  Normal numbers were introduced by Borel. Normality is certainly a weak notion
of randomness; for instance, there are computable numbers which are absolutely
normal. In the present paper, we introduce a relativization of normality to a
fixed representation system. When we require normality with respect to large
sets of such systems, we find variants of normality that imply randomness
notions much stronger than absolute normality.
  The primary classes of numbers investigated in this paper are the supernormal
numbers and the highly normal numbers, which we will define. These are
relativizations of normality which are robust to all reasonable changes of
representation.
  Among other results, we give a proof that the highly normal numbers are
exactly those of computable dimension 1, which we think gives a more natural
characterization than was previously known of this interesting class.
