Kerry writes: << 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?

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Roth's Theorem (cf. MathWorld) says this: All algebraic numbers x satisfy, for any eps > 0, | x - p/q | < 1/q^(2+eps) for only finitely many rationals p/q. This led to an irrationality measure (cf. MathWorld again) due to I'm not sure whom, as follows: ----------------------------------------------------------------------- Given x real, define irr(x) := inf over all u such that (*) |x - p/q| < 1/q^(u+eps) for all eps > 0 and for only finitely many rationals p/q. Quoting from < http://mathworld.wolfram.com/IrrationalityMeasure.html >: << mu(x) = = 1 if x is rational; = 2 if x is algebraic of degree > 1;

= 2 if x is transcendental.

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This article includes some computed values (or bounds) for mu(x) for some familiar x's: (The table looks terrible to me in Arial (my only earthLlink webmail font choice), but it was created in Courier New, and I hope that font still makes it look good.) | | x | mu(x) >= | mu(x) <= | | ---------------------------------------------- | | e | 2 | 2 | | | | L | oo | oo | | | | pi | 2 | 8.0161 | | | | pi^2 | 2 | 5.4412 | | | | ln(2) | 2 | 3.8913 | | | | zeta(3) | 2 | 5.5138 ---------------------------------------------- where L := Liouville constant = sum_n=1^oo}1/10^(n!). (Note: mu(L) = oo means that there is no u for which (*) is true.) SO . . . mu(x) may be a good measure of "how transcendental" x is. --Dan