is irrational for
We here present a proof of the following theorem:
is irrational for all
To begin with, note that it is sufficient to show that is irrational for any positive integer (http://planetmath.org/NaturalNumber)11In this entry, and . (for if were rational, so would ). Next, we look at some simple properties of polynomial :
, with for all .
and are integers for all : as is a root (http://planetmath.org/Root) of order , unless , in which case , an integer. Since , the same is true for .
For all we have .
Now we can readily prove the theorem:
The result could also easily have been obtained by starting with and integrating by parts times. Note also that much stronger statements are known, such as “ is transcendental for all ”22 denotes the set of algebraic numbers.. We conclude this entry with the following evident corollary:
For all is irrational.
- 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3 edition (2004), Springer-Verlag, 30–31.
- 2 G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers, 5 edition (1979), Oxford University Press, 46–47.
|Title||is irrational for|
|Date of creation||2013-03-22 15:07:46|
|Last modified on||2013-03-22 15:07:46|
|Last modified by||Cosmin (8605)|
|Synonym||is irrational for non-zero rational r|
|Synonym||irrationality of the exponential function on|