Transcendental number
In mathematics, 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
where n ≥ 1 and the coefficients ai are integers (or, equivalently, rationals), not all 0.
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.
The existence of transcendental numbers was first proved in 1844 by Joseph Liouville, who exhibited examples, including the Liouville constant:
in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. The first number to be proved transcendental without having been specifically constructed to achieve this was e, by Charles Hermite in 1873. In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. In 1874, Georg Cantor found the argument described above establishing the ubiquity of transcendental numbers.
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, the Gel'fond-Schneider constant, or more generally ab where a ≠ 0,1 is algebraic and b is algebraic but not rational (Gel'fond-Schneider theorem). The general case of Hilbert's seventh problem, where b is not algebraic, remains open.
- sin(1)
- ln(a) if a is positive, rational and ≠ 1
- Γ(1/3), Γ(1/4), and Γ(1/6) (see gamma function).
- where
is the floor function. For example if β = 2 then this number is 0.11010001000000010000000000000001000...
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.