example of transcendental number
The following is a classical application of Liouville’s approximation theorem. For completeness, we state Liouville’s result here:
Clearly, the number is well defined, i.e. the series converges. Indeed,
and . Thus, by the comparison test, the series converges and .
where is the constant given by the theorem above. Let be such that . Then, in fact, we will show that there are infinitely many rationals with such that
For all we define a rational number by:
then and are relatively prime integers and we have:
where in the last inequality we have used the fact that . Therefore, all rationals satisfy the desired inequality, which leads to the contradiction with the theorem above. Thus cannot be algebraic and it must be transcendental. ∎
Many other similar transcendental numbers can be constructed in this fashion.
|Title||example of transcendental number|
|Date of creation||2013-03-22 15:02:45|
|Last modified on||2013-03-22 15:02:45|
|Last modified by||alozano (2414)|