Transcendental number
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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
- <math>a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_1 x^1 + a_0 = 0<math>
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:
- <math>
\sum_{k=1}^\infty 10^{-k!} = 0.110001000000000000000001000....
<math> 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).
- <math>\sum_{k=0}^\infty 10^{-\lfloor \beta^{k} \rfloor};\qquad \beta > 1\; , <math>
- where <math>\beta\mapsto\lfloor \beta \rfloor<math> 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.de:Transzendente Zahl es:Número trascendente eu:zenbaki transzendente fr:Nombre transcendant ko:초월수 it:Numero trascendente nl:Transcendent getal ja:超越数 pl:Liczba przestępna pt:Número transcendente ru:Трансцендентное число sl:Transcendentno število zh:超越數 th:จำนวนอดิศัย