# Definition:Transcendental Number

## Definition

A number (either real or complex) is transcendental if and only if it is not algebraic.

### Transcendental Number over Field

Some sources define a transcendental number over a more general field:

Let $F$ be a field.

Let $z$ be a complex number.

$z$ is a transcendental number over $F$ if and only if $z$ cannot be expressed as a root of a polynomial with coefficients in $F$.

## Historical Note

The first numbers that were demonstrated as being transcendental were presented by Joseph Liouville in $1851$.

He did this with the aid of what is now known as Liouville's theorem.