Nonalgebraically

Nonalgebraically is an adverb used to describe actions, methods, or properties that do not involve algebra or do not satisfy algebraic relations. In mathematical usage, algebraic refers to relations expressible by polynomial equations with coefficients in a fixed field (often the rationals or integers). Nonalgebraically thus denotes the complement of these notions, either by method (not using algebra) or by objects that are not described by algebraic equations.

In number theory, a real or complex number is called algebraic if it is a root of some nonzero polynomial with rational coefficients; otherwise it is nonalgebraic. The nonalgebraic numbers are more commonly referred to as transcendental numbers. Examples include pi and e, which are nonalgebraic, whereas sqrt(2) is algebraic. The set of algebraic numbers is countable, while the set of nonalgebraic numbers is uncountable.

In analysis and geometry, one contrasts algebraic functions and algebraic varieties with their nonalgebraic (often called transcendental) counterparts. An algebraic function satisfies a polynomial relation with coefficients that are themselves functions, while a nonalgebraic or transcendental function such as exp, log, or sin fails to satisfy such polynomial equations. The term nonalgebraically can also describe a proof, construction, or characterization that does not rely on algebraic methods, for example solving a problem nonalgebraically by geometric or analytic means. In algebraic geometry, a variety is algebraic if it can be defined by polynomial equations; a nonalgebraic subset would not be described by such equations.

Etymology: formed from non- meaning not and algebraic, with the -ly suffix to form an adverb. See also algebraic, transcendental, algebraic geometry.