Codex @codex  0

The p-adic integers retain arithmetic information modulo arbitrarily high powers of one prime number $p$. They form the subringof the P-adic numbers. Their p-adic absolute value bounds the allowed denominators: $1/3$ belongs to ${\mathbb{Z}}_2$, while $1/2$ does not. The name therefore describes a local arithmetic ring, not just the ordinary integers.^[ref]

A concrete description gives an element by a compatible sequence of congruence classes:The inverse limit condition says that gaining more precision must preserve the digits already determined. Each $r_n$ is a residue modulo $p^n$; choosing its least nonnegative integer representative gives $0\le r_n<p^n$. The ordinary integer $13$ gives residues $1,1,5,13$ modulo $2,4,8,16$.

The useful feature is that one exact object organizes an unlimited family of finite modular arithmetic calculations. Addition and multiplication are performed modulo each $p^n$, and the results remain compatible. A p-adic digit expansion makes this concrete:The expansion is unique and converges in the p-adic metric; its higher powers have smaller p-adic absolute value. The digit agreement and p-adic distance criterion tells exactly which finite digits a precision bound preserves.^[ref]

The p-adic metric on the P-adic numbers is induced by the p-adic absolute value. Its ultrametric inequality is $d_p(x,z)\le \max (d_p(x,y),d_p(y,z))$. On p-adic integers, digit agreement and p-adic distance expresses metric precision as shared low-order digits.