Transcendental numbers are numbers that cannot be the root of a nonzero polynomial with rational coefficients (i.e., not an algebraic number). Examples of such numbers are $\pi$ and $e$.
Let $\mathbb{A}\subset \mathbb{C}$ be the set of algebraic numbers: $$ \mathbb{A} = \{ z\in \mathbb{C}: p(z)=0, p(z)\in \mathbb{Z}[x] \} $$The set of transcendental numbers is $\mathbb{A}^c$. The first construction of a transcendental number, $\sum_{n=1}^{\infty} 10^{-n!}$, was due to Liouville in 1851.
If we fix a degree of polynomials in $\mathbb{Z}[x]$, there are only countably many such polynomials, each with finitely many roots. Then $\mathbb{A}$ is countable, whence $\mathbb{A}^c$ is uncountable. Put crudely, 'most' complex numbers are transcendental, though showing a particular number is transcendental is usually rather difficult.
All transcendental numbers have an irrationality measure of at least 2; see irrationality-measure for more information.
