Numbers whose digits run forever to the left, not the right
Ordinary decimals trail off endlessly after the point. The p-adic numbers flip that picture: pick a prime p, write in base p, and let the digits stretch infinitely leftwards instead. The result is a whole alternative to the real numbers, built from the same fractions but measuring closeness in a completely different way.
Kurt Hensel first described these numbers in 1897, although some of Ernst Kummer's earlier work can, looking back, be read as using them without naming them. The motivation came from modular arithmetic, where every integer is replaced by its remainder after division by some n. Reducing an equation modulo a prime is a handy trick for studying Diophantine problems, but it throws information away. Working modulo higher powers such as p squared or p cubed keeps more detail, yet those systems are not fields, so much of the useful algebra disappears.
Hensel's answer was to start with a prime modulus and use what is now called Hensel's lemma to lift a solution modulo p to one modulo p squared, then p cubed, and onward. That produces an endless chain of residues, and the chain's limit is what counts as a p-adic number. In effect it lets a mathematician work modulo every power of p at once, while still being allowed to divide by p. Ordinary integers slot in without two of them collapsing into the same p-adic number, so little is lost.
In practice, p-adic integers look like base-p numerals with no leftmost digit. Fifty, for instance, is 1212 in base 3. Adding two such numbers means summing digit by digit and carrying from right to left, and multiplication follows the familiar long method, which makes the p-adic integers a ring. Some fractions sneak in as p-adic integers too: one fifth has a repeating 3-adic expansion. Others cannot be written that way, which is why the idea is widened to full p-adic numbers.
Every rational number has exactly one such series expansion. The series usually fail to converge in the everyday sense but do converge under the p-adic absolute value, so the p-adic numbers are the completion of the rationals for that measure, just as the reals complete them for the usual one. For each prime they form a field. Allowing p to be composite mostly breaks this, so those variants rarely appear.
Source: P-adic number
