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:
  1. |a*b| = |a|*|b|
  2. |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...

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