Fix some prime number p
. The p-adic valuation
(which you really
should read about!) vp
can be used to define a norm
on the integer
s and on the rational
s (but it's not quite the norm
defined on vector space
s; still, it's pretty close).
Just define |a/b|p = pvp(b)-vp(a).
Denoting for convenience this norm by just |.|, note that the properties given in the p-adic valuation node
immediately imply these properties:
- |a*b| = |a|*|b|
- |a+b| <= max(|a|,|b|)
In particular, it follows that |x-y
| is a metric
; this metric is very different from the metric defined by the usual absolute value norm
. It measures how many digits past the "decimal point
" in the base
p expansions of x
are the same, but counting from the rightmost digit. And property (2) even means it's an ultrametric
This makes for some very neat mathematics...