Many algebra textbooks give the definition of a Euclidean domain as an integral domain $R$ equipped with a Euclidean function/map (let's call it $\nu$). What I don't understand is the significance of the property frequently stated as:
$\forall a,b\in R \colon\ \nu(a) \leq \nu(ab)$.
I understand the other property as it allows the Euclidean algorithm to be performed, but I don't understand the purpose of this particular property.
Thanks.