p-adic absolute value

Noun

 * 1)  A norm for the rational numbers, with some prime number p as parameter, such that any rational number of the form pk(a/b) &mdash; where a, b and k are integers and a, b and p are coprime &mdash; is mapped to the rational number p-k and 0 is mapped to 0. (Note: any nonzero rational number can be reduced to such a form.)

Usage notes

 * A notation for the p-adic absolute value of rational number x is $$|x|_p $$.
 * The function is actually from the set of rational numbers to the set of real numbers, because it is used to construct/define a completion of the set of real numbers, namely, the field of p-adic numbers, and this field inherits this p-adic absolute value and extends it to apply to p-adic irrationals, which could well be mapped to real numbers in general (not merely rationals).

Synonyms

 * p-adic norm