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<!DOCTYPE HTML>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Introduction</title>
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<style type="text/css">
/** Table of Contents taken from html 5.2 spec
stop flicker in toc **/
.toc a {
/* More spacing; use padding to make it part of the click target. */
padding-top: 0.1rem;
/* Larger, more consistently-sized click target */
display: block;
/* Reverse color scheme */
color: black;
border-color: #3980B5;
border-bottom-width: 3px !important;
margin-bottom: 0px !important;
}
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
div.graphic{margin-left:2em}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
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text-align: left;
}
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code { font-family: monospace; }
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div.note,
div.notice { margin-left: 2em; }
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margin-bottom: 0.3em; }
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margin: 0em; padding: 0em}
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border-bottom-color: #d3d3d3;
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margin: 4px}
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border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
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h5 { text-align: left;
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</style>
<link rel="stylesheet" type="text/css" href="https://www.w3.org/StyleSheets/TR/2016/W3C-ED.css"><script src="//www.w3.org/scripts/TR/2016/fixup.js"> </script></head>
<body>
<h1><a id="intro"></a>1 Introduction</h1>
<!-- TOP NAVIGATION BAR -->
<nav id="toc">
<ol class="toc">
<li><a href="Overview.html"><span class="secno"></span> Mathematical Markup Language (MathML) Version 4.0</a></li>
<li><a href="chapter2.html"><span class="secno">2 </span> MathML Fundamentals</a></li>
<li><a href="chapter1.html"><span class="secno">1 </span> Introduction</a>
<ol class="toc">
<li><a href="chapter1.html#intro.notation"><span class="secno">1.1 </span> Mathematics and its Notation</a>
</li>
<li><a href="chapter1.html#intro.origin"><span class="secno">1.2 </span> Origins and Goals</a>
<ol class="toc">
<li><a href="chapter1.html#intro.goals"><span class="secno">1.2.1 </span> Design Goals of MathML</a>
</li>
</ol>
</li>
<li><a href="chapter1.html#intro.overview"><span class="secno">1.3 </span> Overview</a>
</li>
<li><a href="chapter1.html#intro.example"><span class="secno">1.4 </span> A First Example</a>
</li>
</ol>
</li>
</ol>
</nav>
<div class="div1">
<div class="div2">
<h2><a id="intro.notation"></a>1.1 Mathematics and its Notation</h2>
<p>A distinguishing feature of mathematics is the use of a complex and
highly evolved system of two-dimensional symbolic notation. As
J. R. Pierce writes in his book on communication theory,
mathematics and its notation should not be viewed as one and the same
thing <a href="appendixh.html#Pierce1961">[Pierce1961]</a>. Mathematical ideas can exist independently of
the notation that represents them. However, the relation between meaning
and notation is subtle, and part of the power of mathematics to describe
and analyze derives from its ability to represent and manipulate ideas in
symbolic form. The challenge before a Mathematical Markup Language (MathML)
in enabling mathematics on the World Wide Web
is to capture both notation and content (that is, its meaning) in such a way
that documents can utilize the highly evolved notation of written
and printed mathematics as well as the new potential for interconnectivity in
electronic media.</p>
<p>Mathematical notation evolves constantly as people continue
to innovate in ways of approaching and expressing ideas. Even
the common notation of arithmetic has gone through an amazing
variety of styles, including many defunct ones advocated by leading
mathematical figures of their day <a href="appendixh.html#Cajori1928">[Cajori1928]</a>. Modern
mathematical notation is the product of centuries of refinement, and
the notational conventions for high-quality typesetting are quite
complicated and subtle. For example, variables and letters which stand for
numbers are usually typeset today in a special mathematical italic
font subtly distinct from the usual text italic; this seems to have been
introduced in Europe in the late sixteenth century. Spacing around
symbols for operations such as +, -, × and / is slightly
different from that of text, to reflect conventions about operator
precedence that have evolved over centuries.
Entire books have been devoted to the conventions of
mathematical typesetting, from the alignment of superscripts and
subscripts, to rules for choosing parenthesis sizes, and on to
specialized notational practices for subfields of mathematics. The
manuals describing the nuances of present-day computer typesetting and
composition systems can run to hundreds of pages.</p>
<p>Notational conventions in mathematics, and in printed text in general,
guide the eye and make printed expressions much easier to read and
understand. Though we usually take them for granted, we, as modern readers,
rely on
numerous conventions such as paragraphs, capital letters, font
families and cases, and even the device of decimal-like numbering of
sections such as is used in this document.
Such notational conventions are perhaps even more important
for electronic media, where one must contend with the difficulties of
on-screen reading. Appropriate standards coupled with computers
enable a broadening of access to mathematics beyond the world of
print. The markup methods for mathematics in use
just before the Web rose to prominence importantly included T<span style="vertical-align:-.4ex; margin-left:-.15ex; margin-right:-.15ex">E</span>X
(also written <code>TeX</code>) <a href="appendixh.html#Knuth1986">[Knuth1986]</a>
and approaches based on SGML (<a href="appendixh.html#AAP-math">[AAP-math]</a>, <a href="appendixh.html#Poppelier1992">[Poppelier1992]</a> and <a href="appendixh.html#ISO-12083">[ISO-12083]</a>).
</p>
<p>It is remarkable how widespread the current
conventions of mathematical notation have become. The general
two-dimensional layout, and most of the same symbols, are used
in all modern mathematical communications,
whether the participants are, say, European, writing left-to-right,
or Middle-Eastern, writing right-to-left. Of course, conventions
for the symbols used, particularly those naming functions and
variables, may tend to favor a local language and script.
The largest variation from the most common is a form used in
some Arabic-speaking communities which lays out the entire
mathematical notation from right-to-left, roughly in mirror image
of the European tradition.</p>
<p>However, there is more to putting mathematics on the Web
than merely finding ways of displaying traditional
mathematical notation in a Web browser. The Web represents a
fundamental change in the underlying metaphor for knowledge
storage, a change in which <em>interconnection</em>
plays a central role. It has become important to
find ways of communicating mathematics which facilitate
automatic processing, searching and indexing, and reuse in
other mathematical applications and contexts. With this
advance in communication technology, there is an opportunity
to expand our ability to represent, encode, and ultimately to
communicate our mathematical insights and understanding with
each other. We believe that MathML as specified below is an important step in
developing mathematics on the Web.</p>
</div>
<div class="div2">
<h2><a id="intro.origin"></a>1.2 Origins and Goals</h2>
<div class="div3">
<h3><a id="intro.goals"></a>1.2.1 Design Goals of MathML</h3>
<p>MathML has been designed from the beginning with the following
ultimate goals in mind.</p>
<p>MathML should ideally:
</p>
<ul>
<li>
<p>Encode mathematical material suitable for all educational
and scientific communication.
</p>
</li>
<li>
<p>Encode both mathematical notation and mathematical meaning.</p>
</li>
<li>
<p>Facilitate conversion to and from other mathematical
formats, both presentational and semantic. Output formats should include:
</p>
<ul>
<li>
<p>graphical displays</p>
</li>
<li>
<p>speech synthesizers</p>
</li>
<li>
<p>input for computer algebra systems</p>
</li>
<li>
<p>other mathematics typesetting languages, such as T<span style="vertical-align:-.4ex; margin-left:-.15ex; margin-right:-.15ex">E</span>X</p>
</li>
<li>
<p>plain text displays, e.g. VT100 emulators</p>
</li>
<li>
<p>international print media, including braille</p>
</li>
</ul>
<p>It is recognized that conversion to and from other notational
systems or media may entail loss of information in the process.</p>
</li>
<li>
<p>Allow the passing of information intended for
specific renderers and applications.</p>
</li>
<li>
<p>Support efficient browsing of lengthy expressions.</p>
</li>
<li>
<p>Provide for extensibility.</p>
</li>
<li>
<p>Be well suited to templates and other common techniques for editing formulas.</p>
</li>
<li>
<p>Be legible to humans, and simple for software to generate and process.</p>
</li>
</ul>
<p>No matter how successfully MathML achieves its goals as
a markup language, it is clear that MathML is useful only
if it is implemented well. The W3C Math Working
Group has identified a short list of additional implementation
goals. These goals attempt to describe concisely the minimal
functionality MathML rendering and processing software should
try to provide.
</p>
<ul>
<li>
<p>MathML expressions in HTML (and XHTML) pages should render
properly in popular Web browsers, in accordance with reader and author
viewing preferences, and at the highest quality possible given the
capabilities of the platform.</p>
</li>
<li>
<p>HTML (and XHTML) documents containing MathML expressions should
print properly and at high-quality printer resolutions.</p>
</li>
<li>
<p>MathML expressions in Web pages should be able to react to
user gestures, such those as with a mouse, and to coordinate
communication with other applications through the
browser.</p>
</li>
<li>
<p>Mathematical expression editors and converters should be developed
to facilitate the creation of Web pages containing MathML
expressions.</p>
</li>
</ul>
<p>The extent to which these goals are ultimately met depends on the
cooperation and support of browser vendors and other developers. The W3C
Math Working Group has continued to work with other working groups of the W3C,
and outside the W3C, to ensure that the
needs of the scientific community will be met.
MathML 2 and its implementations showed considerable progress in this area
over the situation that obtained at
the time of the MathML 1.0 Recommendation (April 1998) <a href="appendixh.html#MathML1">[MathML1]</a>. MathML3 and the developing Web are expected to
allow much more.</p>
</div>
</div>
<div class="div2">
<h2><a id="intro.overview"></a>1.3 Overview</h2>
<p>
MathML is a markup language for describing mathematics.
It is usually expressed in XML syntax, although HTML and other syntaxes are possible.
A special aspect of MathML is that there are two main strains of markup:
Presentation markup, discussed in <a href="chapter3.html">Chapter 3 Presentation Markup</a>,
is used to display mathematical expressions;
and Content markup, discussed in <a href="chapter4.html">Chapter 4 Content Markup</a>,
is used to convey mathematical meaning.
Content markup is specified in particular detail.
This specification makes use of
an XML
format called Content Dictionaries
This format has been developed by the
OpenMath Society, <a href="appendixg.html#OpenMath2004">[OpenMath2004]</a> with the dictionaries being used by this specification involving joint development
by the OpenMath Society and the W3C Math Working Group.
</p>
<p>Fundamentals common to both strains of markup
are covered in <a href="chapter2.html">Chapter 2 MathML Fundamentals</a>,
while the means for combining these strains,
as well as external markup, into single MathML objects
are discussed in <a href="chapter5.html">Chapter 5 Mixing Markup Languages for Mathematical Expressions</a>.
How MathML interacts with applications is covered
in <a href="chapter6.html">Chapter 6 Interactions with the Host Environment</a>.
Finally, a discussion of special symbols,
and issues regarding characters, entities and fonts,
is given in <a href="chapter7.html">Chapter 7 Characters, Entities and Fonts</a>.</p>
</div>
<div class="div2">
<h2><a id="intro.example"></a>1.4 A First Example</h2>
<p>
The quadratic formula provides a
simple but instructive illustration of MathML markup.
</p>
<div class="graphic">
<p><img src="image/f1002.gif" alt="x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}"></p>
</div>
<p>MathML offers two flavors of markup of this formula. The first is
the style which emphasizes the actual presentation of a formula, the
two-dimensional layout in which the symbols are arranged. An example
of this type is given just below. The second flavor
emphasizes the mathematical content and an example of it follows the first one.
</p>
<pre class="mathml">
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>-</mo>
<mi>b</mi>
</mrow>
<mo><span title="PlusMinus">&#xB1;</span><span class="uname"><!--plus-minus sign--></span></mo>
<msqrt>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mrow>
<mn>4</mn>
<mo><span title="InvisibleTimes">&#x2062;</span><span class="uname"><!--invisible times--></span></mo>
<mi>a</mi>
<mo><span title="InvisibleTimes">&#x2062;</span><span class="uname"><!--invisible times--></span></mo>
<mi>c</mi>
</mrow>
</mrow>
</msqrt>
</mrow>
<mrow>
<mn>2</mn>
<mo><span title="InvisibleTimes">&#x2062;</span><span class="uname"><!--invisible times--></span></mo>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
</pre>
<p>
Consider the superscript 2 in this formula. It represents the squaring operation
here, but the meaning of a superscript in other situations depends on
the context. A letter with a superscript can be used to signify a particular
component of a vector, or maybe the superscript just labels a different
type of some structure. Similarly two letters written one just after the
other could signify two variables multiplied together, as they do in the
quadratic formula, or they could be two letters making up the name of a
single variable. What is called Content Markup in MathML allows closer
specification of the mathematical meaning of many common formulas. The
quadratic formula given in this style of markup is as follows.
</p>
<pre class="mathml">
<apply>
<eq/>
<ci>x</ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<minus/>
<ci>b</ci>
</apply>
<apply>
<root/>
<apply>
<minus/>
<apply>
<power/>
<ci>b</ci>
<cn>2</cn>
</apply>
<apply>
<times/>
<cn>4</cn>
<ci>a</ci>
<ci>c</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<cn>2</cn>
<ci>a</ci>
</apply>
</apply>
</apply>
</pre>
</div>
</div>
<div class="minitoc">
Overview: <a href="Overview.html">Mathematical Markup Language (MathML) Version 4.0</a><br>
Next: 2 <a href="chapter2.html">MathML Fundamentals</a></div>
</body>
</html>