# Point 1

*The sample mean is never exactly equal to the true mean when the true mean is
 irrational.*

# Example:

Let $D$ be any distribution over the integers. Suppose that the first moment,
$\mu$, exists for $D$ and is an irrational number
(i.e. $\mu \not\in \mathbb{Q}$).

In this case, the sample mean of $n$ IID draws from $D$ is never exactly equal
to the true mean because the sample mean is always a rational number. This is
true for all $n$ as $n \to \infty$.

# Note

This example is really more illustrative than problematic: the discrepancy
between the rational approximation to $\mu$ and the true, irrational value of
$\mu$ becomes arbitrarily small as $n$ grows. The genuine issue with sample
means always being rational is that it can induce strange discretization
problems.
