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The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems
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Classification of submultiplicative ring norms on $\mathbb Q$
Let $R$ be a ring with identity. I call a non-negative real valued function $N: R \to \mathbb R_{\geq 0}$ a ring norm, if it has the following properties:
$N(r) = 0$ iff $r = 0$
$N(r+s) \leq N(r) + N …