I like that using the sum-of-reciprocal-factorials definition of e, one can immediately see why it must be irrational. (As made clear in that paper of Sondow's, at < http://arxiv.org/pdf/0704.1282v2 >.) For, any rational number is equal to L/N! for some integers L, N. But any partial sum Sum_{k=1..N} 1/k! of e is of the form J/N!, and so adding the next term 1/(N+1)! means dividing the interval [J/N!, (J+1)/N!] into N+1 equal intervals and ensuring that e must lie somewhere in the *second* one of them. Because second is neither the first nor the last, this immediately implies that e cannot be of the form L/N!, and since N is arbitrary, we're done. --Dan ________________________________________________________________________________________ It goes without saying that .