Googling for 'e irrational' turns up a 2006 MAA article by Ed Sandifer that credits Euler in 1737 for proving the irrationality of e and e^2. The method is somewhat roundabout, proving the continued fractions from the Ricatti equation, and noting that the infinite CF -> irrational. My (admittedly rusty) memory is that the CF for tanh(x), and I1/I0, can be established directly by doing GCD-like steps on the power series for sinh/cosh; the subsequent steps in the GCD process have relatively clean power series. For full rigor, you'd need to prove some convergence results. But this makes the results a little less mysterious. Rich --- Quoting Dan Asimov <dasimov@earthlink.net>:
I think I just found some article that claims this representation was invented by Cantor.
I've read (in a Monthly article by Jonathan Sondow) that the fact Warren states below (which has a 0-line proof) was used by Cantor to give perhaps the first proof that e is irrational, as an immediate corollary.
--Dan
Warren wrote: << . . . . . . the FracFac expansion of x terminates iff x is rational. . . .
________________________________________________________________________________________ It goes without saying that .
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