What are p-adic numbers?

Euclidean geometry relies on the Archimedean axiom. This says that every straight line segment of finite length can be surpassed by N segments of unit length, if only N is sufficiently large. Thus the Euclidean norm fulfills the Archimedean axiom:

    |x| < |N|

for suitable N. The geometry of hierarchies is non-Archimedean. Here, the strict triangle inequality

    d(x,y) ≤ max {d(x,z),d(z,y)}. 

holds true. Such a metric d is also called ultrametric. An example for non-Archimedean geometry is given  by p-adic expansion 

    x = ∑_v a_v p^v

of numbers. Here, p is a prime number, v = 0,1,2, ... and a_v is a "digit" from {0, ... , p-1}. The p-adic norm is 

    |x|_p  =   p ^{-v (x)},

where v(x) is the first "digit" a_v ≠ 0 in the p-adic expansion. The p-adic numbers are given by p-adic expansion with infinitely many of these "digits" and can be hierachically arranged in an infinite regular tree T_p, the Bruhat-Tits-tree. Further information to p-adic numbers can be found on the very recommendable Wikipedia pages on the topic. The pages in German and English. complement each other nicely.