Not to be confused with Transcendental integer.

A transcendental number is a number which is not an algebraic number[1]. 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:

External links


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