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
- 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.