In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.

For example, comparing the expansion of the rational number ${\displaystyle {\tfrac {1}{5}}}$ in base 3 vs. the 3-adic expansion, $${\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}$$

Formally, given a prime number p, a p-adic number can be defined as a series $${\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }$$ where k is an integer (possibly negative), and each ${\displaystyle a_{i}}$ is an integer such that ${\displaystyle 0\leq a_{i}<p.}$ A p-adic integer is a p-adic number such that ${\displaystyle k\geq 0.}$

In general the series that represents a p-adic number is not convergent in the usual sense, but it is convergent for the p-adic absolute value ${\displaystyle |s|_{p}=p^{-k},}$ where k is the least integer i such that ${\displaystyle a_{i}\neq 0}$ (if all ${\displaystyle a_{i}}$ are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value).

Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.

p-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers.

Roughly speaking, modular arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n.

When studying Diophantine equations, it's sometimes useful to reduce the equation modulo a prime p, since this usually provides more insight about the equation itself. Unfortunately, doing this loses some information because the reduction ${\displaystyle \mathbb {Z} \twoheadrightarrow \mathbb {Z} /p}$ is not injective.