Motivation of irrationality measure

I have a question about the irrationality of e:

In proving the irrationality of e, one can prove the irrationality of e1 by using the series ex=1+x+x22!++xnn!+ So the series for e1 is sn=nk=0(1)kk! so that 0<e1s2k1<1(2k)! or 0<(2k1)!(e1s2k1)<12k12 If e1 were rational then we would have a difference of two integers which is not an integer (i.e. between 0 and 12).

Question: Is this "irrationality of e1 by alternating series" proof what motivated the definition of irrationality measure? It seems that this was born out of using alternating series.

Does the same method work for π ?


The motivation for the definition of irrationality measure came from results in diophantine approximation. E.g., Dirichlet's Theorem states that for every real irrational x there are infinitely many integer pairs p,q with |x(p/q)|<q2. If x is a real quadratic irrational, then there's a positive constant C such that for every integer pair p,q we have |x(p/q)|>Cq2. Liouville's Theorem was mentioned in the first comment. These can all be stated in terms of irrationality measure, and form an obvious motivation for that concept.

Source : Link , Question Author : Damien , Answer Author : Gerry Myerson

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