/IEC 40314] since 2015.
=ith six children in tow, Catherine raced to the airport departing gate. This wasn't an easy task as the children had other priorities than to get to the gate. She knew that she was tight on time and the frustration came out as she yelled at the kids to keep up. They continued to test her, pretending not to listen and to move in directions that only slowed them down. They had no idea the wrath they were about to receive when Catherine made it to the gate only to be informed that they had all missed the plane.
==History==
He sat across from her trying to imagine it was the first time. It wasn't. Had it been a hundred? It quite possibly could have been. Two hundred? Probably not. His mind wandered until he caught himself and again tried to imagine it was the first time.
MathML 1 was released as a W3C recommendation in April 1998 as the first [[XML]] language to be recommended by the [[W3C]]. Version 1.01 of the format was released in July 1999 and version 2.0 appeared in February 2001.
Have you ever wondered about toes? Why 10 toes and not 12. Why are some bigger than others? Some people can use their toes to pick up things while others can barely move them on command. Some toes are nice to look at while others are definitely not something you want to look at. Toes can be stubbed and make us scream. Toes help us balance and walk. 10 toes are just something to ponder.
Sleep deprivation causes all sorts of challenges and problems. When one doesn’t get enough sleep one’s mind doesn’t work clearly. Studies have shown that after staying awake for 24 hours one’s ability to do simple math is greatly impaired. Driving tired has been shown to be as bad as driving drunk. Moods change, depression, anxiety, and mania can be induced by lack of sleep. As much as people try to do without enough sleep it is a wonder more crazy things don’t happen in this world.
In October 2003, the second edition of MathML Version 2.0 was published as the final release by the W3C Math Working Group.
I'm meant to be writing at this moment. What I mean is, I'm meant to be writing something else at this moment. The document I'm meant to be writing is, of course, open in another program on my computer and is patiently awaiting my attention. Yet here I am plonking down senseless sentiments in this paragraph because it's easier to do than to work on anything particularly meaningful. I am grateful for the distraction.
Brenda never wanted to be famous. While most of her friends dreamed about being famous, she could see the negative aspects that those who wanted to be famous seemed to ignore. The fact that you could never do anything in public without being mobbed and the complete lack of privacy was something that she never wanted to experience. She also had no desire to have strangers speculating about every aspect of her life and what each thing she did was supposed to mean. Brenda was perfectly happy with her anonymous life where she could do exactly as she wanted without anyone else giving a damn. Thus, her overnight Internet celebrity was not something she was thrilled about as her friends told her how lucky she was.
MathML was originally designed before the finalization of [[XML namespace]]s. However, it was assigned a namespace immediately after the Namespace Recommendation was completed, and for XML use, the elements should be in the namespace with namespace URL <nowiki>http://www.w3.org/1998/Math/MathML</nowiki>. When MathML is used in HTML (as opposed to XML) this namespace is automatically inferred by the HTML parser and need not be specified in the document.
Do you think you're living an ordinary life? You are so mistaken it's difficult to even explain. The mere fact that you exist makes you extraordinary. The odds of you existing are less than winning the lottery, but here you are. Are you going to let this extraordinary opportunity pass?
Patrick didn't want to go. The fact that she was insisting they must go made him want to go even less. He had no desire to make small talk with strangers he would never again see just to be polite. But she insisted that Patrick go, and she would soon find out that this would be the biggest mistake she could make in their relationship.
===MathML version 3===
It was always the Monday mornings. It never seemed to happen on Tuesday morning, Wednesday morning, or any other morning during the week. But it happened every Monday morning like clockwork. He mentally prepared himself to once again deal with what was about to happen, but this time he also placed a knife in his pocket just in case.
Version 3 of the MathML specification was released as a [[World Wide Web Consortium#W3C recommendation (REC)|W3C recommendation]] on 20 October 2010. A recommendation of ''A MathML for CSS Profile'' was later released on 7 June 2011;<ref name=CSSProfile>{{cite web|title=A MathML for CSS Profile|url=http://www.w3.org/TR/mathml-for-css/|publisher=W3C|access-date=25 July 2013|date=7 June 2011}}</ref> this is a subset of MathML suitable for CSS formatting. Another subset, ''Strict Content MathML'', provides a subset of content MathML with a uniform structure and is designed to be compatible with [[OpenMath]]. Other content elements are defined in terms of a transformation to the strict subset. New content elements include {{tag|bind|o}} which associates bound variables ({{tag|bvar|o}}) to expressions, for example a summation index. The new {{tag|share|o}} element allows structure sharing.<ref name="V3"/>
The computer wouldn't start. She banged on the side and tried again. Nothing. She lifted it up and dropped it to the table. Still nothing. She banged her closed fist against the top. It was at this moment she saw the irony of trying to fix the machine with violence.
There once lived an old man and an old woman who were peasants and had to work hard to earn their daily bread. The old man used to go to fix fences and do other odd jobs for the farmers around, and while he was gone the old woman, his wife, did the work of the house and worked in their own little plot of land.
The development of MathML 3.0 went through a number of stages. In June 2006 the W3C rechartered the MathML Working Group to produce a MathML 3 Recommendation until February 2008 and in November 2008 extended the charter to April 2010. A sixth Working Draft of the MathML 3 revision was published in June 2009. On 10 August 2010 version 3 graduated to become a "Proposed Recommendation" rather than a draft.<ref name="V3">[http://www.w3.org/TR/MathML3/ Mathematical Markup Language Version 3.0 W3C Recommendation]. W3.org. Retrieved on 9 May 2012.</ref>
Betty was a creature of habit and she thought she liked it that way. That was until Dave showed up in her life. She now had a choice to make and it would determine whether her lie remained the same or if it would change forever.
It was a question of which of the two she preferred. On the one hand, the choice seemed simple. The more expensive one with a brand name would be the choice of most. It was the easy choice. The safe choice. But she wasn't sure she actually preferred it.
The Second Edition of MathML 3.0 was published as a W3C Recommendation on 10 April 2014.<ref name="w3c-standard-2nd-ed" /> The specification was approved as an [[International Electrotechnical Commission|ISO/IEC]] international standard 40314:2015 on 23 June 2015.<ref>{{cite web|url=https://www.w3.org/2015/06/mathmlpas.html.en |title=W3C MathML 3.0 Approved as ISO/IEC International Standard |website=W3.org |date=23 June 2015 |access-date=12 July 2015}}</ref>
There was a time when this wouldn't have bothered her. The fact that it did actually bother her bothered her even more. What had changed in her life that such a small thing could annoy her so much for the entire day? She knew it was ridiculous that she even took notice of it, yet she was still obsessing over it as she tried to fall asleep.
The opened package of potato chips held the answer to the mystery. Both detectives looked at it but failed to realize it was the key to solve the crime. They passed by it assuming it was random trash ensuring that the case would never be solved.
==Presentation and semantics==
One dollar and eighty-seven cents. That was all. And sixty cents of it was in pennies. Pennies saved one and two at a time by bulldozing the grocer and the vegetable man and the butcher until one’s cheeks burned with the silent imputation of parsimony that such close dealing implied. One dollar and eighty-seven cents. And the next day would be Christmas...
MathML deals not only with the ''presentation'' but also the ''meaning'' of formula components (the latter part of MathML is known as "Content MathML"). Because the meaning of the equation is preserved separate from the presentation, how the content is communicated can be left up to the user. For example, web pages with MathML embedded in them can be viewed as normal web pages with many browsers, but visually impaired users can also have the same MathML read to them through the use of [[screen readers]] (e.g. using the [[MathPlayer]] [[Plug-in (computing)|plugin]] for [[Internet Explorer]] or [[Firefox]], [[Opera (web browser)|Opera]] 9.50 build 9656+ or the Fire Vox extension for Firefox). Newer versions of [[JAWS (screen reader)|JAWS]] support MathML voicing as well as braille output.<ref>{{Cite web|title=JAWS Version 16|url=https://support.freedomscientific.com/downloads/jaws/JAWSWhatsNew?version=16|url-status=live}}</ref>
Twenty-five hours had passed since the incident. It seemed to be a lot longer than that. That twenty-five hours seemed more like a week in her mind. The fact that she still was having trouble comprehending exactly what took place wasn't helping the matter. She thought if she could just get a little rest the entire incident might make a little more sense.
She put the pen to paper but she couldn't bring herself to actually write anything. She just stared at the blank card and wondered what words she could write that would help in even a small way. She thought of a dozen ways to begin but none seemed to do justice to the situation. There were no words that could help and she knew it.
===Presentation MathML===
You're going to make a choice today that will have a direct impact on where you are five years from now. The truth is, you'll make choice like that every day of your life. The problem is that on most days, you won't know the choice you make will have such a huge impact on your life in the future. So if you want to end up in a certain place in the future, you need to be careful of the choices you make today.
Presentation MathML focuses on the display of an equation, and has about 30 elements. The elements' names all begin with {{code|m}}. A Presentation MathML expression is built up out of ''tokens'' that are combined using higher-level elements, which control their layout (there are also about 50 attributes, which mainly control fine details).
The tree missed the days the kids used to come by and play. It still wore the tire swing the kids had put up in its branches years ago although both the tire and the rope had seen better days. The tree had watched all the kids in the neighborhood grow up and leave, and it wondered if there would ever be a time when another child played and laughed again under its branches. That was the hope that the tree wished every day as the swing gently swung empty in the wind.
I checked in for the night at Out O The Way motel. What a bad choice that was. First I took a shower and a spider crawled out of the drain. Next, the towel rack fell down when I reached for the one small bath towel. This allowed the towel to fall halfway into the toilet. I tried to watch a movie, but the remote control was sticky and wouldn’t stop scrolling through the channels. I gave up for the night and crawled into bed. I stretched out my leg and felt something furry by my foot. Filled with fear, I reached down and to my surprise, I pulled out a raccoon skin pair of underwear. After my initial relief that it wasn’t alive, the image of a fat, ugly businessman wearing raccoon skin briefs filled my brain. I jumped out of the bed, threw my toothbrush into my bag, and sprinted towards my car.
Token elements generally only contain characters (not other elements). They include:
They needed to find a place to eat. The kids were beginning to get grumpy in the back seat and if they didn't find them food soon, it was just a matter of time before they were faced with a complete meltdown. Even knowing this, the solution wasn't easy. Everyone in the car had a different opinion on where the best place to eat would be with nobody agreeing with the suggestions of the others. It seemed to be an impossible no-win situation where not everyone would be happy no matter where they decided to eat which in itself would lead to a meltdown. Yet a decision needed to be made and it needed to be made quickly.
* {{code|<mi>x</mi>}} – identifiers;
It wasn't supposed to end that way. The plan had been meticulously thought out and practiced again and again. There was only one possible result once it had been implemented, but as they stood there the result wasn't anything close to what it should have been. They all blankly looked at each wondering how this could have happened. In their minds, they all began to blame the other members of the group as to why they had failed.
* {{code|<mo>+</mo>}} – operators;
Brock would have never dared to do it on his own he thought to himself. That is why Kenneth and he had become such good friends. Kenneth forced Brock out of his comfort zone and made him try new things he'd never imagine doing otherwise. Up to this point, this had been a good thing. It had expanded Brock's experiences and given him a new appreciation for life. Now that both of them were in the back of a police car, all Brock could think was that he would have never dared do it except for the influence of Kenneth.
* {{code|<mn>2</mn>}} – numbers.
It was their first date and she had been looking forward to it the entire week. She had her eyes on him for months, and it had taken a convoluted scheme with several friends to make it happen, but he'd finally taken the hint and asked her out. After all the time and effort she'd invested into it, she never thought that it would be anything but wonderful. It goes without saying that things didn't work out quite as she expected.
* {{code|<mtext>non zero</mtext>}} – text.
The thing that's great about this job is the time sourcing the items involves no traveling. I just look online to buy it. It's really as simple as that. While everyone else is searching for what they can sell, I sit in front of my computer and buy better stuff for less money and spend a fraction of the time doing it.
Indescribable oppression, which seemed to generate in some unfamiliar part of her consciousness, filled her whole being with a vague anguish. It was like a shadow, like a mist passing across her soul's summer day. It was strange and unfamiliar; it was a mood. She did not sit there inwardly upbraiding her husband, lamenting at Fate, which had directed her footsteps to the path which they had taken. She was just having a good cry all to herself. The mosquitoes made merry over her, biting her firm, round arms and nipping at her bare insteps.
Note, however, that these token elements may be used as extension points, allowing markup in host languages.
The boxed moved. That was a problem. Peter had packed the box three hours before and there was nothing inside that should make it move. The question now was whether or not Peter was going to open it up and look inside to see why it had moved. The answer to that question was obvious. Peter dropped the package into the mailbox so he would never have to see it again.
MathML in [[HTML5]] allows most inline HTML markup in mtext, and {{code|<mtext><b>non</b> zero</mtext>}} is conforming, with the HTML markup being used within the MathML to mark up the embedded text (making the first word bold in this example).
The shoes had been there for as long as anyone could remember. In fact, it was difficult for anyone to come up with a date they had first appeared. It had seemed they'd always been there and yet they seemed so out of place. Why nobody had removed them was a question that had been asked time and again, but while they all thought it, nobody had ever found the energy to actually do it. So, the shoes remained on the steps, out of place in one sense, but perfectly normal in another.
The light blinded him. It was dark and he thought he was the only one in the area, but the light shining in his eyes proved him wrong. It came from about 100 feet away and was shining so directly into his eyes he couldn't make out anything about the person holding the light. There was only one thing to do in this situation. He reached into his pocket and pulled out a flashlight of his own that was much stronger than the one currently blinding him. He turned it on and pointed it into the stranger's eyes.
These are combined using layout elements, that generally contain only elements. They include:
It was a simple green chair. There was nothing extraordinary about it or so it seemed. It was the type of chair one would pass without even noticing it was there, let alone what the actual color of it was. It was due to this common and unassuming appearance that few people actually stopped to sit in it and discover its magical powers.
* {{tag|mrow|o}} – a horizontal row of items;
Lori lived her life through the lens of a camera. She never realized this until this very moment as she scrolled through thousands of images on your computer. She could remember the exact moment each photo was taken. She could remember where she had been, what she was thinking as she tried to get the shot, the smells of the surrounding area, and even the emotions that she felt taking the photo, yet she had trouble remembering what she had for breakfast.
* {{tag|msup|o}}, {{tag|munderover|o}}, and others – superscripts, limits over and under operators like sums, etc.;
The time to take action was now. All three men knew in their hearts this was the case, yet none of them moved a muscle to try. They were all watching and waiting for one of the others to make the first move so they could follow a step or two behind and help. The situation demanded a leader and all three men were followers.
* {{tag|mfrac|o}} – fractions;
The headache wouldn't go away. She's taken medicine but even that didn't help. The monstrous throbbing in her head continued. She had this happen to her only once before in her life and she realized that only one thing could be happening.
* {{tag|msqrt|o}} and {{tag|mroot|o}} – roots;
It was hidden under the log beside the stream. It had been there for as long as Jerry had been alive. He wasn't sure if anyone besides him and his friends knew of its existence. He knew that anyone could potentially find it, but it was well enough hidden that it seemed unlikely to happen. The fact that it had been there for more than 30 years attested to this. So it was quite a surprise when he found the item was missing.
* {{tag|mfenced|o}} – surrounding content with fences, such as parentheses.
There were a variety of ways to win the game. James had played it long enough to know most of them and he could see what his opponent was trying to do. There was a simple counterattack that James could use and the game should be his. He began deploying it with the confidence of a veteran player who had been in this situation a thousand times in the past. So, it was with great surprise when his opponent used a move he had never before seen or anticipated to easily defeat him in the game.
Her breath exited her mouth in big puffs as if she were smoking a cigarette. The morning dew had made her clothes damp and she shivered from the chill in the air. There was only one thing that could get her up and out this early in the morning.
As usual in HTML and XML, many [[character entity reference|entities]] are available for specifying special symbols by name, such as {{code|π}} and {{code|→}}. An interesting feature of MathML is that entities also exist to express normally-invisible operators, such as {{code|⁢}} (or the shorthand {{code|⁢}}) for implicit multiplication. They are:
Wandering down the path to the pond had become a daily routine. Even when the weather wasn't cooperating like today with the wind and rain, Jerry still took the morning stroll down the path until he reached the pond. Although there didn't seem to be a particular reason Jerry did this to anyone looking in from the outside, those who knew him well knew exactly what was going on. It could all be traced back to a specific incident that happened exactly 5 years previously.
* {{unichar|2061|FUNCTION APPLICATION}};
What have you noticed today? I noticed that if you outline the eyes, nose, and mouth on your face with your finger, you make an "I" which makes perfect sense, but is something I never noticed before. What have you noticed today?
* {{unichar|2062|INVISIBLE TIMES}};
It really doesn't matter what she thinks as it isn't her problem to solve. That's what he kept trying to convince himself. She was trying to insert her opinion where it wasn't wanted or welcome. He already had a plan and even though that plan didn't correspond with what she wanted him to do or what should be done, it wasn't her decision to make. The question now became whether he w
* {{unichar|2063|INVISIBLE SEPARATOR}};
* {{unichar|2064|INVISIBLE PLUS}}.
The full specification of MathML entities<ref>{{Cite web|url=http://www.w3.org/TR/MathML3/chapter7.html|title=Characters, Entities and Fonts|website=www.w3.org}}</ref> is closely coordinated with the corresponding specifications for use with HTML and XML in general.<ref>{{Cite web|url=http://www.w3.org/TR/xml-entity-names/Overview.html|title=XML Entity Definitions for Characters (2nd Edition)|website=www.w3.org}}</ref>
Thus, the expression <math forcemathmode=mathml>a x^2+b x+c</math> requires two layout elements: one to create the overall horizontal row and one for the superscripted exponent. Including only the layout elements and the (not yet marked up) bare tokens, the structure looks like this:
<syntaxhighlight lang="xml">
<mrow>
a ⁢ <msup>x 2</msup>
+ b ⁢ x
+ c
</mrow>
</syntaxhighlight>
However, the individual tokens also have to be identified as identifiers (mi), operators (mo), or numbers (mn). Adding the token markup, the full form ends up as
<syntaxhighlight lang="xml">
<mrow>
<mi>a</mi> <mo>⁢</mo> <msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo><mi>b</mi><mo>⁢</mo><mi>x</mi>
<mo>+</mo><mi>c</mi>
</mrow>
</syntaxhighlight>
A valid MathML document typically consists of the XML declaration, [[Document type declaration|DOCTYPE]] declaration, and document element.
The document body then contains MathML expressions which appear in <math> elements as needed in the document. Often, MathML will be embedded in more general documents, such as [[HTML]], [[DocBook]], or other [[XML]] schemas. A complete document that consists of just the MathML example above, is shown here:
<syntaxhighlight lang="xml">
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE math PUBLIC "-//W3C//DTD MathML 2.0//EN"
"http://www.w3.org/Math/DTD/mathml2/mathml2.dtd">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>a</mi> <mo>⁢</mo> <msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo><mi>b</mi><mo>⁢</mo><mi>x</mi>
<mo>+</mo><mi>c</mi>
</mrow>
</math>
</syntaxhighlight>
===Content MathML===
Content MathML focuses on the semantics, or meaning, of the expression rather than its layout. Central to Content MathML is the {{tag|apply|o}} element that represents function application. The function being applied is the first child element under {{tag|apply|o}}, and its operands or parameters are the remaining child elements. Content MathML uses only a few attributes.
Tokens such as identifiers and numbers are individually marked up, much as for Presentation MathML, but with elements such as {{code|ci}} and {{code|cn}}. Rather than being merely another type of token, operators are represented by specific elements, whose mathematical semantics are known to MathML: {{code|times}}, {{code|power}}, etc. There are over a hundred different elements for different functions and operators.<ref>{{Cite web|url=http://www.w3.org/TR/MathML3/chapter4.html#contm.opel|title=Content Markup|website=www.w3.org}}</ref>
For example, {{code|2=xml|<apply><sin/><ci>x</ci></apply>}} represents <math>\sin(x)</math> and {{code|2=xml|<apply><plus/><ci>x</ci><cn>5</cn></apply>}} represents <math>x+5</math>. The elements representing operators and functions are empty elements, because their operands are the other elements under the containing {{tag|apply|o}}.
The expression <math>a x^2+b x+c</math> could be represented as
<syntaxhighlight lang="xml">
<math>
<apply>
<plus/>
<apply>
<times/>
<ci>a</ci>
<apply>
<power/>
<ci>x</ci>
<cn>2</cn>
</apply>
</apply>
<apply>
<times/>
<ci>b</ci>
<ci>x</ci>
</apply>
<ci>c</ci>
</apply>
</math>
</syntaxhighlight>
Content MathML is nearly isomorphic to [[Binary expression tree|expressions]] in a [[Functional programming|functional language]] such as [[Scheme (programming language)|Scheme]]. {{tag|apply}} amounts to Scheme's {{code|(...)}}, and the many operator and function elements amount to Scheme functions. With this trivial literal transformation, plus un-tagging the individual tokens, the example above becomes:
<syntaxhighlight lang="scheme">
(plus
(times a (power x 2))
(times b x)
c)
</syntaxhighlight>
This reflects the long-known close relationship between XML element structures, and [[Lisp (programming language)|LISP]] or Scheme [[S-expressions]].<ref>Steven DeRose. The SGML FAQ Book: Understanding the Relationship of SGML and XML, Kluwer Academic Publishers, 1997. {{isbn|978-0-7923-9943-8}}.</ref><ref>[[Canonical S-expressions#cite note-0]]</ref>
====Wikidata annotation in Content MathML====
According to the OM Society,<ref name="OpenMath">{{Cite web|url=https://www.openmath.org/|title=OpenMath Home · OpenMath|website=www.openmath.org}}</ref> OpenMath Content Dictionaries can be employed as collections of symbols and identifiers with declarations of their semantics{{snd}}names, descriptions and rules. A 2018 paper presented at the [[Special Interest Group on Information Retrieval|SIGIR]] conference<ref name="SchubotzScharpfGipp2018">{{cite journal|author1=Moritz Schubotz|author2=Philipp Scharpf|author3=Bela Gipp|title=Representing Mathematical Formulae in Content MathML using Wikidata.|url=http://ceur-ws.org/Vol-2132/paper5.pdf|date=2018|journal=Birndl@sigir}}</ref> proposed that the semantic knowledge base [[Wikidata]] could be used as an OpenMath Content Dictionary to link semantic elements of a mathematical formula to unique and language-independent Wikidata items.
==Example and comparison to other formats==
The well-known [[quadratic formula]]:
: <math>x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}</math>
would be marked up using [[LaTeX]] syntax like this:
<syntaxhighlight lang="latex">
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
</syntaxhighlight>
in [[eqn (software)|troff/eqn]] like this:
x = {-b +- sqrt{b sup 2 – 4ac}} over 2a
in [[Apache OpenOffice Math]] and [[LibreOffice Math]] like this (all three are valid):
x = {-b plusminus sqrt {b^2 – 4 ac}} over {2 a}
x = {-b +- sqrt {b^2 – 4ac}} over {2a}
x = {-b ± sqrt {b^2 – 4ac}} over {2a}
in [[AsciiMath]] like this:
x = (-b +- sqrt(b^2 - 4ac))/(2a)
The above equation could be represented in Presentation MathML as an expression tree made up from layout elements like ''mfrac'' or ''msqrt'' elements:
<syntaxhighlight lang="xml">
<math mode="display" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo form="prefix">−</mo>
<mi>b</mi>
<mo>±</mo>
<msqrt>
<msup><mi>b</mi><mn>2</mn></msup>
<mo>−</mo>
<mn>4</mn><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>c</mi>
</msqrt>
</mrow>
<mrow>
<mn>2</mn>
<mo>⁢</mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<annotation encoding="application/x-tex"><!-- TeX -->
x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}
</annotation>
<annotation encoding="StarMath 5.0">
x = {-b plusminus sqrt {b^2 - 4 ac}} over {2 a}
</annotation>
<!-- More annotations can be written: application/x-troff-eqn for eqn, application/x-asciimath for AsciiMath... -->
<!-- Semantic MathML go under <annotation-xml encoding="MathML-Content">. -->
</semantics>
</math>
</syntaxhighlight>
This example uses the {{tag|annotation|open}} element, which can be used to embed a semantic annotation in non-XML format, for example to store the formula in the format used by an equation editor such as [[StarMath]] or the markup using [[LaTeX]] syntax. The {{code|encoding}} field is usually a [[MIME type]], although most of the equation encodings don't have such a registration; freeform text may be used in such case.
Although less compact than TeX, the XML structuring promises to make it widely usable, allows instant display in applications such as [[Web browser]]s, and facilitates an interpretation of its meaning in mathematical software products. MathML is not intended to be written or edited directly by humans.<ref>{{cite web
| url = http://www.w3.org/TR/REC-MathML/#abstract
| title = Mathematical Markup Language (MathML) 1.01 Specification (Abstract)
| access-date =26 September 2006
| last = Buswell
| first = Steven
| author2 = Devitt, Stan
| author3 = Diaz, Angel
| display-authors=etal
| date = 7 July 1999
| quote = While MathML is human-readable it is anticipated that, in all but the simplest cases, authors will use equation editors, conversion programs, and other specialized software tools to generate MathML.
}}</ref>
==Embedding MathML in HTML/XHTML files==
MathML, being XML, can be embedded inside other XML files such as [[XHTML]] files using XML namespaces. Browsers such as Firefox 3+ and Opera 9.6+ (support incomplete) can display Presentation MathML embedded in XHTML.
<syntaxhighlight lang="xml">
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
"http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>Example of MathML embedded in an XHTML file</title>
<meta name="description" content="Example of MathML embedded in an XHTML file"/>
</head>
<body>
<h1>Example of MathML embedded in an XHTML file</h1>
<p>
The area of a circle is
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>π<!-- π --></mi>
<mo>⁢<!-- ⁢ --></mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</math>.
</p>
</body>
</html>
</syntaxhighlight>
[[File:MathMLxhtml.png|thumb|435px|A rendering of the formula for a circle in MathML+XHTML using Firefox 22 on Mac OS X]]
Inline MathML is also supported in [[HTML5]] files in the current versions of [[WebKit]] ([[Safari (web browser)|Safari]] and [[JavaFX|JavaFX/WebView]] ), [[Gecko (software)|Gecko]] ([[Firefox]]). There is no need to specify namespaces like in the XHTML.
{{clear}}
<syntaxhighlight lang="html5">
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Example of MathML embedded in an HTML5 file</title>
</head>
<body>
<h1>Example of MathML embedded in an HTML5 file</h1>
<p>
The area of a circle is
<math>
<mi>π</mi>
<mo>⁢</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</math>.
</p>
</body>
</html>
</syntaxhighlight>
==Browser support==
{{outdated section|date=September 2021}}
Of the major [[web browser]]s, [[Gecko (software)|Gecko]]-based browsers (e.g., [[Firefox]]) have the most complete native support for MathML.<ref>{{citation |url=https://developer.mozilla.org/en-US/docs/Mozilla_MathML_Project/Status |title=Mozilla MathML Status |first=Frédéric |last=Wang |publisher=Mozilla}}</ref><ref>{{citation |url=https://www.mozilla.org/projects/mathml/authoring.html |title=Authoring MathML for Mozilla |first=Roger B. |last=Sidje |publisher=Mozilla}}</ref>
While the [[WebKit]] layout engine has a development version of MathML,<ref>{{citation |url=https://bugs.webkit.org/show_bug.cgi?id=3251 |title=Bug 3251 – Implement MathML (master bug) |publisher=WebKit}} . . . (However, "see also") https://trac.webkit.org/wiki/MathML%20Status</ref> this feature is only available in version 5.1 and higher of [[Safari (web browser)|Safari]]. It was available in Chrome up through <ref>{{citation|url=http://www.leancrew.com/all-this/2011/07/safari-5-1-and-math/|title=Safari 5.1 and math}}</ref> [[Google Chrome|Chrome]] 24<ref>{{Citation|title=MathML Support Coming To Chrome|url=http://browserfame.com/900/mathml-in-chrome|publisher=BrowserFame|access-date=16 October 2012|archive-url=https://web.archive.org/web/20121014191355/http://browserfame.com/900/mathml-in-chrome|archive-date=14 October 2012|url-status=dead}}</ref><ref>{{Citation|title=Stable Chrome 24 supports MathML and closes security holes |url=http://www.h-online.com/open/news/item/Stable-Chrome-24-supports-MathML-and-closes-security-holes-1781648.html|publisher=The H Online|access-date=11 January 2013}}</ref> at which point it was removed.<ref>{{Citation|title=Stable Channel Update |url=http://googlechromereleases.blogspot.ca/2013/02/stable-channel-update_21.html|access-date=7 April 2013}}</ref> At the time, Google removed support of MathML claiming architectural security issues and low usage did not justify their engineering time.<ref name="CNET">{{cite web|url=https://www.cnet.com/news/google-subtracts-mathml-from-chrome-and-anger-multiplies/|title=Google subtracts MathML from Chrome, and anger multiplies|work=CNET|access-date=22 November 2014}}</ref> In January 2023, support for MathML was reintroduced to Chrome in version 109 due to work by Igalia.<ref>{{cite web|url=https://developer.chrome.com/blog/new-in-chrome-109/|title=New in Chrome 109 |date=10 January 2023 |access-date=2023-01-10 |publisher=Chrome Developers}}</ref><ref>{{cite web|url=https://www.igalia.com/2023/01/10/Igalia-Brings-MathML-Back-to-Chromium.html|title=Igalia Brings MathML Back to Chromium|date=10 January 2023 |access-date=2023-01-10 |publisher=Igalia News}}</ref> {{As of|October 2013}}, the WebKit/Safari implementation has numerous bugs.<ref>{{cite web|url=http://www.maths-informatique-jeux.com/blog/frederic/?post/2013/10/12/Funding-MathML-Developments-in-Gecko-and-WebKit|title=Funding MathML Developments in Gecko and WebKit - Blog de Frédéric|author=Frédéric Wang|work=maths-informatique-jeux.com|access-date=22 November 2014|archive-url=https://web.archive.org/web/20141214030114/http://www.maths-informatique-jeux.com/blog/frederic/?post%2F2013%2F10%2F12%2FFunding-MathML-Developments-in-Gecko-and-WebKit|archive-date=14 December 2014|url-status=dead}}</ref>
[[JavaFX|JavaFX/WebView]]. Also based on WebKit, the JavaFX embedded web browser supports MathML starting with JavaFX 8 Update 192 and JavaFX 11 versions. Support is broken, in JavaFX 8 previous versions, JavaFX 9 and JavaFX 10.
[[Opera (web browser)|Opera]], between version 9.5 and 12, supports MathML for CSS profile,<ref>{{citation |url=http://dev.opera.com/articles/view/can-kestrels-do-math-mathml-support-in/ |title=Can Kestrels do Math? MathML support in Opera Kestrel |first=Charles |last=McCathieNevile |date=27 September 2007 |publisher=Opera}}</ref><ref>{{citation |url=http://my.opera.com/desktopteam/blog/2007/11/16/even-more-work |title=Even more work |first=Tommy A. |last=Olsen |date=16 November 2007 |publisher=Opera|archive-url=https://web.archive.org/web/20071118141611/http://my.opera.com/desktopteam/blog/2007/11/16/even-more-work |archive-date=18 November 2007}}</ref> but is unable to position diacritical marks properly.<ref>[https://web.archive.org/web/20081211143506/http://my.opera.com/mathml/blog/show.dml/1460837#comments MathML – The Opera MathML blog]. My.opera.com (1 November 2007). Retrieved on 9 May 2012.</ref> Prior to version 9.5 it required User JavaScript or custom stylesheets to emulate MathML support.<ref>[https://web.archive.org/web/20080928225043/http://my.opera.com/community/forums/topic.dml?id=124033 UserJS for MathML 2.0]. My.opera.com. Retrieved on 9 May 2012.</ref> Starting with [[Opera (web browser)|Opera]] 14, Opera drops support for MathML by switching to the Chromium 25 engine.<ref>{{citation|url=http://my.opera.com/ODIN/blog/2013/03/05/opera-14-beta-for-android-is-out|title=Opera 14 beta for Android is out|archive-url=https://web.archive.org/web/20130308155256/http://my.opera.com/ODIN/blog/2013/03/05/opera-14-beta-for-android-is-out |archive-date=2013-03-08}}</ref>
[[Microsoft Edge]] does not support MathML, nor are there plans to add support.<ref>{{citation |url=https://developer.microsoft.com/en-us/microsoft-edge/status/mathml/?q=MathML |title=Microsoft Edge Platform Status |access-date=7 April 2021}}</ref>
[[Internet Explorer]] does not support MathML natively. Support for [[Internet Explorer 6|IE6]] through [[Internet Explorer 9|IE9]] can be added by installing the [[MathPlayer]] plugin.<ref>{{citation |url=http://www.dessci.com/en/products/mathplayer/versionhistory.htm |title=MathPlayer – version history}}</ref> IE10 has some crashing bugs with MathPlayer and Microsoft decided to completely disable in IE11 the binary plug-in interface that MathPlayer needs.<ref>{{cite web|url=http://news.dessci.com/2013/10/microsoft-cripples-display-math-ie10-11.html|title=Microsoft cripples the display of math in IE10 & 11|work=dessci.com|access-date=22 November 2014|archive-url=https://web.archive.org/web/20140701235623/http://news.dessci.com/2013/10/microsoft-cripples-display-math-ie10-11.html|archive-date=1 July 2014|url-status=dead}}</ref> MathPlayer has a license that may limit its use or distribution in commercial webpages and software. Using or distributing the MathPlayer plugin to display HTML content via the WebBrowser control in commercial software may also be forbidden by this license.
The [[KHTML]]-based [[Konqueror]] currently does not provide support for MathML.<ref>{{citation |url=http://bugs.kde.org/show_bug.cgi?id=30526 |title=Bug 30526 – MathML support for Konqi}}</ref>
The quality of rendering of MathML in a browser depends on the installed fonts. The [[STIX Fonts project]] have released a comprehensive set of mathematical fonts under an open license. The [[Cambria Math]] font supplied with Microsoft Windows had a slightly more limited support.<ref name="Vismor">{{Citation|url=https://vismor.com/documents/site_implementation/viewing_mathematics/viewing_mathematics.php|title=Viewing Mathematics on the Internet|access-date=13 April 2011|first=Timothy|last=Vismor}}</ref>
According to a member of the MathJax team, none of the major browser makers paid any of their developers for any MathML-rendering work; whatever support exists is overwhelmingly the result of unpaid volunteer time/work.<ref>{{cite web|url=http://radar.oreilly.com/2013/11/mathml-forges-on.html|title=MathML forges on|author=Peter Krautzberger|work=oreilly.com|access-date=22 November 2014}}</ref>
In 2015 the MathML Association was founded to support the adoption of the MathML standard.<ref>{{citation|url=http://mathml-association.org/about.html|title=Mondial Association for Tools Handling MathML|access-date=20 June 2016|author1=Deyan Ginev |author2=Michael Kohlhase |author3=Moritz Schubotz |author4=Raniere Silva |author5=Frédéric Wang }}</ref>
{| class="wikitable sortable" style="font-size: 85%; text-align: center; width: auto;"
|-
! style="width:12em" | Browser
! MathML
|-
! {{rh2}} | [[Amaya (web browser)|Amaya]]
| {{Yes}}
|-
! {{rh2}} | [[AOL Explorer]]
| {{No}} <ref group="note" name="IEMathML">Available with the MathPlayer plugin.</ref>
|-
! {{rh}} | [[Avant Browser|Avant]]
| {{No}}
|-
! {{rh2}} | [[Arora (browser)|Arora]]
| {{Yes}}
|-
! {{rh}} | [[Basilisk (web browser)|Basilisk]]
| {{Yes}}
|-
! {{rh2}} | [[Camino (web browser)|Camino]]
| {{Yes}}
|-
! {{rh}} | [[Chromium (web browser)|Chromium]], [[Google Chrome|Chrome]]
| {{Yes}}<ref>{{cite web|url=https://developer.chrome.com/blog/new-in-chrome-109/|title=New in Chrome 109 |date=10 January 2023 |access-date=2023-01-10 |publisher=Chrome Developers}}</ref>
|-
! {{rh2}} | [[Dillo]]
| {{No}}
|-
! {{rh}} | [[Dooble (web browser)|Dooble]]
| {{Yes}}
|-
! {{rh2}} | [[ELinks]]
| {{No}}
|-
! {{rh}} | [[Falkon]]
| {{No}}
|-
! {{rh2}} | [[Flock (web browser)|Flock]]
| {{Yes}}
|-
! {{rh2}} | [[Galeon]]
| {{Yes}}
|-
! {{rh}} | [[iCab]]
| {{No}}
|-
! {{rh}} | [[Internet Explorer]]
| {{No}} <ref group="note" name="IEMathML"/>
|-
! {{rh2}} | [[Internet Explorer for Mac]]
| {{No}}
|-
! {{rh}} | [[K-Meleon]]
| {{Yes}}
|-
! {{rh}} | [[Konqueror]]
| {{No}}
|-
! {{rh}} | [[Links (web browser)|Links]]
| {{No}}
|-
! {{rh}} | [[Lunascape (web browser)|Lunascape]]
| {{Yes}} <ref name="luna">Depends on the [[Browser engine|layout engine]] which is chosen: Trident or Gecko or WebKit.</ref>
|-
! {{rh}} | [[Lynx (web browser)|Lynx]]
| {{No}}
|-
! {{rh}} | [[Maxthon]]
| {{No}}
|-
! {{rh}} | [[Microsoft Edge]]
| {{Yes}}
|-
! {{rh}} | [[Midori (web browser)|Midori]]
| {{No}}
|-
! {{rh2}} | [[Mosaic (web browser)|Mosaic]]
| {{No}}
|-
! {{rh2}} | [[Mozilla Application Suite|Mozilla]]
| {{Yes}}
|-
! {{rh}} | [[Firefox|Mozilla Firefox]]
| {{Yes}}
|-
! {{rh2}} | [[Netscape (web browser)|Netscape]]
| {{Yes}}
|-
! {{rh2}} | [[Netscape Browser]]
| {{depends}} <ref name="ns8">Depends on the [[Browser engine|layout engine]] which is chosen: Trident or Gecko.</ref>
|-
! {{rh2}} | [[Netscape Navigator]]
| {{No}}
|-
! {{rh2}} | [[Netscape Navigator 9]]
| {{Yes}}
|-
! {{rh}} | [[NetSurf]]
| {{No}}
|-
! {{rh}} | [[OmniWeb]]
| {{No}}
|-
! {{rh}} | [[Opera (web browser)|Opera]]
| {{Yes}}
|-
! {{rh}} | [[Otter Browser]]
| {{Yes}}
|-
! {{rh}} | [[Pale Moon (web browser)|Pale Moon]]
| {{Yes}}
|-
! {{rh}} | Polarity
| {{No}}
|-
! {{rh}} | [[qutebrowser]]
| {{No}}
|-
! {{rh}} | [[Safari (web browser)|Safari]]
| {{Yes}}
|-
! {{rh}} | [[SeaMonkey]]
| {{Yes}}
|-
! {{rh2}} | [[Shiira]]
| {{No}}
|-
! {{rh}} | [[Sleipnir (web browser)|Sleipnir]]
| {{No}} <ref group="note" name="IEMathML"/>
|-
! {{rh}} | [[surf (web browser)|surf]]
| {{No}}
|-
! {{rh}} | [[Torch browser|Torch Browser]]
| {{No}}
|-
! {{rh}} | [[Web (web browser)|Web]]
| {{Yes}}
|-
! {{rh2}} | [[WorldWideWeb]]
| {{No}}
|-
! {{rh}} | [[w3m]]
| {{No}}
|}
==Other standards==
Another standard called [[OpenMath]] that has been designed (largely by the same people who devised Content MathML) more specifically for storing formulae semantically can also be used to complement MathML. OpenMath data can be embedded in MathML using the {{code|1=<annotation-xml encoding="OpenMath">}} element. OpenMath ''content dictionaries'' can be used to define the meaning of {{code|<csymbol>}} elements. The following would define ''P''<sub>1</sub>(''x'') to be the first [[Legendre polynomial]]
<syntaxhighlight lang="xml">
<apply>
<csymbol encoding="OpenMath" definitionURL="http://www.openmath.org/cd/contrib/cd/orthpoly1.xhtml#legendreP">
<msub><mi>P</mi><mn>1</mn></msub>
</csymbol>
<ci>x</ci>
</apply>
</syntaxhighlight>
The [[OMDoc]] format has been created for markup of larger mathematical structures than formulae, from statements like definitions, theorems, proofs, or example, to theories and text books. Formulae in OMDoc documents can either be written in Content MathML or in OpenMath; for presentation, they are converted to Presentation MathML.
The [[International Organization for Standardization|ISO]]/[[International Electrotechnical Commission|IEC]] standard [[Office Open XML]] (OOXML) defines a different XML math syntax, derived from [[Microsoft Office]] products. However, it is partially compatible<ref>{{Citation| url=http://dpcarlisle.blogspot.com/2007/04/xhtml-and-mathml-from-office-20007.html
| 