Definitions for p-adic number
p-adic num·ber

WiktionaryRate this definition:0.0 / 0 votes

1. p-adic numbernoun

   An element of a completion of the field of rational numbers which has a p-adic ultrametric as its metric.

WikipediaRate this definition:0.0 / 0 votes

1. P-adic number

   In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way 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 "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.These 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. 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 the p-adic order, 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 that gives the p-adic number systems their power and utility. The p in "p-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "p-adic" comes from the ending found in words such as dyadic or triadic.

WikidataRate this definition:3.0 / 1 vote

1. p-adic number

   In mathematics 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 "closeness" or absolute value. In particular, p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p – the higher the power the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. p-adic numbers were first described by Kurt Hensel in 1897, though with hindsight some of Kummer's earlier work can be interpreted as implicitly using p-adic numbers. 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.

How to pronounce p-adic number?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say p-adic number in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of p-adic number in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of p-adic number in Pythagorean Numerology is: 7

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/P-adic_Number
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/P-adic_Number
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=p-adic number