Today we learned about Euclidean domains in class but I don't understand why we need one of the conditions stated in the definition. We called an integral domain $R$ a Euclidean domain if there exists a function $f$ from $R$ to strictly positive integers such that:
1) For $a,b$ non zero in $R$, $f(ab)\ge f(a)$.
2) If $a,b\in R$, $b\neq 0$, then we can write $a=bq+r$ with $q,r\in R$ such that either $r=0$ or $f(r)<f(b)$.
So from what I understand the whole point of a Euclidean domain is to be able to define a Euclidean algorithm, but I don't see why (1) is needed.
Furthermore later in the class we proved a Euclidean domain is a principal ideal domain and in the proof we didn't use the property (1), so my question is:
Why do we need (1) in the definition?
Thanks in advance.