Transcendental

Transcendental is an adjective used in philosophy and mathematics to describe something that goes beyond the boundaries of ordinary experience or cannot be described by simple or finite rules. The word derives from Latin transcendere, “to go beyond.”

In Kantian philosophy, the term denotes the a priori conditions that make possible experience and knowledge. Transcendental aesthetics concerns the forms of sensibility such as space and time; transcendental logic studies the conditions for the possibility of knowledge itself. In more general usage, “transcendental” can describe ideas, methods, or phenomena that exceed empirical observation or ordinary experience, but the term is often kept distinct from religious or mystical connotations.

In number theory, a real or complex number is called transcendental if it is not a root of any nonzero polynomial with integer coefficients (equivalently rational coefficients). Algebraic numbers are those that are roots of such polynomials. The set of algebraic numbers is countable, while the transcendental numbers are uncountable and far more numerous. Famous examples of transcendental numbers are pi and e. Other known results include Liouville’s constant, which provided the first explicit example of a transcendental number, and e^π, which is transcendental by the Gelfond–Schneider theorem.

In the theory of functions, a function is called transcendental when it is not algebraic, meaning it does not satisfy a polynomial equation with polynomial coefficients P(x, y) = 0. Examples include the exponential function e^x and the trigonometric functions sin x and cos x.