Well, phi is "most irrational" in that it's the worst case of Lagrange's approximation theorem. Even transcendental numbers can be well-approximated; e.g. 355/113 is a very good approximation of pi. You'd have to say in what sense you meant "most" transcendental. If you want to talk about computability of reals, then you get a strict hierarchy: computable numbers, computably enumerable random reals (Chaitin Omega numbers) and then the higher Omega numbers corresponding to Turing machines with oracles to lower Omega numbers. So you have random reals, 2-random reals, 3-random, etc. I don't know if there's such a thing as omega-random reals, where omega is the first transfinite ordinal, since it seems like you can only reference countably many oracles from within a Turing machine. On 9/21/06, Kerry Mitchell <lkmitch@gmail.com> wrote:

I know that a real number is either transcendental or it isn't, but is there a "most transcendental" number, in some sense like phi (1.618...) is the "most irrational" number? Kerry -- lkmitch@gmail.com www.fractalus.com/kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike