p-adic digit ($b_j\in \{0,\dots ,p-1\}$)

The p-adic digits are the unique coefficients in $x={\sum }_{j\ge 0}{b}_j{p}^j$ for a p-adic integer $x$. Taking least nonnegative representatives of its congruence classes,so each new residue determines one more p-adic digit. For $13$ in ${\mathbb{Z}}_2$, the low-to-high digits are $1,0,1,1,0,\dots$, giving $13=1+0\cdot 2+1\cdot 4+1\cdot 8$. A finite prefix describes a whole coset, not a unique infinite expansion.

Two p-adic integers agree in their first $n$ low-order p-adic digits precisely when their difference is divisible by $p^n$. Equivalently,If the first differing digit has index $j$, the remaining leading coefficient is nonzero modulo $p$, giving exact P-adic valuation $j$ and p-adic metric $p^{-j}$. If no digit differs, the two elements are equal and their distance is zero. Truncating at $r_n$ therefore gives $|x-r_n{|}_p\le p^{-n}$, with a stricter bound possible when further digits are zero.