Transendental

Transendental, usually spelled transcendental, is an adjective used in mathematics to describe concepts that go beyond algebraic relations. In practice, it applies to numbers, constants, and functions that are not solutions of polynomial equations with rational coefficients.

A transcendental number is a real or complex number that is not a root of any non-zero polynomial with rational coefficients. Algebraic numbers are those that are roots of such polynomials. The first explicit transcendental number was constructed by Liouville in 1844, and Liouville’s constant serves as an early example. The Lindemann–Weierstrass theorem shows that numbers such as pi and e are transcendental. There are infinitely many transcendental numbers, and since the algebraic numbers are countable while the real numbers are uncountable, most real numbers are transcendental.

A transcendental function is a function that is not algebraic, meaning it does not satisfy a polynomial equation P(x, f(x)) = 0 with coefficients from a given field. Classic examples include the exponential function exp(x) and the trigonometric functions sin(x) and cos(x). By contrast, algebraic functions satisfy such a polynomial relation; for example, y = sqrt(x) is algebraic since it satisfies y^2 − x = 0.

In addition to mathematics, the term transcendental appears in philosophy to describe a priori conditions of knowledge, but in mathematical usage it primarily denotes the distinction between algebraic and non-algebraic objects. Note that “transendental” is a common misspelling of the standard term “transcendental.”