  In math.NT/0307308 we defined the irrationality base of an irrational number
and, assuming a stronger hypothesis than the irrationality of Euler's constant,
gave a conditional upper bound on its irrationality base. Here we develop the
general theory of the irrationality exponent and base, giving formulas and
bounds for them using continued fractions and the Fibonacci sequence. A theorem
of Jarnik on Diophantine approximation yields numbers with prescribed
irrationality measure. By another method we explicitly construct series with
prescribed irrationality base. Many examples are given.
