P-adic number

related topics
{math, number, function}

In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of absolute value.

First described by Kurt Hensel in 1897[1], the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.

More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility.

The p in p-adic is a variable and may be replaced with a constant (yielding, for instance, "the 2-adic numbers") or another placeholder variable (for expressions such as "the l-adic numbers").


Full article ▸

related documents
Lambda calculus
Lebesgue integration
Travelling salesman problem
Formal power series
Binomial coefficient
Discrete cosine transform
Grothendieck topology
Banach–Tarski paradox
Pythagorean triple
Original proof of Gödel's completeness theorem
Red-black tree
Μ-recursive function
Riemann integral
Group theory
Lie group
Big O notation
Class (computer science)
Linear programming
Combinatory logic
Relational model
Dedekind domain
Orthogonal matrix
Wikipedia:Free On-line Dictionary of Computing/R - S
Hilbert's tenth problem
System of linear equations
Laplace transform
Quadratic reciprocity