Transcendental number

A transcendental number is a number which is not an MathWorld, Transcendental Number. This means that an \(r \in \mathbb{R}\) is a transcendental number if there exists no non-zero polynomial \(f(x)\) with coefficients in \(\mathbb{Q}\) such that \(f(r) = 0\).
The notion is obviously extended to other systems of numbers such as \(\mathbb{C}\). One of various theorems called Liouville's theorem gives a useful criterion to determine whether a given complex number is a transcendental number or not. An analogous result for \(\mathbb{Q}_p\) is also known, while it has less usability.
== Common mistakes ==
The notion of a transcendental number is sometimes misunderstood. For example, the description "it cannot be solved by any polynomial" is wrong, as the transcendental number \(\pi\) is the solution of the first-degree polynomial equation \(x-\pi=0\) and the zero-degree polynomial equation \(0 = 0\). In other words, \(\pi\) is a root of \(x-\pi\) and \(0\).
== In googology ==
There are several transcendental numbers relating to googology:
* Champernowne constant = 0.12345678910... Kurt Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), pp. 421-428, Satz 1
* Ramanujan constant = \(e^{\pi\sqrt{163} }\) (due to , because it can be written as \(a^b\) for \(a=e^{i\pi}\) and \(b=-i\sqrt{163}\))
* Phiplex = 10φ (due to )
== External links ==
* in wikipedia
== Sources ==
Category:Numbers
Category:Transcendental numbers