The purpose of this paper is to present some results and observations regarding the
beta transformations introduced by Rényi [8]. Given 
, the
beta transformation 
is defined for 
by 
. Parry [7] defined 
to be a beta number if the orbit
is finite. If 
for some n, then 
is a simple
beta number. If 
is a beta number which is not simple, then there is some
smallest 
(the preperiod length) and 
(the period
length) for which 
. For a simple beta number we define m = 0
and p to be the smallest integer with 
. Notice that m+p is the size
of the orbit 
.
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