Modular inversion converges to a p-adic inverse ($r_n\equiv a^{-1}(\mathrm{mod}{p}^n)⟹r_n\to a^{-1}$)

Let $a$ be an integer not divisible by $p$. Its modular inverse modulo $p^n$ has a unique least nonnegative representative $r_n$. Reducing the inverse equation modulo $p^n$ shows that the representatives are compatible. Since $a$ is a p-adic unit,so the sequence converges to the inverse $a^{-1}$ in ${\mathbb{Z}}_p$. This computes a fraction through finite integer operations.

For $a=3$ and $p=2$, the first eight representatives are $1,3,3,11,11,43,43,171$. Each satisfies $3r_n\equiv 1(\mathrm{mod}{2}^n)$. The resulting expansion isThe geometric series has ratio $4$, whose 2-adic absolute value is $1/4$, and its sum is $1+2/(1-4)=1/3$. In the ordinary real number absolute value, the representatives grow. Different metrics impose different meanings of a good approximation. The periodic digits here are a specific instance of the rational expansions studied in Conrad's authored note.^[ref]