---
tag: 079EGSD32Y
kind: definition
name: Transcendental number
msc: [11J81]
defines: 0VTQXTCR2G
uses: [0MB4TGMXG6]
sources:
  - url: https://en.wikipedia.org/wiki/Transcendental_number
    locator: lead section
---

## Definition

A complex number is transcendental if it is not an [[algebraic number|0MB4TGMXG6]].
That is, it is not a root of any nonzero polynomial with rational coefficients.

## Remarks

The numbers $e$ and $\pi$ are transcendental, by the [[Lindemann–Weierstrass theorem|0JNCBFMPXR]].
Each [[Liouville number|0J1R5NP36S]] is transcendental, by [[Liouville's theorem|0N3WFZEFPM]].
The algebraic numbers are countable, so almost all real numbers are transcendental.
Joseph Liouville first proved that transcendental numbers exist, in 1844.

## Explanations

### L1 human

A transcendental number solves no polynomial equation with integer coefficients.
It is usually hard to prove that a given number is transcendental, even though most numbers are.