This is a bit skew, but for a direct matrix product conversion of the (non regular!) CF of x/atan x to y/(1-y^2)/the taylor series for atan y, where atan x = 2 atan y, see bottom of p14, p15, http://www.tweedledum.com/rwg/stanfordn3.pdf . We should look for a direct matrix CF/series conversion for tanh. --rwg rcs> 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