  A real number \alpha is called recursively enumerable if there exists a
computable, increasing sequence of rational numbers which converges to \alpha.
The randomness of a recursively enumerable real \alpha can be characterized in
various ways using each of the notions; program-size complexity, Martin-L\"{o}f
test, Chaitin's \Omega number, the domination and \Omega-likeness of \alpha,
the universality of a computable, increasing sequence of rational numbers 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 number T\in(0,1]. We thus
present several equivalent characterizations of partial randomness for a
recursively enumerable real number.
