Ultrametrics and p-adic Numbers

Abstract

The author recalls some measures of the difference between two objects: similarity, dissimilarities, metrics, ultrametrics, ultramines. Ultrametric is a dissimilarity index, ultramine is a similarity index. p-adic integers and p-adic numbers are introduced. p-adic numbers are non-archimedean. p-adic numbers were introduced by Hensel in 1897 after Kronecker’s work in Numbers Theory. The p-adic distance associated with a pair (a, b) of p-adic numbers satisfies all conditions of distance function and the ultrametric inequality.KeywordsPrime NumberMultivariate Data AnalysisDissimilarity IndexUltrametric SpaceAdic IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.