p-adic number

plural p-adic numbers

1. mathematicsAn element of a completion of the field of rational numbers with respect to a p-adic ultrametric.

   The expansion (21)2121ₚ is equal to the rational p-adic number #92;textstyle#123;2p#43;1#92;overp²-1#125;.

   4 more examples

   In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set x|∃n∈ℤ.,x=3n+1. This closed ball partitions into exactly three smaller closed balls of radius 1/9: x|∃n∈ℤ.,x=1+9n, x|∃n∈ℤ.,x=4+9n, and x|∃n∈ℤ.,x=7+9n. Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning 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, x|∃n∈ℤ.,x=1+n/3, which is one out of three closed balls forming a closed ball of radius 9, and so on.

   3. In his recent book Professor Hensel has developed a theory of logarithms of the rational p-adic numbers, and from this he has shown how all such numbers can be written in the form p#92;alpha#92;omega#92;betae#92;gamma.

   1914, Bulletin of the American Mathematical Society, page 452

   p-Adic numbers were introduced in mathematics by K. Hensel, and this invention led to substantial developments in number theory, where p-adic numbers are now as natural as ordinary real numbers.[…]Bleher noticed in [19] that the set of purely fractional p-adic numbers is an example of hierarchical lattice.

   1991, M. D. Missarov, “Renormalization Group and Renormalization Theory in p-Adic and Adelic Scalar Models”, in Ya. G. Sinaĭ, editor, Dynamical Systems and Statistical Mechanics: From the Seminar on Statistical Physics held at Moscow State University, American Mathematical Society, page 143

   #92;Q#95;p is called the p-adic number field, and its elements are called p-adic numbers. In this section we introduce the p-adic number fields, which are very important objects in number theory. The p-adic numbers were originally introduced by Hensel around 1900.

   2000, Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, translated by Masato Kuwata, Number Theory: Fermat's dream, American Mathematical Society, page 58