Overview
MathML's API consists of two main groups: presentation elements and content elements. Presentation elements describe a mathematical notation's visually oriented two-dimensional structure while content elements describe what the math means.

Presentation Markup
MathML has about 30 presentation elements that accept 50 attributes. Most elements represent templates or patterns for laying out subexpressions.

Elements
Presentation elements can be broadly grouped under the following five classes of elements.

1.   Token elements: Token elements represent the most basic structures in mathematics. For example:

There are also a few presentation elements that are empty elements, and are used mostly in connection with alignment.

2.   General layout: General layout elements describe the nature of the layout. For example:

3.   Scripts and limits: Scripts and embellishments to symbols are common in mathematical notation. The elements in this section address this requirement. For example:

4.   Tables: Matrices, arrays, and table-like mathematical notations are also represented using tags in MathML. For example:

5.   Actions: The maction element is in a category by itself, and allows coding of various kinds of actions on notation, such as occur in an expression which toggles between two pieces of notation.

It is important to note that the order of elements is important in MathML. This is the only way that the MathML processor can understand the mathematical constructs. For example, the first child of an mfrac element is the numerator and the second child is the denominator.

Attributes
Presentation elements can take up to 50 different attributes. Attributes generally specify additional optional information about the element. Each attribute has a name and a value. Using attributes, it's possible to precisely control how an expression will look when displayed. For example, the mfrac element has an attribute called linethickness and is described below:

attribute: linethickness
values: number [ v-unit ] | thin | medium | thick
default: 1 (rule thickness)

Content Markup
There are about 120 content elements and they accept close to a dozen attributes. Content markup facilitates applications other than display, like computer algebra and speech synthesis. The scope of content markup includes arithmetic, algebra, logic, relations, set theory, calculus, sequences, series, functions, statistics, linear algebra, and vector calculus.

Elements
There are many elements in the content markup part of MathML. The best way to think about elements is to categorize them into six buckets as follows.

1.   Containers: Container elements are used to indicate the basic units of mathematical content such as mathematical identifiers, numbers, and symbols. For example:

2.   Operators, qualifiers, and functions: These are tags used to define different mathematical and statistical functions and operators. For example:

3.   Relations: Relations are characterized by the fact that, if an external application were to evaluate them, they would typically return a truth value. For example:

4.   Conditions: The <condition> element is used to define the "such that" construct in mathematical expressions. The condition element is always used together with one or more bvar elements and the interpretation depends on the context.

5.   Syntax and semantics: This class provides additional information required for mathematical processing. For example:

6.   Constants and symbols: These are a collection of predefined constants and symbols which represent frequently encountered concepts. For example:

The <apply> element is perhaps the single most important content element. It is used to apply a function or operation to a collection of arguments. For example:

<mrow>
<apply>
<minus/>
<ci>a</ci>
<ci>b</ci>
</apply>
</mrow>

Attributes
There are about 12 attributes for the content presentation markup in MathML. These are used to add information about the content. For example:

<cn type="real"> 12345.7 </cn>

Creating MathML
There are a few Web browsers that support MathML, including Netscape 7.0, Amaya, and Mozilla. Amaya is the W3C's editor and browser and is used to demonstrate and test many of the new developments in Web protocols and data formats. In the simple example shown in Listing 1, I will use Amaya. Amaya can be downloaded from www.w3.org/Amaya/User/BinDist.html.

Here we have the normal structure of an XHTML document. It begins with the start tag <html> embellished with an XML namespace declaration and language assertions. A head element contains a title as is customary. Then the <body> beginning also has a namespace declaration of an abbreviative prefix letter m to be used for the standard MathML namespace. Next comes a simple paragraph. Finally we get the math element which also has a namespace association declared. Inside the math element is MathML markup as we are beginning to be used to it. It is rendered in Amaya as shown in Figure 1

Amaya and Mathematica are two examples of editors that can be used for editing the content of MathML. The W3C MathML page also mentions a LaTeX to MathML converter known as WeM, which is a MathML editor that converts a subset LaTeX to MathML. It can be tested online and is also available for download (GPL, requires PHP). Several other available tools can also be found on the MathML Web site (www.w3.org/Math).

Summary
MathML is a powerful XML-based markup language for publishing mathematics on the Web. MathML makes it possible to develop Web-based applications for displaying, searching, indexing, archiving, and evaluating mathematical content. It consists of two basic structures: presentation markup for describing math notation, and content markup for describing mathematical objects and functions. The motivation for MathML is a world in which individuals and companies want to move away from proprietary data formats and use XML to express, interchange, and share information. With increasing tool and browser support (both Microsoft and Netscape have publicly declared support for the XML recommendation), MathML's importance and prevalence will continue to grow.