Notations

Notations are systems of symbols and signs used to express ideas, quantities, relations, and operations within a domain. They provide precise, compact representations that facilitate reasoning, verification, and communication among practitioners. Notation evolves with the disciplines it serves, balancing ambiguity and expressiveness to support manipulation, calculation, and inference.

In mathematics and related fields, common notation includes variables and function notation such as f(x), operators, brackets, and specialized constructs like summation ∑, product ∏, limits lim, and integrals ∫. Vector and matrix notation, often with bold symbols or arrows, encodes geometric and linear-algebra relations; set notation uses symbols like ∈, ⊆, and ∅. Notation can vary by tradition and region, so new symbols are typically defined when introduced.

Historically, notation matured from verbal descriptions to symbols: early algebra relied on words, Descartes popularized symbolic algebra with x and y, Leibniz contributed differential notation, and Georg Cantor and Frege helped formalize logic and set theory. The development of standardized conventions, such as those in calculus, geometry, and later Unicode or LaTeX representations, improved readability and interoperability.

Notational systems may be overloaded across domains, making context essential for interpretation. Standards and style guides reduce confusion, but learners must learn the prevailing conventions of their field. Beyond symbol strings, visual notations such as graphs, trees, and flowcharts convey structure that complements symbolic notation and supports understanding for non-specialists.