  A real \alpha is called recursively enumerable ("r.e." for short) if there
exists a computable, increasing sequence of rationals which converges to
\alpha. It is known that the randomness of an r.e. real \alpha can be
characterized in various ways using each of the notions; program-size
complexity, Martin-L\"{o}f test, Chaitin \Omega number, the domination and
\Omega-likeness of \alpha, the universality of a computable, increasing
sequence of rationals which converges to \alpha, and universal probability. In
this paper, we generalize these characterizations of randomness over the notion
of partial randomness by parameterizing each of the notions above by a real T
in (0,1], where the notion of partial randomness is a stronger representation
of the compression rate by means of program-size complexity. As a result, we
present ten equivalent characterizations of the partial randomness of an r.e.
real. The resultant characterizations of partial randomness are powerful and
have many important applications. One of them is to present equivalent
characterizations of the dimension of an individual r.e. real. The equivalence
between the notion of Hausdorff dimension and compression rate by program-size
complexity (or partial randomness) has been established at present by a series
of works of many researchers over the last two decades. We present ten
equivalent characterizations of the dimension of an individual r.e. real.
