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example of transcendental number
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(Example)
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The following is a classical application of Liouville's approximation theorem. For completeness, we state Liouville's result here:
Next we use the theorem to construct a transcendental number.
Proof. Clearly, the number  is well defined, i.e. the series converges. Indeed,
and
 . Thus, by the comparison test, the series converges and
 .
Suppose, for a contradiction, that is algebraic of degree . We will construct infinitely many rationals such that
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.
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"example of transcendental number" is owned by alozano.
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Cross-references: similar, inequality, integers, relatively prime, rational number, algebraic, contradiction, comparison test, converges, series, well defined, number, transcendental, real number, transcendental number, rationals, degree, algebraic number, Liouville's approximation theorem, application
There are 2 references to this entry.
This is version 4 of example of transcendental number, born on 2005-02-16, modified 2005-02-16.
Object id is 6760, canonical name is ExampleOfTranscendentalNumber.
Accessed 2664 times total.
Classification:
| AMS MSC: | 11J81 (Number theory :: Diophantine approximation, transcendental number theory :: Transcendence ) | | | 11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence) |
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Pending Errata and Addenda
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