Yahoo Answers: Answers and Comments for What are padic numbers;what are some of their uses in mathematics? [Mathematics]
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Yahoo Answers: Answers and Comments for What are padic numbers;what are some of their uses in mathematics? [Mathematics]
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From Cheetah: In mathematics, and chiefly number theory, the...
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In mathematics, and chiefly number theory, the padic 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 padic 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 padic analysis essentially provides an alternative form of calculus.
More formally, for a given prime p, the field Qp of padic 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 padic number systems their power and utility.
The p in padic is a variable and may be replaced with a constant (yielding, for instance, "the 2adic numbers") or another placeholder variable (for expressions such as "the ℓadic numbers").
Hehner and Horspool proposed in 1979 the use of a padic representation for rational numbers on computers.[13] The primary advantage of such a representation is that addition, subtraction, and multiplication can be done in a straightforward manner analogous to similar methods for binary integers; and division is even simpler, resembling multiplication. However, it has the disadvantage that representations can be much larger than simply storing the numerator and denominator in binary; for example, if 2n−1 is a Mersenne prime, its reciprocal will require 2n−1 bits to represent.
The reals and the padic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way.
Helmut Hasse's localglobal principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the padic numbers for every prime p.