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A 2-adic integer is an element of the p-adic integers for $p=2$, equivalently a compatible sequence of residues modulo $2^r$. Its binary p-adic digits begin with its parity. The highlighted residue-tree path for $13$ begins $1,0,1,1$, while modular inversion converges to a p-adic inverse computes the infinite expansion of $1/3$.

A 7-adic integer is an element of the p-adic integers for $p=7$, equivalently a compatible sequence of residues modulo $7^r$. Its p-adic digits lie in $\{0,1,\dots ,6\}$. For example, $10=3+1\cdot 7$ has low-to-high digits $3,1,0,\dots$.

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.

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.

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]

A p-adic unit is an invertible element of the ring ${\mathbb{Z}}_p$. For a p-adic integer $x$, this is equivalent to $v_p(x)=0$, $|x{|}_p=1$, or a nonzero initial p-adic digit $b_0$. A rational number $a/b$ is a p-adic unit when neither its reduced numerator nor its denominator is divisible by $p$.

Thus $3$ and $1/3$ are p-adic units at $p=2$, while $2$ is a p-adic integer with no inverse inside ${\mathbb{Z}}_2$. The modular inversion converges to a p-adic inverse construction computes the inverse of an integer unit to arbitrary finite precision.