Transcendental number

A transcendental number is any real or complex number that is not an algebraic number, i.e., it is not the solution of any polynomial equation of the form

a_nxⁿ + a_n-1x^n-1 + ... + a₁x¹ + a₀ = 0

where n >= 1 and the coefficients a_i are integers (or, equivalently, rationals), not all 0.

The set of algebraic numbers is countable while the set of transcendental numbers is uncountable, so in a very real sense there are many more transcendental numbers than algebraic ones. However, only few classes of transcendental numbers are known and proving that a given number is transcendental can be extrememely difficult.

The first numbers to be proved transcendental were the [Liouville numbers]?, by [Joseph Liouville]? in 1844. This was also the first proof that transcendental numbers exist. The first important number to be proved transcendental was e, by [Charles Hermite]? in 1873. Other known transcendental numbers include:

• e^a if a is algebraic and nonzero
• π
• e^Π
• 2^√2 or more generally a^b where a ≠ 0,1 is algebraic and b is algebraic but not rational. The general case of Hilbert's seventh problem, namely to determine whether a^b is transcendental whenever a ≠ 0,1 is algebraic and b is irrational, remains unresolved.
• sin(1) (see trigonometric function)
• ln(a) if a is positive, rational and ≠ 1 (see natural logarithm)
• Γ(1/3) and Γ(1/4) (see Gamma function).