example of transcendental number

The following is a classical application of Liouville’s approximation theorem. For completeness, we state Liouville’s result here:

Theorem 1.

For any algebraic numberMathworldPlanetmath α with degree m>1, there exists a constant c=c(α)>0 such that:


for all rationals p/q (with q>0).

Next we use the theorem to construct a transcendental numberMathworldPlanetmath.

Corollary 1.

The real number


is transcendental.


Clearly, the number ψ is well defined, i.e. the series convergesPlanetmathPlanetmath. Indeed,


and n=110-n=1/9. Thus, by the comparison test, the series converges and 0<ψ<1/9.

Suppose, for a contradictionMathworldPlanetmathPlanetmath, that ψ is algebraic of degree m. We will construct infinitely many rationals p/q such that


where c=c(ψ) is the constant given by the theorem above. Let k be such that 1/2k<c. Then, in fact, we will show that there are infinitely many rationals p/q with q2 such that


For all j>k+m we define a rational number pj/qj by:


then pj and qj are relatively prime integers and we have:

|ψ-pjqj| = n=j+1110n!
< 110(j+1)!(1+110+1102+)
= 10/91qj(j+1)
< 1qjj
< 1qj(k+m)

where in the last inequalityMathworldPlanetmath we have used the fact that j>k+m. Therefore, all rationals {pj/qj}j=k+m+1 satisfy the desired inequality, which leads to the contradiction with the theorem above. Thus ψ cannot be algebraic and it must be transcendental. ∎

Many other similar transcendental numbers can be constructed in this fashion.

Title example of transcendental number
Canonical name ExampleOfTranscendentalNumber
Date of creation 2013-03-22 15:02:45
Last modified on 2013-03-22 15:02:45
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Example
Classification msc 11J82
Classification msc 11J81
Related topic LiouvillesTheorem
Related topic RothsTheorem