  We show that rational data of bounded input length are uniformly distributed
with respect to condition numbers of numerical analysis. We deal both with
condition numbers of Linear Algebra and with condition numbers for systems of
multivariate polynomial equations. For instance, we show that for any $w>1$ and
for any $n\times n$ rational matrix $M$ of bit length $O(n^4\log n) + \log w$,
the condition number $k(M)$ satisfies $k(M) \leq w n^{5/2}$ with probability at
least $1-2w^{-1}$. Similar estimates are shown for the condition number
$\mu_{norm}$ of M. Shub and S. Smale when applied to systems of multivariate
homogeneous polynomial equations of bounded input length. Finally we apply
these techniques to show the probability distribution of the precision (number
of bits of the denominator) required to write down approximate zeros of affine
systems of multivariate polynomial equations of bounded input length.
