determinateindeterminate

Determinate and indeterminate describe a contrast used across several fields to indicate whether a quantity, problem, or expression has a definite value or remains unresolved. In mathematics, the terms are often applied to solvability or specification: a determinate situation yields a unique answer or value, while an indeterminate one does not, either because multiple outcomes are possible or because the data are insufficient to fix a single result.

In linear algebra and systems of equations, a determinate system has a unique solution. This occurs when the coefficient matrix has full rank equal to the number of unknowns. An indeterminate (or underdetermined) system has infinitely many solutions because there are fewer independent equations than unknowns. If the equations are inconsistent, there is no solution at all.

In algebra and calculus, indeterminacy also appears with variables and limits. An indeterminate in algebra refers to a symbol used to represent a variable within polynomials, such as x, without yet assigning a value. Once a value is specified, the quantity becomes determinate. In calculus, indeterminate forms such as 0/0 or ∞/∞ signal that a limit cannot be evaluated from the form alone and requires further techniques.

Beyond equations, determinacy relates to reference and meaning in philosophy and linguistics, where it concerns whether a term or statement fixes a particular reference or interpretation. In set theory and mathematical logic, determinacy describes whether a game or logical situation has a winning strategy for a player.

Overall, determinate and indeterminate function as a broad, cross-disciplinary dichotomy between fixed, resolvable outcomes and unresolved or non-unique ones.