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
integers and on the
rationals (but it's not quite the
norm defined on
vector spaces; 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 and
y 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...