P-adic Number

Given a prime number p, the p-adic numbers form an extension of the rational numbers distinct from the real numbers, though sharing some similar properties. A p-adic number can be written in a form similar to a possibly infinite decimal, but with digits based on the prime p rather than ten, and extending to the left rather than to the right. Formally, a p-adic number is a series of terms, each a digit between zero, inclusive, and p, exclusive, multiplied by a power of p, starting from some integer power that may be negative; a p-adic number is called a p-adic integer when that starting power is zero or more. Such a series generally does not converge in the usual sense, but does converge with respect to the p-adic absolute value, under which every rational number can be uniquely expressed as such a series, allowing the p-adic numbers to be defined as the completion of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rationals for the ordinary 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 read as implicitly using them.