p-adic number

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group

Similar picture for p = 3 (click to enlarge) shows three closed balls of radius 1/3, where each consists of 3 balls of radius 1/9

A scheme showing the topology of the dyadic (or indeed p-adic) integers. Each clump is an open set made up of other clumps. The numbers in the left-most quarter (containing 1) are all the odd numbers. The next group to the right is the even numbers not divisible by 4.

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.

单词	Dyadic number
释义	Dyadic number 英语百科 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.
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Dyadic number

p-adic number