analysis

Analysis is the systematic examination of a complex object or phenomenon by breaking it down into its constituent parts to understand structure, function, and relationships. In general usage, analysis involves careful observation, evaluation, and the identification of patterns, causes, and effects, often leading to explanations or judgments. The term is used across disciplines, with domain-specific methods and goals.

In mathematics, analysis is a branch concerned with limits, continuity, differentiation, integration, sequences, and functions, and with the rigorous justification of results. It includes real analysis, complex analysis, functional analysis, harmonic analysis, and numerical analysis. It provides foundations for much of science by formalizing change, approximation, and measurement, and it underpins proofs and theoretical models.

In the sciences and social sciences, analysis refers to the examination of data and observations to extract information, test hypotheses, and build models. Techniques include statistics, data mining, machine learning, and qualitative analysis. Outputs may include conclusions, predictions, or decisions, as well as explanations of mechanisms or relationships.

Analyses often proceed by decomposition, abstraction, comparison, and synthesis. Analysts may use critical thinking, quantitative metrics, or textual interpretation. Ethical considerations and transparency are increasingly emphasized, including clear documentation of data, assumptions, and uncertainty.

Other uses include literary analysis, where text is interpreted for themes and meaning; chemical analysis to determine composition; and financial analysis to assess investments. Historically, analysis emerged from Greek philosophy and developed through algebra, calculus, and statistics, expanding into many specialized subfields.