I am trying to discern some mathematical term known as a $n$-adic expansion. For example the book I am reading says: write every $x \in[0, 1]$ its "$4$-adic" expansion \begin{equation} x = \sum\limits_{i=1}^\infty\dfrac{x_i}{4^i} = \sum\limits_{i=1}^k\dfrac{x_i} {4^i} + r_k(x) \end{equation} for $(x_i \in \{0, 1, 2, 3\}, 1 \leq i < \infty)$
Could someone explainwhat this $4$-adic expansion means and why it is useful? Every google result search brings up $p$-adic expansions where $p$ is a prime.
As explained in comments: $n$-adic is sometimes used as a synonym of "base $n$", although this use is old-fashioned and potentially confusing. These days one is more likely to hear binary expansion than dyadic expansion. Nonetheless, an interval of $\mathbb R$ formed by all numbers with a given initial segment of binary expansion is usually called a dyadic interval.