The Van de Corput sequence can be generated using the following formula:
$\phi_b(n) = \sum_{i=1}^N { a_j \over b^{i-1}}.$
where this can be defined as the "one-dimensional sequence defined by the radical-inverse function $\phi$ in base b"
and where the coefficients $a_i$ are given by the digit expansion in base b of n (n is a positive integer):
$n = \sum_{i=0}^N a_i b^i.$
This is all good and I understand the overall concept.
The thing I would like to learn/understand is why it is called a radical inverse. The inverse comes from the fact that to compute the final number in the sequence, the digit is divided by the base raised to the power $i$ but I'd like to have an explanation on the term radical (a radical function is a function that contains a variable inside a root? but there's no root in these formulas?).
Thank you.
hint: is that because the function can be written as $\phi_b(n) = \sum_{i=0}^N a_i \sqrt[i-i]{b}$? in which case yes this function has a variable inside a root? and because the magnitude of the root is negative we call it inverse?
"Radical" in this context has nothing to do with $\sqrt{}\ $. Rather, it's the adjective from radix, which is another word for base, as in base-10, base-2, and so on. You're taking the expansion to base $b$ --- the expansion with radix $b$ --- and writing it backwards --- "inverting" it.