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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.

Mathematical typesetting arranges symbolic expressions in two dimensions. It must coordinate computer fonts, glyph positioning, typesetting boxes and text baselines while preserving the expression's visible structure. A typeset fraction positions its components above and below a bar; superscripts, subscripts, square roots and mathematical delimiters impose other geometric relationships.^[ref]

For example,contains a superscript inside the upper component and a square root inside the lower component. Their dimensions help determine the height of the complete typeset fraction. A larger expression may then require larger surrounding mathematical delimiters. Laying out the smallest parts first makes this a recursive geometric construction.

An OpenType MATH table supplies font-specific information such as placement constants and suitable glyph variants. The table supplies data for layout; different implementations can use those data in different algorithms. TeX and LaTeX provide established authoring systems for expressing such mathematical structure.^[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 is a prepared collection of definitions and initialization state used with TeX. A TeX format file, conventionally ending in .fmt, stores internal structures so they can be loaded efficiently instead of reconstructed from the original initialization input.^[ref]

Plain TeX supplies one such foundation; LaTeX supplies another, with document-level conventions and facilities. The same mathematical input $x^2$ can be embedded in either system, but an authoring command defined by one format need not be available in the other.

Keeping the layers separate helps diagnose a failed document. The TeX implementation provides TeX primitives; the format makes further TeX macros and defaults available; a document invokes those definitions. Loading a different format can change the meaning or availability of commands without changing their spelling in the input file.