Rational approximation of transcendental and algebraic numbers I had a discussion with a friend that say that 

transcendental numbers can be better approximated by rationals than
  algebraic (irrational) numbers, so, is some sense, they are more
  ''near'' to rationals that the algebraic numbers.

I don't understand in what sense this statement can be true (and I suspect that it is false).
I know that the way to estimate how a irrational number is approximated by rational is the irrationality measure, and that this measure is $1$ for rational numbers, $2$ for algebraic numbers, $\ge 2$ for transcendental numbers and $\infty$ for Liouville numbers.  
I interpret this as the Liouville numbers are the numbers better approximated by rationals (correct?) and we know that these numbers are transcendental. But I also know that the set of Liouville numbers has measure $0$, so almost all transcendental numbers have irrationality measure $\ge 2$ but $< \infty$.  But we cannot prove that almost all have irrationality measure $>2$, so we cannot say that thay are generally better approximated by rationals. Or there is some proof?
Anyway, i think that the idea of ''near to'' cannot be applied in this situation since it is obviously false in standard topology of reals and I don't see a topology in which it can be true. Or there is someone?

A proof is given in Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193. With theorem statement on page 246 and proof reaching page 247. Naturally, page 246 is not part of the online preview. The really good pages never are. It seems it is in Appendix E.  

