padic number
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English[edit]
Noun[edit]
padic number (plural padic numbers)
 (mathematics) An element of a completion of the field of rational numbers which has a padic ultrametric as its metric.^{[1]}
 The expansion (21)2121_{p} is equal to the rational padic number .
 In the set of 3adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set . This closed ball partitions into exactly three smaller closed balls of radius 1/9: , , and . Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the subpartitioning can be continued indefinitely, in a fractal manner.
Likewise, going upwards in the hierarchy, B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, , which is one out of three closed balls forming a closed ball of radius 9, and so on.
Usage notes[edit]
 The 'p' in "padic" is a parameter which stands for a positive integer, preferably a prime number.
 For a fixed prime value of p, a padic number is a member of the field which is a completion of the set of rational numbers.
 For a composite value of p, a padic number is a member of a ring which is an extension of the field of rational numbers.
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See also[edit]
References[edit]
 ^ 2008, Jacqui Ramagge, Unreal numbers: The story of padic numbers