Transcendental numbermathematics, a transcendental number is any irrational number that is not an algebraic number, i.e., it is not the solution of any polynomial equation of the form
The set of algebraic numbers is countable while the set of all real numbers is uncountable; this implies that the set of all transcendental numbers is also uncountable, so in a very real sense there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult. Another property of the normality of one number might also help to distinguish it to be transcendental.
See also Lindemann-Weierstrass theorem.
Here is a list of some numbers known to be transcendental:
- ea if a is algebraic and nonzero. In particular, e itself is transcendental.
- 2√2 or more generally ab 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 ab is transcendental whenever a ≠ 0,1 is algebraic and b is irrational, remains unresolved.
- ln(a) if a is positive, rational and ≠ 1
- Γ(1/3) and Γ(1/4) (see gamma function).
- Ω, Chaitin's constant.
The discovery of transcendental numbers allowed the proof of the impossibility of several ancient geometric problems involving ruler-and-compass construction; the most famous one, squaring the circle, is impossible because π is transcendental.