nonalgebraic

Nonalgebraic is a term used in mathematics to describe objects that are not algebraic. An object is algebraic, relative to a chosen field such as the rational numbers or the complex numbers, if it satisfies a polynomial equation with coefficients in that field. Conversely, nonalgebraic objects do not satisfy any such polynomial equation.

In numbers, this yields the familiar division between algebraic and transcendental numbers. An algebraic number is a root of a nonzero polynomial with rational coefficients; a nonalgebraic number is one that is not such a root, i.e., a transcendental number. There are countably many algebraic numbers, while the nonalgebraic ones are uncountably infinite, so almost all real numbers are nonalgebraic. Notable nonalgebraic numbers include π and e, while √2 is algebraic since it satisfies x^2 − 2 = 0.

Beyond numbers, algebraicity depends on the field. A function is algebraic over a field if it satisfies a polynomial equation with coefficients in that field; many analytic functions are nonalgebraic. In algebraic geometry, algebraic varieties are the common zeros of polynomial equations; sets defined by transcendental relations are nonalgebraic.

The term is thus relative and forms part of broader study in transcendence theory and the comparison between algebraic and analytic objects.