and are irrational
and are irrational.
For let us define
We have that and if or . But, if , then
an integer. Hence and all its derivates take integral values at .Since , the same is true at
so that and are integers. We have
witch is an integer. But by equation 1,
For a large enough , we obtain a contradiction.
For any integer , if is irrational then a is irrational http://planetmath.org/?op=getobj&from=objects&id=5779(proof), and since is irrational is also irrational. ∎
The irrationality of was Proved by Lambert in 1761. The above proof is not the original proof due to Lambert.
- 1 G.H.Hardy and E.M.Wright An Introduction to the Theory of Numbers, Oxford University Press, 1959
The MacTutor History of Mathematics Archive, http://www-gap.dcs.st-and.ac.uk/ history/HistTopics/Pi_through_the_ages.htmlA history of Pi
The MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lambert.htmlJohann Heinrich Lambert
|Title||and are irrational|
|Date of creation||2013-03-22 14:44:00|
|Last modified on||2013-03-22 14:44:00|
|Last modified by||mathcam (2727)|