Transcendental Numbers

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 = 0.


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

  • ea if a is algebraic and non-zero
  • π
  • eπ
  • ab where a,b are algebraic, but a ≠ 0,1
    • in particular, 2^{\sqrt {2}}, the Gelfond-Schenider Constant
  • Continued Fraction Constant{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{5+{\cfrac {1}{6+\ddots }}}}}}}}}}}

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


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