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_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x^{1} + a_{0} = 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).