A transcendental number is a number that is real or complex, but it not algebraic, meaning that it is not the root of a polynomial with non-zero integer coefficients. For example, √2 is algebraic as it is the solution to the polynomial equation **x^{2 }– 2 = 0**.

### History

In 1844, Joseph Liouville proved the existence of transcendental numbers and in 1851 he gave the first example of such a number:

**= 0.11000100000000000000000100……**

(i.e. the nth digit after the decimal point is 1 if n = k! for some k, and 0 otherwise). This number is now known as the **Liouville constant**.

Only in 1873 was the first ‘non-constructed’ number shown to be transcendental when Charles Hermite proved that **e** was transcendental. Then, in 1882, Ferdinand von Lindemann proved that **π** was transcendental.

In fact, proving a number is transcendental is extremely challenging, even though they are known to be very common.

**Why are they very common?**

The algebraic numbers are countable (*the set of algebraic numbers is the countable union of countable sets and so is therefore countable*). However, the real numbers are uncountable. Therefore, since every real number is either algebraic or transcendental, the transcendentals must be uncountable. This implies that there are a lot more transcendental numbers than algebraic numbers.

### Examples of Transcendental Numbers

**e**if a is algebraic and non-zero^{a}**π***e*^{π}**a**^{b}where a,b are algebraic, but a ≠ 0,1^{ }- in particular, the
**Gelfond-Schenider Constant**

- in particular, the
- Continued Fraction Constant

If you want to find out more examples, click here.

Would you like to see a blog post specifically on Liouville numbers? M x