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LaTeX is a document preparation system built on TeX technology. It lets an author specify structure such as sections and references while reusable definitions control presentation. Its facilities support mathematical typesetting, figures, bibliographies and cross-references.^[ref]

For example, a minimal mathematical article can contain \documentclass{article}\begin{document}$x^2+1$\end{document}. The source describes the document class and content; the selected LaTeX definitions and computer fonts help determine the resulting geometry. The same separation lets one change a document's design while retaining its logical section structure.

Leslie Lamport developed the system; his LaTeX: A Document Preparation System by Leslie Lamport is listed in his bibliography as an Addison-Wesley publication from 1986.^[ref] A TeX format makes the system's definitions available to a compatible TeX implementation.

A mathematical delimiter is a symbol used to enclose or separate parts of a mathematical expression, such as parentheses, brackets or vertical bars. Its visible size should suit the enclosed material. Mathematical typesetting can select a larger mathematical glyph variant or construct a mathematical glyph assembly when a default glyph is too small.^[ref]

For example, compare the grouping inThe parentheses enclose the complete typeset fraction, so its height and depth inform their layout. The superscript applies to the grouped expression. In TeX, \left( and \right) request paired size-sensitive delimiters around the enclosed material. The stored parenthesis characters can therefore yield different glyphs in different formula contexts.

The Math axis provides a useful alignment reference for symmetric enclosing shapes. A font's choice of larger variants or assembly parts can preserve appropriate stroke weights and proportions while the expression grows.

A mathematical font constant is a font-specific parameter used in mathematical typesetting. The MathConstants portion of an OpenType MATH table supplies distances, percentages and other parameters, including the Math axis offset and script scaling.^[ref]

For an invented distance example, suppose a computer font uses 1000 font units per em and specifies fractionRuleThickness as 40 font units. At 25 pixels per em, the ideal typeset fraction bar thickness isAt 50 pixels per em, the same stored value gives 2 pixels before rasterization adjustments. A percentage value such as 80, however, is a ratio and must not be converted as though it meant 80 font units.

Keeping the units explicit prevents plausible-looking scale errors. Suggested values in the specification guide font design; they are not universal dimensions shared by every computer font.

An OpenType MATH table stores font-specific information for mathematical typesetting. Its major components supply mathematical font constants, per-glyph placement information, and data for mathematical glyph variants or mathematical glyph assemblies.^[ref]

For example, a tall typeset fraction can require larger surrounding mathematical delimiters. The font may offer several ready-made parentheses and parts for assembling still taller ones. An implementation can use those data together with the expression's measured dimensions. Merely scaling one outline uniformly could change its stroke weight along with its height; the supplied variants and parts give the font designer more control over the result.

The same general division of labour applies to subscripts and superscripts: a font supplies relevant dimensions and placement information, while the layout process combines them with the actual expression. The table is a data interface used by layout algorithms. It does not uniquely prescribe the complete appearance of every formula.^[ref]

Plain TeX is Donald E. Knuth's foundational collection of TeX macros, parameters and conventions, described in The TeXbook by Donald E. Knuth. It makes a usable document interface from the lower-level facilities of TeX.^[ref]

For example, $x^2+1$\bye requests an inline mathematical expression and ends the document. The \bye macro handles paragraph completion and final output before ending the run.^[ref] The example uses the format's existing definitions together with TeX primitives and mathematical typesetting rules.

Plain TeX is useful when studying how relatively small definitions control layout. LaTeX provides a different TeX format with more document-level structure. Choosing a format determines which definitions an input file can expect to find already loaded.

TeX is Donald E. Knuth's programmable system for typography, particularly mathematical typesetting. A document supplies text and instructions; TeX macros and TeX primitives build material whose dimensions determine the eventual page layout. Computer fonts supply glyphs and font metrics, while line breaking selects how a text paragraph is divided. The Knuth-Plass line-breaking algorithm is one of the system's central layout methods.^[ref]

A TeX format supplies a prepared collection of definitions. Plain TeX and LaTeX provide different document interfaces built on TeX technology. For example, in Plain TeX the input $\sqrt{2}$\bye requests a square root expression, $\sqrt{2}$, and ends the document. The computer font, glyph positioning and mathematical typesetting rules determine the printed geometry; the source characters alone are not a picture of the result.

The TeXbook by Donald E. Knuth is the system's user and reference manual. TeX source code by Donald E. Knuth exposes the implementation as a documented computer program, making the paragraph algorithm and its numerical conventions inspectable.^[ref]

A TeX format file stores initialized internal structures for a TeX format, usually under a .fmt filename. Loading it reconstructs prepared definitions and related state more quickly than rereading the initialization input. Creating it uses the initialization facility and the \dump command.^[ref]

This is an example of serialization. Suppose an initialization input defines \def\banner{First} and its state is saved. Changing the original input to define Second does not by itself replace the already saved state. The format must be regenerated, or the loaded definition explicitly changed, for subsequent processing to use that new initialization.

A loaded TeX format file initializes future document processing. The document can then add further TeX macros, change parameters and request mathematical typesetting. The relevant implementation must be able to read the saved representation; a .fmt suffix alone does not establish compatibility between different implementations.

A TeX macro associates a TeX control sequence with a replacement list of TeX tokens, possibly containing parameters. During TeX expansion, arguments are substituted into that list. Character tokens retain their TeX category codes, so this mechanism has more structure than unrestricted replacement of raw text strings.^[ref]

For a simple example, \def\dup#1{#1+#1} defines a macro with one parameter. In mathematics, $\dup{x}$ yields the token sequence for $x+x$. With $\dup{u+v}$, the expansion gives $u+v+u+v$. The replacement duplicates the supplied tokens; it does not evaluate or simplify the mathematical expression.

This makes TeX macros useful for reusable notation. A document can express an intention through one command and change the associated presentation centrally. The actual printed result still depends on subsequent execution, computer fonts and mathematical typesetting.