  For a finite set $X \subset \mathbb{Z}^d$ that can be represented as $X = Q
\cap \mathbb{Z}^d$ for some polyhedron $Q$, we call $Q$ a relaxation of $X$ and
define the relaxation complexity $rc(X)$ of $X$ as the least number of facets
among all possible relaxations $Q$ of $X$. The rational relaxation complexity
$rc_\mathbb{Q}(X)$ restricts the definition of $rc(X)$ to rational polyhedra
$Q$. In this article, we focus on $X = \Delta_d$, the vertex set of the
standard simplex, which consists of the null vector and the standard unit
vectors in $\mathbb{R}^d$. We show that $rc(\Delta_d) \leq d$ for every $d \geq
5$. That is, since $rc_{\mathbb{Q}}(\Delta_d)=d+1$, irrationality can reduce
the minimal size of relaxations. This answers an open question posed by Kaibel
and Weltge (Lower bounds on the size of integer programs without additional
variables, Mathematical Programming, 154(1):407-425, 2015). Moreover, we prove
the asymptotic statement $rc(\Delta_d) \in O(\frac{d}{\sqrt{\log(d)}})$, which
shows that the ratio $rc(\Delta_d)/rc_{\mathbb{Q}}(\Delta_d)$ goes to $0$, as
$d\to \infty$.
