  The family of lattice simplices in $\mathbb{R}^n$ formed by the convex hull
of the standard basis vectors together with a weakly decreasing vector of
negative integers include simplices that play a central role in problems in
enumerative algebraic geometry and mirror symmetry. From this perspective, it
is useful to have formulae for their discrete volumes via Ehrhart
$h^\ast$-polynomials. Here we show, via an association with numeral systems,
that such simplices yield $h^\ast$-polynomials with properties that are also
desirable from a combinatorial perspective. First, we identify $n$-simplices in
this family that associate via their normalized volume to the $n^{th}$ place
value of a positional numeral system. We then observe that their
$h^\ast$-polynomials admit combinatorial formula via descent-like statistics on
the numeral strings encoding the nonnegative integers within the system. With
these methods, we recover ubiquitous $h^\ast$-polynomials including the
Eulerian polynomials and the binomial coefficients arising from the factoradic
and binary numeral systems, respectively. We generalize the binary case to
base-$r$ numeral systems for all $r\geq2$, and prove that the associated
$h^\ast$-polynomials are real-rooted and unimodal for $r\geq2$ and $n\geq1$.
