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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:

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