can anyone help me proof this
Elementary proof that e is irrational. I suppose that the reason it is easier
to show the irrationality of e is that e is defined analytically by
e = 1+ 1/1!+1/2!+...+1/b!+!/(b+1)!+..... =a/b in case e is
rational which gives (b-1)!a =b!(1+1/1!+...+1/b!) +b!(1/(b+1)!+...)
and the first term on the right hand side is an integer and the second term
on the right hand side is less that 1. This means that their sum is not an integer.
Yet the sum is the integer (b-1)!a, which is a contradiction. Therefore
e is irrational.
For pi the proof is more involved using some calculus and is in
Hardy and Wright's Intro to number theory among other books
and is probably on some pi websites.
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Elementary proof that e is irrational. I suppose that the reason it is easier
to show the irrationality of e is that e is defined analytically by
e = 1+ 1/1!+1/2!+...+1/b!+!/(b+1)!+..... =a/b in case e is
rational which gives (b-1)!a =b!(1+1/1!+...+1/b!) +b!(1/(b+1)!+...)
and the first term on the right hand side is an integer and the second term
on the right hand side is less that 1. This means that their sum is not an integer.
Yet the sum is the integer (b-1)!a, which is a contradiction. Therefore
e is irrational.
For pi the proof is more involved using some calculus and is in
Hardy and Wright's Intro to number theory among other books
and is probably on some pi websites.