  Here we show that by expressing a van der Waerden number $W(r, k)$ by its
radix polynomial representation, it not only is possible to locate each proper
subset on $\mathbb{R}$ in which the van der Waerden number lies, but also to
show that conditions exist for which the logarithm of the van der Waerden
number necessarily is bounded above by the square of the number of terms $k$ in
the arithmetic progression. Furthermore we also use the method to find a
mathematical expression or formula for the ratio of two "consecutive" van der
Waerden numbers of the kind $W(r, k)$, $W(r, k + 1)$.
