It is well known that the nontorsion part of the unit group of a real
quadratic field $\K$ is cyclic. With no loss of generality we may assume
that it has a generator $\eps_{0} > 1$, called the fundamental unit of $\K$.
The natural logarithm of $\eps_{0}$ is called the regulator R of $\K$.
This paper considers the following problems:
How large, and how small, can R get? And how often?
¶ The answer is simple for the problem of how small R can be, but
seems to be extremely difficult for the question of how large R can
get. In order to investigate this, we conducted several large-scale
numerical experiments, involving the Extended Riemann Hypothesis and
the Cohen--Lenstra class number heuristics. These experiments provide
numerical confirmation for what is currently believed about the
magnitude of R.