Meaning of p-adic number | Babel Free

Definitions

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

Equivalents

Examples

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

“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.”

“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.”

“#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.”

CEFR level